GEOMETRIC INTEGRATORS FORmypages.iit.edu/~mdixon7/thesis/OfficialCopyPhDThesisDixon_Jul07.pdfof...

GEOMETRIC INTEGRATORS FOR

CONTINUUM DYNAMICS

by

MATTHEW F DIXON

A thesis presented for the degree of

Doctor of Philosophy of the University of London

and the Diploma of Imperial College

Department of Mathematics

Imperial College

Huxley Building

180 Queens Gate

London SW7 2BZ

United Kingdom

JULY 2007

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ii

Abstract

We develop a unified computational framework for deriving geometric integrators which

implement the conservation laws of Hamiltonian continuum dynamics in the convective

and spatial representations. In an abstract setting, these representations of continuum

dynamics correspond to the body and spatial representations of the rigid body (Holm

et al. 1986) - the latter of which has no spatial dependence. We apply the discrete

Clebsch approach (Cotter & Holm 2006) to the body representation of the rigid body

to give a momentum map through which aMoser-Veselov (MV) integrator is recovered.

The discrete Clebsch approach also gives a momentum map through which a MV in-

tegrator for motion in the spatial representation, with an advected quantity, is defined.

We then apply the discrete Clebsch approach to ellipsoidal and elastic rod continuum

motions. In each case, the discrete Clebsch approach gives momentum maps encoding

discrete conservation laws corresponding to those of the continuous system, which we

verify by numerical experiment.

Practitioners seek to implement geometric integrators for fluids through a unified

discrete framework. We turn to free-Lagrange methods, which compute the spatial

variables using a dynamically generated mesh. We investigate a variational formulation

of the free-Lagrange method for shallow water and show that the semi-discrete shallow

water equations conserve energy and have an associated divergence law. We, however,

obtain an evolution equation for the potential vorticity with a non-zero right hand side

attributed to the discrete curl of the discrete gradient operator. We apply symplectic

integrators to the semi-discrete scheme and present numerical results demonstrating

that the variational free-Lagrange method for rotating shallow water conserves energy

over long-time simulations and exhibits the geostrophic adjustment mechanism.

Finally, in Chapter 5 we describe the implementation of boundary conditions using

geometric integrators for fluid dynamics to address the problem of preserving sym-

plectic structure of a Hamiltonian particle-mesh method (Frank et al. 2002) for shallow

water with spatial velocity free-slip boundary conditions. We formulate the boundary

condition in terms of ghost particles and show by numerical experiment that this ap-

proach gives energy-conserving numerical approximations of rotating shallow water in

a bounded domain.

Acknowledgements

I would like to thank my supervisors, Darryl Holm and Sebastian Reich for their dir-

ection and extensive assistance with understanding the theory of geometric mechanics

and the development of the computational skills needed to produce the numerical res-

ults in this thesis. Thanks is extended to Colin Cotter for providing an abundant source

of ideas, assistance and encouragement. Others that deserve special mention are Mark

Petersen, Matthew Hecht, Beth Wingate, Todd Ringler and Jean-Luc Thiffeault.

I declare that the material presented in this thesis is my own work and any material

which is not my own has been acknowledged.

Signed: Matthew Francis Dixon Date: 4th July, 2007

iv

Contents

Copyright ii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Extended Synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Literature Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Important related works . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Contributions of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 A computational framework . . . . . . . . . . . . . . . . . . . . . 14

1.4.2 Development of new DMV algorithms . . . . . . . . . . . . . . . 15

1.4.3 Geometric integrators for shallow water . . . . . . . . . . . . . . 16

1.5 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Moser-Veselov Integrators for Spatial and Body Representations of

Rigid Body Motions 20

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 The Free Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2.1 Discrete velocities . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Discrete Constrained Variational Principle . . . . . . . . . . . . . . . . . 28

2.4 Symmetry Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Clebsch Potentials and Momentum Maps . . . . . . . . . . . . . . . . . 32

2.5.1 Geometric preliminaries . . . . . . . . . . . . . . . . . . . . . . . 32

2.5.2 The Clebsch approach . . . . . . . . . . . . . . . . . . . . . . . . 34

2.5.3 The discrete Clebsch approach . . . . . . . . . . . . . . . . . . . 36

2.5.4 The body representation . . . . . . . . . . . . . . . . . . . . . . . 41

2.5.5 The spatial representation . . . . . . . . . . . . . . . . . . . . . . 46

2.6 Poisson Brackets on Semidirect Products . . . . . . . . . . . . . . . . . . 51

2.7 The Heavy Top . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54


2.8 The Coupled Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . 59

v

2.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


2.9 The Cayley-Klein Parameterisation of Rigid Body Motion . . . . . . . . 66

2.9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

2.9.2 Momentum maps and Hopf fibrations . . . . . . . . . . . . . . . 69

2.10 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.10.1 Body and spatial DMV algorithms for the rigid body . . . . . . . 72

2.10.2 Body DMV algorithm for the heavy top and coupled rigid body 78

2.10.3 Comparison with a Lie-Poisson integrator based on splitting . . . 82

2.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3 Moser-Veselov Integrators for Elastic Body and Rod Motions 88

3.0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.1 Free Ellipsoidal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.1.2 Convective and spatial representations of discrete ellipsoidal motion 94

3.2 The Pseudo-Rigid Body . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

3.2.1 Polar decomposition of discrete pseudo-rigid body motion . . . . 99

3.2.2 Conservation of circulation . . . . . . . . . . . . . . . . . . . . . 101

3.2.3 MV integrators for Mooney-Rivlin materials . . . . . . . . . . . . 103

3.3 Elastic Rod Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.3.1 The discrete Kirchhoff rod analogy . . . . . . . . . . . . . . . . . 105

3.3.2 Time dependent discrete Kirchhoff rod models . . . . . . . . . . 108

3.4 The Geometrically Exact Elastic Rod Model . . . . . . . . . . . . . . . . 112

3.4.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.4.2 The variational formulation of the geometrically exact elastic rod

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.5 The Discrete Geometrically Exact Elastic Rod Model . . . . . . . . . . . 115


3.6.1 DMV algorithms for pseudo-rigid bodies and elastic rods . . . . 118

3.6.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

3.7.1 Future research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3.7.2 Proceeding Chapters . . . . . . . . . . . . . . . . . . . . . . . . . 132

4 A Variational Free-Lagrange Method for Shallow Water 134

4.1 The Lagrangian Description of Shallow Water . . . . . . . . . . . . . . . 135

4.2 The Variational Free-Lagrange Method for Shallow Water . . . . . . . . 137

4.2.1 Mass conservation . . . . . . . . . . . . . . . . . . . . . . . . . . 139

vi

4.3 The Variational Free-Lagrange Equations for 1D Shallow Water . . . . . 139

4.3.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.4 A Variational Free-Lagrange method for 2D Shallow Water . . . . . . . 141

4.4.1 Energy Conservation . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.5 The Shallow Water Vorticity Equation . . . . . . . . . . . . . . . . . . . 144

4.6 The Semi-Discrete Divergence Form of the Shallow Water Equations . . 148

4.7 A Symplectic Time Stepping Scheme . . . . . . . . . . . . . . . . . . . . 149

4.8 1D Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 151

4.8.1 Experiment 1: conservative properties of VFL . . . . . . . . . . . 151

4.8.2 Experiment 2: geostrophic adjustment . . . . . . . . . . . . . . . 154

4.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5 A Hamiltonian Particle Mesh Method for ShallowWater in a Bounded

Domain 161

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.2 The Hamiltonian Particle Mesh Approximation . . . . . . . . . . . . . . 163

5.2.1 Symplectic time stepping . . . . . . . . . . . . . . . . . . . . . . 167

5.3 Layer Depth Smoothing on a Bounded Domain . . . . . . . . . . . . . . 168

5.3.1 A smoothing operator for 1D shallow water . . . . . . . . . . . . 169

5.3.2 A smoothing operator for 2D shallow water . . . . . . . . . . . . 170


5.4.1 HPM for 1D (non-rotating) shallow water . . . . . . . . . . . . . 172

5.4.2 The 2D (non-rotating) shallow water equations in a channel . . . 173

5.4.3 Rotating shallow water in a channel . . . . . . . . . . . . . . . . 176

5.5 Summary and Further Research . . . . . . . . . . . . . . . . . . . . . . . 178

6 Summary 179

6.1 Contribution of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.1.1 Development of a unified framework . . . . . . . . . . . . . . . . 179

6.1.2 Development of new DMV algorithms . . . . . . . . . . . . . . . 181

6.1.3 Geometric integrators for shallow water . . . . . . . . . . . . . . 182

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.2.1 Future directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

A Properties of MV Integrators 193

A.1 Body and Spatial Representations in Continuous

and Discrete Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

A.2 MV Integrators for the Cayley-Klein Parameterisation of Rigid Body

Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

vii

A.3 The Spatial DMV Algorithm for the Rigid Body . . . . . . . . . . . . . 196

A.4 A DMV Algorithm for Coupled Rigid Body Motion . . . . . . . . . . . . 196

B The Variational Description of Elastic Body and Rod Models 198

B.1 The Anisotropic Pseudo-Rigid Body . . . . . . . . . . . . . . . . . . . . 198

B.2 The Geometrically Exact Elastic Rod Model . . . . . . . . . . . . . . . . 200

C Additional aspects of the Variational Free-Lagrange Method 204

C.1 Canonical Formulation of the Variational Free-Lagrange Method . . . . 204

C.2 Representation on a Fixed Grid . . . . . . . . . . . . . . . . . . . . . . . 205

viii

Chapter 1

Introduction

1.1 Motivation

Geometric integrators transfer powerful concepts in geometric mechanics to compu-

tational continuum dynamics by preserving properties of the continuous system such

as the geometric structure, symmetries and phase space volume. There are numerous

reasons for using geometric integrators to preserve these properties in computer simu-

lations. Intuitively, one might expect that integrators which preserve properties of the

continuum produce solutions that capture the qualitative features of the continuum dy-

namics too. An increasingly diverse range of continuum models would indeed confirm

this to be the case.

Zero-Helicity flows For a captivating example of this, we refer the reader to the

work of Holm & Kimura (1991) who apply a volume preserving integrator to the in-

tegrable and zero-Helicity Chandrasekhar flow. This flow is used to the study of the

onset of Rayleigh Benard convection in incompressible fluids. The intricate spatially

periodic heteroclinic network of saddle-focus connections which is present analytically

is also exhibited by the numerical solution.

Liquid Crystal Devices LCDs operate on the principle that a suitable applied

field will change the orientation of the liquid crystals, while conserving the pointwise-

norm. Lewis & Nigam (2003) presents a norm-preserving integrator for micromagnetic

applications which have the same constraints. By constraining the geometric integrator

to the sphere, the authors are able to demonstrate a number of versatile and robust

algorithms for modelling micromagnetic applications which are equally applicable to

LCDs.

1

Elastic rods Elastic rods are a core model for studying a wide range of applications

in engineering science and increasingly biomechanics. Idealised rod models are used

to study dynamical properties of rods leading to critical insight into their failure and

fatigue mechanisms under dynamic loads. Although the implications of symmetries

in these models on their dynamics is not fully understood, the preservation of these

symmetries under discretisation has motivated the development of a class of geometric

integrators referred to as Mechanical integrators (see Barth, Leimkuhler & Reich 1999,

Gonzalez & Simo 1996, Marsden, O’Reilly, Wicklin & Zombro 1991). Most crucially,

geometric integrators enable us to understand the role of symmetries and correspond-

ing conservation laws in the dynamics of elastic rods and in the very least provide a

benchmark for more established, classical, numerical methods for rod models in which

numerical dissipation hinders the integrity and efficiency of simulations.

Geophysical fluid dynamics The free evolution of many idealised geophysical flu-

ids are constrained by the existence of conservation laws. Ripa (1981) points out that

these are most easily found in the Lagrangian description because they correspond to

symmetries of Hamilton’s principle. For example a particle relabelling symmetry in

shallow water models gives Ertel’s theorem for potential vorticity conservation. In con-

trast, use of a beta-plane approximation in Hamilton’s principle breaks homogeneity in

the meridional direction and isotropy in the horizontal plane. Consequently a conser-

vation law for Enstrophy is not present as is in an f-plane approximation. Hamilton’s

principle is not only a good starting point for finding conservation laws under various

geophysical approximations but also for deriving new geophysical models (see Salmon

1983, for an introduction to a balanced model now referred to as L1 dynamics). A

discrete variational approach would appear to be the only way for a numerical analyst

to formulate a computational analogue to many of these models without ”exercising

his bias” (Salmon 1983).

Numerical stability Conservation of energy is often regarded as a strong manifest-

ation of unconditional numerical stability (Lewis & Simo 1994). The ever diversifying

set of computational models which crucially rely on the ability to tractably simulate

long-time dynamics, drives the need to develop new computationally competitive geo-

metric integrators. Implementers of global climate models, for example, are challenged

by the need to resolve complex geophysical fluid dynamics over long periods. The be-

nefits of geometric integrators are four-fold. Firstly, by conserving energy, geometric

integrators enable the simulation to be run at larger time-steps offering potential com-

putational savings. To illustrate this point, the reader should refer to the description of

the Stormer Verlets integrated particle shallow water (Frank, Gottwald & Reich 2002,

Frank & Reich 2004) and particle Euler Slice (Dixon & Reich 2004) models.

2

Conservation laws Secondly, geometric integrators for continuum dynamics provide

the capability to selectively resolve the dynamical features which are manifestations of

the conservation law. For example, one can introduce a discrete Bjerknes’ circulation

theorem (an extension of Kelvin’s circulation theorem to include forcing terms) in an

idealised oceanographic model in contrast to the traditional and brute-force approach

of adaptively resolving dynamical features by grid refinement (without an explicit rep-

resentation of a discrete circulation theorem). This paradigm is a direct extension of

geometric integrators for ordinary differential equations, where symplectic integrators

have largely superceeded adaptive generic Runge-Kutta methods as a more efficient nu-

merical approach to simulating Hamiltonian dynamics. Certainly, the inadequacy and

inability of existing global climate prediction models to recover observable features, such

as the western boundary current intensification (Stommel 1948), suggests the need for a

new generation of numerical methods. This generation of numerical methods shall im-

plement a conferred mathematical understanding of the dynamics into computational

models, in turn providing new scientific insight.

Unified computational framework Of course, this idea is far from new, but the

idea of deriving geometric integrators for continuum dynamics in a unified and consist-

ent way has arguably been a more recent pursuit (see Lewis & Simo 1994). Starting with

the priviledged mathematical insight that geometric mechanics has provided, one can

systematically derive variational integrators by discretising Hamilton’s action principle

and using standard methods of variational calculus. The theoretical investigations into

this approach are partially listed later in this Chapter, but the real hurdle is convincing

practitioners to adopt this approach - for numerous good reasons too.

Firstly, there is the need to compound the literature on geometric continuum dy-

namics and geometric integrators to provide an accessible and pedagogical context

for implementors of geometric integrators for continuum dynamics. In particular this

should address how to systematically recover geometric integrators for the various rep-

resentations of continuum dynamics, paying close attention to the classification of geo-

metric structure that is preserved when physical quantities are advected by the flow.

Secondly, there is the need to understand how to cast existing numerical methods

for continuum dynamics into a variational framework to derive geometric variants of

established approaches. The reason for doing this might be to simply improve the

stability of the scheme through application of a symplectic integrator to a Hamiltonian

construction of it, or to transfer concepts in geometric mechanics to the computational

model.

Thirdly, numerous computational aspects remain unaddressed, but the implement-

ation of boundary conditions in a framework for computational dynamics must surely

3

be one of the most fundamental of these. Finally, a much neglected aspect of numer-

ical methods for continuum dynamics is their ability to systematically be verified upon

implementation. Conserved quantities provide additional measures with which to as-

sess the validity of implementation. Moreover, a variational framework systematically

provides the exact form of these discrete conserved quantities.

1.2 Extended Synopsis

Holm, Marsden & Ratiu (1986) present a unified geometric approach for the study of

idealised Hamiltonian continuum models (fluids, plasmas, elasticity, etc.) in the ma-

terial, inverse material, spatial and convective representations. This approach is based

on maps, referred to as momentum maps, which carry the Poisson brackets in one

representation into another. In this thesis, we investigate a corresponding computa-

tional variational framework for the convective and spatial representations of continuum

dynamics in which momentum maps are fundamental to the derivation of geometric

integrators which preserve Lie-Poisson structure1.

Holm et al. (1986) show that the convective and spatial representations of continuum

dynamics correspond to the body and spatial representations of rigid body dynamics.

The rigid body therefore serves as a suitable starting point for developing a unified

computational framework.

The discrete Clebsch approach We begin by applying the discrete Clebsch ap-

proach of Cotter & Holm (2006) to formulate geometric integrators for the body and

spatial representation of rigid body dynamics. The distinguishing feature of the dis-

crete Clebsch approach arises from its construction in a reduced discrete Hamilton’s

action principle. By adding so called Clebsch constraints for the reconstruction of the

dynamics on the full phase space, the discrete Clebsch approach gives the momentum

maps for the cotangent lifted actions to the reduced phase space. These momentum

maps are the conserved quantities associated with the Noether symmetries, see Noether

(1918). Through the momentum map associated with symmetry reduction to the body

representation of the reduced dynamics, the discrete flow on phase space is the same

integrable discretisation of rigid body dynamics discovered by Moser and Veselov in

1991. These integrators are commonly referred to as Moser-Veselov integrators and are

well established for the body representation of the rigid body.

The spatial representation The spatial representation of the dynamics of the rigid

body differs from the body representation by the presence of an advected quantity- the

inertia matrix rotates in the spatial frame. Following Holm et al. (1986), we augment

1All of this terminology is conventional in geometric mechanics (see Marsden & Ratiu 1999)

4

the Lagrangian with the advected quantity and apply the discrete Clebsch again to give

a momentum map through which the MV integrator for the spatial representation is

defined. We will show that the spatial representation of the MV integrator matches the

form of the discrete Euler-Poincare equations derived by Bobenko & Suris (1999a) in a

discrete Euler-Poincare framework. Bobenko & Suris (1999a) showed that the discrete

EP equations for the spatial representation of the dynamics are Lie-Poisson w.r.t. to

the dual of a semi-direct product Lie algebra, a result which we verify for the spatial

MV integrators.

Spatial versus body discrete Moser-Veselov (DMV) algorithms Informally

put, semi-direct product group actions define how advected quantities feedback on the

body dynamics and are fundamental to the Lagrangian description of dynamics with

advected quantities. Given the significance of semi-direct product in continuum dynam-

ics, we show how to modify the DMV algorithm, developed by McLachlan & Zanna

(2005) for the body representation of rigid bodies, to solve the spatial MV integrat-

ors. We provide several numerical experiments to study the comparative conservative

properties of the spatial DMV algorithm with the body DMV algorithm.

Heavy tops and coupled rigid bodies We apply the discrete Clebsch approach to

give MV integrators for the body representation of heavy tops and coupled rigid bodies

respectively as these provide examples of motions with potential and coupling terms

and, in the latter case, can only be solved by adapting the DMV algorithms. These

examples prepare us for more challenging examples in elastic dynamics which feature

potential forces and coupling motions.

Cayley-Klein parameters Before applying the discrete Clebsch approach to the

convective and spatial representations of elastic dynamics, we will show that the dis-

crete Clebsch approach also gives MV integrators for the Cayley-Klein parameterisa-

tion of the rigid body (see Whittaker 1944). MV integrators for rigid body motions

formulated as SO(3) matrices, represent the attitude of the rigid body as a three-stage

rotation by each Euler angle about its corresponding principal axis. One could para-

meterise the rigid body dynamics in a minimal representation in terms of these angles.

This parameterisation is well know to exhibit singularities however (see Leimkuhler &

Reich 2005). Parameterisation of the rotation in terms of four Cayley-Klein parameters

provides a double covering of the rigid body configuration to avoid singularities. These

parameters constitute the group of unit quaternions which is isomorphic to the matrix

group SU(2). Leimkuhler & Reich (2005) point out that it is better to use rotation

5

matrices for parameterisation, despite the introduction of redundant variables together

with constraints, because this leads to a global parameterisation.

We will show that the discrete Clebsch approach can be easily cast in terms of SU(2)

matrices to give a MV integrator for the Cayley-Klein parameterisation of rigid body

motion together with the conserved momentum maps corresponding to symmetries of

the discrete Lagrangian.

Ellipsoidal motion The rigid body example serves only to present the geometric

principles governing our framework. We will further the development of our computa-

tional framework by applying the discrete Clebsch approach to give MV integrators for

the convective and spatial representations of ellipsoidal motion on the group GL(n)+.

In this generalised model, either the shape matrix, describing the shape of the ellipsoid

in an Eulerian frame, or the right Cauchy-Green matrix, describing the shape of the

Eulerian frame in the body, are advected quantities depending on whether the motion

is in the spatial or convective representation. In each case, the MV integrators define

a co-adjoint action on the dual of a semi-direct product Lie-algebra and preserve the

Lie-Poisson bracket defined on the dual of this Lie-algebra.

The Pseudo-rigid body In the absence of a generalised DMV algorithm for MV

integrators on GL(n)+, we will apply a polar decomposition to a special case of ellips-

oidal motion on GL(3)+, referred to as isotropic pseudo-rigid body motion, in which

the body is initially spherical. Applying the discrete Clebsch approach gives MV in-

tegrators for the polar components of pseudo-rigid body motion. These components

describe the orientation, stretching and internal circulation of the rigid body, where the

internal circulation is coupled to the orientation through a Coriolis term. The adapted

DMV algorithm for solving these coupled polar component MV integrators is based on

the DMV algorithm which we develop for the coupled rigid body.

Conservation Laws The polar decomposed isotropic pseudo-rigid body is invariant

under left and right actions of SO(3). The discrete Clebsch approach gives the left and

right conserved momentum maps corresponding to these Noether symmetries. These

momentum maps are the angular momentum and vorticity of the internal circulation.

We will show that the conservation of vorticity is the discrete Kelvin circulation law

for ellipsoidal motions. This law corresponds to a relabelling symmetry and is central

to the geometric description of idealised continuum dynamics.

Kirchhoff elastic Rods So far, we have only described the application of our frame-

work to the spatial and convective representations of discrete time finite dimensional

motions, yet, continuum motion is infinite dimensional. One of the simplest examples

6

of such motions is that of the Kirchhoff rod which is defined by a continuum of or-

thonormal frames describing the orientation of an inextensible rod. It is well known

that the static configuration of the Kirchhoff rod is in one-to-one correspondence with

the motion of the Lagrange top, by a result known as the Kirchhoff kinetic analogy.

We will show that the analogy only holds in the discrete case, i.e. when there are

a finite number of orthonormal frames and time steps, if the spatial discretisation of

the rod corresponds to the temporal discretisation of the Lagrange top. This analogy

extends to the correspondence between dynamical Kirchhoff rods and sequences of

elastically coupled Lagrange tops. It follows that the discrete rod model then exhibits

a discrete compatibility equation in the same form as the discrete auxiliary equation for

the relative orientation matrix in the discrete coupled rigid body model.

Geometrically exact elastic rod models Following Krishnaprasad, Marsden &

Simo (1988), we will apply our computational framework to the convective represent-

ation of a fully non-linear rod model which bends, twists, shears and stretches. This

elastic rod model is referred to by Krishnaprasad et al. (1988) as a geometrically exact

elastic rod model. This model represents the motion of the rod as an elastic coupling

of its line of centroids and orthonormal frames. We will show that the discrete Clebsch

approach gives the conserved momentum map for the total spatial angular momentum

and use numerical experiments to show that the DMV algorithm is conservative.

Geometric integrators for Hamiltonian fluids Implementation of geometric in-

tegrators seeks to transfer powerful concepts in the convective and spatial representa-

tions of Hamiltonian fluid dynamics using a unified framework. The configuration space

for fluids is the infinite dimensional group of diffeomorphisms. Symmetry reduction by

this group action to the convective and spatial representations is challenging in the

discrete case.

This thesis considers geometric integrators for Hamiltonian fluids which preserve the

canonical symplectic structure of a finite dimensional system of fluid particles and are

expressed in terms of the material velocity and pressure field. The challenging problem

of developing a unified discrete framework for fluids based on the finite dimensional

representation of the group of diffeomorphisms developed by Zhong & Scovel (1994)

and Zeitlin (2004) is not considered here.

Variational free-Lagrange method We will consider Hamiltonian shallow water

as a model for furthering application of our framework. This model describes the

motion of an incompressible inviscid layer of fluid, under the influence of gravity and

Coriolis forces. Following the method described by Augenbaum (1984), we spatially

discretise the shallow water action principle for a free-Lagrange method. This method

7

represents the material velocity at the position of point particles and the layer depth

on a mesh, dynamically generated from the position of the particles, referred to as

a Voronoi diagram. The use of a Voronoi diagram to represent the layer depth is a

computationally tractable and stable approach.

Stationarity of the discrete action principle gives a semi-discrete method for shallow

water which conserves energy. We also formulate a semi-discrete divergence conserva-

tion law and an evolution equation for the potential vorticity with a non-zero right

hand side. We will finally apply symplectic integrators to the semi-discrete scheme

and present numerical results demonstrating that the VFL method for rotating shallow

water exhibits no secular drift in energy over long-time simulations and the geostrophic

adjustment mechanism of rotating shallow water in a f-plane.

Boundary conditions Finally, we implement boundary conditions using geometric

integrators for Hamiltonian fluid dynamics to address the problem of preserving sym-

plectic structure of a particle-mesh formulation of shallow water with spatial velocity

free-slip boundary conditions. For convenience, we extend the Hamiltonian particle-

mesh method for shallow water in a periodic domain, developed by Frank, Gottwald

& Reich (2002), to bounded domains. We formulate the boundary condition in terms

of ghost particles and show by numerical experiment that this approach gives energy

conservative numerical approximations of rotating shallow water in a bounded domain.

1.3 Literature Survey

Holm et al. (1986) presented the Hamiltonian structure of continuum mechanics in the

material, inverse material, spatial and convective representations.

Body and Spatial Representations of Rigid Body Motions Their work iden-

tified the body and spatial representations of rigid body motions as prototypes for the

respective convective and spatial representations of continuum dynamics. Their com-

parison of the spatial and convective representations also put the Hamiltonian treat-

ments of elasticity by Holm & Kupershmidt (1983) and by Marsden, Ratiu & Weinstein

(1984) into a unified framework.

The convective representation The convective and also the inverse material (aug-

mented Eulerian) representations offer alternative descriptions of continuum models.

The motivation for the convective representation of the continuum arose from a num-

ber of sources in the 1980s, including the study of relativistic adiabatic fluids by Holm

8

(1985), stability analysis of the coupled rigid body-beam and plate models of Krish-

naprasad & Marsden (1987) and the geometrically exact rod and plate models of Krish-

naprasad et al. (1988).

Semi-direct products Holm, Marsden & Ratiu (1998) derived the Euler-Poincare

(EP) formulation of the Eulerian fluid equations for an ideal fluid by applying sym-

plectic reduction to Hamilton’s principle for fluids. Legendre-transforming the EP

theory recovered the semidirect-product Lie-Poisson Hamiltonian theory that had been

discovered and applied earlier for nonlinear stability analysis by Holm, Marsden, Ratiu

& Weinstein (1998). A key step in the analysis of nonlinear stability of fluid equilibria

relies on the existence of Casimirs – quantities whose Lie-Poisson bracket vanishes with

all Eulerian (spatial) fluid variables because of right-invariance of the Eulerian variables

under reparameterisation of the Lagrangian labels. Because their Poisson brackets with

the Hamiltonian vanish, the Casimirs are conserved quantities.

Circulation theorems Fluid mechanics literature widely refers to the reparamet-

erisation of labels as fluid parcel relabelling and attributes the existence of the Kelvin

circulation theorem for ideal flow to the application of Noether’s theorem for the particle

relabelling symmetry group. Holm, Marsden & Ratiu (1998) showed that when advec-

ted quantities are present, a corollary of the EP framework is a geometric form of the

Kelvin circulation theorem referred to as the Kelvin Noether theorem. In this frame-

work, Holm et al. (1986) and Holm, Marsden & Ratiu (1998) further revealed the utility

of simple finite dimensional examples, such as the heavy top, by demonstrating that

they also exhibit a Kelvin Noether theorem. This theorem together with the EP equa-

tions form an essential ingredient in the geometric description of idealised continuum

dynamics.

Variational integrators Geometric numerical methods seek to transfer these power-

ful concepts in geometric mechanics to computational models. The pioneering work of

Moser & Veselov (1991) revealed integrable classical mechanical systems which have in-

tegrable discrete time counterparts. They considered the free rigid body as one example

and derived a discrete analogue to the Euler-Arnold equations for rigid body motion in

the body description. These integrators, referred to by McLachlan & Scovel (1995) as

Moser-Veselov (MV) integrators, conserve the rigid body energy to an arbitrary order

of the time step size and angular momentum to numerical round off.

Moser’s and Veselov’s key step was to form a discrete Hamilton’s action principle

9

and then derive variational integrators by deriving the discrete Euler-Lagrange equa-

tions. Although the number of contributions that later followed this approach are too

extensive to list here, the reader may follow some important aspects of its development

in Bobenko & Suris (1999a), Marsden, Pekarsky & Shkoller (1999), Marsden & West

(2001), McLachlan & Scovel (1995), Wendlandt & Marsden (1997), Leok, Marsden &

Weinstein (2004) who provide a differential geometric foundation for variational integ-

rators applied to mechanical systems. The number of numerical studies supporting the

theory appears less extensive, however.

Explicit integrators Moser-Veselov integrators are solved using an explicit algorithm,

referred to by McLachlan & Scovel (1995) as a DMV algorithm. Cardoso & Leite (2001)

cast the expression for the discrete angular momentum of Moser’s and Veselov’s rigid

body into a matrix Ricatti equation and solved it by Schur decomposing the Hamilto-

nian matrix. With the exception of the Schur decomposition, this DMV algorithm is

explicit. McLachlan & Zanna (2005) provide a more detailed description of this DMV

algorithm and demonstrate how to avoid the costly computation of the Schur form by

using an explicit spectral decomposition of the Hamiltonian instead. The Hamiltonian

can be decomposed in this way whenever its characteristic polynomial can be solved

analytically. The simple models considered herein do not require this optimisation step.

Pseudo-rigid bodies The pseudo-rigid body is a useful prototype for the geometric

description of homogeneous elasticity. Sousa-Dias (2002) applied the EP theory to

the polar decomposed pseudo-rigid body and showed that the motion of the isotropic

pseudo-rigid body is described by two coupled Lax equations for the angular momentum

and vorticity and a second order differential equation on the set of diagonal matrices

with positive determinant.

The geometrically exact rod model Krishnaprasad et al. (1988) applied the ma-

terial, spatial and convective description of continuum mechanics by Holm et al. (1986)

to the study of the Hamiltonian structure of non-linear elasticity. Krishnaprasad et al.

(1988) considered the Poisson structure and the reduced Poisson (Lie-Poisson) struc-

ture of non-linear elastic media before formulating the spatial, material and convective

representations of a geometrically exact rod and plate model. This rod model is a

generalisation of Antman’s director approach for the classical Kirchoff-Love elasticity

model, described in Antman (1995), to allow for finite extension and shear. This is

most conveniently achieved when the kinematics of the rod are described in terms of

orthogonal rotating frames(or orthogonal directors whose origins are fixed along the

rod centroid. Then, the configuration of the rod is described entirely by the attitude

10

of the rotating frames and the rod centroid position.

Simo’s rod model Simo & Vu-Quoc (1986, 1988) formulated a discrete geometrically

exact rod model. Their approach was novel in that it avoided making approximations

which resulted in artificial numerical features such as damping of shear and compres-

sion waves. Their approach combined the Rodrigues formula to compute infinitesimal

rotations and the Newmark variational integrator to compute the displacement of a

finite element discretisation of the rod. Their approach is not structure preserving,

however, nor is it explicit.

Other discrete rod models Symplectic methods for the discrete Kirchhoff rod have

been developed to enforce the rod inextensibility constraint. We note the Impetus-

striction method (Dichmann & Maddocks 1996) and holonomically constrained rod

model (Barth et al. 1999), both of which address the difficult numerical problem of

enforcing inextensibility in Kirchhoff rod simulations. Neither of these approaches

formulate the rod motion in the reduced representation and do not, therefore, provide

expressions for the conserved momentum maps corresponding to rotational symmetries.

Variational free-Lagrange methods Augenbaum (1984) derived a variational free-

Lagrange (VFL) method from a semi-discrete Hamilton’s action principle. The novel

feature of this method is the use of a Voronoi diagram to construct the layer depth

over. The method represents the material velocity field as tangent vectors of particle

positions and the layer depth as piecewise constant functions defined over Voronoi cells

of fixed mass moving with the (single) particle that each cell contains. The cell vertices

are defined at the mid-points between neighbouring particles. A variety of cell polygons

can then be constructed from the particle positions in this way.

Augenbaum’s work did not establish the conservative properties of this method nor

did it detail the use of symplectic time integrators to solve the Hamiltonian ordinary

differential equations describing Voronoi-cell trajectories on phase space

The VFL approach is conceptually similar to the particle moving-grid method

(Nishiguchi & Yabe 1982) which was developed to eliminate the presence of spurious

modes exhibited by the particle-in-cell method developed by Harlow (1964). The ori-

ginal particle-in-cell method represents the fluid mass in fixed cells and tracks particle

cell boundary interaction. An interesting feature of the moving grid variant is that

few particles are needed to resolve large pressure gradients. The concept of discretising

mass into moving mass packets was first developed by Pasta & Ulam (1959) and more

11

recently formed the basis of the energy conserving finite mass method (Gauger et al.

2000). Unlike these approaches, however, the VFL method does not exhibit mesh-

tangling because a Voronoi diagram is generated at each time step (see Augenbaum

1984) rather than moving a mesh of fixed connectivity .

Mesh tangling The use of the Voronoi diagram is not the only means of avoiding

mesh-tangling. Harlen et al. (1995) present a split Eulerian-Lagrangian scheme for

viscoelastic fluids that retains the nodes as material points and reconnects them to

produce the optimal Delaunay triangulation. This offers an advantage over the free-

Lagrange method in that the discrete fluid equations can be solved using standard finite

elements. This avoids one complication with the Voronoi diagram -the property that

the number of sides of a cell may vary over each time interval.

Petera & Nassehi (1996) presents a Galerkin-Lagrange finite element approximation

of a shallow water model for tidal flow which are challenged by fluctuations between dry

and wetted regions. Their basis for node adjustment is a physical one. The nodes of the

mesh are only moved if the neighbourhood of the node changes state between wetted

and dried. This approach avoids excessive mesh distortions but requires a smoothing

operator to eliminate numerical oscillations introduced by succesive modifications of

the mesh in very shallow regions.

Other variational methods Buneman (1982) describes a two-dimensional Eulerian

model based upon the Clebsch representation of Hamilton’s principle. The stability

of the approach appears to be contingent on the invariance of the discrete Hamilton’s

principle under particle relabelling. Salmon (1983) presents a numerical analogue of a

variational shallow water blob model. The curious feature of the design is the flexibility

to allow the blobs to subdivide which appears to be motivated by its application to

the study of the ocean’s main thermocline. A relevant outcome of his work is the

observation that the potential vorticity conservation is difficult to enforce numerically.

Variational formulations of smooth particle hydrodynamics (SPH) methods have

also been developed (see Bonet & Rodriguez-Paz 2005, and the references by the same

first author therein). The authors use the variational principle to derive spatially

dependent smoothing lengths - a limiting feature in conventional SPH methods is the

use of a fixed smoothing length. In particular, there appears to be a notable advantage

in using a variational SPH for very compressible fluids.

Poisson bracket methods An alternative approach to discretising Hamilton’s ac-

tion principle is to formulate numerical schemes which satisfy a Poisson-bracket. By

12

doing so, this approach captures symmetries of the continuum motion and correspond-

ing conservation laws. Strauss & Longcope (1998) construct an adaptive unstructured

zero-residual Galerkin finite element approximation of the vorticity stream function

formulation of the incompressible 2D Magnetohydrodynamical equations. This ap-

proximation exhibits the same anti-symmetry property of the Poisson bracket, proving

critical to magnetic flux and energy conservation. The authors do not, however, at-

tempt the more difficult problem of formulating approximations which also satisfy an

additional property of the Poisson bracket, namely, the Jacobi identity. This has been

the pursuit of Salmon (2004) in the context of shallow water leading to a scheme which

conserves energy and potential enstrophy. The expressions are somewhat complex,

however, perhaps to the detriment of flexibility of the approach.

Boundary Conditions Cotter, Frank & Reich (2004), Frank, Gottwald & Reich

(2002) and Frank & Reich (2004) have successfully applied a geometric numerical

method, referred to as the Hamiltonian Particle-Mesh (HPM) method, to numerous

shallow water models on the torus and the sphere, but did not consider boundary con-

ditions. Several approaches to implementing boundary conditions for smooth particle

hydrodynamics (SPH) methods (Monaghan 2002) with particle approximations of the

Euler equations are discussed by Vila (1999), namely, (i) image or ghost particles (ii)

boundary particles and forces, and (iii) semi-analytic techniques. We note a potential

extension of this approach to variational SPH methods (see Bonet & Rodriguez-Paz

2005). The image particle approach is, however, successfully used by the Hamiltonian

finite mass method of Klinger, Leinen & Yserentant (2005) and we shall consider the

implementation of the image particle approach in the HPM method.

1.3.1 Important related works

Simo’s computational framework Juan Carlos Simo investigated a unified ap-

proach for deriving mechanical integrators on Lie groups for continuum dynamics, as

described in Lewis & Simo (1994). Mechanical integrators are geometric integrators

which conserve any two of energy, symplectic structure or momentum, but not all three.

Simo showed that, given any Hamiltonian flow on a Lie group, the integrated con-

strained canonical Hamiltonian equations on phase space are equivalent to the Co-

adjoint orbits on the reduced Poisson manifold through the momentum map. This

result is the principle governing Simo’s framework, in which momentum preserving

integrators on Lie groups are derived from a discrete time update of the momentum

map.

Simo then used an algorithmic exponential to construct multiplicative integrators

on canonical phase space, which in the case when the group action is SO(3), take the

13

form

Λk+1 = Λkcay(Θ), Λk ∈ SO(3), Θ ∈ R3

Pk+1 = cay(−Θ)Pk, Pk ∈ T∗ΛkSO(3),

(1.1)

where cay : R3 → SO(3) denotes the Cayley transform and Θ is arbitrary at this stage

but can be constrained so that the discrete flow on the phase space is a mechanical

integrator.

1.4 Contributions of this Thesis

The results and partial results that are presented in this thesis are the culmination of

key independent ideas and developments which are now stated.

1.4.1 A computational framework

This thesis pursues the development of a unified computational framework for deriving

geometric integrators for the convective and spatial representation of continuum dy-

namics. This computational framework transfers powerful concepts given by the unified

framework of Holm et al. (1986), for the convective and spatial Hamiltonian continuum

dynamics, to computational continuum dynamics. Specifically,

1 Holm et al. (1986) show that the group action for passing between the represent-

ations generates an infinitesimally equivariant momentum map which carry the

Poisson brackets in one representation to those of the other. Using the discrete

Clebsch approach (Cotter & Holm 2006), we give the corresponding (diagonal)

group actions for passing between the representations and their momentum maps

from the cotangent bundle to the dual of the Lie algebra of the group.

2 Holm et al. (1986) show that the equations of continuum motion with advected

quantities are coadjoint orbits for the action of a semi-direct product Lie-algebra

on the dual of a semi-direct product Lie algebra. These orbits are symplectic foli-

ations of the Poisson manifold P defined by the augmented cotangent bundle. We

show that the discrete Clebsch approach gives discrete equations of motion with

advected quantities which define co-adjoint orbits for the action of the semi-direct

product group on the dual of its Lie-algebra. These orbits are also symplectic

foliations of P . The co-adjoint actions preserves the ± Lie-Poisson brackets on

the dual of this semi-direct product Lie algebra.

14

3 Holm et al. (1986) show through various examples, that these momentum maps

encode fundamental conservation laws of Hamiltonian continuum dynamics. These

conserved momentummaps are generated from the Noether symmetries for passing

to the spatial and convective representations, the latter of which is referred to

as a particle relabelling symmetry. We show that the discrete Clebsch approach

gives

I conserved momentum maps for the polar decomposed pseudo-rigid body

which are the conserved spatial angular momentum and the discrete Kelvin

circulation theorem generated by the respective rotational and material re-

labelling symmetries.

II conserved momentum maps for the geometrically exact elastic rods which are

the total spatial angular momentum generated by the rotational symmetries.

For the latter case, there is a fundamental difference between the form of the

momentum maps derived from the continuum framework and our computational

framework, however. Our computational framework represents the continuum

as a finite dimensional system of particles. The conserved momentum map then

takes the form of a discrete sum over all particle labels and only recovers the form

of the conserved momentum map for the continuum, in the continuum limit of

the particle system.

4 Our computational framework gives a prototype MV integrator for the convective

and spatial representations of compressible fluids.

Metric tensors are central to the theory of continuum mechanics. Holm et al.

(1986) consider a compressible fluid flow, in which the passage to the convective

and spatial representation is by reduction under the group of diffeomorphisms,

and show how the metric tensor and densities respectively transform in the differ-

ent representations. Analogously, we consider ellipsoidal motion and demonstrate

how our framework transforms the metric tensor (the Cauchy-Green matrix) and

shape matrices under reduction by the group GL(n,R)+ to the convective and

spatial representations. The discrete Clebsch approach gives MV integrators for

these representations in which the metric and shape matrix are respectively the

advected quantities.

1.4.2 Development of new DMV algorithms

This thesis shows that, under a forward in-time finite difference approximation of the

continuous Clebsch constrained action principle for the body representation of the rigid

body, that the discrete Clebsch approach recovers the Moser-Veselov integrator. MV

15

integrators are computed using the explicit DMV algorithm developed by McLachlan

& Zanna (2005) for solving the associated matrix Ricatti equation. In parallel with the

development of our computational framework, as described above, we give new DMV

algorithms to solve for the MV integrators and verify their conservative properties by

numerical experiment.

• Rigid body motions in the spatial representation We modify the DMV algorithm,

for the body representation of rigid bodies, to solve the spatial MV integrat-

ors. We then provide several numerical experiments to study the comparative

conservative properties of the spatial DMV algorithm with the DMV algorithm.

The results are largely conclusive and show that, in general, the spatial angular

momentum error profiles differ by a factor of 103.

• Rigidly Coupled motions We develop a DMV algorithm for solving the coupled

matrix Ricatti equation. This equation arises from coupled rigid body motion,

examples of which include the free rigid body motions of the coupled rigid body

and the circulatory and rotational polar components of the pseudo-rigid body

motion. We implement a model of a Mooney-Rivlin (Mooney & Rivlin 1977)

type pseudo-rigid body to describe the stretching and rotational components of

the motion and show that the DMV algorithm conserves angular momentum and

vorticity (relative to the Lagrangian frame) to an order of 10−15 and the energy

error exhibits no secular drift with a mean to the order of 10−3 (over 104 time

steps).

• Elastically coupled motions We solve a system of elastically coupled MV integrat-

ors for the elastically coupled director motions of the geometrically exact elastic

rod. The DMV algorithm for an elastically coupled rigid body differs from that

of the rigidly coupled rigid body. In the former case, the coupling is only through

the source term and not, as in the latter case, through the Coriolis term and re-

quires minor modification. The discrete Clebsch approach also gives a variational

integrator for the material representation of the rod centroid positions which is

equivalent to a Stormer-Verlet symplectic integrator. The DMV algorithm for a

rod of 50 sections conserves total spatial angular momentum to an order of 10−8,

linear momentum to an order of 10−11 and energy levels exhibit no secular drift

with a mean error to the order of 10−2 (after 104 time steps).

1.4.3 Geometric integrators for shallow water

In the absence of a general theory for the finite dimensional representation of the

group of diffeomorphisms, this thesis considers geometric integrators for shallow water

which preserve the canonical symplectic structure of a finite dimensional system of

16

fluid particles and are expressed in terms of the material velocity and the Eulerian

layer depth.

• Variational free-Lagrange method A variational free-Lagrange method for rotat-

ing shallow water with bottom topography is presented. We will establish the

conservative properties of the semi-discrete shallow water equations and derive

the semi-discrete shallow water divergence conservation law and potential vorti-

city evolution equations. The semi-discrete divergence equation is given by the

extrema of the discrete action principle. The semi-discrete potential vorticity

equation, however, is formed through a choice of a discrete curl operator which

is not resolved from the extrema of the action principle. This suggests the need

for an additional constraint in the action principle to constrain the form of the

discrete curl operator so that a semi-discrete potential vorticity conservation law

is also exhibited by the VFL method. Numerical results are also presented which

show that the VFL method for 1D rotating shallow water conserves energy to

an order of the integrator and exhibits the geostrophic adjustment mechanism of

rotating shallow water.

• Boundary conditions in the HPM method We extend the HPMmethod to bounded

rotating shallow water flows by implicity introducing ghost or image particles. We

demonstrate that the HPM approximation of rotating shallow water in a bounded

domain conserves mass, exhibits no secular drift in the energy level and remains

stable over long-time simulations. We also simulate the motion of a vortex pair

in a channel of rotating shallow water and show the motion of the vortex pair as

it reaches the channel wall.

1.5 Overview

Chapters 2 and 3 both start with largely abstract details and end with specific examples

and details of numerical experiments.

Chapter 2 We begin with a review of the geometric description of the free rigid body

and the necessary preliminaries of geometric mechanics. We derive Moser-Veselov in-

tegrators for body and spatial representations of rigid body motions using a discrete

variational, referred to as the discrete Clebsch approach (Cotter & Holm 2006). The

continuous time Clebsch approach provides a systematic means of deriving the Euler

Poincare equations from a symmetry reduced Hamilton’s action principle and the mo-

mentum maps (see Marsden & Ratiu 1999) associated with this symmetry reduction.

Through examples of a top and a coupled rigid body we develop a geometric de-

scription of the discrete time motion in both representations and then compare these

17

descriptions with their well-known continuous counterparts. Where appropriate, we

verify that the momentum maps of the integrators are conserved and use numerical ex-

periment primarily to (i) validate the conservative properties of the DMV algorithms,

(ii) compute the Casimirs and (iii) in the case of the rigid body, compare the numer-

ical with the analytic solution. We assess the comparative performance of the DMV

algorithm with the explicit Lie-Poisson integrator of McLachlan (1993) which is based

on a splitting of the rigid body Hamiltonian.

We also demonstrate the utility of the discrete Clebsch approach by deriving a MV

integrator for motions on SU(2) which corresponds to the Cayley-Klein parameterisa-

tion of discrete time Lagrange top motion. This parameterisation avoids the notorious

problem of Gimble-lock which occurs under Euler angle parameterisations.

Chapter 3 further applies the discrete Clebsch approach to derive Moser-Veselov

integrators for the convective and spatial representations of ellipsoidal motion. In Sec-

tion 3.1.1, we then derive a MV integrator for the polar decomposed pseudo-rigid body

motion on GL(3)+. We show that the discrete Clebsch approach gives two conserved

momentum maps representing the angular momentum and encoding a Kelvin circula-

tion theorem, given in Section 3.2.2. We also describe a new algorithm for solving the

coupled MV integrators for the internal circulatory and rotational motions.

We then develop a discrete form of the Kirchhoff rod analogy theorem which states

that the motion of a Lagrange top in discrete time is in one-to-one correspondence

with the frames of an isotropic symmetric inextensible elastic rod at equilibrium. By

extending this analogy further, we show that the dynamical discrete rod motion is in

one-to-one correspond with an elastically coupled series of Lagrange tops. This rod

motion exhibits a discrete compatibility equation of the form of the discrete auxiliary

equation for the relative orientation matrix in the coupled rigid body model. We then

derive a MV integrator for an extensible (and shearable) elastic rod model, referred to

by Krishnaprasad et al. (1988) as a geometrically exact elastic rod model. Numerical

results show the conservative properties of the DMV algorithms for solving the MV

integrators for the pseudo-rigid body and geometrically exact elastic rod model.

Chapter 4 We turn to rather more implementational aspects of variational integ-

rators for canonical Hamiltonian fluid dynamics. We consider a rotating shallow water

model since it is ubiquitous in geophysical fluid dynamics and extend geometric numer-

ical methods to represent the layer-depth in a Voronoi diagram. The use of a Voronoi

diagram is a key step in developing variational moving-mesh based numerical methods

which avoid the difficulty associated with tangling. We start by deriving a variational

free-Lagrange method for rotating shallow water with bottom topography and then

validate conservative properties by analysis and numerical experiment. The method

18

generates a Voronoi diagram from the position of particles at each time step. Numer-

ical experiments of 1D rotating shallow water demonstrate conservation of energy to

the order of the integrator. This approach is well suited to long-time simulations on

the sphere and seeks application in global climate models.

Chapter 5 We finally address the formulation of boundary conditions in geomet-

ric integrators for rotating shallow water by extending the Hamiltonian particle mesh

(HPM) method to bounded shallow water models. By treating planar boundaries as a

particle symmetry, we show how image or ghost particles may be implicitly introduced

into the HPM method. We also derive a finite element approximation of the Helm-

holtz operator which is used to dispersively regularise the layer depth and thus relaxes

the CFL stability constraint. A Neumann boundary condition on the regularised layer

depth is imposed naturally so that the resulting Helmholtz matrix is symmetric.

Appendix A gives a series of tables comparing the MV integrators with the continu-

ous Euler-Poincare equations for various rigid body motions, MV integrators for the

Cayley-Klein parameterisation of the rigid body formulated as SU(2) matrices and qua-

ternions. Appendix A also gives the DMV algorithms for solving MV integrators for the

spatial representation of the rigid body and the coupled rigid body. Appendix B gives

the Euler-Poincare description of the anisotropic pseudo-rigid body and the Lagrange-

Poincare description of the geometrically exact elastic rod. Finally, Appendix C gives

a summary of the Hamiltonian aspects of the variational free-Lagrange method and a

useful modelling step, referred to rezoning for computing the material variables over a

fixed mesh.

19

Chapter 2

Moser-Veselov Integrators for

Spatial and Body

Representations of Rigid Body

Motions

Synopsis In this review Chapter, we consider the problem of formulating Moser-

Veselov (MV) integrators for the body and spatial representations of rigid body motions

as a prototype for a unified variational framework for the convective and spatial repres-

entations of continuum dynamics. We apply the discrete Clebsch approach of Cotter &

Holm (2006) to (i) derive conserved momentum maps corresponding to symmetries of

the discrete Lagrangian and then (ii) discrete Euler equations for discrete rigid body

motions. The body representation of these discrete Euler equations match the integ-

rable discretisation of rigid body motion discovered by Moser & Veselov (1991). For

this reason, McLachlan & Zanna (2005) refer to these discrete Euler equations as a

Moser-Veselov (MV) integrator. The discrete Euler equations are also equivalent to

the discrete Euler-Poincare equations later given by Bobenko & Suris (1999a). For

consistency with McLachlan & Zanna (2005) , we will refer, however, to these discrete

Euler equations as MV integrators as this is more natural terminology for discussing

the computational aspects of these equations.

In the spatial representation, the framework gives a spatial variant of the MV

integrator which has an additional equation for the advection of the inertia matrix.

This integrator, again, matches the discrete Euler-Poincare equations with an advected

quantity given by Bobenko & Suris (1999a).

Using the discrete Clebsch approach, we use examples of heavy tops and coupled

rigid bodies to derive body MV integrators for motion driven by a potential and coupled

20

motion and give their corresponding DMV algorithms. We also observe that a MV

integrator for the Cayley-Klein parameterised rigid body follows from our framework.

In Section 2.10, we demonstrate the conservative properties of the DMV algorithms

for the heavy top and coupled rigid body by numerical experiment. This Chapter

reviews the preliminary theory and computations necessary to address MV integrators

and DMV algorithms for pseudo-rigid body and elastic rod motions in Chapter 3.

2.1 Introduction

Holm et al. (1986) showed that the body and spatial representations of rigid body mo-

tion correspond, respectively, to the convective and spatial representations of continuum

dynamics. The convective representation expresses the kinematics of the continuum in

terms of the convective velocity and the spatial or, equivalently, Eulerian representa-

tion expresses the kinematics in terms of the spatial velocity. In order to define these

velocities, we shall briefly review the terminology and notations of geometric continuum

dynamics given by Holm et al. (1986).

Continuum dynamics preliminaries The configuration of a material point (or

label) ` ∈ B, where B is the continuum reference space, is a diffeomorphic map g ∈

Diff(B) (a smooth invertible map with a smooth inverse) to a spatial point in the

continuum container (taken to be the Euclidean three space R3)

x(`) = g ∙ `. (2.1)

The motion of a material point ` is a time dependent curve g(t) ∈ Diff(B) defining

a trajectory of the material point in the container

x(t, `) = g(t) ∙ `. (2.2)

Definition 2.1.0.1 (The spatial velocity). The spatial velocity is the time derivative

of the motion evaluated at a fixed spatial point and takes the right invariant form

u(x, t) = g(t)g−1(t) ∙ x. (2.3)

Conversely, the configuration of a (fixed) spatial point x ∈ R3 is the inverse map

g−1 ∈ Diff(R3) to a material point ` in the continuum reference space

` = g−1 ∙ x. (2.4)

The motion of a (fixed) spatial point x, is a time dependent curve g−1(t) ∈ Diff(R3)

defining a trajectory of the spatial point in the reference space

21

`(t) = g−1(t) ∙ x. (2.5)

Definition 2.1.0.2 (The convective velocity). The convective velocity is the time de-

rivative of the motion of a spatial point, evaluated at a fixed material coordinate and

takes the left invariant form

V(`, t) = −g−1(t)g(t) ∙ `. (2.6)

We will review the correspondence between the spatial and convective representa-

tions of continuum mechanics and the spatial and body representations of rigid body

dynamics (Holm et al. 1986) once we have revisited the standard description of rigid

body dynamics in the next Section.

MV integrators With a view to developing a unified computational framework for

both spatial and convective representations, we will apply the discrete Clebsch approach

(Cotter & Holm 2006), developed for continuum dynamics, to derive the conserved

momentum maps associated with the Noether symmetries of the discrete Lagrangian for

body and spatial representations of rigid body dynamics in discrete time. Through these

momentum maps, we will show that the discrete Clebsch approach yields a known class

of variational integrators which were discovered by Moser & Veselov (1991). Following

McLachlan & Scovel (1995), these integrators have become commonly referred to as

Moser-Veselov (MV) integrators.

Body MV integrators It is well known that the body representation of these in-

tegrators has two distinguishing features:

1 Firstly, the time discretisation preserves the integrability of the body represent-

ation of the continuous rigid body motion. Integrable motions exhibit the same

number of invariants as their number of degrees of freedom (Moser & Veselov

1991). Motion in discrete time is consequently on the intersection of the level

sets of the same invariants and gives dynamics which are consistent with the con-

tinuous motion. We discuss the geometric properties of the MV integrator at the

end of Section 2.10.

2 Secondly, discrete Moser-Veselov integrators are solved using explicit algorithms

and hence avoid iterative computations with conditional convergence criteria. We

will review the performance of these algorithms by numerical experiments, the

details of which are provided in Section 2.10.

22

Spatial MV integrators

1 We will show that the application of the discrete Clebsch approach in the spatial

representation gives a spatial MV integrator with an (auxiliary) equation for the

advection of the Inertia matrix. These equations are the same as the discrete

Euler-Poincare equations with an advected quantity derived by Bobenko & Suris

(1999a). Bobenko & Suris (1999a) show these discrete Euler-Poincare equations

are Lie-Poisson w.r.t. to the dual of a semi-direct product Lie algebra, a result,

which we confirm for the spatial MV integrators for the rigid body.

2 The spatial DMV algorithms for solving the spatial MV integrators are also ex-

plicit. We give an algorithm for solving the spatial MV integrator in Section

A.3 of the Appendix and provide several numerical experiments to compare the

conservative properties of the spatial with the standard body DMV algorithm in

Section 2.10.

Caveat 2.1.0.3. Preservation of the Lie-Poisson structure on the dual of the semi-

direct product Lie algebra and demonstration of the conservative properties of the al-

gorithm by numerical experiment render the spatial MV integrator for the rigid body

a suitable prototype for the geometric integration of Hamiltonian continuum dynamics

with advected quantities.

Heavy tops and coupled rigid bodies We apply the discrete Clebsch approach

to give MV integrators for the body representation of both rigid body motions in a

potential field and coupled rigid body motions, referred to as heavy tops and coupled

rigid bodies respectively. We also consider the application of the DMV algorithms to

these integrators. These examples prepare us for the modelling of more challenging

models in Chapter 3 which exhibit potential and motion coupling terms.

Cayley-Klein parameters With a view to developing a unified computational vari-

ational framework, we will show that the discrete Clebsch approach also gives MV

integrators for the Cayley-Klein parameterisation of the rigid body. This Chapter

largely reviews MV integrators for rigid body motions formulated as SO(3) matrices.

These matrices represent the attitude of the rigid body as a three-stage rotation by

each Euler angle about its corresponding principal axis. Parameterisation of the rigid

body dynamics in terms of these angles and their momenta gives rigid body equations

which exhibit singularities (see Leimkuhler & Reich 2005). These singularities cause

Gimble-lock, a loss of degree of freedom of motion when one of the three stages of

rotation maps a principal axis to the previous position of another principal axis. The

combination of Euler-angles resulting in a singularity must be excluded and for this

reason, the Euler-angle parameterisation is referred to as non-global.

23

It is preferable to represent the rotation in terms of four Cayley-Klein parameters,

which form a global parameterisation (see Whittaker 1944). The Cayley-Klein para-

meters constitute the group of unit quaternions and are isomorphic to the matrix group

SU(2) consisting of all 2-by-2 unitary matrices with unit determinant. The discrete

Clebsch approach can be easily cast in terms of SU(2) matrices to give a MV integrator

for the Cayley-Klein parameterisation of rigid body motion together with the conserved

momentum maps corresponding to symmetries of the discrete Lagrangian. In doing so,

the framework gives a three-way correspondence between the equations of motion and

momentum maps formulated as matrices of SO(3), SU(2) and quaternions. The cor-

respondence between the first two can be observed by comparing Tables (A.1, pg. 193)

and (A.2, pg. 195) of Appendix A. Table A.2 compares the SU(2) and quaternionic for-

mulation of the MV integrator for the Cayley-Klein parameterised rigid body equations

and the conserved momentum maps.

Numerical experiments Finally, Section 2.10 presents numerical experiments which

demonstrate the conservative properties, computational efficiency and accuracy of the

numerical solutions of the body and spatial representations of the SO(3) formulated

MV integrators. We now begin by recalling the geometric description of the free rigid

body given by Marsden & Ratiu (1999).

2.2 The Free Rigid Body

In this Section, we review the geometric description and notation for the free rigid

body given in (Marsden & Ratiu 1999, Chapter 15) as a discrete time problem in

both the body and the spatial representations. Much of these follow the notations and

conventions reviewed in the previous Section and given in Holm et al. (1986).

Consider a free rigid body as a solid body, occupying a reference configuration

B ⊂ R3, which is free to move in R3 by rotations about a fixed point. Material points

` ∈ B are position vectors whose components, relative to a fixed orthonormal basis

(E1,E2,E3) in B, are the material coordinates. For the rigid body, a configuration of B

is a C1, invertible and orientation preserving map ψ : B → R3 from material points to

spatial points in R3. The spatial points are position vectors whose components, relative

to (e1, e2, e3), the right-handed orthonormal basis of R3 are spatial coordinates. This

basis is commonly referred to as the spatial (or Eulerian) frame.

Adapting the terminology for the discrete time case, we define a discrete motion

ψk := ψ(tk) of B as a family of discrete time dependent configurations of B. The discrete

motion gives a k parameterised sequence of spatial points representing the position of

a material point at time tk

xk = ψk ∙ `, k ∈ Z+. (2.7)

24

The discrete motion satisfies ψ0 ∙ ` = `. Marsden & Ratiu (1999) explain how the last

property together with rigidity of the body and continuity of the motion imply that the

configuration of B may be identified with the matrix SO(3) and the k parameterised

sequence of spatial points is given by

xk = Λk`, Λk ∈ G = SO(3), (2.8)

where, for notational convenience, Λk := Λ(tk). Λ is commonly referred to as the

attitude of the body.

The body coordinates of a material position vector are its components relative to a

time-dependent basis (ξ1, ξ2, ξ3)(tk) which is defined by

ξi(tk) = ΛkEi, i := 1→ 3, (2.9)

and hence is attached to the rigid body that rotates about the origin. This basis is

commonly referred to as the body frame.

The spatial position vector relative to the spatial frame is equal to the material position

vector relative to the body frame

xk = x(tk)i︸︷︷︸spatial

ei = Λk `i︸︷︷︸material

Ei = `i︸︷︷︸body

ξi(tk). (2.10)

Remark 2.2.0.4 (The continuous angular velocities). We recall from the geometric

description of the continuous rigid body given by Marsden et al. (1999), that the body

angular velocity Ω and spatial angular velocity ω are respectively given by

Ω = ΛTt Λt, ω = ΛtΛT , Λ ∈ SO(3). (2.11)

The body angular velocity is left invariant and corresponds (up to a minus sign) to the

convective velocity of continuum dynamics given in equation (2.6). The spatial angular

velocity is right invariant and corresponds to the spatial velocity of continuum dynamics

given in equation (2.3).

2.2.1 Discrete velocities

Moser & Veselov (1991) introduce the notion of a discrete velocity. In order to define

this, we first recall from the definition of the motion that the position of material points

in the container at time tk is given by

xk = Λk`. (2.12)

25

Conversely, the position of spatial points in the reference space at time tk is given by

`k = ΛTk x. (2.13)

Substituting equation (2.13) into equation (2.12) evaluated at time tk+1, gives the

following relation between spatial points at consecutive times

xk+1 = ωk+1xk, (2.14)

where ωk+1 = Λk+1ΛTk is referred to by Moser & Veselov (1991) as the discrete spatial

angular velocity.

Similarly, substituting equation (2.12) into equation (2.13) evaluated at time tk+1,

gives the following relation between material points at consecutive times

`k+1 = ΩTk+1`k, (2.15)

where Ωk+1 = ΛTkΛk+1 is referred to by Moser & Veselov (1991) as the discrete body

angular velocity. The two velocities are related to each other by a rotation

Ωk+1 = ΛTk ωk+1Λk. (2.16)

Order of the discrete velocities

The reason why Moser & Veselov (1991) refers to Ωk+1 and ωk+1 as discrete velocities is

because they approximate their continuous velocity counterparts to O(Δt2). To show

this, we revisit once more the geometric description of the (continuous) free rigid body

given by Marsden & Ratiu (1999), which describes the dynamics of the rigid body

on velocity phase space in terms of the body angular velocity by the reconstruction

formula

Λ(t) = Λ(t)Ω. (2.17)

The solution to this equation is given by a t-parameterised curve on G = SO(3)

Λ(t) = Λ0exp{Ωt}. (2.18)

Expressing this equation as an incremental solution

Λ(t+Δt) = Λ(t)exp{ΩΔt}, (2.19)

26

and then taking the step described by Lewis & Nigam (2003) of replacing the exponen-

tial function with the Cayley transform

cay : so(3)→ SO(3) , cay(Ω) = (Id +Ω

2)(Id −

Ω

2)−1, (2.20)

where Id denotes the identity matrix, gives

Λ(t+Δt) = Λ(t)(Id +ΩΔt

2)(Id − Ω

Δt

2)−1. (2.21)

Keeping only the first two terms of the binomial expansion of the denominator and

neglecting the remaining O(Δt2) term gives

Λ(t+Δt) = Λ(t)(Id +ΩΔt). (2.22)

Comparing this equation evaluated at time t = kΔt with the definition of Ωk+1 and

recalling the definition Λk := Λ(tk) gives the O(Δt2) approximation of Ω

Ωk+1 = Id +ΩΔt, (2.23)

or, equivalently, the finite difference approximation

Ω ≈ΛTkΔt(Λk+1 − Λk), (2.24)

which satisfies the definition of Ωk+1, given in equation (2.16), when substituted into

equation (2.23). By analogy with the reconstruction formula given in equation (2.17)

we refer to the equation

Λk+1 = ΛkΩk+1, (2.25)

as the discrete reconstruction formula. This formula reconstructs the dynamics on

G×G from Ωk+1.

This explanation holds for the spatial representation also. We note that the (con-

tinuous) body and spatial angular velocities are right and left invariant respectively, so

too are the corresponding body and spatial discrete angular velocities.

With the spatial and body discrete angular velocities defined, we now consider the

geometric mechanics of rigid body motion in discrete time by following Moser & Veselov

(1991).

27

2.3 Discrete Constrained Variational Principle

Moser & Veselov (1991) consider a functional

Sd =∑

k

L(Λk,Λk+1), (2.26)

where the discrete Lagrangian L : G×G→ R is a smooth function defined as

Lk := L(Λk,Λk+1) = Tr(ΛkI0ΛTk+1)︸︷︷︸

Kinetic energy

, (2.27)

in which I0 is a positive definite, symmetric and constant matrix referred to, in the

context of rigid body mechanics, as the inertia matrix.

Remark 2.3.0.1. The discrete Lagrangian is the first order finite difference approxim-

ation of the continuous Lagrangian Tr2

(ΛI0Λ

t). We show this by substituting Λ(tk) ≈

(Λk+1−Λk)h into the continuous Lagrangian to give

Tr

h2((Λk+1 − Λk) I0

(ΛTk+1 − Λ

Tk

))=Tr

h2(Λk+1I0Λ

Tk+1 + ΛkI0Λ

Tk − Λk+1I0Λ

Tk − ΛkI0Λ

Tk+1

)

=Tr

h2(I0 − Λk+1I0Λ

Tk

)

= −Tr

h2(ΛkI0Λ

Tk+1

)+Tr(I0)

h2,

(2.28)

where we have made use of the properties that (i) the trace operator is both invariant

under cyclic permutations Tr(AB) = Tr(BA) and transposition Tr(AB) = Tr(BTAT )

and (ii) orthogonality of the attitude matrix ΛTkΛk = Id. The last termTr(I0)h2is ignored

because it is constant in the body frame.

With the use of the following definition given by Wendlandt & Marsden (1997)

(which is stated in a general form for any Lie group G), we may state the invariance

properties of this Lagrangian.

Definition 2.3.0.2 (Diagonal action (Wendlandt & Marsden 1997)). The (left) diag-

onal action of G on G×G is defined as Ψ : G× (G×G)→ G×G | Ψ(f, (g, h)) =

f ∙ (g, h) = (fg, fh). Denote by Ψf , for all f ∈ G, the continuous linear transformation

Ψf : G×G→ G×G given by (g, h)→ Ψ(f, (g, h)).

The discrete time Lagrangian in equation (2.27) is invariant under the (left) di-

agonal action of Ψg. For this reason the group action of G on itself is referred to

as a symmetry of the discrete Lagrangian. In the continuous case, it is well known

28

that this symmetry gives a new reduced Lagrangian by a process referred to as sym-

metry reduction (Marsden & Ratiu 1999). The reduced Lagrangian for the rigid body

is expressed solely in terms of the body angular velocity (rather than velocity phase

space variables) and by Euler-Poincare reduction (Marsden & Ratiu 1999), gives the

Euler-Poincare equations defining the reduced dynamics.

It is natural to question the analogous process of symmetry reduction in the discrete

case, if only to obtain sufficient intuition to distinguish this process between the discrete

and continuous cases. We begin in the next Section by using recent work by Leok,

Marsden and Weinstein Leok et al. (2004) to help us visualise the otherwise largely

abstract notions described in the remainder of this Chapter.

2.4 Symmetry Reduction

Differential geometric aside Leok, Marsden & Weinstein (2004) show that a prin-

cipal G-Bundle furnishes the geometric description of the symmetry reduction of the

discrete Lagrangian to the body representation. A principal G-bundle is a bundle

(Q,S, π) where G acts freely on a bundle space B by left translation and is isomorphic

to (Q,Q/G, πQ/G). For the rigid body discrete Lagrangian, the bundle space Q = G×G,

the shape space is G ' G × G/G and the projection π : Q → S is isomorphic to the

natural projection πQ/G : Q → Q/G. At time tk, the natural projection is given by

the invariant action Ψ(ΛTk , (Λk,Λk+1)). The bundle space and natural projection of

the principal G-bundle furnishing the geometric description of the symmetry reduction

of the rigid body Lagrangian to the body representation are illustrated in Figure 2.1

below.

Body representation Following Moser & Veselov (1991), we reduce the discrete

Lagrangian defined on G×G to the body reduced Lagrangian l : G→ R defined on G

and given in body variables by

l(Ωk+1) = Tr(Ωk+1I0) (2.29)

The lowercase l denotes that the Lagrangian has been reduced from G×G to G.

Remark 2.4.0.3. This Lagrangian can be also obtained from the reduced continuous

Lagrangian by substituting the finite difference approximation Ω(tk) ≈ΛTkh (Λk+1 − Λk)

into the continuous time Lagrangian Tr2

(ΩI0Ω

T). As shown in the previous Section,

this finite difference approximation is consistent with a second order approximation of

the Cayley transform of ΩΔt which provides an algorithmically convenient form of the

exponential function (Lewis & Nigam 2003).

29

Ωk+1

e

G=SO(3)

G=SO(3)

ΨΛTk

ΨΛTk

Λk

Λk+1

Figure 2.1: This Figure shows the principal G-Bundle (Q,S, π) furnishing the description of symmetryreduction of the discrete Lagrangian for the rigid body at time tk (Leok et al. 2004). This bundle consists ofa bundle space Q = G × G, a shape space S = G ' G × G/G (not shown on the Figure) and a projectionπ : Q → S which is isomorphic to the natural projection πQ/G = Q → G × G/G. At time tk, this naturalprojection is defined by the diagonal action of ΛTk on (Λk,Λk+1) and is illustrated by the two curved arrows.

The constrained coordinate formulation Still following Moser & Veselov (1991),

we take one further step and embed G in the linear space of real matrices V (the

symmetric part of which is denoted V ) and use holonomic constraints in the form of

matrix Lagrange multipliers Θk+1 to constrain the family of curves satisfying δSd = 0

to G. This formulation is explained in Wendlandt & Marsden (1997) in which it is

referred to as the constrained coordinate formulation. An alternative formulation (which

is shown by these authors to be equivalent), is referred to as the generalised coordinate

formulation. This formulation uses a coordinate chart to extremise the discrete action

principle directly on any G. We do not pursue this alternative approach here but

instead consider the holonomically constrained Lagrangian lc : V → R defined in body

variables as

lc(Ωk+1) = Tr(Ωk+1I0)− Tr(Θk+1(Ωk+1Ω

Tk+1 − Id)

)(2.30)

30

in which the trace operator gives the pairing between elements of V and V ∗, and Θk+1

is a symmetric matrix Lagrange multiplier enforcing the orthogonality constraint on

Ωk+1. Note that Id denotes the identity matrix and the superscript c on lc denotes that

the reduced discrete Lagrangian is holonomically constrained. We shall now derive the

reduced discrete Lagrangian in the spatial representation.

Spatial representation The spatial representation distinguishes itself from the body

representation by not only expressing the dynamics in terms of discrete spatial angular

velocities but also by having an additional dynamical variable, the inertia matrix. The

inertia matrix, which is fixed in the body frame, rotates relative to the spatial frame

of the body. Holm et al. (1986) show that the rotation of the inertia matrix, as the

rigid body moves relative to the spatial frame, provides an example of a much more

general physical process in continuum dynamics, referred to as advection. Our purpose

here, is to show how the approach taken by Moser & Veselov (1991) can be extended

to include advected quantities. The spatial representation of the rigid body provides a

convenient example to explain this.

To transform to spatial variables, we follow Holm et al. (1986) and define a Lag-

rangian on the augmented space,

LI0 : G×G→ R, (2.31)

given by (Λk,Λk+1)→ L(Λk,Λk+1, I0) where

L(Λk,Λk+1, I0) = Tr(ΛkI0ΛTk+1). (2.32)

The notation LI0 denotes that I0 ∈ V ∗ becomes a dynamical variable in the spatial

representation. To discuss the symmetries of this Lagrangian, we must first define how

G acts on V ∗. Let φLg (I0) be the left group translation on V∗ given by the map

φLg : V∗ → V ∗ | φLg (I0) = g ∙ I0 = gI0g

T . (2.33)

The corresponding right group translation is given by the map

φRg∗ : V ∗ → V ∗ | φR(I0) = I0 ∙ g = g

−1I0g−T , (2.34)

which is equal to the left translation g−1 ∙I0. We use the definition of these translations

to describe how G acts on the augmented state space G×G× V ∗.

Definition 2.4.0.4 (Augmented diagonal action). The (right) augmented diagonal

31

action of G on G × G × V ∗ is defined as Ψ : (G × G × V ∗) × G → G × G ×

V ∗ | Ψ((g, h, a), f) = (g, h, a) ∙ f = (gf, hf, f−1af−T ). The continuous linear trans-

formation Ψf : G×G× V ∗ → G×G× V ∗ is given by (g, h, a)→ Ψ((g, h, a), f).

The right translation Ψg leaves the discrete time Lagrangian LI invariant

LI0∙g(Λkg,Λk+1g) = LI0(Λk,Λk+1). (2.35)

We observe this property from the form of LI0

LI0∙g(Λkg,Λk+1g) = Tr(Λkgg−1I0g

−T gTΛTk+1). (2.36)

Symmetry reduction of the Lagrangian by the augmented diagonal action gives the

reduced Lagrangian lI : G→ R given in spatial variables as

lIk(ωk+1) = Tr(ωk+1Ik). (2.37)

Analogous to the body reduced Lagrangian, we shall take one further step and define

a holonomically constrained Lagrangian lcI : V → R given by

lcIk(ωk+1) = Tr(ωk+1Ik)− Tr(Θk+1(ω

Tk+1ωk+1 − Id)

). (2.38)

This equation defines the spatial representation of the discrete Lagrangian considered

by Moser & Veselov (1991) which is extended to include the advected inertia matrix.

In order to transfer concepts in continuous geometric mechanics to the discrete

formulation, we need to review how the body and spatial representations of the discrete

Lagrangians give corresponding equations of motion and conservation laws using the

discrete Clebsch approach (Cotter & Holm 2006).

2.5 Clebsch Potentials and Momentum Maps

We shall give a brief introduction to an approach for obtaining encodings of the mo-

mentum conservation laws in continuous time, referred to as the Clebsch approach. In

order to do so, we must define some standard terminology of geometric mechanics. This

is best done in the context of an example.

2.5.1 Geometric preliminaries

We revisit the classical continuous time rigid body once again, only this time we will

review the construction of vector fields on the configuration space Q = G, where G =

SO(3).

32

The trajectories of the rigid body are described by the left translation of G on itself

Φg : G → G, for all g ∈ G, given by h → Φ(g, h) = gh. Recall that the motion of

the rigid body Λ(t) defines a continuous t-parameterised path with the property that

Λ(0) = Id and Λ(0) = ζ, ζ ∈ g = so(3). The infinitesimal generator of Φg in the

direction of ζ is the vector field ζG on G defined by

ζG(g) =d

dt

∣∣t=0ΦΛ(t)(g). (2.39)

The path Λ(t) = exp(tζ) is an integral curve of ζG(g) and by definition

Λ(t) = ζG(Λ(t)) = ζΛ(t), (2.40)

where Λ(t) ∈ TΛ(t)G. The Lagrangian for this motion is the smooth function

L : TG→ R, (2.41)

that takes a form for which Φg is a symmetry transformation of the Lagrangian- its

tangent lift on TG leaves L invariant. Analogously, Φg is a symmetry of the Hamiltonian

H : T ∗G→ R if the cotangent lift of Φg on T ∗G leaves H invariant. For example, when

the symmetry transformation is given by the left action of G, the cotangent lift of this

translation Φ′g([Λ, P ]) = (gΛ, g−TP ) and Φ′g(Λ) is a symmetry of H if H ◦ Φg(z) =

H(z), z ∈ T ∗G. The cotangent bundle is an example of a more general manifold

referred to as a Poisson manifold (P, {, }), equipped with a Poisson bracket. It is

conventional to simply denote the Poisson manifold as P.

The action Φ : G × P → P is referred to as canonical if Φg is a Poisson map for

every g ∈ G, that is

{F1 ◦ Φg, F2 ◦ Φg} = {F1, F2} ◦ Φg, (2.42)

for every smooth scalar function F1, F2 ∈ F(P).

Recall that the infinitesimal generator of Φ on G in the direction of ζ is the vector

field ζG on G given by equation (2.40). The corresponding infinitesimal generator for

the action of Φ on P in the direction of ζ is the vector field ζP on P defined by

ζP =d

dt

∣∣t=0Φg(t)(z). (2.43)

A Hamiltonian vector field on P is the vector field XH defined by

XH [F ] = {F,H}, (2.44)

for all smooth scalar functions F ∈ F(P). It follows from the property of the Poisson

bracket that XH [H] = 0 meaning that the Hamiltonian vector field conserves the

33

Hamiltonian. We are interested in the case when ζP is globally Hamiltonian, i.e. there

is some global Hamiltonian, a linear map Jζ : P → F(P), such that XJζ = ζP. We

will now clarify why we are interested in a global Hamiltonian of this form through the

following definition.

Definition 2.5.1.1 (Momentum Map (Marsden & Ratiu 1999)). Let G act canonically

on a Poisson manifold P. A momentum map for this action is a map J : P→ g∗ such

that the map Jζ : P→ F(P) : Jζ(z) = 〈J(z), ζ〉 satisfies

XJζ = ζP, ∀ζ. (2.45)

Momentum maps are conserved quantities when they correspond to symmetries.

The following fundamental theorem of geometric mechanics formalises this statement.

Theorem 2.5.1.2 (Noether’s theorem (Marsden & Ratiu 1999)). The Hamiltonian

version of Noether’s theorem states that if Φ acts canonically on P with a momentum

map J such that H is G invariant, then J is conserved by the flow of XH .

Theorem 2.5.1.3 (Momentum maps for lifted actions (Marsden & Ratiu 1999)). If

ζP is a vector field on P = T∗G given by the cotangent lift of the infinitesimal action of

G to T ∗G, then ζP is Hamiltonian with an infinitesimally equivariant momentum map

J : P→ g∗ given by

〈J([Λ, P ]), ζ〉 = 〈P, ζG(Λ)〉. (2.46)

The brackets on the right hand-side denote the pairing of T ∗ΛG with TΛG and on

the left hand-side, the pairing of g∗ with g.

By the theorem of Canonical Momentum maps (Marsden & Ratiu 1999), it fol-

lows that all infinitesimally equivariant momentum maps are Poisson. This property

is fundamental to the development of geometric integrators for (reduced) continuum

dynamics, since the existence of a momentum map in a discrete geometric framework

implies preservation of Poisson structure in the reduced phase space g∗.

2.5.2 The Clebsch approach

The naming of this approach is misleading since Clebsch did not pioneer this approach

as the name suggests. In fact, there a complex history behind this approach which

we do not attempt to detail here but rather refer the reader to Seliger & Whitham

(1968). The most crucial step appears to have originated from the difficulties that Lin

(1963) encountered when formulating a variational principle for continuum mechanics

in Eulerian variables because of the non-canonical form of Hamilton’s principle. Lin

(1963) was able to derive the equations of isentropic fluid flow by using the Clebsch

34

representation (Clebsch 1859) in which the Eulerian velocity is, in the most basic case,

expressed in terms of Clebsch potentials :

u = ∇χ+ λ∇μ. (2.47)

So by expressing the velocity field in a general form, Lin (1963) was able to recover the

equations of motion for the Clebsch potentials from a Hamilton’s principle. The con-

nection was also made between the potential representation of Electromagnetic fields

and the variational principle for Maxwell’s equations. This connection was soon exten-

ded to a wider range of idealised flows by numerous authors (see, for example, Seliger

& Whitham 1968, Bretherton 1970). Seliger & Whitham (1968) pointed out that Lin’s

approach remained somewhat of a mystery as a mathematical device. Moreover, at

that time, no general methodology existed for deriving equations of continuum motion

in the Eulerian representation from the canonical Hamilton’s principle, expressed in

the material representation.

Marsden & Weinstein (1974) addressed this by explicating a geometric approach for

deriving Euler-Poincare equations of motion using abstract Clebsch variables, the theory

of symmetry reduction and canonical maps. In the context of fluid mechanics, the

Clebsch variables take the form of potentials but in other contexts may not necessarily

do so. Cendra & Marsden (1987) refer to this abstract Clebsch setting as the Clebsch

approach for deriving variational principles for symmetry reduced motions and illustrate

it with the example of the free rigid body.

Example: free rigid body motion Given the left invariant Lagrangian `(Ω) for

the body representation of the free rigid body, Cendra & Marsden (1987) define a new

Lagrangian, which in our notation, is given by ˜ : T (G × V × V∗) → R, for a linear

vector space V ⊂ Rn×n. For the rigid body, n = 3 and G = SO(3) and ˜ takes the form

˜(A,Λ, Λ, P, P ) = `(Ω) + 〈P, Λ− ΩG(Λ)〉, (2.48)

where Ω = AT A, P and Λ are the Clebsch variables, ΩG(Λ) is the infinitesimal generator

of the action of G on Λ in the direction of Ω and the equation paired with P , is the Lin

constraint (we will refer to this constraint from hereon as the Clebsch constraint only

for simplicity of terminology). Note that the˜ symbol over ` denotes the addition of

a Clebsch constraint. In this context, the Lagrange multiplier P denotes the canonical

rigid body momentum. Ω is a solution to the Euler equations if (Ω,Λ, P ) is a critical

point of Hamilton’s action principle

S =

∫ t2

t1

`′dt. (2.49)

35

By the standard procedure of calculus of variations, we obtain the Clebsch representa-

tion for the body representation of the free rigid body

Ωˆ1 =(ΛTP )23 − (ΛTP )32

2I1, Ωˆ2 =

(ΛTP )31 − (ΛTP )132I2

, Ωˆ3 =(ΛTP )12 − (ΛTP )21

2I3,

(2.50)

where Ij are the principal moments of inertia.

Example: perfect incompressible fluid motion To connect this example with

the standard notion of Clebsch potentials, consider the Hamilton’s action principle for

Euler’s equations for a perfect incompressible fluid which takes the form

S =

∫ t2

t1

∫C1

2||u(`, t)||2 + 〈P (`, t), ∂ta(`, t) + u(`, t) ∙ ∇a(`, t)〉d

3`dt, (2.51)

where a and P are the passively advected Clebsch potentials. A special case which eases

comparison between the forms of the Clebsch constraint for perfect fluids and the rigid

body is when a(`, t) = X(`, t) is a fluid parcel position and P (`, t) is the corresponding

conjugate momenta.

Stationarity of S gives the familiar form of the Clebsch representation of the Eu-

lerian velocity in terms of these Clebsch potentials

ui = −∂`ia(`, t) ∙P(`, t) + ∂`iφ, (2.52)

where φ is a Lagrange multiplier paired with an incompressibility condition. Equation

2.52 is the perfect fluid analogue of the rigid body form of the Clebsch potential rep-

resentation given in equation 2.50. So, although the form of the Clebsch representation

is context dependent, the procedure for its derivation is systematic and amenable to

formulation of a discrete analogue, the discrete Clebsch approach of Cotter & Holm

(2006).

2.5.3 The discrete Clebsch approach

The discrete Clebsch approach (Cotter & Holm 2006), which we will now briefly in-

troduce, provides a systematic means of deriving momentum maps for cotangent lifted

actions in the discrete Lagrangian framework. Consider first the left reduced continuous

Lagrangian on g defined as l : g→ R. A Clebsch (Lin) constrained reduced Lagrangian

l, is given by

l = l(Ω) + 〈P, Λ− ΛΩ〉︸︷︷︸Clebsch constraint

, (2.53)

36

in which 〈∙, ∙〉 denotes the pairing of elements in T ∗ΛG with TΛG and Ω is the body

angular velocity.

Just as the discrete reduced body Lagrangian given by equation (2.29) follows from

a finite difference approximation of l, so too does the discrete form of l. To show this,

we approximate the reconstruction formula for describing the dynamics on velocity

phase space from the body angular velocity Ω

Λ(t) = Λ(t)Ω, (2.54)

with finite differences to give

Λ(tk)− Λ(tk)Ω =1

h

((Λk+1 − Λk)− ΛkΛ

Tk (Λk+1 − Λk)

)+O(h2)

=1

h(Λk+1 − ΛkΩk+1) +O(h

2).

(2.55)

This equation evaluates to the discrete reconstruction formula for the body represent-

ation of the reduced rigid body dynamics in discrete time given in equation (2.25).

We then formulate this discrete reconstruction formula in constrained coordinates by

embedding G in V, the linear space of all 3-by-3 real matrices as described in Wend-

landt & Marsden (1997). The Clebsch constraint is reparameterised by h to give

Pk := hP (tk) ∈ T ∗ΛkV. This constraint represents the (reparameterised) canonical mo-

mentum of the rigid body at time tk. The Clebsch constrained body reduced discrete

Lagrangian thus takes the form

lk+1 = lc(Ωk+1) + 〈Pk+1,Λk+1 − ΛkΩk+1〉, (2.56)

where the superscript k on lk is used to denote that the Lagrangian is evaluated at

time tk.

Stationarity of the discrete action principle can be expressed in terms of variations

in the dynamical variables

δSd =∑

k

δlk+1 =∑

k

〈∇Λk lk+1, δΛk〉

+ 〈∇Λk lk+1, δΛk+1〉

+ 〈∇Pk+1 lk+1, δPk+1〉

+ 〈∇Ωk+1 lk+1, δΩk+1〉 = 0,

(2.57)

where ∇v lk : V → V∗ denotes the gradient of lk with respect to the linear matrix v ∈ V ,

and is defined in Bobenko & Suris (1999a) by the formula

37

〈∇v lk, w〉 =

d

dε

∣∣ε=0

lk(v + εw), ∀w ∈ V . (2.58)

The term in equation (2.57) paired with δΛk is the discrete Euler-Lagrange equation

〈∇Λk lk+1 +∇Λk l

k, δΛk〉 = 0, (2.59)

which evaluates to

〈−Pk+1ΩTk+1 + Pk, δΛk〉 = 0, (2.60)

giving the discrete flow of Pk as

Pk+1 = PkΩk+1. (2.61)

In the same way, the stationary discrete action principle also recovers the discrete

reconstruction formula

Λk+1 = ΛkΩk+1, (2.62)

from the expression paired with variations Pk+1.

Symplectic flow on the cotangent bundle Equations (2.61) and (2.62) define

the (discrete) flow map on zk := [Λk, Pk] as a cotangent lifted right translation (see

Marsden & Ratiu 1999) ΦΩk+1(Λk) of the form zk+1 = Φ′Δt(zk) := Φ

′Ωk+1(zk) :=

(ΛkΩk+1, PkΩ−Tk+1).

The discrete flow on the cotangent bundle given by equations (2.76) preserves the

symplectic two-form dPk ∧ dΛk.

Proof.

dPk+1 ∧ dΛk+1 = d(PkΩk+1) ∧ Ωk+1dΛk

= −Ωk+1dΛk ∧ dPkΩk+1

= −dΛk ∧ dPkΩk+1ΩTk+1

= dPk ∧ dΛk.

(2.63)

The discrete flow on the cotangent bundle preserves the symplectic two-form. Re-

ferring to Hairer et al. (2002) we point out that this property is equivalent to the

statement that the flow is Poisson w.r.t. the canonical Poisson bracket {, } on the

cotangent bundle meaning that

38

{F1 ◦ φ, F2 ◦ φ}(zk) = {F1, F2}(zk+1). (2.64)

This is a convenient point at which to define the fibre derivative for the discrete

Lagrangian. This definition will be used to prove the proceeding discrete Noether’s

theorem given by Wendlandt & Marsden (1997) in generalised coordinates.

Definition 2.5.3.1 (Discrete Time Fibre Derivative (Wendlandt & Marsden 1997)).

The fibre derivative for the discrete (unreduced) Lagrangian on G×G is a smooth map

of the form

FL : G×G→ T ∗G , (g1, g0) 7→ (g0,∇g0L(g0, g1)) , (2.65)

where ∇gL : G→ T ∗G denotes the gradient of L with respect to the group g ∈ G, and

is defined in Bobenko & Suris (1999a) by the formula

〈∇gL, g〉 =d

dε

∣∣ε=0

L(g(ε)). (2.66)

We shall now state the discrete Noether’s theorem which is adapted from Wendlandt

& Marsden (1997) for the discrete Clebsch approach.

Theorem 2.5.3.2 (Discrete Noether’s Theorem). If the diagonal lift of the left action

Φ to the diagonal action Ψ, is a symmetry of the discrete Lagrangian L : G × G →

R, then the corresponding symplectic flow Φ′Δt (corresponding to the discrete Euler-

Lagrange equations) on the cotangent bundle preserves the (left) momentum map JLk+1◦

Φ′Δt = JLk , where J

Lk+1 := J

L([Λk, Pk]).

Proof. We follow the general proof, given in (Wendlandt & Marsden 1997, Section

3.3), for any Lie group g ∈ G. The proof is most conveniently formulated in generalised

coordinates. Invariance of Lk := L(Λk,Λk+1) under the diagonal (left) action Ψ implies

invariance of the discrete action principle

d

dε

∣∣ε=0

Sd =∑

k

d

dε

∣∣ε=0

L (exp(Ωε)Λk, exp(Ωε)Λk+1) =∑

k

d

dε

∣∣ε=0

Lk = 0. (2.67)

Using the chain rule we express the derivative of Lk w.r.t. to ε in terms of derivatives

of Lk in Λk and Λk+1 giving

d

dε

∣∣ε=0Lk = 〈∇ΛkL

k, ζG(Λk)〉+ 〈∇Λk+1Lk, ζG(Λk+1)〉 = 0. (2.68)

Substitution of this expression into the discrete action principle gives, after a shift of

the time parameterisation index, the discrete Euler-Lagrange equations paired with

39

ζG(Λk+1) which take the form

〈∇Λk+1Lk+1 +∇Λk+1L

k, ζG(Λk+1)〉 = 0. (2.69)

Subtracting the expression in equation (2.68) from this last equation gives

〈∇Λk+1Lk+1, ζG(Λk+1)〉 = 〈∇ΛkL

k, ζG(Λk)〉. (2.70)

Substituting the definition of Pk = ∇ΛkLk, from the definition of the fibre derivative

given above for the discrete Lagrangian Lk, and the definition for the infinitesimal

generator for the left action of G on itself into the above expression gives

〈Pk+1, ζΛk+1)〉 = 〈Pk, ζΛk〉. (2.71)

Substituting the discrete reconstruction formula Λk+1 = ΛkΩk+1 from equation

(2.25) into the above expression gives

〈Pk+1 − PkΩ−Tk+1, ζG(Λk)〉 = 0. (2.72)

The image of the fibre derivatives given by the discrete Euler-Lagrange equations is

the discrete symplectic flow (see lemma 2.5.3 on pg. 38 for a proof that this flow is

symplectic) on T ∗G given by

Λk+1 = ΛkΩk+1,

Pk+1 = PkΩ−Tk+1.

(2.73)

Equation (2.71) gives the statement of conservation of the infinitesimally equivariant

left momentum map

〈skew(Pk+1ΛTk+1)− skew(PkΛ

Tk ), ζ〉 = 0. (2.74)

So the symplectic flow on the cotangent bundle corresponding to the left invariant

discrete Lagrangian conserves the right infinitesimally equivariant momentum map.

We will now use the discrete Clebsch approach in the body and spatial representa-

tions to derive momentum maps and the equations of motions.

40

2.5.4 The body representation

The Clebsch constrained reduced discrete Lagrangian given in equation (2.56) takes

the form

lk+1 = Tr (I0Ωk+1) +Tr

2

(P Tk+1(Λk+1 − ΛkΩk+1)

)− Tr

(Θk+1(Ωk+1Ω

Tk+1 − Id)

),

(2.75)

where the second term is the Clebsch constraint for the discrete auxiliary equation and

the last term is the holonomic constraint on Ωk+1. Recall that the stationary discrete

action principle gives the discrete symplectic flow on the cotangent bundle

Pk+1 = PkΩk+1,

Λk+1 = Λk Ωk+1.(2.76)

The momentum maps gives the corresponding preserved geometric structure on the

reduced phase space of the rigid body so(3)∗.

Derivation of momentum maps The derivative ∇Ωk+1 l, paired with the variation

δΩk+1 in the discrete action principle given by equation (2.57) is a Clebsch relation

which evaluates to

I0 −1

2ΛTk Pk+1 = Θk+1Ωk+1. (2.77)

We use the symmetry property of Θk+1 and the expression for Pk+1 given by equa-

tion (2.76) to give

I0ΩTk+1 −

1

2ΛTk Pk = Ωk+1I0 −

1

2P Tk Λk. (2.78)

This equation can be written as

JRk+1 = skew(ΛTk Pk), (2.79)

which by the theorem of momentum maps for lifted actions (see 2.5.1.3) satisfies the

definition of a right momentum map

〈JRk+1, ζ〉 = 〈skew(ΛTk Pk), ζ〉

= 〈Pk,Λkζ〉

= 〈Pk, ζG(Λk)〉.

(2.80)

Holm, Marsden & Ratiu (1998) show that the momentum map for cotangent lifted

actions takes the form

41

JRk+1 = Pk � Λk, (2.81)

where the bilinear operator � : V∗ × V → g∗ is defined by the pairing

〈Pk � Λk, ζ〉 = 〈Pk, ζG(Λk)〉, (2.82)

where T ∗ΛG ⊂ V . The diamond operator, as notation, thus makes explicit the property

that equation (2.81) is a momentum map for cotangent lifted actions.

The spatial angular momentum is conserved The discrete Noether’s theorem

states that the left momentum map JLk+1 is preserved by the symplectic flow on the

cotangent bundle corresponding to the left invariant form of the discrete Euler-Lagrange

equations. The spatial angular momentum is the left momentum map

mk+1 = ΛkMk+1ΛTk = Λkskew(Λ

Tk Pk)Λ

Tk = skew(PkΛ

Tk ) = J

Lk+1. (2.83)

Substitution of equations (2.76) defining the discrete symplectic flow on the cotangent

bundle T ∗G into the right momentum map given in equation (2.81) gives the recursion

relation on the cotangent bundle

ΛTk Pk = ΩTkΛ

Tk−1Pk−1Ωk. (2.84)

The right momentum map projects the skew-symmetric component of this equation

onto so(3)∗, giving the discrete Euler equation for the body representation of rigid

body motion

Mk+1 = Ad∗ΩkMk, Mk+1 = I0Ω

Tk+1 − Ωk+1I0, (2.85)

where the body angular momentum Mk+1 ∈ so(3)∗ is defined as

Mk+1 := 2skew(∇Ωk+1 lkΩT ), (2.86)

in which ∇g lk : G → T ∗G denotes the gradient of lk with respect to the group G =

SO(3), and is given by equation (2.66).

Equation (2.85) defines a Moser-Veselov integrator on the dual of the Lie algebra

so(3)∗ as a co-adjoint action of G. The co-adjoint action is defined by the pairing

(Marsden & Ratiu 1999)

〈Ad∗gα, ζ〉 = 〈α,Adgζ〉, (2.87)

42

where ζ ∈ g, α ∈ g∗and Ad : G × g → g : Adgζ = gζg−1. Substituting the form of

Adg into this pairing

〈gTαg−T , ζ〉 = 〈α, gζg−1〉, (2.88)

implying that Ad∗g takes the form Ad∗gα = gTαg−T .

Notational remark 2.5.4.1. Moser & Veselov (1991) define the Moser-Veselov in-

tegrator in slightly different notation. In their notation, Mk = ωTk J − Jωk, where

ωk := XTk Xk−1 is the discrete body angular velocity, Xk is the attitude of the body

and J is the fixed inertia matrix. ωk is not to be mistaken with our definition of

the discrete spatial angular velocity. A comparison of Mk, given by equation (2.85),

with the definition in Moser & Veselov (1991) gives the relation between the different

notations for the discrete body angular momentum Ωk = −ωTk . Table A.1 of Appendix

A provides a summary comparing our notation with Moser & Veselov (1991). Our

notation is chosen to be as consistent as possible with the convention of continuous

geometric mechanics (see Marsden & Ratiu 1999), for the purpose of developing a uni-

fied discrete framework for transferring concepts in continuous geometric mechanics to

computational models. In particular, the use of Ωk and ωk to denote the respective body

and spatial angular discrete velocities is necessary for exposition of a discrete unified

computational framework. We also find it convenient to define Ωk and ωk so that the

discrete time reconstruction formulae

Λk = Λk−1Ωk Λk = ωkΛk−1, (2.89)

take a similar form to the continuous reconstruction formulae

Λ = ΛΩ Λ = ωΛ. (2.90)

Remark 2.5.4.2 (Relation to Bobenko’s and Suris’s discrete Euler-Poincare equa-

tions). Bobenko & Suris (1999a) give, in our notation, the discrete Euler-Poincare

equation on G as

Mk+1 = Ad∗ΩkMk, (2.91)

where the body angular momentum Mk+1 = dΩk+1 lk is expressed in terms of left Lie

derivatives of lk w.r.t. to Ωk+1, and dgl : G→ g∗ denotes the (left) Lie derivative of L

w.r.t. to g and is given in Bobenko & Suris (1999a) by the formula

〈dgl, ζ〉 =d

dε

∣∣ε=0

l(exp(εζ)g), ∀ζ ∈ g. (2.92)

The relation between the gradient ∇gl and the Lie derivative of l w.r.t. to g is given in

43

Bobenko & Suris (1999a) by

〈T ∗Rg∇gl, ζ〉 = 〈dgl, ζ〉, ∀ζ, (2.93)

where T ∗Rg is the cotangent lift of right translations in the group given by the pairing

〈T ∗RgP,A〉 = 〈P, TRgA〉, P ∈ T ∗gG,A ∈ TgG, (2.94)

and TRgA is the tangent lift of the right translation by g on A given by TRgA = Ag.

Using the expression for the tangent lift of the right translation of g, equation (2.94)

evaluates to

〈PgT , A〉 = 〈P,Ag〉. (2.95)

It follows from the definition of T ∗Rg and equation (2.93) that

〈dΩk+1 lk, ζ〉 = 〈∇Ωk+1 l

kΩTk+1, ζ〉 (2.96)

and the discrete EP equation in equation (2.91) is therefore (up to a factor of 2) the

Moser-Veselov integrator (2.85) that we recovered by application of the discrete Clebsch

approach.

We now review the properties of MV integrators.

Moser-Veselov integrators

Poisson integrators MV integrators for the body representation of the rigid body

belong to a class of Poisson structure preserving integrators referred to as Poisson

integrators. Hairer et al. (2002) give the following definition of a Poisson integrator.

Definition 2.5.4.3 (Poisson Integrator (Hairer et al. 2002)). A discrete flow map φΔt

defined on a Poisson manifold (M, {, }) is Poisson with respect to the Poisson bracket

{, } if

{F1, F2} ◦ φΔt = {F1 ◦ φΔt, F2 ◦ φΔt}, (2.97)

where F1 and F2 are scalar functions. If ΦΔt defines a path on the level sets of Casimir

functions C (which Poisson commute with any function F , {C,F}=0) of this Pois-

son bracket, then ΦΔt is referred to as a Poisson integrator. (Lie-Poisson integrators

preserve the Lie-Poisson structure {, } on the dual of the Lie algebra).

Definition 2.5.4.4 (The so(3)∗ Lie-Poisson bracket for the rigid body (Moser &

Veselov 1991)). The so(3)∗(' R3) Lie-Poisson bracket for the body representation of

the (continuous) rigid body is given by

{F1, F2}(M) = −Tr(∇TMC[∇MF1,∇MF2]), (2.98)

44

where F1 and F2 are scalar functions, ∇MF1 denotes the skew-symmetric matrix of

partial derivatives ∂F1/∂Mij, C =||M ||222 is the Casimir of this bracket and M is the

body angular momentum of the rigid body in so(3)∗.

Definition 2.5.4.5 (Body MV flow map (Moser & Veselov 1991)). Equation (2.85)

defines a discrete flow Mk+1 = ΦΔt(Mk) where ΦΔt is the discrete flow map taking the

form of the co-adjoint action

ΦΔt(Mk) = ΦΩk(Mk) := Ad∗ΩkMk. (2.99)

ΦΔt defines a Lie-Poisson integrator with respect to the so(3)∗ Lie-Poisson bracket

for the rigid body (Moser & Veselov 1991).

Proof. We verify that M ′ = Φg(M) is Poisson w.r.t. to the Lie-Poisson bracket given

by equation (2.100). From the definition of the so(3)∗ Poisson bracket for the rigid

body

{F1 ◦ Φg, F2 ◦ Φg}(M) = −Tr(∇TMC[g∇M ′F1g

T , g∇M ′F2gT ])

= −Tr(g(∇TM ′C

)gT [g∇M ′F1g

T , g∇M ′F2gT ])

= −Tr((∇TM ′C

)[∇M ′F1,∇M ′F2]) = {F1, F2}(M

′),

(2.100)

which proves that Φg is Poisson.

ΦΔt preserves the Casimir ||M ||2 of the rigid body bracket and it follows that ΦΔtsatisfies the definition of a Lie-Poisson integrator.

Remark 2.5.4.6. Rigid body Lie-Poisson integrators hence define a discrete flow on

the intersection of the momentum sphere (the Casimir of this bracket) and approximate

energy ellipsoids (approximated to within an order of Δt).

DMV algorithm Cardoso & Leite (2001) cast equation (2.85) into a discrete matrix

Ricatti equation and solve the eigenvalue problem using the Hamiltonian for this Ricatti

equation. McLachlan & Zanna (2005) provide a detailed description of the explicit

algorithm for implementing these steps and propose an optimised DMV algorithm. This

optimised algorithm avoids the use of a Schur factorisation to compute the eigenvalues

and eigenvectors of the Hamiltonian Ricatti equation. The numerical experiments

presented in Section 2.10 are performed using the unoptimised version of the DMV

algorithm, however.

45

2.5.5 The spatial representation

We now derive the equations of motion in the spatial representation by applying the

discrete Clebsch approach. Recall that the motivation for deriving these equations in

the spatial representation is to formulate a prototype MV integrator for the spatial

representation of continuum dynamics, in which quantities are advected. Following

Holm et al. (1986), who show that the spatial representation of the rigid body (in which

the inertia matrix is advected) corresponds to the spatial representation of continuum

dynamics, we pursue a MV integrator for the spatial representation of the rigid body.

In spatial variables, Ik and ωk := ΛkΛTk−1, the Clebsch constrained discrete Lagrangian

lIk is given by adding Clebsch constraints to the reduced discrete Lagrangian given in

equation (2.38)

lIk := Tr (Ikωk+1) +Tr

2(P Tk+1(Λk+1 − ωk+1Λk)

︸︷︷︸Clebsch constraint for Λk

+Tr

2(Jk+1(Ik+1 − ωk+1Ikω

Tk+1))

︸︷︷︸Clebsch constraint for Ik

−Tr(Θk+1(ωk+1ω

Tk+1 − Id)

).

(2.101)

Remark 2.5.5.1. The description of the Clebsch constraint for the evolution of Λk

is analogous to the description given in Section 2.5.3. The Clebsch constraint for the

evolution of Ik follows from expressing Ik = ΛkI0ΛTk as a recursion in Ik.

Stationarity of the Clebsch constrained discrete action principle

δSd =∑

k

lIk = 0, (2.102)

gives, as terms paired with variations in δΛk and δIk, the respective equations

[∇Λk lIk(Λk−1,Λk)] + [∇Λk lIk+1(Λk,Λk+1)] = 0,

[∇Ik lIk(Ik−1, Ik)] + [∇Ik lIk+1(Ik, Ik+1)] = 0.(2.103)

Evaluation of the derivatives of lIk in each equation w.r.t. Λk and Ik gives

[Pk/2] + [−ωTk+1Pk+1/2] = 0,

[Jk/2] + [∇Ik lIk − ωTk+1Jk+1ωk+1/2] = 0.

(2.104)

where the matrix derivative ∇Ik lIk evaluates to ∇Ik lIk = sym(ωk+1). Subsequent re-

arrangement of these equations gives,

46

Pk+1 = ω−Tk+1Pk,

Jk+1 = ωk+1(2∇Ik lIk + Jk)ωTk+1.

(2.105)

These two equations together with the discrete reconstruction formula for Λk and the

discrete auxiliary equation for Ik define the discrete flow on the augmented cotangent

bundle T ∗(G× V ∗).

Definition 2.5.5.2 (The discrete flow on the augmented cotangent bundle). The

discrete flow Φ′Δt : T∗(G × V ∗) → T ∗(G × V ∗) on the augmented cotangent bundle

zk ∈ T ∗(G× V ∗) is a smooth map zk+1 = Φ′Δt(zk) given by

Λk+1 = ωk+1Λk,

Pk+1 = ω−Tk+1Pk,

Ik+1 = φωk+1(Ik),

Jk+1 = φω−Tk+1(2∇Ik lIk + Jk).

(2.106)

The Clebsch relation, the expression paired with δωk+1, gives

(Ik − P

Tk+1Λ

Tk /2− Jk+1ωk+1Ik

)ωTk+1 = Θk+1. (2.107)

Using the symmetry property of the holonomic constraint Θk+1, the above equation

gives the map

mk+1 = skew(Pk+1ΛTk+1) + [Jk+1, Ik+1], (2.108)

where the spatial angular momentum mk+1 ∈ so(3)∗ is defined as

mk+1 := IkωTk+1 − ωk+1Ik. (2.109)

Using this definition of mk+1, we show that the spatial angular momentum is related

to the body angular momentum Mk+1 by the transformation

mk+1 = ΛkI0ΛTkΛkΛ

Tk+1 − Λk+1Λ

TkΛkI0Λ

Tk

= Λk(I0ΛTk+1Λk − Λ

TkΛk+1I0)Λ

Tk

= Λk(I0ΩTk+1 − Ωk+1I0)Λ

Tk

= ΛkMk+1ΛTk .

(2.110)

We shall now show that the map given by equation (2.108) is a left momentum map.

47

Lemma 2.5.5.3. The map mk : T∗(G× V ∗)→ g∗ given by

mk = skew(PkΛTk ) + [Jk, Ik] (2.111)

is an infinitesimally equivariant left momentum map of the general form

JL = Λ � P + J � I, (2.112)

for cotangent lifted left actions of ζ ∈ g on the augmented cotangent bundle T ∗(G×V ∗).

Proof. We first state the form of the left action of ζ on G×V ∗ and then use the pairing

between the augmented cotangent and tangent space to verify the form of the map

given by equation (2.111).

Consider the case when Λ is an element of any finite dimensional Lie group G. The

left translation on G× V ∗, for all g ∈ G, is given by the map

φg : G× V∗ → G× V ∗ : φg(Λ, I0) := (gΛ, gI0g

T ). (2.113)

The infinitesimal generator of φg on G × V ∗ is the left action of the Lie algebra ζ on

G× V ∗ given by

ζG×V ∗(Λ, I) := (ζΛ, ζI + IζT ), (2.114)

where I = gI0gT . The pairing between elements of the augmented cotangent space

T ∗(Λ,I)(G× V∗) and the augmented tangent space T(Λ,I)(G× V

∗) is defined as

〈(P, J), (ζΛ, ζI + IζT )〉 = 〈P, ζΛ〉+ 〈J, ζI + IζT 〉, (2.115)

where the first term on the right hand side pairs elements of T ∗ΛG and TΛG and the

second term pairs elements of the symmetric matrices V and TvV ' V ∗. Expressing

the last equation as a pairing of elements of g and g∗ gives

〈PΛT + 2JI, ζ〉 = 〈(P, J), (ζΛ, ζI + IζT )〉. (2.116)

By the theorem of momentum maps for lifted actions (Marsden & Ratiu 1999) given,

for convenience, on page 34, the expression on g∗ is an infinitesimally equivariant left

momentum map

JL = Λ � P + J � I. (2.117)

The bilinear operator � pairing elements in the first term of this expression is defined

as � : V ×V∗ → g∗, where T ∗ΛG ⊂ V∗, G ⊂ V and in the second term, � : V × V ∗ → g∗.

48

Rotations For the case when g ∈ G = SO(3) and g = so(3)∗, equation (2.116) takes

the form

〈skew(PΛT ) + [J, I], ζ〉 = 〈(P, J), (ζΛ, [ζ, I])〉, (2.118)

and setting Λ = Λk, P = Pk, I = Ik and J = Jk in this expression finishes the proof.

Discrete equations of motion Substituting the discrete flow equations on the aug-

mented co-tangent bundle into the left momentum map gives the spatial representation

of the discrete EP equations with an advected parameter

mk+1 = skew(Pk+1ΛTk+1) + [Jk+1, Ik+1]

= Ad∗ωTk+1

(skew(PkΛ

Tk ) + [Jk + 2∇Ik lIk , Ik]

)

= Ad∗ωTk+1(mk + 2[∇Ik lIk , Ik]) ,

(2.119)

which, together with the discrete auxiliary equation for Ik give the spatial MV integ-

rator

Ad∗ωk+1mk+1 = mk + 2∇Ik lIk � Ik,

Ik+1 = φωk+1(Ik),(2.120)

where the bilinear operator � is defined as � : V × V ∗ → so(3)∗.

These equations define the spatial representation of the MV integrator and are dis-

tinguished from the body representation by a discrete auxiliary equation for Ik and

an additional term 2∇Ik lIk(ωk+1) � Ik contributing to the expression for mk in so(3)∗.

These equations match the discrete Euler-Poincare equations for the spatial represent-

ation of the rigid body given by Bobenko & Suris (1999a) (up to a factor of 2 in front

of the derivative of lIk).

Using the invariance property of ∇Ik lIk(ωk+1) under transformations

∇Ik lIk = φ∗ωTk+1(∇Ik lIk) = ωk+1∇Ik lIkω

Tk+1, (2.121)

where φ∗g : V → V denotes the left translation on V by g, for all g ∈ G, defined by the

pairing between elements of V and V ∗

〈φ∗g(v), I〉 = 〈v, φg(I)〉, (2.122)

49

the spatial MV integrator, given by equations (2.120), may be written as

mk+1 = Ad∗ωTk+1

mk + 2∇Ik lIk � φωk+1Ik,

Ik+1 = φωk+1(Ik).(2.123)

The relation of the spatial and body MV integrators to their continuous time coun-

terparts are summarised in Figure 2.5.5.

Ad∗Λ−1(m, I)

M = ad∗ΩM←−−→ m = ad∗ωm+∇I lI � I

Ad∗Λ(M, I0)

↓ Ω ≈ΛTkh (Λk+1 − Λk) ω ≈ (Λk+1 − Λk)

ΛTkh ↓

Ad∗(Λk)−1(mk, Ik)

Mk+1 = Ad∗ΩkMk

←−−→ mk+1 = Ad

∗ωTk+1(mk + 2∇Ik lIk � Ik)

Ad∗(Λk)(Mk, I0)

Figure 2.2: The relation between the continuous body and spatial representations andthe body and spatial representations of the rigid body dynamics. Bobenko & Suris(1999a) show that the spatial representation of the discrete Euler Poincare equationfor the rigid body is also integrable.

Conservation of spatial angular momentum Before reviewing the geometric

properties of the spatial MV equations, we firstly verify, through the momentum map

given by equation (2.111), that the spatial angular momentum is conserved by the

discrete flow on the augmented cotangent bundle.

Lemma 2.5.5.4 (Conservation of spatial angular momentum). The spatial MV integ-

rator given by equations (2.120) conserves the spatial angular momentum mk.

Proof. Evaluating the expression for ∇Ik lIk and using the form of the bilinear operator

a � b = [a, b] gives

2∇Ik lIk(ωk+1) � Ik = [ωk+1 + ωTk+1, Ik]

= ωTk+1Ik − Ikωk+1 − (IkωTk+1 − ω

Tk+1Ik)

= Ad∗ωk+1(IkωTk+1 − ωk+1Ik)−mk+1

= Ad∗ωk+1mk+1 −mk+1.

(2.124)

50

Substituting this relation into equation (2.120) gives

Ad∗ωk+1mk+1 = mk +Ad∗ωk+1

mk+1 −mk+1, (2.125)

which implies the statement of conservation of spatial angular momentum mk+1 =

mk.

In the next Section, we will show that the spatial MV integrator is a co-adjoint

action in the dual of a semi-direct product Lie-algebra and preserves the Lie-Poisson

structure on this Lie-algebra.

2.6 Poisson Brackets on Semidirect Products

We will show that the spatial representation of the MV integrator is Lie-Poisson w.r.t.

to the same Lie-Poisson bracket on the dual of the semi-direct product algebra. This

result was independently shown by Bobenko & Suris (1999a) for the right reduced

discrete EP equations. We begin by introducing the following terminology.

Definition 2.6.0.5 (Lie-Poisson bracket on s∗ (Holm et al. 1986)). The (±) Lie-

Poisson brackets [s∗ = g n V ]± on the dual of the semi-direct product algebra s∗ are

given by (Holm et al. 1986, equation (3.11L), pg 59) as

{F1, F2}±(m, I) = ±〈m, [∇mF1,∇mF2]〉 ± 〈I,∇mF1 ∙ ∇IF2 −∇mF2 ∙ ∇IF1〉, (2.126)

where F1 and F2 are scalar functions defined on s∗ and ∇mF1 ∙ ∇IF2 denotes the left

action of ∇mF1 ∈ g on ∇IF2 ∈ V given by the pairing

〈ζ ∙ v, I〉 = 〈v,LζI〉. (2.127)

Lζv denotes the Lie derivative of ζ ∈ g on I ∈ V ∗, the vector field on V ∗ defined by the

left action of ζ on V ∗

Lζv = ζT v + vζ. (2.128)

Definition 2.6.0.6 (CoAdjoint actions of semi-direct products (Holm et al. 1986)).

The co-adjoint action of the semi-direct product (g, v) ∈ S = G n V on (M, I) ∈ s∗ =

g∗ n V ∗ is given by

Ad∗(g,v)−1(m, I) = (Ad∗gTm+ v � φg(I), φg(I)), (2.129)

where g acts on m by the co-adjoint action and on I by the left action. The bilinear

operator is defined as � : V × V ∗ → g∗.

51

Definition 2.6.0.7 (Spatial MV flow map). Equation (2.120) defines a discrete flow

(mk+1, Ik+1) = ΦΔt(mk, Ik) on s∗ = g∗nV ∗, where ΦΔt is the discrete flow map taking

the form of the co-adjoint action of (ωk+1, 2∇Ik lIk) ∈ Gn V on (Mk, Ik) ∈ s∗ given by

ΦΔt(mk, Ik) = Ad∗(ωk+1,2∇Ik lIk )

−1(mk, Ik) = (Ad∗ωTk+1

mk+2∇Ik lIk �φωk+1(Ik), φωk+1(Ik)).

(2.130)

ΦΔt defines a Lie-Poisson integrator with respect to the (±) Lie-Poisson brackets

[s∗]± on the dual of the semi-direct product Lie algebra s∗.

Proof. Recall that a map Φ is Poisson w.r.t. to [s∗]± if

{F1 ◦ Φ, F2 ◦ Φ}(m, I) = {F1, F2}(Φ(m, I)). (2.131)

We first note how derivatives of a function F on s∗ transform under the map (m′, I ′) =

Φ(m, I) := Ad∗(g,v)(m, I) by application of the chain rule

∇mF = AdgT∇m′F, ∇IF = φ∗g(∇I′F − [v,∇m′F ]). (2.132)

Using these expressions, the left hand side of equation of (2.131) evaluates to

{F1 ◦ Φ, F2 ◦ Φ}(m, I) = ±〈m,AdgT [∇′mF1,∇

′mF2]〉

+±〈I,AdgT (∇m′F1) ∙(φ∗g(∇I′F2 +∇m′F2 ∙ v)

)

−AdgT (∇m′F2) ∙(φ∗g(∇I′F1 +∇m′F1 ∙ v)

)〉,

(2.133)

On expressing m in terms of m′ and I in terms of I ′, the first bracket becomes

〈Ad∗g−T (m′ − v � I ′), AdgT [∇

′mF1,∇

′mF2]〉 = 〈m

′ − v � I ′, [∇′mF1,∇′mF2]〉. (2.134)

where from the definition of the diamond operator

〈v � I ′, [∇′mF1,∇′mF2]〉 = 〈I

′, [∇′mF1,∇′mF2] ∙ v〉. (2.135)

The second bracket becomes

〈φg−1(I′), AdgT (∇m′F1) ∙φ

∗g(∇I′F2+∇m′F2 ∙v)−AdgT (∇m′F2) ∙φ

∗g(∇I′F1+∇m′F1 ∙v)〉,

(2.136)

which simplifies to

〈I ′,∇m′F1 ∙ (∇I′F2 +∇m′F2 ∙ v)−∇m′F2 ∙ (∇I′F1 +∇m′F1 ∙ v)〉. (2.137)

52

This bracket rearranges to the form

〈I ′, [∇m′F1,∇m′F2] ∙ v〉, (2.138)

which cancels with the bracket given in equation (2.135). The discrete flow map there-

fore satisfies equation (2.131).

Holm et al. (1986) show that the Casimir functions on the Poisson manifold [s∗]± are

the set of functions invariant under the co-adjoint action of the Lie group S = GnV ∗ for

any finite dimensional Lie group G. It follows from the definition of ΦΔt as a co-adjoint

action of S, that the discrete flow defines a Poisson integrator.

Remark 2.6.0.8. For the case when G = SO(3), it is trivial to show that co-adjoint

actions of S preserve the Casimirs, the family of functions of ||m|| and ||I||.

Numerical experiments In Section 2.10, we implement the spatial MV integrator

using an adaptation of the DMV algorithm given by McLachlan & Zanna (2005). This

spatial variant of the DMV algorithm, referred to as the spatial DMV algorithm is

provided in Appendix A.3. We then compare the conservative properties of the spatial

DMV algorithm with the body DMV algorithm. These numerical experiments provide

insight into how the presence of an advected quantity affects the conservative properties

of the DMV algorithm.

In the next Section, we apply the discrete Clebsch approach to formulate the body

representations for two further examples of rigid body motion. The first example is

the heavy top and shows how body MV integrators are formulated using the discrete

Clebsch approach when there is a potential field. The second example is a coupled

rigid body, and shows how MV integrators are coupled. The formulation of coupled

MV integrators gives insight into how to apply MV integrators to elastic rod models,

as explicated in the next Chapter.

Each of these examples exhibit an advected quantity in the body representation.

The spatial representation provides little further insight into how to formulate MV

integrators for continuum dynamics with advected quantities and we do not belabour

its formulation.

For convenience, a summary of the main terms describing the MV integrator in the

body representation for these examples is given in a series of Tables in Appendix A.1.

Alongside each term in the Tables, we also provide its continuous (time) counterpart.

53

2.7 The Heavy Top

In this Section, we extend the review of the spatial and body representations of the rigid

body, presented thus far, to the heavy top. We consider the kinematics and symmetries

of the heavy top, described in the body frame by the orientation Γk of the vertical axis

z and the body angular velocity (see Figure 2.3). In the spatial representation, the

position of center of mass χk rotates relative to the point of support of the top and,

as previously described for the spatial representation of the free rigid body, the inertia

matrix also rotates.


Figure 2.3: The heavy top as viewed in the body frame. The top is attached to the spatial frame at anarbitrary point (in this diagram this point is the origin). The motion of the top is composed of two components,precession and spinning. The unit vector z, representing the direction of gravity, rotates about each axis of theheavy top with body angular velocity Ω. Spatial angular momentum is only conserved for motions purely aboutE3, however. The body frame also spins about its centreline axis but this is only observable in the spatial frame.The vector χ0 from the point of support to the centre of mass of the top (c.o.m.), which lies on the centrelineof the top, remains fixed in the body frame.

The heavy top differs from the rigid body in the body representation by the presence

of an advected quantity, the body unit vector Γk, representing the direction of the unit

54

gravity vector z in the body frame given by

Γk = ΛTk z. (2.139)

The discrete auxiliary equation for Γk is given by substitution of the rearranged defin-

ition of Γk

z = ΛkΓk, (2.140)

into the definition of Γk+1 to give the discrete auxiliary equation for Γk of the form

Γk+1 = ΛTk+1z = Λ

Tk+1ΛkΓk = Ω

Tk+1Γk. (2.141)

The holonomically constrained discrete Lagrangian lcΓk : V → R for heavy top

motion in the body representation is defined as

lcΓk(Ωk+1) = Tr (I0Ωk+1) +mgh2〈Γk, χ0〉

− Tr(Θk+1(Ωk+1Ω

Tk+1 − Id)

),

(2.142)

where the first term is the kinetic energy, the second term is the gravitational potential,

the last term is the holonomic constraint restricting Ωk+1 to G = SO(3) and h is

the fixed time interval. The corresponding Clebsch constrained discrete Lagrangian is

defined as

lΓk = lcΓk+Tr

2

(P Tk+1(Λk+1 − ΛkΩk+1)

)+ 〈Jk+1,Γk+1 − Ω

Tk+1Γk〉, (2.143)

where the second and third terms are the Clebsch constraints for the discrete recon-

struction formula and the discrete auxiliary equation for Γk.

The stationary discrete action principle gives the following expressions paired with

Λk and Γk respectively

∇Λk lΓk−1(Λk−1,Λk) +∇Λk lΓk(Λk,Λk+1) = 0,

∇Γk lΓk−1(Γk−1,Γk) +∇Γk lΓk(Γk,Γk+1) = 0.(2.144)

Evaluation of the derivatives of lΓk

Pk/2− ΩTk+1Pk+1/2 = 0,

mgh2χ0 + Jk − Ωk+1Jk+1 = 0,(2.145)

where the second equation is derived by expressing the vector pairing as matrix pairings,

55

a ∙ b = Tr(abT ), and abT is a tensor product a⊗ b, gives

〈Jk+1,Γk+1 − ΩTk+1Γk〉 = Tr

(Jk+1(Γ

Tk+1 − Γ

TkΩk+1)

). (2.146)

Subsequent rearrangement of equations (2.145) gives,

Pk+1 = PkΩk+1,

Jk+1 = ΩTk+1 (Jk +∇Γk lΓk)︸︷︷︸

Gk

, (2.147)

where the derivative ∇Γk lΓk evaluates to ∇Γk lΓk = mgh2χ0.

The Clebsch relation, following from the discrete action principle, takes the form

I0 − ΛTk Pk+1/2− ΓkJ

Tk+1 = Θk+1Ωk+1. (2.148)

Using the symmetry property of the Lagrange multiplier Θk+1 and the recursions given

by equations (2.147) gives

I0ΩTk+1 − Λ

Tk Pk/2− ΓkG

Tk = Ωk+1I0 − P

Tk Λk/2−GkΓ

Tk . (2.149)

Using the tensor relation involving the hap map ˆ

abT − baT = [bˆ,aˆ], (2.150)

we write equation 2.149 as the map

Mk+1 = JRk+1 = skew(Λ

Tk Pk) + [Gkˆ,Γkˆ], (2.151)

where Mk+1 := I0ΩTk+1 − Ωk+1I0.

Recall from the theorem of momentum maps for tangent lifted actions (see 2.5.1.3

on pg. 34) that the right infinitesimally equivariant momentum map for left actions on

G× R3 is given by

〈JRk+1, ζ〉 = 〈Pk, ζG(Λk)〉+ 〈Gkˆ,−[ζ,Γkˆ]〉, (2.152)

in which JRk+1 is of the form

JRk+1 = Pk � Λk +Gk � Γk, (2.153)

where Gk = Jk +∇Γk lΓk , the � operator in the first term is the same form as for the

free rigid body and the � operator in the second term � : R3 ×R3 → so(3)∗ is given by

a � b = (a× b)ˆ = [aˆ, bˆ].

56

Substitution of the two expressions for Pk and Jk given by equation (2.147) into

equation (2.153) for the right momentum map gives

Mk+1 = ΩTkΛ

Tk−1Pk−1Ωk + [(Jk +∇Γk lΓk)ˆ,Γkˆ]

= ΩTk (Mk − [Gk−1ˆ,Γk−1ˆ])Ωk + [(Jk +∇Γk lΓk)ˆ,Γkˆ]

= Ad∗ΩTkMk − [Jkˆ,Γkˆ] + [(Jk +∇Γk lΓk)ˆ,Γkˆ]

= Ad∗ΩTkMk + [∇Γk lΓkˆ,Γkˆ],

(2.154)

which is the definition of the body representation of the MV integrator for the heavy

top on s∗ = so∗(3)nR3

Mk+1 = Ad∗ΩTkMk +∇Γk lΓk � Γk,

Γk = ΩkΓk−1,(2.155)

with the body angular momentum defined as

Mk+1 :=2

h2skew

((∇Ωk+1 lΓk)Ω

Tk+1

)= I0Ω

Tk+1 − Ωk+1I0. (2.156)

It follows from the previous Section, that the MV integrator for the heavy top, given by

equation (2.155), defines a co-adjoint action of S = SO(3)n R3 on s∗ = (so(3)× R3)∗

of the form

Ad∗(Ωk,∇Γk lΓk )−1(Mk,Γk) = (Ad

∗ΩTkMk +∇Γk lΓk � φΩk(Γk−1), φΩk(Γk−1)), (2.157)

and is Lie-Poisson w.r.t. to the ± Lie-Poisson brackets on s∗ given by

{F1, F2}±(Mk,Γk) = −1

2Tr(±MT

k [∇MkF1,∇MkF2])

± Γk ∙ (Φ(∇MkF1)∇ΓkF2 − Φ(∇MkF2)∇ΓkF1) .(2.158)

In Section 2.10, we verify the conservative properties of these equations by numerical

experiment.

This example demonstrates the formulation of a MV integrator for advected quant-

ities arising from the potential energy. In the next example, we shall consider the

formulation of MV integrators for coupled rigid body motion. The coupled rigid body

57

serves as a simple example of more complex multi-body systems considered in the next

Chapter.

58

2.8 The Coupled Rigid Body

Figure 2.4: The coupled rigid body as viewed in the frame of body 1. Each body is attached from its centreof mass (c.o.m.) to an ideal spherical joint. In the R3 reduced configuration space, the origin of the spatialframe is the centre of mass (C.O.M.) of the coupled rigid body. In the frame of body 1, the motion is composedof two components, precession and spinning. The vector Λd02, representing the position of the centre of mass ofbody 2 in the frame of body 1, rotates about the origin with relative orientation Λ = ΛT1 Λ2. φ and ψ denotethe angles between the body axes E1 and the vertical and θ denotes the angle between the body attachmentsat the joint. Each body also spins about its axes, but only the spin of body 2 is observable.

2.8.1 Preliminaries

Following Patrick (1989), we outline the Lagrangian description of the free motion of

two rigid bodies coupled with an ideal spherical joint, placed in a container whose origin

is fixed at the center of mass (c.o.m.) of the coupled body (as shown in Figure 2.4)

and Λ1,Λ2 ∈ G = SO(3) denote the attitude of body 1 and 2 respectively. The basic

configuration space, under the assumption that the centre of mass of the coupled body

is stationary, is C = G×G.

Patrick (1989) denotes the total mass as m = m1 +m2, the position of the center

of mass of each body as di and the attitude of body 2 in the frame of body 1, referred

to by Grossman, Krishnaprasad & Marsden (1988) as the relative orientation matrix,

as Λ = ΛT1 Λ2. ε =m1m2m denotes the reduced mass.

59

Patrick (1989) considers the Lagrangian L : TC → R which takes the form of the

kinetic energy of the coupled rigid body

L =2∑

i=1

1

2Tr(ΛiI1Λ

Ti )+

ε

2||Λidi|| − ε〈Λ1d1, Λ2d2〉.

(2.159)

which (except for the case when g1 = g2) is not invariant under tangent lifted left actions

of (g1, g2) ∈ G ×G on TC on account of an addition term gT1 g2 ∈ SO(3) appearing in

the coupling component of the kinetic energy, which for the case when g1 = Λ1 and

g2 = Λ2 is the relative orientation matrix.

Following Holm et al. (1986), we consider the Lagrangian

LΛ0 : TC→ R, (2.160)

which is given by the map Λ→ L(Λi, Λi,Λ), where L is defined on the augmented space

TC× SO(3). The variable Λ0 = Id denotes that Λ becomes a dynamical parameter in

the reduced configuration.

Under the tangent lift of the linear translation Φ(g1,g2) of g1 and g2 on SO(3)3

defined as

Φ(g1,g2)(Λ1,Λ2,Λ0) := Φg1 ◦ Φg2 := (g1Λ1, g2Λ2, g1Λ

0gT2 ), (2.161)

the Lagrangian LΛ0 becomes

LΛ(g1Λ1, g1Λ1; g2Λ2, g2Λ2) =2∑

i=1

1

2Tr(giΛiI1Λ

Ti g

Ti ) +

ε

2||giΛidi||

− εdT1 ΛT1 g

T1 Λg2Λ2d2,

(2.162)

where Λ = g1Λ0gT2 .


To express the kinetic energy of the coupled rigid body as a quadratic form in the body

angular velocities, Grossman et al. (1988) introduce a modified inertia matrix, which

we denote as Ii (Grossman et al. (1988) denote this as J), defined as Ii = Ii + ε(dˆi)2,

where (∙) denotes the hat map ˆ : R3 → so(3).

Grossman et al. (1988) give the following reduced Lagrangian lΛ : so(3)×so(3)→ R

for the body representation given as the quadratic form

60

lΛ =1

2〈Ω,JΩ〉, (2.163)

where Ω := [Ω1,Ω2]. Ωi is the left invariant body angular velocity of body i and J is

the symmetric positive definite metric which Grossman et al. (1988) define as

J =

(I1 εA

εAT I2

)

where A := d1ˆTΛd2ˆ.

Expanding the quadratic form in equation (2.163) gives

lΛ =1

2

2∑

i=1

Tr(ΩTi IiΩi)− εTr(ΩT1AΩ2), (2.165)

where the first term is the kinetic energy of each free rigid body motion and the second

term is the kinetic energy of the rigid coupling motion. The modified inertia matrices

Ii and the position vectors di are fixed in the body frame but the relative orientation

matrix Λ is an advected quantity.

The discrete coupled rigid body

Repeating the procedure described for deriving the discrete Lagrangian for the body

representation of the free rigid body, we substitute the finite difference approximation

of the body angular velocities Ωi(tk) ≈ 1hΛ

iT

k (Λik+1 − Λ

ik) into the Lagrangian defined

in equation (2.165) to give the body representation of the holonomically constrained

discrete Lagrangian lcg : V2 → R given by

lcΛk =2∑

i=1

Tr(IiΩ

k+1i

)+ εTr

((Id − Ω

k+1T

1 )Ak(Id − Ωk+12 )

)

− Tr(Θk+1i (Ωk+1i Ωk+1

T

i − Id)),

(2.166)

where the first term is the approximation of the kinetic energy of the free motion of

each rigid body and the second term is the approximation of the coupling component

of the kinetic energy1. The third term is the holonomic constraint for restricting each

Ωk+1i to SO(3).

We obtain the discrete auxiliary equation for Λk by formulating the definition of

1The discrete Lagrangian has been factored by − 1h2which explains the change in sign of the coupling

component under discretisation of the Lagrangian.

61

Λk = ΛkT

1 Λk2 as a recursion

Λk+1 = Λk+1T

1 Λk1ΛkΛk

T

2 Λk+12 = Ωk+1

T

1 ΛkΩk+12 . (2.167)

The corresponding Clebsch constrained discrete Lagrangian is given by

lΛk = lcΛk +

2∑

i=1

Tr

2

(P k+1

T

i (Λk+1i − ΛkiΩk+1i )

)+ Tr

(Jk+1

T

(Λk+1 − Ωk+1T

1 ΛkΩk+12 )),

(2.168)

where the second and third terms are the Clebsch constraints for the discrete recon-

struction formula and the relative orientation matrix Λk ∈ V .

Under stationarity of the discrete action principle, the discrete Clebsch approach

gives the following two discrete Euler-Lagrange equations, paired with variations in Λkiand Λk respectively

∇Λki lΛk−1(Λk−1i ,Λki ) +∇Λki lΛk(Λ

ki ,Λ

k+1i ) = 0, i ∈ {1, 2},

∇Λk lΛk−1(Λk−1,Λk) +∇Λk lΛk(Λ

k,Λk+1) = 0.(2.169)

Evaluation of the derivatives of lΛk w.r.t. Λki and subsequent rearrangement gives,

P k+1i = P ki Ωk+1i . (2.170)

The derivative of lΛk w.r.t. Λk evaluates to

−Jk +Ωk+11 Jk+1Ωk+1T

2 +∇Λk lΛk = 0, (2.171)

where the derivative ∇Λk lΛk evaluates to

∇Λk lΛk = εd1ˆ(Id − Ωk+11 )(Id − Ω

k+1T

2 )d2ˆT . (2.172)

Equation (2.173) rearranges to

Jk+1 = Ωk+1T

1 (∇Λk lΛk + Jk)

︸︷︷︸Gk

Ωk+12 . (2.173)

The two Clebsch relations paired with δΩk+1i evaluate respectively to

Ck+11 − ΛkT

1 P k+11 /2− ΛkΩk+12 Jk+1T

= Θk+11 Ωk+11 ,

Ck+12 − ΛkT

2 P k+12 /2− ΛkT

Ωk+11 Jk+1 = Θk+12 Ωk+12 ,(2.174)

62

where Ck+1i := ∇Ωk+1ilΛk take the form

Ck+11 = I1 + εAk(Ωk+12 − Id),

Ck+12 = I2 + εAkT (Ωk+11 − Id).

(2.175)

The recursions for P ki and Jk given by equations (2.173) and (2.170), together with the

symmetry property of the Lagrange multipliers Θk+1i , yield the maps:

Mk+11 := Ck+11 Ωk+1

T

1 − Ωk+11 Ck+1T

1 = skew(ΛkT

1 P k1 + ΛkGk

T

),

Mk+12 := Ck+12 Ωk+1

T

2 − Ωk+12 Ck+1T

2 = skew(ΛkT

2 P k2 + ΛkTGk),

(2.176)

where Gk := ∇Λk lΛk + Jk.

From the theorem for momentum maps of lifted actions, the momentum maps for

the respective cotangent lifted left actions of ζ1 and ζ2 on P = T ∗C× SO(3) defined as

JRi : P → g∗ = so(3)∗, i ∈ {1, 2}, (2.177)

are infinitesimally equivariant momentum maps given by

〈JR1 , ζ〉 = 〈Pk1 , ζG(Λ

k1)〉+ 〈G

k,Λkζ〉,

〈JR2 , ζ〉 = 〈Pk2 , ζG(Λ

k2)〉+ 〈G

k,−ζΛk〉,(2.178)

and take the form

JR1 = Pk1 � Λ

k1 +G

k � Λk,

JR2 = Pk2 � Λ

k2 +G

k � Λk,(2.179)

where the second term is paired with the bilinear operator � : V∗×V . These expressions

evaluate to

JR1 = skew(ΛkT

1 P k1 + ΛkGk

T

),

JR2 = skew(ΛkT

2 P k2 + ΛkTGk),

(2.180)

verifying that equations (2.176) give the momentum maps associated with the reduction

to body representation, by the actions Φg1 and Φg2 defined in equation (2.161).

Conservation of spatial angular momentum The following calculation verifies

that the total spatial angular momentum is conserved by the discrete flow on the

augmented cotangent bundle through the momentum map given by equation (2.179).

63

The total spatial angular momentum is

mk+1 = mk+11 +mk+1

2 =2∑

i=1

ΛkiMk+1i Λk

T

i

=2∑

i=1

skew(ΛkiΛkT

i P ki ΛkT

i + Λk1Λ

kGkT

ΛkT

1 + Λk2Λ

kTGkΛkT

2 )

=2∑

i=1

skew(P k−1i ΩkiΩkT

i Λk−1Ti + Λk2G

kTΛkT

1 + Λk1G

kΛkT

2 )

=2∑

i=1

skew(P k−1i Λk−1T

i + Λk−12 Gk−1T

Λk−1T

1 + Λk−11 Gk−1Λk−1T

2 )

= mk1 +m

k2 = m

k.

(2.181)

Substituting the expressions for the discrete flow on the augmented cotangent bundle,

given by the discrete auxiliary equation, the discrete construction formulae and equa-

tions (2.170) and (2.173), into the right momentum map given in equation (2.179)

gives

Mk+11 = Ad∗

Ωk1(skew(Λk−1

T

1 P k−11 )) + skew(Λk(JkT

+∇TΛk lΛk)),

= Ad∗Ωk1

(Mk1 − skew(Λ

k−1Gk−1T

))+ skew(Λk(Jk

T

+∇TΛk lΛk)),

= Ad∗Ωk1

(Mk1 − skew(Λ

k−1Ωk2JkTΩk

T

1 ))+ skew(Λk(Jk

T

+∇TΛk lΛk)),

= Ad∗Ωk1Mk1 − skew(Λ

kJkT

) + skew(Λk(JkT

+∇TΛk lΛk)),

= Ad∗Ωk1Mk1 + skew(Λ

k∇TΛk lΛk),

(2.182)

and by an identical procedure

Mk+12 = Ad∗

Ωk2Mk2 + skew(Λ

kT∇Λk lΛk), (2.183)

which defines the body representation of the MV integrator for the coupled rigid body

defined on so(3)∗ × so(3)∗ × SO(3)

Mk+1i = Ad∗

ΩkiMki +∇Λk lΛk � Λ

k, i ∈ {1, 2}, (2.184)

together with the discrete auxiliary equation for the relative orientation matrix Λk in

64

the frame of body 1

Λk = φ(Ωk1 ,Ωk2)(Λk−1) = Ωk

T

1 Λk−1Ωk2. (2.185)

The MV integrator for the coupled rigid body is co-adjoint action of the Lie group

S = SO(3)2 × V∗ on the dual of the semi-direct product Lie algebra s∗ = so(3)∗ ×

so(3)∗ × SO(3) given by

Ad∗(Ωk1 ,Ω

k2 ,∇Λk lΛk )

−1(Mk1 ,M

k2 ,Λ

k) =

2∑

i=1

Ad∗ΩkiTM

ki +∇Λk lΛk � φ(Ωk1 ,Ωk2)(Λ

k−1), φ(Ωk1 ,Ωk2)(Λk−1).

(2.186)

A DMV algorithm for coupled rigid body motion The form of the momenta

given by equations (2.176) can be cast into a coupled matrix Ricatti equation

M1 =M′1 + J(Ω2)Ω

T1 − Ω1J(Ω2)

T ,

M2 =M′2 + J(Ω1)Ω

T2 − Ω2J(Ω1)

T ,(2.187)

where M ′i is an uncoupled term of the form M ′i = IiΩTi − ΩiIi and J(Ωi) is a function

of Ωi which takes the form

J(Ω1) = εAkT (Ωk+11 − Id),

J(Ω2) = εAk(Ωk+12 − Id).

(2.188)

Numerical experiments We provide an algorithm in Appendix A.4 for solving

equations (2.187) for the discrete angular velocities Ωk+11 and Ωk+12 . Preliminary nu-

merical experiments showing the conservative properties of the MV integrator for the

coupled rigid body are presented in Section 2.10. Grossman et al. (1988) give the Lie-

Poisson bracket for the coupled rigid body on so(3)∗×so(3)∗×SO(3). This bracket has

a Casimir of the form C = ||M1+ΛM2ΛT ||2 which we verify by numerical experiment.

Tables 5 and 6 of Appendix A summarise the main results of this Section.

We now consider how to adapt the discrete Clebsch approach to give the Cayley-

Klein parameterisation of the rigid body.

65

2.9 The Cayley-Klein Parameterisation of Rigid BodyMo-

tion

2.9.1 Background

It is well known that the parameterisation using Euler-angles of rigid body motion is

not global. This has important practical consequences, most familiar of which is the

problem of Gimble-lock. The Cayley-Klein parameterisation of rigid body motion is,

however, not only singularity free but equal to simple combinations of the components

of the quaternions. We shall therefore briefly review some relevant definitions before

deriving Moser-Veselov integrators for Cayley-Klein parameterisations of the rigid body.

Following Kosenko (1998), we consider the map from the quaternions to SU(2),

whose elements consist of all 2 × 2 unimodular unitary matrices,

h : q 7→1

|q|

(q0 − ıq3 −q2 − ıq1q2 + ıq1 q0 + ıq3

)

(2.189)

is a diffeomorphism on S3. When the quaternions have unit norm, they form a multi-

plicative group Q and h defines a group isomorphism h : Q → SU(2).

The matrix elements of SU(2) are the components of a quaternion q = q01+ q1i1+

q2i2+ q3i3 and |q|2 = 1 which, by definition, describes a rotation θ about a unit vector

[k1, k2, k3] given by qj = kjsin(θ2

), j := 1→ 3 and q0 = cos

(θ2

).

Elements of the Lie algebra su(2), all traceless skew-Hermitian matrices, may be

identified with the pure quaternions (R3,×) by the isomorphism

x 7→ x =1

2ıσjxj , (2.190)

whereσj2ı are the Pauli spin matrices which form the basis of su(2). In this basis, the

natural pairing between elements of x ∈ su(2) and its dual y ∈ su(2)∗ can be written

as

〈x, y〉 = −2Tr(xy) = x ∙ y. (2.191)

This pairing defines the map ˇ which identifies elements of su(2)∗ with R3. The dual of

su(2) thus also consists of all traceless skew-Hermitian matrices.

The MV integrators are now adapted from the Euler-angle to the Cayley-Klein

parameterisation by identifying the configuration space with SU(2) instead of SO(3).

Accordingly, the discrete body angular ”velocity” becomes

Ωk+1 = Λ†kΛk+1, (2.192)

where † denotes the adjoint (conjugate transpose) of the matrix and the Lagrangian is

66

then a function SU(2)× SU(2)→ R given by

L = Tr(Ω†k+1J(Ωk+1)

). (2.193)

The operator J takes the form J(Ωk+1) = Id − (1 − Ii)Ωk+1Tr2(E†iΩk+1

). So for

the special case when the principal moments of inertia I1 = I2 = 1 and I3 6= 1, J takes

the form J(Ωk+1) = Id − (1 − I3)Ωk+1Tr2(E†3Ωk+1

). Recall that this is a rigid body

motion with an axis of symmetry about the body axis E3. The discrete Lagrangian

is invariant under the left action of SU(2) and, when two of the principal moments of

inertia are equal, is also right invariant under the subgroup consisting of all matrices

of the form

g =

(α ıβ

ıβ α

)

, α, β ∈ R. (2.194)

This subgroup defines rotations θ about the E3 axis, the axis of symmetry of the

rigid body. This rotation is equivalent to the quaternion q01+ 0ı1 + 0ı2 + q3ı3, where

q0 = cos(θ/2) and q3 = sin(θ/2) which is mapped to a double covering of SO(3) of the

form

g =

q20 − q23 −2q0q3 0

2q0q3 q20 − q23 0

0 0 q20 + q23

. (2.195)

Now let a Hermitian matrix Uk+1 = Ωk+1Ω†k+1 and denote the real and imaginary

parts of a complex matrix as <{} and ={}. By analogy with equation (2.75), the

Clebsch constrained discrete Lagrangian lk+1 (where˜denotes the addition of Clebsch

constraints) for the body representation of the rigid body becomes

lk+1 = Tr(Ω†k+1J(Ωk+1)

)−Tr

2

(P†k+1(Λk+1 − ΛkΩk+1)

)

− Tr(<{Θ†k+1}(<{Uk+1} − Id)

)+ Tr

(={Θ†k+1}={Uk+1}

).

(2.196)

The last two terms in the definition of lk+1 are simply the real and imaginary parts

of the unitary constraint when Θk+1 is Hermitian. To show this, first split Θk+1 and

Uk+1 into its real and imaginary components

67

Tr(Θ†k+1(Uk+1 − Id)

)= Tr

(<{Θ†k+1}(<{Uk+1} − Id)

)+ ıT r

(<{Θ†k+1}={Uk+1}

)

+ ıT r(={Θ†k+1}(<{Uk+1} − Id)

)− Tr

(={Θ†k+1}={Uk+1}

).

(2.197)

Since any Hermitian matrix U has the property, <{U} = <{U}T and ={U} = −={U}T ,

it follows by the property of the trace operator that when the <{Θ†k+1} is symmetric

and ={Θ†k+1} is anti-symmetric, the unitary constraint on the left hand side of equa-

tion (2.197) is equivalent to the constraint in the discrete Lagrangian when Θk+1 is a

Hermitian matrix. We therefore use the unitary constraint with a Hermitian Lagrange

multiplier Θk+1 from hereon. The derivation of the Euler-Lagrange equations is then

a trivial generalisation of the procedure outlined in equation 2.5 to complex matrices.

The Clebsch relation paired with the variation δΩk+1 in the discrete stationary

action principle gives the expression

JRk+1 = skew(Λ†kPk), (2.198)

where skew now denotes the skew-Hermitian component. This expression satisfies the

definition of an equivariant right momentum map for cotangent lifted left actions of

SU(2)

〈JRk+1, ζ〉 = 〈Pk, ζGΛk〉

= 〈Pk � Λk, ζ〉,(2.199)

where the bilinear operator � is given by � : V∗ × V → su(2)∗. The image of this

map is the body angular momentum Mk+1 := Ak+1Ω†k+1 − Ωk+1A

†k+1 ∈ su(2)

∗, where

Ak+1 is defined as Ak+1 = ∇Ωk+1 lk+1, which from the definition of J takes the form

Id − IiTr(Ω†k+1Ei

)Ei.

Substitution of the following discrete flow equations on the cotangent bundle T ∗SU(2)

Pk+1 = PkΩk+1,

Λk+1 = ΛkΩk+1,(2.200)

into the right momentum map gives the discrete basic EP equation for the co-adjoint

orbits in su(2)∗ describing rigid body motion in the body representation

Mk+1 = Ad∗Ω†kMk. (2.201)

68

2.9.2 Momentum maps and Hopf fibrations

We now consider the form of the right momentum map for cotangent lifted right actions

of ζ ∈ su(2) on T∗G. Substitution of the equations for the discrete flow on the cotangent

bundle into the right momentum map gives

Λ†k+1Pk+1 = Ω†k+1Λ

†kPkΩk+1, (2.202)

which leaves the determinant of Λ†kPk invariant. When scaled by the constant1|P0|, this

product is a special unitary matrix. The value of this constant is obtained by equating

the expression for the body angular momentum with the momentum map to give

Pk = Λk+1Ak+1,1

|Ak+1|Ak+1 ∈ SU(2). (2.203)

where |Pk| = |Ak+1| = |A0|. The pre-image of the momentum map is therefore iso-

morphic to S3 with radius√|A0|. In the case of a rigid body with distinct principal

moments of inertia, the radius is√1 + I2i Ω

2i (t0), i := 1 → 3. The momentum map is

the Hopf fibration (Marsden & Ratiu 1999)

JRk+1 : S3 := {z ∈ C2| |z|2 = 1 + I2i Ω

2i (t0)} −→ S2 := {x ∈ R3| |x|2 =

3∑

i=1

M2i (t0)}.

(2.204)

This result is consistent with the continuous time geometric description of the rigid

body for which the momentum map for the SU(2) action on C2, the Cayley-Klein

parameters and the family of Hopf fibrations on concentric three-spheres in C2 are all

equivalent. We refer the reader to Table A.2 in Appendix A which compares the main

expressions for the MV integrators in terms of SU(2) matrices and quaternions.

2.10 Numerical Experiments

This Section presents results demonstrating the conservative properties, computa-

tional efficiency and accuracy of the rigid body, heavy top and coupled rigid body

integrators.

The components of the body momentum are compared with the analytic solution

for the rigid body only, and the Matlab Ode45 integration of the Euler-Arnold ordinary

differential equations and their variants for the heavy top and coupled rigid body. The

tolerance of the Ode45 routine is set to 10−15. The components of the quaternions and

69

the body angular momentum computed by the DMV algorithm for the Cayley-Klein

parameterisation of the symmetric rigid body is also presented.

The time step for all numerical experiments is Δt = 0.1. Although the Figure

captions give details of each experiment, we point out a few general features here.

Rigid body experiments

• Firstly, the choice of initial parameters in each experiment avoids intersection

of the body momenta with fixed points. It is well known that the co-adjoint

orbits of the classical rigid body with distinct moments of inertia have saddle

points at (0,±π, 0) (which are connected by four heteroclinic orbits) and centers

at (±π, 0, 0) and (0, 0,±π). We find that the numerical solution does not become

unstable, however, provided the time step is no larger than approximately 0.5.

• Figures 2.5 and 2.6 show that there is a good agreement between the numerical

results and the analytic solution for the rigid body and confirm conservation of

spatial angular momentum and the Casimirs ||M ||22 and (det(I), ||I||) for the body

and spatial representations respectively.

• The numerical round-off error in each representation depends upon the principle

moments of inertia. Figure 2.7 shows the comparative errors in the spatial angular

momentum and energy after 104 time steps for the case when I1 = I2 > I3. The

error in the (i) spatial angular momentum is O(10−8) and O(10−11) and (ii) energy

is O(10−7) and O(10−10) for the respective body and spatial representations.

Figure 2.8 shows the comparative errors in the spatial angular momentum and

energy after 104 time steps for the case when I1 > I2 > I3. The errors in the

(i) spatial angular momentum are O(10−14) and O(10−11) and (ii) energy are

O(10−13) and O(10−10) for the respective body and spatial algorithms.

• Each of these graphs also include the results of the explicit Runge-Kutta method

of Dormand & Prince (1980) (implemented in the Matlab ode45 routine). These

graphs do not indicate comparative performance of the DMV algorithms with

the Ode45 method. In each experiment, the ode45 solver was run at the smallest

time step possible (tol=10−15) purely to provide a quantitative benchmark for

the DMV algorithm.

• The comparative computational performance of the DMV algorithm with an ex-

plicit Lie-Poisson integrator based on splitting of the rigid body Hamiltonian

(McLachlan 1993) is shown in Section 2.10.3. Although this Lie-Poisson method

70

is not derived in a unified discrete framework it does provide a performance bench-

mark for the DMV algorithm of McLachlan & Zanna (2005). Both methods are

explicit and conserve the Hamiltonian to O(Δt)2 but the error in the Casimir and

spatial angular momentum conservation and computational efficiency differ. This

is because the DMV algorithm is formulated as matrices and uses a Schur fac-

torisation to solve the matrix Ricatti equation. The explicit Lie-Poisson method

based on splitting is formulated in vectors and performs only matrix vector multi-

plications. McLachlan (1993) present an optimised DMV algorithm, which avoids

the Schur factorisation, but we do not pursue this here.

Figures 2.9 and 2.10 show the respective components of the quaternions and the

body angular momentum computed by the DMV algorithm for the Cayley-Klein

parameterised rigid body motion. The group of quaternions forms a double cov-

ering of SO(3) and consequently the components of the quaternions are observed

to vary over two time scales. Only the first two components of the body angular

momentum vary periodically because the moments of inertia I1 = I2.

Heavy top and coupled rigid body experiments

• Figure 2.11 shows the components of the body angular momentum of heavy top

motion and the error in the Casimir 〈M,Γ〉 of the heavy top Lie-Poisson bracket.

We observe that whenever the first and second components of the body angular

momentum are non-zero, heavy top motion breaks the S1 symmetry about the

vertical axis. The experiment confirms that the Casimir 〈M,Γ〉 is conserved.

• We perform two coupled rigid body experiments in which two identical bodies are

subject to the same initial conditions (i) but are initially at right angles to each

other and (ii) are initially aligned with each other. In the first experiment, shown

in Figure 2.12, we observe non-periodic behaviour in the body angular momentum

components caused by exchanges of momentum between the two bodies. The

second component of the momentum changes the most, ranging from −1 to 0.8.

In the second experiment, shown in Figure 2.13, we recover a rigid body motion

similar to that shown in Figure 2.5, except that M3 varies.

71

2.10.1 Body and spatial DMV algorithms for the rigid body

0 100 200 300 400 500 600 700 800 900 1000-0.02

0

0.02

M1-M

* 1

time steps

0 100 200 300 400 500 600 700 800 900 1000-0.02

0

0.02

time steps

M2-M

* 2

0 100 200 300 400 500 600 700 800 900 1000-2

0

2x 10

-9

time steps

M3-M

* 3

0 100 200 300 400 500 600 700 800 900 10000

2x 10

-4

time steps

||M-M

* ||2 2

0 100 200 300 400 500 600 700 800 900 1000-2

0

2x 10

-12

time steps

Spatial DMVBody DMVOde45

|| Ik ||

2 - || I

0 ||

2det(I

k)-det(I

0)

Figure 2.5: This Figure compares numerical simulations of a rigid body over 1000 timesteps for the case when the principal moments of inertia I1 = I2 > I3 (I1 = 2, I2 =2, I3 = 1). The top three graphs show the error in each component of the body angularmomentum of rigid body motion (M∗ denotes the analytic momentum). The bottomtwo graphs show the error in the Casimirs ||M ||22 and ||I||2, det(I) of rigid body flow inthe respective body and spatial representations. The graph labelled (i) ”body DMV”is the solution computed by the body DMV algorithm, (ii) ”spatial DMV” is the bodyframe translated solution computed by the spatial DMV algorithm (which computes theangular momentum and moment of inertia in the spatial representation) (iii) ”ode45” isan explicit Runge-Kutta (4,5) integrated solution of the Euler-Arnold equations usingthe Matlab routine, ode45 and (iv) ”analytic” is the analytical solution. The initialconditions for this simulation are the initial body angular momentum components givenas M1(0) = 0.1, M2(0) = 0, M3(0) = 1. The top three graphs show that the DMVmomentum matches the analytical solution which describes the rolling of a cone ofconstant angle in the body on a second cone of constant angle fixed in space (Marsden &Ratiu 1999). The 2nd from bottom graph shows that the body DMV algorithm preciselycomputes the Casimir ||M ||22 (Marsden & Ratiu 1999) suggesting preservation of therigid body Lie-Poisson structure and consequently that the DMV angular momentumremains on the sphere. The bottom graph shows that the Casimirs ||I||2 and det(I)of the spatial DMV algorithm are correctly computed suggesting that the Lie-Poissonstructure on the dual of the semi-direct product Lie algebra is also preserved.

72

0 100 200 300 400 500 600 700 800 900 1000-0.01

0

0.01

M1-M

* 1

time steps

0 100 200 300 400 500 600 700 800 900 1000-0.02

0

0.02

time steps

M2-M

* 2

0 100 200 300 400 500 600 700 800 900 1000-5

0

5x 10

-3

time steps

M3-M

* 3

0 100 200 300 400 500 600 700 800 900 10000

2x 10

-4

time steps

||M-M

* ||2 2

0 100 200 300 400 500 600 700 800 900 1000-5

0

5x 10

-13

time steps


|| Ik ||

2 - || I

0 ||

2det(I

k)-det(I

0)

Figure 2.6: This Figure compares numerical simulations of a rigid body over 1000time steps for the case when the principal moments of inertia I1 > I2 > I3 (I1 =3.5, I2 = 2.5, I3 = 2). The top three graphs each the error in each component of thebody angular momentum of rigid body motion (M∗ denotes the analytic momentum).The bottom two show the error in the Casimirs ||M ||22 and ||I||2, det(I) of rigid bodyflow in the respective body and spatial representations. The initial conditions forthis simulation are the initial body angular momentum components given as M1(0) =−0.5, M2(0) = 0, M3(0) = 1. The top three graphs show that the DMV momentummatches the analytical solution describing the intersection of energy ellipsoids withco-adjoint orbits of SO(3) which are two-spheres Marsden & Ratiu (1999). Note thatalthough our choice of simulation parameters avoids the flow intersecting either of thetwo saddle points at (0,±||M ||2, 0) or centers at (±||M ||2, 0, 0) and (0, 0,±||M ||2), thesolution does not become unstable provided the time step Δt < 0.5.

73

0 2000 4000 6000 8000 1000010

-20

10-15

10-10

10-5

time steps

|E-E

* |/E*

0 2000 4000 6000 8000 1000010

-20

10-15

10-10

10-5

time steps

||m-m

0|| 2

Spatial DMV Body DMV Ode45

Figure 2.7: This Figure compares the energy and spatial angular momentum error innumerical simulations of rigid body motion over 104 time steps for the case when theprincipal moments of inertia I1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1). The top graphshows the relative energy error growth in the solutions computed by the DMV algorithmand the ode45 integrator and the bottom graph shows the error in the approximatedspatial angular momentum. The initial conditions for this simulation are the initialbody angular momentum components given as M1(0) = 0.1, M2(0) = 0, M3(0) = 1.The graphs show that the error in the energy and spatial angular momentum computedby the body DMV algorithm is higher than the error computed by the spatial DMValgorithm. The bottom graph also shows that the spatial DMV algorithm conservesspatial angular momentum to numerical round off.

74

0 2000 4000 6000 8000 1000010

-20

10-15

10-10

10-5

time steps

|E-E

* |/E*

0 2000 4000 6000 8000 1000010

-16

10-14

10-12

10-10

time steps

||m-m

0|| 2


Figure 2.8: This Figure compares the energy and spatial angular momentum error innumerical simulations of rigid body motion over 104 time steps for the case when theprincipal moments of inertia I1 > I2 > I3 (I1 = 3.5, I2 = 2.5, I3 = 2). The top graphshows the relative energy error growth in the solutions computed by the DMV algorithmand the ode45 integrator and the bottom graph shows the error in the approximatedspatial angular momentum. The initial conditions for this simulation are the initialbody angular momentum components given as M1(0) = −0.5, M2(0) = 0, M3(0) = 1.In contrast with the previous Figure, the graphs show that the error in the energy andspatial angular momentum computed by the spatial DMV algorithm is higher than theerror computed by the body DMV algorithm. The bottom graph also shows that thebody DMV algorithm conserves spatial angular momentum to numerical round off.

75

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1

0

1

time steps

q 0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1

0

1

time steps

q 1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1

0

1

time steps

q 2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1

0

1

time steps

q 3

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1e-11

0

1e-11

time steps

|q|-

12

DMV

Figure 2.9: This Figure shows the components of the quaternions computed by theDMV algorithm for the Cayley-Klein parameterised rigid body motion whose principalmoments of inertia are I1 = I2 = 1 and I3 = 0.5. The group of quaternions providesa double covering of SO(3) and each component is observed to vary over two timescales. The initial conditions for this simulation over 104 time steps are M1(0) =−0.5, M2(0) = 0.1, M3(0) = 0.5.

76

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1

0

1

time steps

M1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1

0

1

time steps

M2

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1e-15

0

1e-15

time steps

M -

13

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-1e-11

0

1e-11

time steps||M|||

-||M

(0)|

|2

222

DMV

Figure 2.10: This Figure shows the components of the angular momentum computedby the DMV algorithm for the Cayley-Klein parameterised rigid body whose principalmoments of inertia are I1 = I2 = 1 and I3 = 0.5 and initial conditions are the same asfor the previous Figure. This Figure verifies that (i) the first two components of thebody angular momentum are periodic, (ii) the third component is conserved and (iii)the norm of the momentum is a Casimir.

77

2.10.2 Body DMV algorithm for the heavy top and coupled rigid

body

0 100 200 300 400 500 600 700 800 900 1000-0.2

0

0.2

M1

0 100 200 300 400 500 600 700 800 900 1000-0.2

0

0.2

M2

0 100 200 300 400 500 600 700 800 900 1000-5

0

5

10x 10

-7

M3-1

0 100 200 300 400 500 600 700 800 900 1000-2

0

2

4x 10

-4

time steps

<M

,Γ>

- <

M0,Γ

0>

Body DMVOde45

Figure 2.11: This Figure compares numerical simulations of the body representationof the heavy top over 1000 time steps for the case when the principal moments ofinertia I1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1). The top three graphs each show acomponent of the body angular momentum of heavy top motion and the bottom graphshows the error in the body DMV and ode45 computed Casimir 〈M,Γ〉 of the heavy topLie-Poisson bracket. The initial conditions for this simulation are the initial (i) bodyangular momentum components and (ii) position of the vertical axis in the body framegiven respectively as M1(0) = 0.1, M2(0) = 0, M3(0) = 1 and Γ = [0, 0, 1]. Wheneverthe first and second components of the body angular momentum are non-zero, heavytop motion breaks the S1 symmetry about the vertical axis. The bottom graph showsthat there is no secular growth in 〈M,Γ〉.

78

0 200 400 600 800 1000-2

0

2

time steps

M1

0 200 400 600 800 1000-1

0

1

time steps

M2

0 200 400 600 800 10000

1

2

time steps

M3

Body 1 Body 2

Figure 2.12: This Figure compares numerical simulations of the coupled rigid body, asseen in the frame of body 1, over 1000 time steps for the case when the two identicalrigid bodies are initially positioned at right angles to each other. The top three graphseach show a component of the body angular momentum of body 1 and 2. The initialconditions for this simulation are the initial (i) body angular momentum components(ii) orientation of the bodies relative to their E3 axes and (iii) angle between themechanical attachments at the ball and socket joint given respectively as M2(0) =M1(0) = [0.5, 0, 1], φ(0) = ψ(0) and θ(0) = π

2 . The principal moments of inertia ofthe two identical rigid bodies are I1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1). The graphsshow that the components of body angular momentum are not periodic and extremevalues are different from those of the single (uncoupled) rigid body shown in Figure2.5, indicating transfer of angular momentum between the two bodies.

79

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.02

-0.01

0

0.01

0.02

M1-M

1ode4

5

time steps

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.01

-0.005

0

0.005

0.01

time steps

M2-M

2ode4

5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-2

-1

0

1

2x 10

-3

time steps

M3-M

3ode4

5

Figure 2.13: This Figure compares numerical simulations of the coupled rigid body, asseen in the frame of body 1, over 10000 time steps for the case when the two identicalrigid bodies are initially aligned with each other. The three graphs each show thedifference of each component of the body angular momentum of body 1 between theDMV algorithm and ode45 computations. The initial conditions for this simulationare the initial (i) body angular momentum components (ii) orientation of the bodiesrelative to their E3 axes and (iii) angle between the mechanical attachments at theball and socket joint given respectively as M2(0) = M1(0) = [0.5, 0, 1], φ(0) = ψ(0)and θ(0) = 0. The principal moments of inertia of the two identical rigid bodies areI1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1).

80

0 200 400 600 800 1000-5

0

5x 10

-12

time steps

||M||2 2

0 200 400 600 800 1000-1

0

1x 10

-3

time steps

Ene

rgy

erro

r

0 200 400 600 800 1000-2

0

2x 10

-11

time steps

||m-m

0|| 2

Figure 2.14: This Figure shows the error in computation of the Casimirs and conservedspatial angular momentum of the coupled rigid body motion, as viewed in the frameof body 1, for the case when both identical bodies are initially positioned at rightangles to each other. The top graph shows the absolute error in DMV and ode45computations of the C.R.B. Casimir ||M ||22 = ||M1+ΛM2Λ

T ||22 (Grossman et al. 1988).The middle and bottom graphs show the comparative error in the energy and spatialangular momentum of the coupled rigid body. The initial conditions for this simulationare the initial (i) body angular momentum components (ii) orientation of the bodiesrelative to their E3 axes and (iii) angle between the mechanical attachments at theball and socket joint given respectively as M2(0) = M1(0) = [0.5, 0, 1], φ(0) = ψ(0)and θ(0) = π

2 . The principal moments of inertia of the two identical rigid bodies areI1 = I2 > I3 (I1 = 2, I2 = 2, I3 = 1).

81

2.10.3 Comparison with a Lie-Poisson integrator based on splitting

Figure 2.15: This Figure shows the comparative error in the Hamiltonian as computedby the (body) DMV algorithm for the rigid body and the splitting integrator, over arange of time steps Δt. The splitting integrator is an explicit Lie-Poisson integratorbased on a splitting of the rigid body Hamiltonian (McLachlan 1993). The Figureshows that both methods are second-order accurate in Δt - their slope coincides withthe slope of the triangle, which represents an O(Δt2) scaling law.

82

Figure 2.16: This Figure shows the comparative cpu time of the (body) DMV algorithmfor the rigid body against the splitting integrator, over a range of total number ofsimulations N . The splitting integrator is an explicit Lie-Poisson algorithm based on asplitting of the rigid body Hamiltonian (McLachlan 1993). Both methods are explicit,but the DMV algorithm is not as efficient for two reasons; firsty the DMV algorithmcomputes the momentum and angular velocities in matrix form where as the splittingintegrator computes the momenta as vectors. Secondly, the DMV algorithm uses aSchur factorisation, which is a computationally expensive step. McLachlan & Zanna(2005) show that this step can be replaced with an explicit expression for computingthe eigenvalues of the Hamiltonian associated with the matrix Ricatti equation, but wedo not implement this here.

83

0 2000 4000 6000 8000 10000-18

-16

-14

-12

-10

-8

-6

-4

-2

0

2x 10

-13

time steps

||M||-

||M0||

DMVSplitting

0 2000 4000 6000 8000 10000-20

-15

-10

-5

0

5x 10

-13

time steps

||m-m

0||

DMVSplitting

Figure 2.17: These graphs show the comparative errors in the rigid body Casimir andspatial angular momentum as computed by the (body) DMV algorithm and the splittingmethod of McLachlan (1993). The time step size is 0.01, the moments of inertia areI1 = 1, I2 = 2, I3 = 3 and the initial conditions are M0 = [−0.5, 0, 1]. The reason whythe error growth computed by the DMV algorithm is larger can be attributed to boththe roundoff error from matrix operations and error in the Schur factorisation.

84

2.11 Summary

We pursue the discrete Clebsch approach by Cotter & Holm (2006) as the basis of a

unified computational framework for deriving geometric integrators for the convective

and spatial description of continuum dynamics. Holm et al. (1986) showed that the

spatial and body representations of rigid body motion correspond to the spatial and

convective representations of continuum dynamics. Under a finite difference approxim-

ation of the continuous Clebsch constrained action principle, we show that the discrete

Clebsch approach gives the same integrator derived by Moser & Veselov (1991) for the

body representation of the free rigid body. We also show that the discrete Clebsch

approach gives a corresponding integrator, referred to as a spatial Moser-Veselov (MV)

integrator, for the spatial description of the rigid body, and takes the form of the dis-

crete Euler-Poincare equations for the spatial representation of the rigid body derived

by Bobenko & Suris (1999a). The discrete EP equations are known to correspond to

the Lie-Poisson equations on the dual of a semi-direct Lie-algebra.

In pursuit of a unified framework, we then apply the discrete Clebsch approach

to give MV integrators for the heavy top and the coupled rigid body. These examples

demonstrate how the discrete Clebsch approach is applied for potential and coupled mo-

tions respectively. Furthering this pursuit still, we also show how the discrete Clebsch

approach gives MV integrators for (the singularity free) Cayley-Klein parameterised

rigid body motion. Finally, we present numerical results which show a comparative

study of the conservative properties of the body and spatial DMV algorithms for the

rigid body and the conservative properties of the body DMV algorithm for the heavy

top and coupled rigid body motions. This Chapter provides the necessary geometric

preliminaries and assessment of the DMV algorithms for extension to elastic bodies, a

subject which we pursue in the next Chapter.

The discrete Clebsch approach The discrete Clebsch approach (Cotter & Holm

2006) gives a (Hamiltonian) discrete Noether’s theorem which states the condition for

the discrete Euler-Lagrange equations to conserve the momentum map. We derived

the momentum maps corresponding to the symmetry reductions of the discrete (time)

Lagrangians and show that the form of the right momentum map is the same as its

continuous form. Through this momentum map, the discrete time body representation

of the dynamics on the cotangent bundle gives the Moser and Veselov integrator. The

key point, here, is that the discrete Clebsch approach only recovers the MV integrator if

the discretisation of the body angular velocity in the reconstruction formula is the same

as that used to derive Moser and Veselov’s discrete Lagrangian from the continuous

Lagrangian.

85

Lie-Poisson structure This Chapter has pursued the relationship between vari-

ational integrators, derived in the discrete Clebsch framework, and existing studies of

discrete integrable rigid body systems. In the body representation, we recover a dis-

crete integrable analogue of the Euler-Arnold equations, first discovered by Moser &

Veselov (1991). In the spatial representation we obtain discrete EP equations with

an advected parameter. These correspond to the Lie-Poisson equations on the dual

space of a semidirect product Lie algebra discovered by Bobenko & Suris (1999a). This

discovery provides a discrete extension to the work by Holm, Marsden & Ratiu (1998)

who developed the theory of EP entirely within a Lagrangian framework so that the

EP equations with advection always correspond to Lie-Poisson Hamiltonian systems on

the dual of a semi-direct product Lie-algebra. Consequently, the DMV equations have

a family of Casimirs associated with the Lie-Poisson bracket for these systems, two of

which we confirmed by numerical simulation of the spatial representation of the rigid

body.

Heavy tops We applied the discrete Clebsch approach to the body representation

of the heavy top to give a MV integrator with a discrete auxiliary equation for the

advection of the gravity vector. This integrator is Lie-Poisson w.r.t. to the dual of the

semi-direct product Lie-algebra s∗ = (so(3)×R3)∗. The MV integrator is solved using

the standard DMV algorithm proposed by McLachlan & Zanna (2005).

Coupled rigid bodies Application of the discrete Clebsch approach to the coupled

rigid body model of Grossman et al. (1988) gives a coupled MV integrator for each

body and an auxiliary equation for the relative orientation matrix. We will refer to

this equation in Chapter 3 as it turns out to be the same as the compatibility equation

for fully discrete elastic rod models. The MV integrator conserves total spatial angular

momentum and is Lie-Poisson w.r.t. to the dual of the semi-direct product Lie-algebra

s∗ = (so(3)×so(3))∗×SO(3). We derive the DMV algorithm for this coupled integrator

from the coupled matrix Ricatti equation. The DMV algorithm for coupled motion is

used in Chapter 3 to solve for the isotropic pseudo-rigid body.

Cayley-Klein parameterisation By simply replacing SO(3) with the matrix group

SU(2) in the discrete action principle, the discrete Clebsch approach gives a DMV al-

gorithm for the Cayley-Klein parameterisation of rigid body motion. It is well known

that this parametrisation is singularity-free in contrast with Euler-angle parameterisa-

tion. The DMV algorithm is explicit when the motion has at least one axis of symmetry,

i.e. is a Lagrange top. Unlike other integrators for the Cayley-Klein parameterised rigid

body (see Hairer et al. 2002, Leimkuhler & Reich 2005, Wendlandt & Marsden 1997)

our approach is unique in that it is systematically derived in a framework for body and

86

spatial representations (although we do not give the spatial representation here). Our

framework provides additional geometric insight into the DMV algorithm. Specifically,

we show that the momentum map for the symmetry reduction by SU(2) is a Hopf

fibration (as it is in the continuous case).

The discrete Clebsch approach gives the correspondence between the quaternionic

formulation and the (standard) SO(3) formulation of DMV integrators. The question

of comparing variational discretisations of rigid body motion formulated in quaternions

with SO(3) formulated MV integrators was posed in Wendlandt & Marsden (1997). We

propose a comparative numerical study of the quaternionic formulation of MV integ-

rators with alternative quaternionic discretisations , such as those given in Wendlandt

& Marsden (1997), as the subject of future research.

Numerical experiments We provide results from several numerical experiments to

demonstrate the conservative properties, computational efficiency and accuracy of the

DMV algorithms for all the rigid body motions considered in this Chapter.

Appendix Appendix A give Tables comparing the MV integrators with the continu-

ous form of the rigid body, heavy top and coupled rigid body. These Tables show that

the discrete and continuous versions of the equations of motion and the momentum

maps from the cotangent bundle to the dual of the Lie algebras are remarkably similar.

Appendix A also provides a Table summarising the SU(2) DMV algorithm for the body

representation of the rigid body in Cayley-Klein parameters and the corresponding for-

mulation in quaternions.

Chapter 3 The rigid body motions considered in this Chapter served only as ex-

amples to (i) explain the geometric principles guiding the development of a unified

computational framework for continuum dynamics and (ii) demonstrate the conser-

vative properties of the DMV algorithms. We continue in the next Chapter by first

generalising the application of the discrete Clebsch approach to give MV integrators

for the convective and spatial representations of ellipsoidal motions and then applying

the approach to more challenging problems in elastic body dynamics.

87

Chapter 3

Moser-Veselov Integrators for

Elastic Body and Rod Motions

Synopsis This Chapter develops a unified computational framework for the convect-

ive and spatial representations of elastic body motion, polar decomposed pseudo-rigid

body motion and geometrically exact elastic rods. In this Chapter, we apply the dis-

crete Clebsch approach of Cotter & Holm (2006) to give Moser-Veselov integrators for

two examples of elastic motions.

We firstly present MV integrators for the spatial and convective representations of free

ellipsoidal motions on GL(n,R)+ (denoted GL(n)+ from hereon) before restricting our

attention to a pseudo-rigid body.

Polar decomposition of the pseudo-rigid body motion gives a MV integrator which (i)

exhibits a discrete momentum and Kelvin circulation law and (ii) is solved using the

DMV algorithm for the coupled rigid body motions given in Appendix A.4, based on

the DMV algorithm for the rigid body given by McLachlan & Zanna (2005). Numerical

simulations of a Mooney-Rivlin material (Mooney & Rivlin 1977) conserve vorticity to

10−15 and exhibit no secular drift in the energy levels.

We then develop MV integrators for continuum elastic rods by using a discrete

Kirchhoff kinetic analogy to apply the MV integrator for the Lagrange top to the static

inextensible rod. Extending this analogy to the time dependent case gives a discrete

compatibility equation for the dynamic elastic rod model which takes the same form as

the discrete auxiliary equation for the relative orientation matrix, given by the discrete

formulation of the coupled rigid body model of Chapter 2. We go on to formulate a

MV integrator for an extensible and shearable elastic rod, which Krishnaprasad et al.

(1988) refer to as a geometrically exact elastic rod. Numerical simulations of this model,

with 50 rod sections, conserve spatial angular momentum to an order of 10−8, linear

momentum to an order of 10−11 and exhibit no secular drift in the energy error whose

88

mean is the order of 10−2 (over 104 time steps).

The main theorems and equations of motion of this Chapter are stated in the Table

below.

Result Equation page

MV integrators for ellipsoidal motion on GL(n)+ (3.27), (3.35) pg. 96 & pg. 98A discrete Kelvin circulation theorem for the pseudo-rigid body (3.53) pg. 102MV integrators for the polar decomposed pseudo-rigid body (3.60) pg. 104A discrete Kirchhoff rod analogy (3.3.1.2) pg. 106MV integrators for the geometrically exact rod (3.106) pg. 117

Table 3.1: A summary of the main theorems and equations of motion in this Chapter.

3.0.1 Overview

In the late 1980’s and early 1990’s Juan Carlos Simo proposed a general approach for

deriving integrators of Hamiltonian continuum systems on manifolds from the Hamilto-

nian on phase space Simo & Wong (1991), Simo et al. (1992), Lewis & Simo (1994),

Simo & Tarnow (1994). This general approach prescribes algorithms which exactly

preserve any two of energy, momentum or symplectic structure. We develop an altern-

ative general approach for deriving continuum integrators from a discrete variational

principle using the discrete Clebsch approach of Cotter & Holm (2006). The purpose

of this is to

• generalise the derivation of Moser-Veselov integrators and the momentum maps

corresponding to cotangent lifted actions of gl(n) for the convective and spatial

representations of ellipsoidal motion on GL(n)+. In this generalised model, either

the shape matrix or the right Cauchy-Green matrix is an advected quantity de-

pending on whether the motion is the spatial or body representation. Just as

we saw in the previous Chapter for the spatial representation of the rigid body

(in which the inertia matrix is an advected quantity), convective and spatial MV

integrators for motion with advected quantities define a co-adjoint action on the

dual of a semi-direct product Lie-algebra and are Poisson w.r.t. to the corres-

ponding Lie-Poisson bracket.

• formulate a MV integrator for the three dimensional isotropic pseudo-rigid body.

This model arises as a special case of ellipsoidal motion when the configuration

is GL(3)+ and the reference configuration is a sphere (the shape matrix is the

identity matrix). Using polar decomposition, we derive MV integrators for the

polar components of pseudo-rigid body motion in Section 3.2.1. These MV in-

tegrators provide a discrete analogue of the Euler-Poincare equations for polar

89

decomposed pseudo-rigid body motions derived by Sousa-Dias (2002). We also

give the left and right momentum maps corresponding to cotangent lifted actions

of so(3) on the cotangent bundle. These maps correspond to conservation of

angular momentum and vorticity.

Relation to fluid dynamics The analogy with fluid dynamics becomes appar-

ent from the latter conserved quantity. The invariance of the Lagrangian under

the action of S is referred to in the context of fluid dynamics as the ”particle re-

labelling symmetry” (see Ripa 1981, Salmon 1982). There, invariance of the vari-

ational principle for idealised fluids under continuous relabelling of fluid particle

labels by the action of the group of diffeomorphisms is associated with conser-

vation of fluid vorticity. Of course, the symmetry transformations in fluids are

both spatially and time dependent. This aside, the analogy with fluids can be

extended - the momentum map corresponding to conserved vorticity is shown to

give a discrete Kelvin circulation theorem. We shall return to the discussion of

particle relabelling symmetries in the next Chapter.

Finally, we implement a model of a Mooney-Rivlin material to describe the

stretching and rotational components of the motion and show that the discrete

Moser-Veselov conserves angular momentum and vorticity (relative to the Lag-

rangian frame) to an order of 10−15 and exhibit no secular drift in the energy

error whose mean is the order of 10−3 (over 104 time steps).

• derive discrete elastic rod models. The key step is the use of a discrete variant of

the Kirchhoff kinetic analogy in the discrete action principle for the discrete Lag-

range top. The discrete Clebsch approach then gives inextensible and extensible,

shearable discrete rod models in Sections 3.3.1 and 3.5 respectively. The latter

model is referred to as a geometrically exact elastic rod model. We implement

a geometrically exact elastic rod (with periodic boundary conditions) using an

explicit discrete Moser-Veselov integrator to compute the orientation of the dir-

ectors and a Stormer-Verlet symplectic integrator to compute the position of the

rod centroid. The discrete Moser-Veselov integrator for 50 rod sections conserves

total spatial angular momentum to an order of 10−8 over 104 time steps and

exhibits no secular drift in energy error whose mean is the order of 10−2. This

establishes a new and explicit geometric integrator for the geometrically exact

rod model.

90

We begin by reviewing the preliminaries of continuous ellipsoidal motion given by

Sousa-Dias (2002) and then formulate the necessary terminology to derive MV integ-

rators for the convective and spatial representations of ellipsoidal motion. Then in

Section 3.2, we consider the formulation of a MV integrator for the polar decomposed

motion of a Pseudo-rigid body and give the discrete conservation laws corresponding

to symmetries of the discrete Lagrangian.

In Section 3.3, we consider the problem of extending the application of MV integ-

rators to elastic rods. We first review Goriely & Nizette (1999), Chouaıeb (2003) for

the preliminaries of elastic rod models. Following the approach of Bobenko & Suris

(1999b), who consider general discrete time Lagrangian mechanics on Lie groups, we

systematically extend the Moser-Veselov integrators for rigid bodies to inextensible

elastic rod models using the discrete Clebsch approach.

We restrict our consideration of the MV integrator for the heavy top, presented

in Chapter 2, to a Lagrange top (I1 = I2). The MV integrator for the Lagrange top

relates to the equilibrium configuration of a static symmetric inextensible rod through a

discrete Kirchhoff kinetic analogy. We then extend this analogy and show that discrete

time motions of the symmetric inextensible rod correspond to discrete time motions of

elastically coupled Lagrange tops.

By relaxing the inextensibility constraint and extending the configuration to include

the position of the rod centroid, we arrive at the discrete geometrically exact elastic

rod model proposed by Krishnaprasad et al. (1988). Sections 3.4 and 3.5 describe

the derivation and discretisation of this model. Finally numerical simulations of the

isotropic pseudo-rigid body and geometrically exact elastic rod model are presented in

Section 3.6.

3.1 Free Ellipsoidal Motion

We follow Sousa-Dias (2002), who consider the geometric description of a free pseudo-

rigid body as a deformable motion in which the configuration space is identified with

the group of all invertible matrices with positive determinant GL(3)+. The free pseudo-

rigid body is a particular case of free ellipsoidal motion on GL(n)+.

We extend MV integrators to this motion by first generalising some relevant results

of Chapter 2 to ellipsoidal motions. For simplicity, it is assumed that the body is free

and there are no external forces. Decomposition of the pseudo-rigid body motion not

only gives insight into the geometry of deformation but is the basis of our proposed

computationally tractable extension of the DMV algorithm for rigid bodies.

We now generalise the body and spatial MV integrators for rigid body motion,

presented in the last Chapter, to the corresponding convective and spatial MV in-

tegrators for compressible ellipsoidal motion, in which the Lie group GL(n)+ is the

91

configuration space. The theory of elasticity describes how homogeneous compressible

elastic bodies deform in terms of ellipsoidal motions.

Ellipsoidal motion has two key properties useful for the development of a unified

computational framework. Firstly, it is the most general form of MV integrators for spa-

tially independent motion on finite dimensional Lie groups (except for the restriction to

real matrices with positive determinant) and secondly, as we will shortly briefly discuss,

the convective and spatial representations of ellipsoidal motion are a useful prototype

for the development of geometric integrators for the corresponding representations of

Hamiltonian compressible fluids.

We shall begin by describing the preliminaries of ellipsoidal motion leading to two

theorems stating the form of the convective and spatial MV integrators for ellipsoidal

motion.

3.1.1 Preliminaries

Consider the ellipsoidal motion in a container C (taken to be the Euclidean three space

R3) of a material point in a body B given by

x(t, `) = Q(t) ∙ `, (3.1)

where Q(t) ∈ GL(n)+ and x(0, `) = `. The kinetic energy of this motion, in the

material representation, is given by

L =1

2

∫

Cρ(x)x ∙ C0xd

nx =1

2

∫

Bρ(`)Q ∙ `, C0Q ∙ `d

n` =1

2Tr(C0QI0Q

T), (3.2)

where C0 ∈ V is a symmetric positive definite Riemannian metric on C defining the

shape of the container and I0 ∈ V ∗ : (I0)ab =∫B ρ(`)`

a`bdn` is a symmetric positive

definite Riemannian metric on B defining the shape of the body.

A Riemannian metric pairs two tangent vectors v,w ∈ Rn producing a real number.

Once a local basis is chosen, the metric takes the form of a constant matrix. The

simplest example is the dot product vigijwj , which is defined by the Riemannian metric

gij . In Rn, the metric is represented as the identity matrix gi,j = (Id)i,j .

The case I0 = Id and C0 = Id respectively describes a spherical reference config-

uration in Rn. The deformation gradient ∂xi/∂`j = Qij(t) is a function only of time.

Following Holm et al. (1986), we define the symmetric positive definite Cauchy-Green

metric as the product of the deformation gradients

Ct = QTC0Q. (3.3)

92

Before proceeding further, we define the notation for how g acts on each metric.

The group G acts on V by the right translation

φ∗g : V → V : φ∗g(C) = gTCgT , (3.4)

and recall that G acts on V ∗ by the left translation

φg : V → V : φg(I) = gIgT . (3.5)

We now express the material representation of the kinetic energy given by equation

(3.2) in the convective and spatial form through the following symmetry actions of

g ∈ GL(n)+ defined respectively as

Lφ∗g(C0)(g−1Q) := L(g−1Q, gTC0g) =

1

2Tr(g−1QI0Q

T g−T gTC0g), (3.6)

and

Lφg(I0)(Qg−1) := L(Qg−1, gI0g

T ) =1

2Tr(Qg−1gI0g

T g−T QTC0). (3.7)

The convective and spatial representations of the Lagrangian follow when g takes

the value Q, giving the reduced Lagrangian with advected parameters in the body

representation lC : g→ R of the form

lCt(Γ) = l(Γ, Ct) =1

2Tr(ΓI0Γ

TCt), (3.8)

where g = gl(n) and Γ is (minus) the left Q invariant convective velocity

Γ := Q−1Q. (3.9)

In the spatial representation, the Lagrangian lI : g→ R takes the form

lIt(γ) = l(Γ, It) =1

2Tr(γItγ

TC0), (3.10)

which is expressed in terms of the right Q invariant spatial velocity

γ := QQ−1. (3.11)

Remark 3.1.1.1. Note how the metrics become advected quantities in either represent-

ation. In the convective representation, the metric Ct (commonly referred to as the right

Cauchy-Green strain tensor), defined on C, is advected, whereas in the spatial repres-

entation, the body shape metric It, defined on B is instead advected. This has a natural

interpretation. In the spatial representation, the body deforms relative to the container,

whereas in the convective representation, the container deforms relative to the reference

93

space. The latter interpretation provides an interpretation of the Cauchy-Green metric

as defining the shape of the container in the reference space. The Cauchy-Green metric

is therefore the analogue of the body shape metric, which defines the shape of the body

in the container.

3.1.2 Convective and spatial representations of discrete ellipsoidal

motion

kinematics We state the discrete time kinematics by generalising the notation and

terminology presented in Chapter 2 for the rigid body. The discrete motion Qk =

Q(tk) ∈ GL(n)+ is a k parameterised sequence of configurations giving the sequence of

spatial points in the container

xk(`) = Qk ∙ `. (3.12)

The discrete convective velocity (up to a minus sign) and discrete spatial velocity are

respectively defined by the recursion relations

`k+1 = Γ−1k+1`k,

xk+1 = γk+1xk,(3.13)

Discrete reconstruction formulae As shown in Chapter 2, substitution of the

finite difference approximations

hΓ(tk) ≈ Λ−1k (Λk+1 − Λk),

hγ(tk) ≈ (Λk+1 − Λk)Λ−1k ,

(3.14)

into the convective and spatial continuous reconstruction formulae respectively given

by

Q = QΓ,

Q = γQ,(3.15)

and using the definition of the discrete velocities gives the convective and spatial rep-

resentations of the discrete reconstruction formula

94

Qk+1 = QkΓk+1,

Qk+1 = γk+1Qk.(3.16)

The finite difference approximation of the convective reduced Lagrangian, given

by equation (3.8), defines (up to a factor of h2) the Ck augmented discrete material

Lagrangian LCk : G×G→ R

LCk(Id, Q−1k Qk+1) :=

1

2Tr(Q−1k (Qk+1 −Qk)I0(Qk+1 −Qk)

TQ−Tk Ck

), (3.17)

where the Cauchy-Green metric Ck is given by Ck = QTkC0Qk. Expressing this equation

in terms of Γk+1, gives the Ck augmented discrete convective Lagrangian lCk : G→ R

lCk(Γk+1) :=1

2Tr((Γk+1 − Id)I0(Γ

Tk+1 − Id)Ck

). (3.18)

Analogously, the finite difference approximation of the spatial reduced Lagrangian,

given by equation (3.10), defines (up to a factor of h2) the Ik augmented discrete

material Lagrangian LIk : G×G→ R

LIk(Id, Qk+1Q−1k ) :=

1

2Tr((Qk+1 −Qk)Q

−1k IkQ

−Tk (Qk+1 −Qk)

TC0

), (3.19)

where the body shape metric Ik is given by Ik = QkI0QTk .

Expressing this equation in terms of γk+1, gives the Ik augmented discrete spatial

Lagrangian lIk : G→ R

lIk(γk+1) :=1

2Tr((γk+1 − Id)Ik(γ

Tk+1 − Id)C0

). (3.20)

In each representation, substitution of the rearranged expressions C0 = Q−Tk CkQTk

and I0 = Q−1k IkQ

−Tk into

Ck+1 = QTk+1C0Qk+1, (3.21)

and

Ik+1 = Qk+1I0QTk+1, (3.22)

gives the discrete auxiliary equations for Ck

Ck+1 = φ∗Γk+1(Ck) = Γ

Tk+1CkΓk+1, (3.23)

95

and for Ik

Ik+1 = φγk+1(Ik) = γk+1IkγTk+1. (3.24)

We seek the left and right momentum maps corresponding to the left and right

augmented diagonal actions of g ∈ G = GL(n)+ on G×G×V ∗ by applying the discrete

Clebsch approach. Recall, from Chapter 2, that we first add Clebsch constraints for

the discrete reconstruction formula and the discrete auxiliary equation to the discrete

Lagrangians lCk and lIk defined in equations (3.8) and (3.10) giving

lCk(Γk+1) = lCk(Γk+1)+ 〈Pk+1, Qk+1−QkΓk+1〉+ 〈Jk+1, Ck+1−ΓTk+1CkΓk+1〉, (3.25)

and

lIk(γk+1) = lIk(γk+1) + 〈Pk+1, Qk+1 − γk+1Qk〉+ 〈Jk+1, Ik+1 − γk+1IkγTk+1〉. (3.26)

Convective representation

Theorem 3.1.2.1 (Convective MV integrators for ellipsoidal motion on GL(n)+). The

MV integrator for the convective representation of (free) ellipsoidal motion on GL(n)+

is given by

Ad∗Γ−1k

Mk+1 =Mk + 2Ck−1 � ∇Ck−1 lCk−1 ,

Ck = φ∗Γk(Ck−1),

(3.27)

where lCk−1 is the convective discrete Lagrangian (evaluated at time tk−1) given by

equation (3.18) and φ∗g is defined by the right action of g on the space of symmetric

matrices V .

Proof. In the convective representation, stationarity of the discrete action principle

δSd =∑

k δlCk = 0 gives the following expressions paired with variations in the dy-

namical variables

〈Pk − Pk+1ΓTk+1, δQk〉 = 0↔ Pk+1 = PkΓ

−Tk+1,

〈Jk − Γk+1Jk+1ΓTk+1 +∇Ck lCk , δCk〉 = 0⇔ Jk+1 = Γ

−1k+1 (Jk +∇Ck lCk) Γ

−Tk+1,

〈∇Γk+1 lCk −QTk Pk+1 − 2CkΓk+1Jk+1, δΓk+1〉 = 0.

(3.28)

We substitute the first two expressions for Pk+1 and Jk+1 into the last to give

96

∇Γk+1 lCk −QTk PkΓ

−Tk+1 − 2CkΓk+1Γ

−1k+1 (Jk +∇Ck lCk) Γ

−Tk+1 = 0. (3.29)

Right multiplication of this expression by ΓTk+1 gives

∇Γk+1 lCkΓTk+1 −Q

Tk Pk − 2CkJk − 2Ck∇Ck l(Ck) = 0. (3.30)

Finally substituting the expressions for the derivatives

Mk+1 := ∇Γk+1 lCk = Ck(Γk+1−Id)I0 and ∇Ck lCk =1

2(Γk+1−Id)I0(Γ

Tk+1−Id), (3.31)

into equation (3.30) simplifies to the expression for the map Mk+1 : T∗(G × V ) → g∗

given by

Mk+1 = QTk Pk + 2CkJk. (3.32)

It follows by the theorem of momentummaps for lifted actions (see theorem 2.5.1.3, pg. 34),

that the map Mk+1 = QTk Pk+2CkJk is an infinitesimally equivariant right momentum

map JR for cotangent lifted left actions of ζ ∈ g on T ∗(G× V ) given by the pairing

〈JR, ζ〉 = 〈Pk, ζG(Qk)〉+ 〈Jk, ζTCk + Ckζ〉. (3.33)

JR takes the form JR = Q � P + C � J , where the bilinear operator � in the first term

is defined as � : V × V∗ → g∗ and in the second term as � : V × V ∗ → g∗, where V ∈ V

is a symmetric matrix.

Substituting the discrete flow equations (3.28), the discrete auxiliary equation (3.23)

and the discrete reconstruction formula (3.16) into the right momentum map given by

equation (3.32) gives

Mk+1 = Ad∗Γk

(QTk−1Pk−1

)+ 2Ck−1

(Jk−1 +∇Ck−1 lCk−1

)

= Ad∗Γk (Mk − 2Ck−1Jk−1) + 2Ck−1(Jk−1 +∇Ck−1 lCk−1

= Ad∗Γk(Mk + 2Ck−1∇Ck−1 lCk−1

).

(3.34)

This expression together with the discrete auxiliary equation give the form of the con-

vective MV integrator in the theorem in which the pairing between elements of V and

V ∗ has been replaced by the operator �.

97

Spatial representation

Theorem 3.1.2.2 (Spatial MV integrators for ellipsoidal motion on GL(n)+). The

MV integrator for the spatial representation of (free) ellipsoidal motion is given by

Ad∗γkmk+1 = mk + 2∇Ik−1 lIk−1 � Ik−1,

Ik = φγk(Ik−1),(3.35)

where lIk−1 is the spatial discrete Lagrangian (evaluated at time tk−1) given by equation

(3.20) and φg is defined by the left action of g on the space of symmetric matrices V∗.

Proof. In the spatial representation, stationarity of the discrete action principle δSd =∑

k δlIk = 0 gives the following expressions paired with variations in the dynamical

variables

〈Pk − γTk+1Pk+1, δQk〉 = 0↔ Pk+1 = Γ

−Tk+1Pk,

〈Jk − γTk+1Jk+1γk+1 +∇Ik lIk , δIk〉 = 0⇔ Jk+1 = γ

−Tk+1 (Jk +∇Ik lIk) γ

−1k+1,

〈∇γk+1 lIk − Pk+1QTk − 2Jk+1γk+1Ik, δγk+1〉 = 0.

(3.36)

We substitute the first two expressions for Pk+1 and Jk+1 into the last expression to

give

∇γk+1 lIk − γ−Tk+1PkQ

Tk − 2γ

−Tk+1 (Jk +∇Ik lIk) Ik = 0. (3.37)

Left multiplication of this expression by γTk+1 gives

γTk+1∇γk+1 lIk − PkQTk − 2 (Jk +∇Ik lIkIk) = 0. (3.38)

Finally substituting the expressions for the derivatives

mk+1 := ∇γk+1 lIk = (γk+1 − Id)Ik and ∇Ik lIk =1

2(γTk+1 − Id)(γk+1 − Id), (3.39)

into equation (3.38) simplifies to the expression for the map mk+1 : T∗(G× V ∗) → g∗

given by

mk+1 = PkQTk + 2JkIk, (3.40)

which is an infinitesimally equivariant left momentum map for cotangent lifted right

actions of ζ ∈ g on T ∗(G × V ∗) (see the right momentum map for the convective

representation for further details).

Substituting the discrete flow equations (3.36), the discrete auxiliary equation (3.24)

98

and the discrete reconstruction formula (3.16) into the left momentum map given by

equation (3.40) gives

mk+1 = Ad∗γ−1k

(Pk−1Q

Tk−1

)+ 2

(Jk−1 +∇Ik−1 lIk−1Ik−1

)

= Ad∗γ−1k(mk − 2Jk−1Ik−1) + 2

(Jk−1 +∇Ik−1 lIk−1Ik−1

)

= Ad∗γ−1k

(mk + 2∇Ik−1 lIk−1Ik−1

).

(3.41)

This expression together with the discrete auxiliary equation give the form of the spatial

MV integrator in the theorem.

DMV algorithms The convective and spatial MV integrators for ellipsoidal motion

can not be solved using the DMV algorithm proposed by McLachlan & Zanna (2005).

Rather than develop a new algorithm, we shall instead restrict our discussion to a spe-

cific homogeneous elasticity model for which McLachlan and Zanna’s DMV algorithm

can be adapted.

3.2 The Pseudo-Rigid Body

We restrict our consideration of MV integrators for ellipsoidal motions on G = GL(3)+.

Following Sousa-Dias (2002), we consider a discrete Lagrangian on the polar decom-

posed configuration space and begin by reviewing the Lagrangian description of the po-

lar decomposed isotropic pseudo-rigid body. This is the simplest case when the reference

configuration is spherical. To stimulate future research, we provide the Euler-Poincare

description of the polar decomposed anisotropic pseudo rigid body in Appendix B.

3.2.1 Polar decomposition of discrete pseudo-rigid body motion

Definition 3.2.1.1 (Polar Decomposition (Sousa-Dias 2002)). The polar decomposition

of any n × n invertible matrix with positive determinant Q ∈ GL(n)+ is Q = RTDS,

where R,S ∈ SO(n)+ and D ∈ D(n) where D(n) is the space of all diagonal matrices

with positive determinant.

Remark 3.2.1.2 (Uniqueness of ”Polar” decomposition). This decomposition is re-

ferred to by Sousa-Dias (2002) as ”bi-polar” decomposition so as to avoid conflict with

the standard definition of polar decomposition as Q = RTU , where R is an orthogonal

matrix and U is a symmetric positive definite matrix (Ciarlet 1988). Further diag-

onalisation of U gives an equivalent decomposition as the definition above. For this

reason, our definition of polar decomposition is only unique when the corresponding

99

diagonalisation of U is unique. Diagonalisation of symmetric positive definite matrices

is unique, up to permutations of the eigenvalues, if the eigenvalues of U are distinct

(see, for example, Horn & Johnson 1991).

Consider a Lagrangian defined on TGL(3)+ of the form

L =Tr

2(QQT ), (3.42)

which is invariant under the left and right actions of SO(3). We polar decompose the

velocity to give

Q = RT (−ΩD + D +Dω)S, (3.43)

where Ω := RRT and ω := SST are the respective right invariant angular velocity and

internal circulation angular velocity. Reduction of L by the left and right actions of

SO(3) gives the reduced Lagrangian l : so(3)2 × TD(3)→ R given by

l =Tr

2

(−Ω2D2 − ω2D2 + 2ΩD + D2

). (3.44)

For the purposes of forming a discrete Kelvin circulation theorem, it is simplest to

neglect the stretching term D here, although we will include stretching in the derivation

of MV integrators for pseudo-rigid bodies in Section 3.2.3.

Discrete motion Consider the discrete action principle for the isotropic (Q = RTS ∈

SO(3)) free pseudo-rigid body

Sd =∑

k

Tr(ΩTk+1ωk+1

)−Tr

2

(P Tk+1(Qk+1 − uk+1Qk)

)

− Tr(ΘRk+1(Ωk+1Ω

Tk+1 − Id)

)− Tr

(ΘSk+1(ωk+1ω

Tk+1 − Id)

),

(3.45)

which is formed by finite difference discretisation of the terms in Ω and ω, where the

discrete velocities are Ωk+1 := Rk+1RTk and ωk+1 = Sk+1S

Tk . Variations in Qk and Pk

give the update equations

Pk+1 = uk+1Pk,

Qk+1 = uk+1Qk,(3.46)

where uk+1 := Qk+1QTk . These equations preserve the symplectic two-form dQk ∧ dPk.

Variations in Ωk+1 give the expression for the momentum map Mk+1 : T∗SO(3) →

100

so(3)∗ of the form

Mk+1 = ωk+1ΩTk+1 − Ωk+1ω

Tk+1 = Sk+1(Q

Tk Pk − P

Tk Qk)S

Tk+1, (3.47)

in which the update equation for Pk+1 and the polar decomposed expression for uk+1

has been used. This expression defines the relation Pk = Qk+1, which is consistent with

the definition of the discrete kinetic energy 〈Qk+1, Qk〉. The angular momentum in the

Lagrangian coordinate frame takes the form

RTk+1Mk+1Rk+1 = PkQTk −QkP

Tk . (3.48)

Similarly, variations in ωk+1 give the expression for the momentum map

Nk+1 = Ωk+1ωTk+1 − ωk+1Ω

Tk+1 = ωk+1Rk(QkP

Tk − PkQ

Tk )R

Tk ω

Tk+1, (3.49)

which is the vorticity in the Lagrangian coordinate frame

STk+1Nk+1Sk+1 = PTk Qk −Q

Tk Pk. (3.50)

An identical proof to that given in Section 2.5.5 of Chapter 2 confirms the con-

servation of mk+1 = RTk+1Mk+1Rk+1 = mk and nk+1 = STk+1Nk+1Sk+1 = nk. The

former conserved quantity is the angular momentum and the latter is the vorticity in

the Lagrangian coordinate frame.

3.2.2 Conservation of circulation

Before stating a discrete Kelvin circulation theorem, it is useful to recall the continuous

Kelvin circulation theorem for the Pseudo-rigid body. In order to do this, we shall

briefly introduce some new notation and terminology.

Background

Recall that the spatial coordinate x of a label in a pseudo-rigid body ` is given by the

motion x = Q(t)`, Q ∈ GL(3)+. It follows from the definition of this motion that the

(right Q invariant) spatial velocity takes the form u = QQ−1x. Kelvin’s circulation

theorem states that the circulation (differential) one-form u ∙ dx around a closed loop

moving with velocity u is conserved. This can be more concisely expressed using the

exterior derivative

d

dtd(u ∙ dx) = 0, along u =

dx

dt. (3.51)

101

Substituting the definition of u into the differential two-form gives

d(QQ−1x ∙ dx) = (QQ−1)ijdxj ∧ dxi

=1

2(Q−T QT − QQ−1)ijdx

i ∧ dxj

=1

2(Q−T QT − QQ−1)ijd(Qia`

a) ∧ d(Qjb`b)

=1

2(Q−T QT − QQ−1)ijQjbQiad`

a ∧ d`b

=1

2(Q−Tim QTmj − QilQ

−1lj )QiaQjbd`

a ∧ d`b

=1

2(δmaQ

TmjQjb − Q

TliQiaδlb)d`

a ∧ d`b

=1

2(QTQ−QT Q)abd`

a ∧ d`b,

(3.52)

where the steps in the second and fourth lines follow from the respective skew-symmetry

and bi-linearity of the differential two-form. The Kelvin circulation theorem is therefore

equivalent to the statement that QTQ−QT Q is a constant of motion along u = dxdt . This

constant of motion is the vorticity of the internal material, relative to the Lagrangian

frame. The vorticity is a conserved momentum map corresponding to invariance of the

continuous Lagrangian under the tangent lifted right action of S on the tangent bundle

TQ. This symmetry is commonly referred to as a ”particle relabelling symmetry”.

Equally, the invariance of the Lagrangian under the tangent lifted right action of R

on the tangent bundle corresponds to another conserved momentum map, the angular

momentum of the pseudo-rigid body.

Discrete Kelvin circulation theorem

Definition 3.2.2.1. It follows from the definition of the forward map that the (right Q

invariant) discrete spatial velocity takes the form uk+1 := Qk+1Q−1k such that xk+1 =

uk+1xk. Consider the circulation (differential) one-form xk+1 ∙dxk around a closed loop

with position xk+1 at time tk+1.

We shall now state the discrete Kelvin circulation theorem and show, analogously

to the continuous case, that it is equivalent to conservation of discrete vorticity.

Theorem 3.2.2.2 (Discrete time Kelvin circulation). The change in the exterior de-

rivative of the circulation one-form about a closed loop c(xk+1) is

Δtd(xk+1 ∙ dxk) = 0 along xk+1 = ukxk. (3.53)

102

Proof. Substituting the reconstruction formula xk+1 = uk+1xk into the differential two-

form gives

d(uk+1xk ∙ dxk) = (Qk+1Q−1k )ijdx

jk ∧ dx

ik

=1

2(Q−Tk QTk+1 −Qk+1Q

−1k )ijdx

ik ∧ dx

jk

=1


−1k )ijd((Qk)ia`

a) ∧ d((Qk)jb`b)

=1


−1k )ij(Qk)ia(Qk)jbd`

a ∧ d`b

=1

2

((Qk)

−Tim (Qk+1)

Tmj − (Qk+1)il(Qk)

−1lj

)(Qk)ia(Qk)jbd`

a ∧ d`b

=1

2

(δma(Qk+1)

Tmj(Qk)jb − (Qk+1)

Tli(Qk)iaδlb

)d`a ∧ d`b

=1

2(QTk+1Qk −Q

TkQk+1)abd`

a ∧ d`b,

(3.54)

where the steps in the second and fourth lines follow from the respective skew-symmetry

and bi-linearity of the differential two-form dxk ∧ dxk. It follows that the discrete

time motion of an isotropic Pseudo-rigid body satisfies a discrete Kelvin circulation

theorem because the discrete vorticity STk+1Nk+1Sk+1 is a momentum map which, upon

substitution of Pk = Qk+1 into its definition in equation (3.50), gives the constant

expression QTk+1Qk − QTkQk+1. Conservation of discrete vorticity corresponds to the

invariance of the discrete Lagrangian under right actions of Sk.

3.2.3 MV integrators for Mooney-Rivlin materials

The MV integrator, with the stretching component Dk, is now derived to include an

elastic potential term. Following Sousa-Dias (2002), we choose to model a rubbery

material classified as a Mooney-Rivlin material. We shall revisit the Lagrangian for

the free polar decomposed pseudo-rigid body in equation (3.44) and now include the

stretching components in the finite difference of ˙RTDS. This approximation gives (up

to a factor of −h2) the holonomically constrained discrete Lagrangian lck : (V×D)2 → R

given by

lck =Tr

2

((4(Ωk+1 + ωk+1)− 6Id)D

2k − 2Ωk+1Dkωk+1Dk − (Dk+1 −Dk)

2)

−Tr

2

(ΘSk+1(ωk+1ω

Tk+1 − Id)

)−Tr

2

(ΘRk+1(Ωk+1Ω

Tk+1 − Id)

),

(3.55)

where (Dk+1 −Dk)/h is referred to as the discrete stretching velocity (in the material

representation) and the last two terms are the holonomic constraints on ωk+1 and Ωk+1

103

respectively.

The Clebsch constrained discrete Lagrangian takes the form

lk = lck + Tr

(PR

T

k+1(Rk+1 − Ωk+1Rk))+ Tr

(PS

T

k+1(Sk+1 − ωk+1Sk)), (3.56)

where the second and third terms are constraints for the update of the pseudo-rigid

body orientation and internal circulation respectively.

Application of the discrete Clebsch approach gives the following Clebsch relations,

paired with δΩk+1 and δωk+1 respectively,

Dk(2Id − ωTk+1)Dk − P

Rk+1R

Tk = Θ

Rk+1Ωk+1,

Dk(2Id − ΩTk+1)Dk − P

Sk+1S

Tk = Θ

Sk+1ωk+1,

(3.57)

and from the symmetry of ΘRk+1 and ΘSk+1, equation (3.57) gives the left momentum

maps for the cotangent lifted actions of SO(3) on T ∗(SO(3)× SO(3)×D)

Mk+1 := J(ωk+1)ΩTk+1 − Ωk+1J

T (ωk+1) = PRk+1R

Tk+1,

Nk+1 := J(Ωk+1)ωTk+1 − ωk+1J

T (Ωk+1) = PSk+1S

Tk+1.

(3.58)

Substituting the discrete flow equations

PRk+1 = Ωk+1PRk ,

PSk+1 = ωk+1PSk ,

(3.59)

into the momentum maps in equation (3.58) gives the MV integrators for the rotational

and internal circulatory momenta

Ad∗Ωk+1Mk+1 =Mk,

Ad∗ωk+1Nk+1 = Nk.(3.60)

Stretching component We add a potential energy term of the form −h2W (Dk) to

the discrete Lagrangian in equation (3.56). The function W (Dk) is given by

W (Dk) = aI1(D2k) + bI2(D

2k) + c|Dk|

2 − dLog(|Dk|), a, b, c, d > 0, (3.61)

104

and characterises a Mooney-Rivlin material (see Sousa-Dias 2002). Note that Ii are

the principal invariants of D2k whose expressions are given in Sousa-Dias (2002). The

discrete Euler-Lagrange equations in Dk define an additive update of the diagonal

matrix Dk given by

Dk+1 = −4πD(−Ωk+1 + 2Id − ωk+1)Dk

+ πD(Ωk+1Dkωk+1 + ωk+1DkΩk+1)−Dk−1 + h2∇DkW (Dk),

(3.62)

where πD denotes projection of the principal diagonal of the matrix. Note that the

elements of Dk are not guaranteed to stay positive. To alleviate this feature, we ensure

that the last term in the potential energy term is sufficiently large to prevent the

determinant of Dk from vanishing during simulation. The simulations of the polar

components of the motion of a Mooney-Rivlin material are provided in Section 3.6.

We shall now consider the formulation of MV integrators for elastic rods and begin

by reviewing the description of the Kirchhoff rod given by Goriely & Nizette (1999).

3.3 Elastic Rod Preliminaries

3.3.1 The discrete Kirchhoff rod analogy

Consider a discrete ribbon, a space curve {φ(Si)}i∈{0,1,...,N} parameterised by arc-length

Si := iΔS, with three smooth orthonormal unit vector-fields, d2(Si) which is orthogonal

to the curve, d3(Si) is aligned with the unit tangent vector to the curve d3(Si) =

t(Si) = φ′(Si) and d1(Si) = d2 × d3. The triad {d1,d2,d3} is related to the Frenet

basis {n,b, t} at each Si by an orthogonal transformation about t(Si). For this reason,

the triad is referred to as an adapted Frenet frame. The Euler angle of rotation about

t(Si) is referred to as the twist angle. The ribbon simplifies to the Bernoulli’s elastica

when the triad coincides with the Frenet basis. When the principal moments of inertia

about the axes d1 and d2 are equal, the ribbon has equal principal bending stiffnesses

and is referred to as a symmetric ribbon.

The definition of the ribbon will now be used to define the discrete elastic rod

(also known as Kirchhoff’s elastica). We refer the reader to Antman (1995), Goriely

& Nizette (1999) for a detailed description of the kinematics of the continuous elastic

rod. The elastic rod is described as symmetric when its ribbon is symmetric.

Definition 3.3.1.1 (Discrete Elastic Rod). In the absence of external torques, it follows

from the definition of a continuum Kirchhoff rod, given by Bobenko & Suris (1999a),

105

Kirchhoff rod Lagrange top

Discrete angular strain at Si Ωi Discrete body angular velocity at time tk ΩkStiffness matrix C0 Inertia matrix I0Rod tension p0 Position of centre of mass χ0

Tangent vector at Si ti Orientation of gravity vector at time tk Γk

Table 3.2: This Table shows the correspondence between the terms used to describethe discrete Lagrange top and those to describe the discrete Kirchhoff rod.

that a discrete elastic rod is an arc-length Si parameterised ribbon of fixed length L

minimising the functional

Fd =∑

i

li(f(Ω)i), (3.63)

where the reduced Lagrangian l : G→ R is given by

li =∑

i

〈f(hΩ)i, C0f(hΩ)i〉︸︷︷︸elastic potential

− 〈p0, ti〉︸︷︷︸inextensibility constraint

, (3.64)

in which Si ∈ {ih, i := 0 → N}, f : so(3) → SO(3) : f(hΩ)i denotes a

transformation of hΩ(Si) into G at position Si, where Ω(Si) is the angular strain at this

position and when the discrete set {Si}i is periodic, h = L/N . p0 is the uniform tension

in the rod which enforces the inextensibility constraint φ(SN )−φ(S1) =∑

i t(Si). C0 is

a spatially independent diagonal matrix comprised of the principal bending and torsional

stiffnesses of the homogeneous rod.

As a result of the following theorem, which is a discrete variant of the Kirchhoff

kinetic analogy given by Bobenko & Suris (1999a), it will be shown that this functional

coincides with the discrete action principle for the body representation of the Lagrange

top under a particular choice of f(hΩ).

Theorem 3.3.1.2 (Discrete Kirchhoff kinetic analogy). The N frames of the arclength

Si parameterised (homogeneous) discrete symmetric elastic rod are in 1-to-1 corres-

pondence with the discrete time motion of the Lagrange top over N time intervals if

f(hΩ)i := Ωki+1 − Id, where the discrete angular strain is given by Ω

ki+1 = Λ

kT

i Λki+1.

Proof. Substitution of f(hΩ)i := ΛkT

i (Λki+1 − Λ

ki ) into the Lagrangian for the discrete

106

rod given by equation (3.64) gives

li =∑

i

Tr((ΛkTi+1 − Λ

kTi )Λ

kiC0Λ

kT

i (Λki+1 − Λ

ki ))− 〈p0, ti〉

=∑

i

Tr((ΛkTi+1Λ

ki − Id)C0(Λ

kT

i Λki+1 − Id)

)− 〈p0, ti〉

=∑

i

Tr(Ωki+1C0)− 〈p0, ti〉.

(3.65)

Now define Pd := {Si, C0, ti,p0}i:=1→N as a subset of variables and constants form-

ing the discrete elastic rod functional Fd and Qd := {tk, I0,mgΓk, χ0}k:=1→N as the

subset of variables and constants forming the discrete Lagrange top action principle

Sd. Forming the discrete elastic rod functional Fd from Qd, instead, gives the reduced

discrete action principle Sd for the Lagrange top given in the form of the first two terms

of equation (2.142) on pg. 55 in Chapter 2.

Replacing the terms in Qd with the those in Pd in the definition of the MV integrator

for the Lagrange top gives

Ni+1 = Ad∗ΩTiNi + p0 � ti,

ti = Ωiti−1,(3.66)

where the dual to the angular strain Ni is given by Ni :=2h2skew

((∇Ωi+1 l)

T Ωi+1

)and

ti := t(Si) is the unit tangent vector to the curve at Si.

Lie-Poisson structure

The Lie-Poisson structure preserved by the discrete elastic rod equations is in one-to-one

correspondence with the structure preserved by the discrete Lagrange top equations,

which are a specialisation of the discrete heavy top equations (2.155) for the case when

the moments of inertia are I1 = I2. It follows from Section 2.7 and the discrete Kirchhoff

rod analogy, that the Lie-Poisson brackets preserved by the discrete elastic rod take

the form

{F1, F2}±(Ni, ti) = −1

2Tr(±NT

i [∇NiF1,∇NiF2])

± ti ∙ (Φ(∇NiF1)∇tiF2 − Φ(∇NiF2)∇tiF1) ,(3.67)

in which the norms of the dual of the angular strain ||Ni|| and the extensibility con-

straints ||ti|| are the Casimirs. Preservation of the Lie-Poisson structure leaves the

107

rod length as an invariant of the discrete flow given by equation (3.66). We shall now

briefly consider the discrete Lagrangian for the discrete time motion of a symmetric

dynamical Kirchhoff rod before turning to the more difficult problem of formulating a

discrete geometrically exact rod.

3.3.2 Time dependent discrete Kirchhoff rod models

We form the discrete Lagrangian for the discrete time motion of the (inextensible)

Kirchhoff rod model by adding a discrete kinetic energy term, for rigid body rotations

of the set of n orthonormal frames, to the discrete Lagrangian given in equation (3.64)

to give

lk =n∑

i=1

Tr(Ωk+1T

i I0)︸︷︷︸kinetic energy

+ Tr(ΩkT

i+1C0)︸︷︷︸elastic potential energy

− 〈p0, tki 〉︸︷︷︸

extensibility constraint

. (3.68)

The essential feature of this model is the relation between the discrete angular

velocities Ωki and the discrete angular strains Ωki which is expressed as a discrete com-

patibility equation.

Lemma 3.3.2.1 (Discrete compatibility equation). The discrete dynamical Kirchhoff

rod exhibits the discrete compatibility equation

Ωk+1i+1 = Ωk+1i+1 Ω

ki+1Ω

k+1T

i . (3.69)

Proof. Temporal recursion of Λki followed by spatial recursion of Λk+1i takes the form

Λk+1i+1 = Λk+1i Ωk+1i+1 = Λ

kiΩ

k+1i Ωk+1i+1 . (3.70)

Conversely, spatial recursion of Λki followed by temporal recursion of Λki+1 gives

Λk+1i+1 = Λki+1Ω

k+1i+1 = Λ

ki Ω

ki+1Ω

k+1i+1 . (3.71)

Equating these two expressions and eliminating Λki gives the discrete compatibility

equation

Ωk+1i+1 = Ωk+1i+1 Ω

ki+1Ω

k+1T

i . (3.72)

Remark 3.3.2.2. The discrete compatibility equation takes the form of the discrete

auxiliary equation for the relative orientation matrix in the discrete coupled rigid body

model given in equation (2.167).

108

Correspondence with the elastically coupled Lagrange top It follows from the

discrete Kirchhoff kinetic analogy that the discrete time motion (over n intervals) of

N frames of the discrete convective representation of the elastic rod are in 1-to-1 cor-

respondence with the discrete time motion (over N intervals) of n elastically coupled

Lagrange tops. From the definition of the discrete Lagrangian for the Lagrange top

given in equation (2.142) and the form of the elastic potential in equation (3.68), the

reduced Lagrangian for n elastically coupled Lagrange tops is given by

lk =n∑

i=1

Tr(Ωk+1T

i I0)︸︷︷︸kinetic energy

+ Tr(ΩkT

i+1C0)︸︷︷︸elastic potential energy

− mg〈χ0i ,Γki 〉︸︷︷︸

gravitational potential energy

. (3.73)

Replacing {tk, I0, C0, χ0i ,Γki } by {Si, C0, I0,p

k0, t

ki } in this discrete Lagrangian gives

the discrete Lagrangian for the discrete dynamical Kirchhoff rod given by equation

(3.68).

The discrete equations of motion

The discrete equations of motion for the dynamical inextensible Kirchhoff rod model

are now derived from the discrete Lagrangian

lk =∑

i

〈Ωk+1i , I0〉+ 〈Ωki+1, C0〉 − 〈p

k0, t

ki 〉. (3.74)

Appending the Clebsch constraints to this Lagrangian gives

lk = lk +∑

i

〈P k+1i

2,Λk+1i − ΛkiΩ

k+1i 〉+ 〈

P ki+12

,Λki+1 − Λki Ω

ki+1〉

+ 〈Jk+1i , tk+1i − Ωk+1iT tki 〉+ 〈J

ki+1, t

ki+1 − Ω

ki+1

T tki 〉

− 〈θk+1i ,Ωk+1i Ωk+1iT − Id〉 − 〈θ

ki+1, Ω

ki+1Ω

ki+1

T − Id〉.

(3.75)

Taking variations in Ωk+1i and Ωki+1 and right multiplying by the transpose of these

matrices respectively gives the Clebsch relations:

I0Ωk+1i

T − ΛkiTP k+1i Ωk+1i

T /2− tki ⊗ Jk+1i Ωk+1i

T = θk+1i , (3.76)

C0Ωki+1

T − ΛkiT P ki+1Ω

ki+1

T /2− tki ⊗ Jki+1Ω

ki+1

T = θki+1. (3.77)

Variations in Λki and tki respectively give

109

P ki − Pk+1i Ωk+1i

T + P ki − Pki+1Ω

ki+1

T = 0,

Jki + Jki +∇tki ltki − Ω

k+1i Jk+1i − Ωki+1J

ki+1 = 0.

(3.78)

Adding equations 3.76 and 3.77 and defining Mk+1i = I0Ω

k+1i

T − Ωk+1i I0 and Nki+1 =

C0Ωki+1

T − Ωki+1C0 gives

Mk+1i +Nk

i+1 + skew((P ki + P

ki )TΛki

)+ 2(Jki + J

ki ) � t

ki + 2∇tki ltki � t

ki = 0. (3.79)

It is convenient to rewrite equation 3.76 as

ΩkiTMk

i Ωki = Ω

kiT(skew(Λk−1i

TP ki ΩkiT ) + 2skew(tk−1i ⊗ JkiΩ

kiT ))Ωki

= Λki � Pki + 2t

ki � J

ki .

(3.80)

Similarly, we can write equation 3.77 as

ΩkiTNk

i Ωki = Ω

kiT(skew(Λki−1

T P ki ΩkiT ) + 2skew(tki−1 ⊗ J

ki Ω

kiT ))Ωki

= Λki � Pki + 2t

ki � J

ki .

(3.81)

Finally, substituting equations 3.80 and 3.81 into equation 3.79 gives the discrete equa-

tions of motion for the dynamical Kirchhoff rod model

Mk+1i +Nk

i+1 −AdΩkiMki −AdΩki

Nki + 2∇tki ltki � t

ki = 0. (3.82)

Note that the only unknowns in this equation at each time step are the body angu-

lar momenta for each node Mk+1i . We also note the symmetry in the equation under

switching Λki+1 with Λk+1i in the definition of Mk

i , Nki ,Ω

ki and Ω

ki . This symmetry of

course only appears because of the choice of discretisation of Ω and Ω. The inextens-

ibility constraint also appears as a forcing term in the above equation of motion.

DMV algorithm In principle, the above equation of motion can be recast as a N

coupled system of matrix Ricatti equations and solved for each Λk+1i with a DMV

algorithm for each matrix Ricatti equation. Once all Λk+1i are known, one can then

compute each Nk+1i from its definition. The presence of the forcing term arising from

the inextensibility constraint renders these equations stiff and a challenge to robustly

solve using the standard form of the DMV algorithm, however. Specifically, we find

that the eigenvalues of the Hamiltonian matrix associated with each of the N matrix

110

Ricatti equations exceed stability constraints. Further research should address how to

regularise the dynamical Kirchhoff rod equations to guarantee stability.

Extension to the geometrically exact elastic rod The discrete Kirchhoff rod

model describes rods which deform by bending and twisting but preserve the length. It

is precisely the imposition of the length constraint which stiffens the system of Matrix

Ricatti equations and renders them challenging to solve. For this reason we consider

an alternative class of elastic rods which deform by shearing and stretching. These are

the class of rod motions which Krishnaprasad et al. (1988) and Simo & Vu-Quoc (1986,

1988) considered leading to the derivation of the geometrically exact elastic rod model.

The purpose of the next Section is to present a MV integrator for this model which can

be solved using a DMV algorithm.

We proceed by reviewing the terminology and notation used by (Krishnaprasad et al.

1988) to describe the geometrically exact rod model. In contrast to the Hamiltonian

approach taken by Krishnaprasad et al. (1988), we formulate the variational descrip-

tion of this rod model. We verify, in Appendix B.2, that the variational description

gives the equations of motion corresponding to the continuum Lie-Poisson equations

given by Krishnaprasad et al. (1988). These equations of motion are referred to, more

generally, as ’Lagrange-Poincare’(LP) equations (see Holm 2002). In general, LP equa-

tions describe partially reduced variational dynamics associated with Lagrangians with

more than one group symmetry. For the geometrically exact rod, the SO(3) reduced

dynamics given by the LP equations are only partially reduced. These equations can

be further reduced by the action of the group of diffeomorphisms Diff(R3), but this

is not a stage which we pursue here because it is only relevant to the continuum rod

model and our interest is pursuing a MV integrator for the discrete rod.

In Section 3.5, we give the MV integrator for the convective angular and linear

momentum equations corresponding to the LP equations in the continuum rod model.

These equations describe the motion of a set of particles, whose position is the rod

centroid at arc-length Sα and is given in coordinates relative to the director frame

at this position. These equations also describe the update of each director frame in

the convective representation. A symplectic integrator is then used to compute the

particle trajectories and the MV integrator, the attitude of the directors positioned on

each particle. Numerical results are presented in Section 3.6 which demonstrate the

conservative properties of the spatial angular momentum, linear momentum (in the

material representation) and energy.

111

3.4 The Geometrically Exact Elastic Rod Model

We begin by reviewing the formulation of the continuum geometrically exact rod model

given by Krishnaprasad et al. (1988).

3.4.1 Preliminaries

Krishnaprasad et al. (1988) consider a rod whose placement in the container C at time

t is defined to be the set

Pt := {x ∈ R3| x = φ(S, t) + ζ(S)αdα(S, t), S ∈ B, (ζ

1(S), ζ2(S)) ∈ A(S), ζ3(S) = 0},

(3.83)

where the family of diffeomorphic maps

φ : B → C, (3.84)

are volume and orientation preserving embeddings of the reference configuration B =

[0, L] in the container. A(S) ⊂ R2 is a (disk-like) cross-sectional area of the rod at a

point in the reference configuration S ∈ [0, L] along a S-parameterised curve, represent-

ing the line of centroids of the rod in its natural (unstressed) reference configuration.

φt(S) = φ(S, t) is the motion of the line of rod centroids given by

φ : [0, L]× R+ → C : φ(S, t0) = S. (3.85)

The special (restricted) theory of Cosserat rods describes the orientation of the

cross-section in the container by the set of director fields {dα(S, t)}α:=1→3, whose third

director d3(S, t) is the normal vector to the cross-sectional area. The directors are

subject to length constraints for all times t and S ∈ [0, L].

||dα(S, t)|| = 1, α = 1, 2; d1(S, t) ∙ d2(S, t) = 0, (3.86)

in addition to the shear limiting constraint

d3(S, t) ∙ φ′(S, t) > 0, d3(S, t) := d1(S, t)× d2(S, t) 6= 0, (3.87)

where ′ = ∂∂S and the set of directors {dα(S, t)}α:=1→3 is a rotating orthonormal frame

whose origin is fixed at the point of the reference line of centroids parameterised by S.

There exists a unique orthogonal transformation

Λt : B → SO(3) : dα(S, t) = Λt(S)Eα(S, t), α := 1→ 3, (3.88)

112

from the inertial frame {Eα}α:=1→3 (the basis of R3) to the set of directors. Thus

{Λt(S)}S∈[0,L] is a S-parameterised curve in SO(3) at each time t.

Adopting the approach taken in Krishnaprasad et al. (1988), which is to regard

the transformations as the state variables rather than the directors themselves, the

configuration space of the rod is the product of smooth maps

C := {Φ = (φ,Λ) : B → R3 × SO(3)}, (3.89)

where the rod motion Φt(S) = Φ(S, t) is a submanifold of SO(3) × R3, which defines

the configuration of the rod at time t. In order to determine the symmetries of the

Lagrangian for this model (which we define in Section 3.4.2), we shall first define the

following SO(3) actions which leave the Lagrangian for this rod invariant.

Lie Group actions We begin by defining the left and right actions of SO(3) on the

rod configuration C for some B ∈ SO(3), A : B → SO(3) which represents the orient-

ation of a director and w ∈Diff(C) which represents the position of the rod centroid

with reference to an arc-length S ∈ B.

ΨL : SO(3)× C → C : ΨL(B, (A(S), w(S))) = B ∙ (A(S), w(S)) := (BA(S), Bw(S)),

(3.90)

ΨR : C × SO(3)→ C : ΨR(B, (A(S), w(S))) = (A(S), w(S)) ∙B := (A(S)B,w(S)),

(3.91)

where the right action of B on w(S) is trivial. Following Krishnaprasad et al. (1988),

we now recall the convective and spatial representations of the velocities and strain

measures arising from symmetry reduction of the tangent lift of the respective group

actions ΨL and ΨR on the tangent bundle TC = {(Φ, Φ) : B → TSO(3)× TC}.

Velocities Associated with the motion of the rod is the material velocity field VΦtdefined by

VΦt(S) := Φt(S) :=(φt(S), Λt(S)

). (3.92)

The spatial velocity of the rod motion is the time derivative of Φt(S) at fixed x ∈ C

uΦt(x) := Φt(S) |x := (uφt(x), ωt(x)) . (3.93)

The pull-back of uΦt by Φt is the convective velocity of the rod motion

VΦt(S) = Φ∗tuΦt(x) = (Φ

∗tuφt ,Φ

∗tωt) = (φ

∗tΛ∗tuφt(x),Λ

∗tωt(x)) := (Vφt(S),Ωt(S)).

(3.94)

113

The definition of the linear and angular velocities in each description is given in

Table 3.3.

component material spatial convective

Linear: Uφt = φt ∈ TφtC, uφt = φt ∙ φ−1t ∈ Tφ0C, Vφ = −Λ

Tt˙(φ−1t ) ◦ φt ∈ Tφ−10

B

Angular: Λt ∈ TΛtSO(3), ωt = ΛtΛTt ∈ so(3), Ωt = Λ

Tt Λt ∈ so(3)

Table 3.3: The material, spatial and convective representations of the velocities.

Strain measures The rod motion describes elastic deformations which result in axial

and torsional strains. The one dimensional deformation gradient of the rod is

Φ′t(S, t) := (φ′t(S, t),Λ

′t(S, t)), (3.95)

where φ′t and Λ′t are the linear (axial) and angular (torsional) strains. The different

representations of these strains are presented in Table 3.4.

Material Convective Spatial

Linear strain measure φ′t − d3 Γ := −ΛTt φ−1t (φ

′t − d3) γ := (φ′t − d3)

′ ∙ φ−1tAngular strain measure Λ′t Ω := ΛTt Λ

′t ω := Λ′tΛ

Tt

Table 3.4: The material, convective and spatial representations of the strain measures.

With all the terms defined, we now describe the variational description of the geo-

metrically exact rod in the next Section.

3.4.2 The variational formulation of the geometrically exact elastic

rod model

Approach Following Krishnaprasad et al. (1988) we consider the material description

of the kinetic energy of the rod which is

K(t) =

∫

BdS

ρ0

2A|φt(S)|

2 +ρ0

2Tr(Λt(S)I0Λt(S)

T ), (3.96)

where ρ0 is the density of the homogeneous rod. The kinetic energy is invariant under

the tangent lift of the actions ΨL and ΨR, the latter of which is the particle relabelling

symmetry. The material description of the potential energy of the rod is the stored

energy

V (t) =

∫

BdS〈(φ′t − d3), C

φ0 (φ

′t − d3)〉+ Tr

(Λ′tC

Λ0 Λ′Tt

). (3.97)

114

It is assumed that along the rod centroid the (i) cross-sectional area, (ii) stiffness

and shear modulus and (iii) elasticity tensor C and moments of inertia are constant.

It is also assumed that the matrix of bending and torsional stiffnesses CΛ0 , the matrix

of shear stiffnesses Cφ0 and the inertia tensor I0 are diagonal and constant in a rotating

frame of directors.

Symmetries Note that invariance of V (t) under the left action of SO(3) is only ob-

tained by imposing that Cφ is isotropic, i.e. the shear stiffnesses in the directions of

each of the directors are the same. Otherwise, rotation of a rod cross-section, para-

meterised by S, breaks the symmetry unless d3(S, t) is aligned with the tangent vector

at φt(S) (this corresponds to the case when there is no rod shearing). V (t) is also

invariant under the right action of SO(3) in the restrictive case when CΛ is isotropic,

i.e., the bending and torsional stiffnesses are the same.

Boundary conditions For simplicity, we shall only consider periodic boundary con-

ditions, i.e. both φt(0) = φt(L) and Λt(0) = Λt(L). With these boundary conditions,

the rod model dynamics are invariant under linear translations and rotations of the rod

in R3.

The Convective Lagrangian The lifted left action of g = ΛTt ∈ SO(3) on the

tangent bundle gives the reduced lagrangian density `(ΛTt φt,ΛTt φt,Λ

Tt φ′t; Ω, Ω) in the

convective representation given by

` =

∫

BdS

ρ0

2A|V|2 +

ρ0

2Tr(ΩI0Ω

T )− Tr(ΩCΛ0 ΩT )− 〈Γ, Cφ0 Γ〉, (3.98)

where V := ΛTt φt and Γ := ΛTt (φ

′t − d3) are the linear material velocity and axial

material strain measure in the rotating frame. All other dynamical variables (in both

the convective and spatial representation) are given in Tables 3.3 and 3.4.

The application of the Clebsch approach (see Holm & Kupershmidt 1983) to derive

the LP equations for the SO(3) reduced rod motion is outlined in Appendix and we

shall not consider the continuum rod further. We shall now consider the application

of the discrete Clebsch approach of Cotter & Holm (2006) to give a MV integrator for

the geometrically exact rod.

3.5 The Discrete Geometrically Exact Elastic Rod Model

Consider the discrete Lagrangian for an extensible and shearable rod in the convective

representation derived by finite difference approximations of the continuous convective

Lagrangian given by equation (3.98)

115

`n+1 =∑

α

ρ02A|Vnα |

2

︸︷︷︸kin. energy of centroid

+ρ02Tr(I0Ω

n+1α

T )︸︷︷︸kin. energy of frames

− Tr(CΛ0 Ωnα+1

T )︸︷︷︸twisting pot. energy

− 〈Γnα, Cφ0 Γ

nα〉︸︷︷︸

shearing pot. energy

,

(3.99)

where the notation is defined in Table 3.5. The second and third term of this discrete

Lagrangian is similar are also present in the discrete Lagrangian for the (inextensible)

time dependent Kirchhoff rod given by equation (3.68). The subscript α denotes a

particle label and the superscript n denotes the nth time increment.

Recall from Chapter 2, that in the constrained coordinate formulation we holonom-

ically constrain Ωn+1α and Ωnα+1 to give the discrete constrained Lagrangian

`n+1c = `n+1 − {∑

α

Tr(Θn+1α (Ωn+1α Ωn+1α

T − Id))+ Tr

(Θnα+1(Ω

nα+1Ω

nα+1

T − Id))}.

(3.100)

The Clebsch constrained Lagrangian density is

ˆn+1 = `n+1c + 〈Pn+1α ,Λn+1α − ΛnαΩn+1α 〉+ 〈J n+1α ,Λnα+1 − Λ

nαΩ

nα+1〉

+ 〈Pn+1α , φn+1α − φnα − hΛnαV

n+1α 〉+ 〈Jnα+1, φ

nα+1 − φ

nα − hΛ

nα(Γ

nα+1 + E3)〉,

(3.101)

where the first and second terms are the Clebsch constraints for the temporal and

spatial update of the rod frame orientations Λnα and the third and fourth terms are the

Clebsch constraints for the temporal and spatial update of the rod centroid positions

φnα.

From the stationary action principle, one gets after rearranging the derived expres-

sions for the functional derivative of ˆ paired with δΛnα and δφnα,

Pn+1α =(Pnα − J

nα + J

nα+1Ω

nα+1

T + hY n+1α+1

)Ωn+1α ,

Pnα − Pn+1α = Jnα+1 − J

nα .

(3.102)

Linear material velocity Vnα :=ΛTαn

h (φn+1α − φnα)

Linear material strain measure Γnα :=ΛTαn

h (φnα+1 − φ

nα − h(d3)

nα)

Convective angular velocity Ωn+1α := ΛTαnΛn+1α

Convective angular strain measure Ωnα+1 := ΛTαnΛnα+1

Table 3.5: This Table gives the expressions for the discrete linear velocities and strainmeasures.

116

where Y n+1α+1 = P

n+1α ⊗ Vn+1α + Jnα+1 ⊗ (Γ

nα+1 + E3).

The left momentum map for the left SO(3) symmetry reduced discrete Lagrangian

density is given by

Jn+1α = 2skew(ΛnαTPnα)− 2skew(Λ

nαTJ nα ) + h skew

(mn+1α ⊗ Vn+1α

)

︸︷︷︸=0

+ hskew(nnα+1 ⊗ (Γ

nα+1 + E3)

),

(3.103)

where the linear angular momentum mn+1α := ∇Vn+1α`n+1 and nnα+1 := ∇Γnα+1`

n+1. The

image of this momentum map is the difference of mn+1α := I0Ω

n+1α

T − Ωn+1α I0 and

nnα+1 := Cα0 Ω

nα+1

T − Ωnα+1Cα0 .

Substituting the expressions in equation (3.102) for Pnα and Jnα into the above equa-

tion gives the explicit discrete Lagrange-Poincare equation for the convective angular

momentum

mn+1α = Ωnα

TmnαΩ

nα + Ω

nαTnnαΩ

nα − n

nα+1 − hskew

(nnα+1 ⊗ (Γ

nα+1 + E3)

). (3.104)

The discrete Clebsch relations also give the following expressions in the rotating

frame for the material linear momentum and dual of the material axial strain measure

mn+1α = hΛnαTPn+1α , nnα+1 = hΛ

nαTJnα+1. (3.105)

In summary the discrete linear and angular momentum equations are respectively

mn+1α = ΩnαTmnα − n

nα+1 + Ω

nαTnnα,

mn+1α = Ad∗ΩnαT

mnα +Ad

∗ΩnαT n

nα − n

nα+1 − hskew

(∇Γnα+1`

n+1 ⊗ (Γnα+1 + E3)).

(3.106)

The discrete linear momentum equation is just the discrete Euler-Lagrange equation

∇φnα`nα(φ

nα, φ

n+1α , φnα+1) +∇φnα`

n−1α (φn−1α , φnα, φ

n−1α+1) +∇φnα`

n−1α−1(φ

nα−1, φ

n+1α−1, φ

nα) = 0,

(3.107)

where `nα denotes the αth term of the discrete Lagrangian given in equation (3.99).

These are the discrete equations of motion which are used to generate the numerical

results in Section 3.6 demonstrating the conservative properties of the DMV algorithm

for this rod model. Just as in the continuous case, the last term of the discrete angular

momentum couples the particle positions with the orientation of the directors and

117

represents shearing motion. Future research should establish whether these discrete

flow maps are Poisson w.r.t. the Lie-Poisson bracket for the geometrically exact rod

model given by Krishnaprasad et al. (1988).

We shall now consider an alternative extension of the MV integrator for rigid body

motion, presented in Chapter 2, to pseudo-rigid body motion. In the next Section we

present the DMV algorithms for solving the MV integrators for the pseudo-rigid body

and geometrically exact rod and assess their conservative properties through numerical

experiments.


3.6.1 DMV algorithms for pseudo-rigid bodies and elastic rods

With the exception of the pseudo-rigid body model, the algorithms used to generate

the numerical results, presented in this Section, are explicit and therefore not subject

to conditions which may affect convergence properties. Details of the iterative step

used to compute the pseudo-rigid body motion are provided in this Section.

The pseudo-rigid body

The discrete internal circulation velocity and rotational velocities of the pseudo-rigid

body are coupled through a Coriolis term. Consequently, the MV integrator for the

pseudo-rigid body casts to a coupled matrix Ricatti equation of the form

Mk+1 =M′k+1 − J(ωk+1)Ω

Tk+1 +Ωk+1J

T (ωk+1),

Nk+1 = N′k+1 − J(Ωk+1)ω

Tk+1 + ωk+1J

T (Ωk+1),(3.108)

where J(Xk+1) := DkXTk+1Dk and M

′k+1 and N

′k+1 are uncoupled matrix Ricatti equa-

tions of the form

M ′k+1 := JkΩTk+1 − Ωk+1Jk,

N ′k+1 := JkωTk+1 − ω

Tk+1Jk,

(3.109)

where Jk = 2D2k.

We solve for the velocity components Ωk+1 and ωk+1 by applying the DMV al-

gorithm for the coupled rigid body motion given in Appendix A.4, to equation (3.108).

Following step 2 of this algorithm, we split equation (3.108) into separate matrix Ric-

atti equations by making the substitution J(Xk+1) ≈ J(Xk) and solve for the velocities

satisfying the split matrix Ricatti equations

118

Ωk+1 = J−1k (−M

′k+1 + S

Rk+1),

ωk+1 = J−1k (−N

′k+1 + S

Sk+1),

(3.110)

where SR and SS are symmetric matrices with expressions determined from the split

matrix Ricatti equations. Following step 3, we recompute M ′k+1 amd N′k+1 using

J(Xk+1) and repeat these steps until Ωk+1 and ωk+1 converge to a specified tolerance

in the matrix two-norm.

In the numerical simulations of a Mooney-Rivlin material, shortly described, only a

few iterations of the DMV algorithm for coupled rigid motions are required on average

for the velocities to converge in the matrix two-norm to an order of 10−15, one order

above the precision of the machine.

Stability of the algorithm Experiments suggest that the DMV algorithm breaks

down for a large initial stretching velocity (Dk+1 − Dk)/h. More precisely, for a suf-

ficiently large stretching velocity, the real parts of the eigenvalues of (−M ′k+1 + SR)T

and (−N ′k+1 + SS)T become negative and no longer satisfy the stability constraint

determined by Cardoso & Leite (2001).

The update of the stretching matrix is explicit but not multiplicative and does there-

fore not ensure that the determinant of Dk remains positive. Consequently, the para-

meterisation of the potential energy is chosen to prevent the determinant vanishing

in numerical simulations. Geometric integrators which preserve positivity in the nu-

merical solution of matrix Riccati equations have been investigated by Dieci & Eirola

(1994) and this model might provide an excellent application of their work.

The geometrically exact rod model

Simo’s computational approach Simo’s computational approach for the geomet-

rically exact rod given by Krishnaprasad et al. (1988) uses an approximation of the

Cayley transform to compute the rotations and the finite element method to spatially

discretise the weakform of the momentum balance equations and the constitutive re-

lations. Nodal incremental rotations and vorticities are computed at each node and

the orientation and position of each frame is computed by interpolation of the nodal

values. This approach, however, requires a Newton-Raphson method to solve for the

new configuration. Simo proposed the use of quaternions to avoid the singularity ex-

hibited by Euler parameterisation and used Spurrier’s algorithm to efficiently compute

the quaternions from an orthonormal matrix.

119

Spurrier’s algorithm provides a singularity free extraction of a Quaternion from a

direction cosine matrix M which is precise for all rotation angles. This algorithm uses

only the largest component of the quaternion (which is greater than or equal to a half)

to compute square roots and the divisor in the computation of the other components.

This step avoids computation of negative square root arguments or division by zero,

both due to numerical imprecision. Essentially, the algorithm determines the largest

of tr(M), Mi,i, i ∈ {1 . . . 3} and computes the components of the quaternions using

different expressions depending upon whether tr(M) is the largest. These expressions

are specified in Spurrier (1977).

Our approach distinguishes itself from Simo’s in two ways

• Our discrete rod model is entirely variational, that is, the discrete model is de-

rived from a discrete Hamilton’s action principle. The configuration update is

performed by two mechanical integrators, a symplectic integrator for the update

of the arc-length parameterised set of rod centroid positions and a Lie-Poisson

integrator for the orientation of the corresponding set of directors. Consequently

energy levels do not drift and angular momentum is conserved to numerical round

off.

• The configuration update is explicit. A Stormer-Verlet second order symplectic

integrator computes the rod centroid positions and a DMV algorithm computes

the orientation of the set of directors.

Details of numerical experiments which demonstrate the conservative properties are

now provided.

3.6.2 Numerical results

Numerical experiments of the pseudo-rigid body and convective representation of the

geometrically exact rod model are performed to demonstrate the conservative properties

of the DMV algorithms and describe the elastic body dynamics. We summarise the

main observations that can be made from the Figures provided in this Section.

The pseudo-rigid body

• The time update of the eigenvalues of the pseudo-rigid body are shown in Figure

3.1. The motion of the Mooney-Rivlin material is observed to pulse with a fre-

quency which increases with the values of the Mooney-Rivlin parameters b = 10

and c = 10. These parameters increase the stiffness of the body. We observe

the determinant of the rigid body configuration varies between approximately 0.8

and 1, thus characterising this motion as quasi-incompressible.

120

• The components of the angular momenta and vorticity relative to the Lagrangian

coordinate frame are shown in Figure 3.2 to be periodic because the Coriolis

coupling between the angular velocity and vorticity is relatively small. The an-

gular momentum and vorticity relative to the Lagrangian coordinate frame are

also shown to remain on the sphere.

• Figure 3.3 shows that the DMV algorithm for the pseudo-rigid body exhibits no

secular drift in the energy error and conserves angular momentum and vorticity

to numerical round-off.

121

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5

1

1.5

λ 1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5

1

1.5

λ 2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5

1

1.5

λ 3

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000.5

1

1.5

|λ|

time steps

Figure 3.1: This Figure shows the eigenvalues of numerical simulations of a Pseudo-rigidbody motion over 104 time steps at a time step Δt = 0.05 for the case when the initialeigenvalues are (d1 = 1, d2 = 0.8, d3 = 1/0.8) and the Mooney-Rivlin parametersare a = 0.1, b = 10, c = 10, d = 50. The top three graphs show, respectively, thefirst, second and third eigenvalues of the body which describe the shape of the ellipsealong the principal axis. The bottom graph shows the determinant of the stretchingmatrix which evaluates to the product of the eigenvalues. The initial conditions forthis simulation are the initial body angular momentum and vorticity components, andstretching velocity given as M1(0) = 0.02, M2(0) = 10

−4, M3(0) = 0.0152, N1(0) =−0.022, N2(0) = −10−4, N3(0) = −0.0172 and d1(0) = 2 × 10−4, d2(0) = 1.6 ×10−4, d3(0) = 2.5× 10−4. The tolerance for the iterative step is 10−10. The first threegraphs show that each eigenvalue evolves in pulses which are approximately of the sameorder of magnitude. The bottom graph shows that the body is nearly incompressible,which is characterised by the large values of the parameters b and c.

122

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.04

-0.02

0

0.02M

1N

1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-5

0

5x 10

-3

M2

N2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.05

0

0.05M

3N

3

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100006

7

8x 10

-4

time steps

||M||||N||

Figure 3.2: This Figure shows the components of the body angular momentum andvorticity in numerical simulations of pseudo-rigid body motion over 104 time steps ata time step Δt = 0.05 for the case when the initial eigenvalues are (d1 = 1, d2 =0.8, d3 = 1/0.8) and the Mooney-Rivlin parameters are a = 0.1, b = 10, c = 10, d = 50.The top three graphs show, respectively, the first, second and third component of thebody angular momentumM and vorticity N . The bottom graph shows the norm of thebody angular momentum and vorticity. The initial conditions for this simulation arethe initial body angular momentum and vorticity components, and stretching velocitygiven as M1(0) = 0.02, M2(0) = 10

−4, M3(0) = 0.0152, N1(0) = −0.022, N2(0) =−10−4, N3(0) = −0.0172 and d1(0) = 2×10−4, d2(0) = 1.6×10−4, d3(0) = 2.5×10−4.The tolerance for the iterative step is 10−10. The first three graphs show that eachcomponent of M and N is periodic with different periods. The bottom graph showsthat the body angular momentum and vorticity remain on the sphere.

123

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-3

E-E

0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

2

4

6

8x 10

-15

||m-m

0||

time steps

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-4

-2

0

2

4

6

8

10x 10

-15

||n-n

0||

time steps

Figure 3.3: This Figure shows the error in the energy, and angular momentum and vor-ticity, relative to the Lagrangian coordinate frame, in numerical simulations of pseudo-rigid body motion over 104 time steps at a time step Δt = 0.05 for the case whenthe initial eigenvalues are (d1 = 1, d2 = 0.8, d3 = 1/0.8) and the Mooney-Rivlinparameters are a = 0.1, b = 10, c = 10, d = 50. The top graph shows the energyerror in the solutions computed by the DMV algorithm and the bottom graph showsthe error in the angular momentum and vorticity relative to the Lagrangian coordinateframe. The initial conditions for this simulation are the initial body angular momentumand vorticity components, and stretching velocity given as M1(0) = 0.02, M2(0) =10−4, M3(0) = 0.0152, N1(0) = −0.022, N2(0) = −10−4, N3(0) = −0.0172 andd1(0) = 2× 10−4, d2(0) = 1.6× 10−4, d3(0) = 2.5× 10−4. The tolerance for the iterat-ive step is 10−10. The first graph shows that the mean energy error does not drift andthe bottom two graphs show that the mean angular momentum and vorticity errors,relative to the Lagrangian coordinate frame, are conserved to numerical round-off.

124

The geometrically exact elastic rod

The experimental parameters for simulation of a geometrically exact elastic rod are

the material properties: the principal moments of inertia {I1 = 2, I2 = 2, I3 = 1},

the respective shear and principal bending stiffnesses along axis t1 and t2 are 0.2 and

[2×10−3, 3×10−3] and the respective axial and torsional stiffnesses are 0.2 and 4×10−3.

The rod centroid is initially perturbed so that φi(t0) = a0{sin(π2Si), 0, sin(π2Si)}, where

the amplitude a0 = 0.1, the arc-length parameterisation Si = i LN−1 , i := 1 → N and

the orientation of the N directors at position Si is chosen so that the N − 1 rod rigid

sections are aligned with the tangent vector of φi(t0). The following Figures show the

conservative properties of the rod motion and provide a qualitative description of the

rod motion and are summarised below

• Figure 3.4 shows that the mean error in total spatial angular momentum for 50

rod sections is of the order of 10−8 after 104 steps. Comparison with the numerical

results for spatial angular momentum conservation of the DMV algorithm for the

rigid body and coupled rigid body, presented in Chapter 2, suggest that the linear

elastic coupling term contributes a marginal error to the momentum.

• Figure 3.5 shows that the mean error in total spatial linear momentum is of the

order of 10−11 and that this error also grows linearly with the number of rod

sections.

• Figure 3.6 shows that the rod model (with 50 sections) exhibits no secular drift

in the mean energy error.

125

0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5x 10

-8

Time steps

Tot

al S

patia

l Ang

ular

Mom

entu

m E

rror

5 sections25 sections50 sections

Figure 3.4: This Figure shows the total spatial angular momentum error of the rodover 104 time steps of size Δt = 0.1 for rods with 5, 25 and 50 frames. The error scaleslinearly with the number of frames.

126

0 2000 4000 6000 8000 100000

0.5

1

1.5

2

2.5x 10

-11

Time steps

Line

ar M

omen

tum

Err

or

12 sections25 sections50 sections100 sections

Figure 3.5: This Figure shows the total material linear momentum error of the rodover 104 time steps of size Δt = 0.1 for rods with 12, 25, 50 and 100 frames. In eachcase, the mean total linear momentum error is to numerical round-off. The numericalround-off error scales linearly with the number of frames.

127

0 2 4 6 8 10

x 104

-0.016

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

Time steps

Ene

rgy

Err

or

Figure 3.6: This Figure shows the energy error of the rod over 104 time steps of sizeΔt = 0.1. There are 50 directors in this model and the initial conditions for thissimulation are given above. The graph shows that the mean energy error is an orderof 10−2 and does not drift.

128

3.7 Summary

In this Chapter, we applied the discrete Clebsch approach of Cotter & Holm (2006) to

derive MV integrators for the convective and spatial representations of free ellipsoidal

models before specialising to a pseudo-rigid body. We also applied the discrete Clebsch

approach to give MV integrators for elastic rod models such as the geometrically exact

elastic rod model.

MV integrators for elastic motions In each case, a critical step is performed in the

formulation of the discrete action principle to give DMV algorithms for these models.

For the pseudo-rigid body, we polar decompose the configuration space GL(3)+ and

derive a MV integrator for the polar components of the rigid body and a corresponding

coupled DMV algorithm. For the elastic rod, a discrete variant of the Kirchhoff kinetic

analogy is used to formulate the discrete Lagrangian from the MV integrator for the

Lagrange top, considered in Chapter 2. The analogy gives a compatibility equation

which takes the form of the discrete auxiliary equation for the relative orientation

matrix given by the MV integrator for the coupled rigid body model, also considered

in Chapter 2. For each model, we show how the discrete Clebsch approach gives

momentum maps which correspond to conservation laws. These conservation laws

are verified by numerical experiment.

Pseudo-rigid Body We present theorems giving MV integrators for the spatial and

convective representations of free ellipsoidal motion on GL(n)+. These integrators

seek a new algorithm to implement them. We then take the critical step of polar

decomposing the motion on GL(3)+ to give MV integrators and DMV algorithms for

the polar components of the pseudo- rigid body corresponding to internal circulation,

stretching and rotation.

We show how to apply the DMV algorithm for coupled rigid body motions to

solve for the internal circulatory and rotational components of the pseudo-rigid body

motion. This is made possible through the observation that the the expressions defining

the momentum and vorticity contain a Coriolis term, coupling the discrete internal

circulatory and rotational velocities, which appears in the same form in the discrete

coupled rigid body model. We cast the expressions for the momentum and vorticity

as a coupled matrix Ricatti equation and solve this using the iterative DMV algorithm

given in Appendix A.4.

We also derive the momentum maps for left and right symmetry reductions which

correspond to conservation of spatial angular momentum and circulation respectively.

This derivation culminates in a discrete Kelvin circulation theorem. Numerical ex-

periments of a Mooney-Rivlin material show that the energy levels exhibit no secular

129

drift and angular momentum and vorticity is conserved to numerical round-off. These

conservative numerical models provide a new and efficient approach to the numerical

modelling of homogeneous elastic materials which exhibit exact conservation laws such

as angular momentum and circulation.

Numerical simulations of the DMV algorithm for the pseudo-rigid body show that it

is only stable for suitable parameterisations of the elastic potential energy term. This

is because the variational integrator for the stretching motion is not constrained to

preserve the strict positivity of the determinant of the diagonal matrix of eigenvalues.

The derivation of a determinant preserving integrator remains a subject for future

research.

Elastic Rod For the elastic rod, we present a discrete variant of the Kirchhoff kinetic

analogy and use it to define the spatial discretisation of the Lagrangian necessary for

the equilibrium configuration of a discrete inextensible elastic rod to be in one-to-

one correspondence with the discrete time motion of the Lagrange top presented in

Chapter 2. We then state the Lie-Poisson structure on the dual of the semi-direct

product Lie algebra which is preserved by the MV integrator. We observe that the

rod inextensibility constraint is intrinsically enforced by preserving the Lie-Poisson

structure.

We then add a discrete kinetic energy term to the discrete action principle and

derive a dynamical inextensible rod model. The critical feature of this model is the

existence of a discrete compatibility equation. We observe that it takes the form of the

discrete auxiliary equation for the relative orientation matrix in the discrete coupled

rigid body model.

Geometrically exact rod model We then extend this discrete model to the (ex-

tensible and shearable) geometrically exact elastic rod of Krishnaprasad et al. (1988),

presented in Section 3.5. In this model, the configuration is split into the position of the

rod centroid and the orientation of the directors. An elastic potential then couples these

two motions. A MV integrator computes the rigid body motions of the rod sections

and an explicit variational integrator computes the particle position of the rod sections

in the body frames at each particle position. Both integrators are explicit. Numerical

simulations of a geometrically exact rod with 50 rod sections conserve spatial angular

momentum to an order of 10−8, linear angular momentum to an order of 10−11 and

exhibit no secular drift in the energy error whose mean is an order of 10−2 (after 104

time steps).

130

3.7.1 Future research

MV integrators for the pseudo-rigid body The stretching component of the dis-

crete pseudo-rigid body motion is currently updated using a discrete Euler-Lagrange

equation. This update does not constrain the stretching matrices to the space of di-

agonal matrices with positive determinant. Further research should consider the for-

mulation of a multiplicative update procedure for the stretching motion. The MV

integrator should also be extended for an anisotropic polar decomposed pseudo-rigid

body in which the shape matrix is advected by both rotations and stretching. In order

to solve this integrator, the DMV algorithm must be extended to solve for stretching

motion.

MV integrators for the geometrically exact rod Future research should address

the extent to which the MV integrator for the geometrically exact rod model is Poisson

with respect to the Lie-Poisson bracket for the continuous SO(3) reduced rod motion

given by Krishnaprasad et al. (1988). The MV integrator for this rod model should

also be adapted to model the director orientations using quaternions, rather than Euler

angles, using the approach described in Chapter 2. The latter is motivated by the need

to model supercoiling and twisting motions of elastic materials such as DNA and other

polymer chains, for which Euler angle parameterisation is not suitable.

The stability of the DMV algorithm for this rod model is conditional upon the

choice of parameterisations of the elastic potential energy term. This problem should

be addressed in two stages. Firstly, the stability of the DMV algorithm for the rigid

body problem should be analysed to determine the bounds on the eigenvalues of the

Hamiltonian for the associated matrix Ricatti equation. Secondly, these bounds should

be expressed in terms of the parameterisations of the elastic energy potential in order

to assess the suitability of the DMV algorithm to rod models. Boundary conditions

should also be investigated.

Compressible fluid dynamics The unified computational framework seeks applic-

ation to the convective and spatial descriptions of fluid dynamics. The convective and

spatial representations of (compressible) ellipsoidal motion provide a basic prototype

for ideal compressible fluid dynamics for reasons which we now state.

• Spatial : Holm et al. (1986) consider a compressible fluid and show that the pas-

sage to the spatial representation is by reduction under the group of diffeomorph-

isms. This group acts on the density by pull-back, but acts trivially on the

metric-tensor on Eulerian space, forming the kinetic energy. Analogously, the

(compressible) ellipsoid reduces to the spatial representation by the right action

131

of GL(n)+. This group acts on the shape matrix by conjugation and trivially on

the right Cauchy-Green Matrix (the metric-tensor on Eulerian space).

• Convective: Conversely, Holm et al. (1986) show that the passage to the con-

vective representation of compressible fluids is by reduction under the group of

spatial diffeomorphisms. This group acts trivially on the density and on the

metric-tensor on Eulerian space by pull-back, forming the kinetic energy. Ana-

logously, the (compressible) ellipsoid reduces to the convective representation by

the left action of GL(n)+. This group acts trivially on the shape matrix and on

the right Cauchy-Green Matrix by conjugation.

In order to extend the MV integrators for ellipsoidal motion to compressible fluids,

we conjecture that two developments are needed, (i) the finite dimensional represent-

ation of the group action of diffeomorphisms on G, where G is a finite dimensional

representation of the group of diffeomorphisms, and (ii) the finite dimensional repres-

entation of the group action of diffeomorphisms on V ∗ by pull-back. Although Zhong &

Scovel (1994), Zeitlin (2004) have pursued the use of a finite dimensional representation

of the group of diffeomorphisms for modelling fluids, it remains an open question as to

how this applies to MV integrators.

3.7.2 Proceeding Chapters

We place the remainder of the thesis in the context of the material explicated thus far on

the derivation of Lie-Poisson integrators in a unified computational framework. Recall

from Chapter 2, that this unified computational framework applies the discrete Clebsch

approach to the Lie symmetry reduced discrete variational principle to give momentum

maps which transfer the canonical Poisson structure to the Lie-Poisson structure on the

(reduced) manifold. This manifold is the dual of the corresponding finite dimensional

Lie algebra. The MV integrators define co-adjoint orbits on the Lie-Poisson manifold

which are non-canonical symplectic foliations.

In Chapters 4 and 5, we restrict our attention to the formulation of geometric in-

tegrators which preserve the canonical symplectic structure on the full (unreduced)

phase space of Hamiltonian particle shallow water models. We will show in the next

Chapter that computational models with favourable long-time conservative properties

can be derived from the material representation of the semi-discrete variational prin-

ciple. Extremising the semi-discrete action principle gives semi-discrete Euler-Lagrange

equations which preserve the canonical symplectic form of the momentum phase space

of the particles. Preliminary numerical results suggest that application of explicit

(canonical) symplectic integrators to these equations gives a viable computational ap-

proach for long time simulations of shallow water. Chapter 5 goes on to demonstrate

132

the preservation of the Hamiltonian structure of particle shallow water models in the

presence of boundary conditions.

We reiterate the need to substantiate the development of a unified computational

framework for computational continuum dynamics by extending the results of this

Chapter. This development entails formulation of a symmetry reduced discrete vari-

ational principle and use of the discrete Clebsch approach to derive momentum maps

transferring the canonical Poisson structure of the particle equations to the reduced

particle phase space in the convective and spatial representations. It remains the sub-

ject of further research to derive integrators which preserve the non-canonical Poisson

structure on this reduced phase space and, by extending the results of this Chapter,

show that they exhibit additional conservation laws (associated with the symmetries of

the discrete variational principle) such as a Kelvin circulation theorem.

133

Chapter 4

A Variational Free-Lagrange

Method for Shallow Water

Synopsis This Chapter derives and investigates a semi-discrete approximation, re-

ferred to as a variational free-Lagrange (VFL) method, for the rotating shallow water

equations. This method is a variational formulation of the free-Lagrange method (see

Fritts 1985). The free-Lagrange method uses a Voronoi diagram to represent the layer-

depth field and is intrinsically locally mass conservative.

The discretisation of the variational principle for shallow water with the free-

Lagrange data structure forms the critical step in the derivation of the VFL shallow

water equations, expressed in terms of the Voronoi cell averaged layer depth and the ma-

terial particle velocities. This Chapter shows that these equations conserve energy and

formulates the corresponding semi-discrete divergence and vorticity equations based on

a discrete divergence and curl operator.

The form of the semi-discrete vorticity equation reveals that the potential vorticity

is not conserved because the discrete curl of the grad operator does not vanish. This

suggests the need for a constraint on the curl operator in the discrete action principle.

This Chapter closes with the presentation of numerical results showing the conservative

properties of the 1D VFL rotating shallow water equations over long time intervals.

Overview We shall begin with a Lagrangian description of the continuum rotating 2D

shallow water equations with varying bottom topography, as derived from a variational

principle. We then define the discretisation of the velocity and layer depth fields and

derive the semi-discrete 1D Euler-Lagrange equations of motion from a semi-discrete

Hamilton’s action principle. We prove that energy is conserved by the semi-discrete

continuity and momentum equations. We then show how this approach can be extended

134

for any two-dimensional Voronoi diagram to yield a weak form of the semi-discrete

Euler-Lagrange equations for shallow water. Using these equations, we derive the

semi-discrete vorticity and divergence equations and show that the potential vorticity

is only conserved if the discrete curl operator is constructed so that the discrete curl of

the gradient operator vanishes.

We give details of the symplectic integrator which we use for numerical simulation

of the 1D rotating shallow water equations on a periodic domain and close the Chapter

with numerical results of the conservative properties of the VFL method. Appendix

C gives the corresponding canonical formulation of the VFL method and outlines a

procedure, referred to as rezoning for approximating the spatial velocity and layer

depth over a fixed grid.

4.1 The Lagrangian Description of Shallow Water

Following Salmon (1983), we recall the Lagrangian description of a rotating fluid, for-

mulated as a continuum of fluid particles. Let each particle be positioned labelled by

some ` ∈ L ⊂ R2 meaning that the label is the initial position of the particle. The

position of a particle in the container R2 is given by a diffeomorphism X = X(`, t)

where

X : L × R+ → R2.

The fluid layer depth over the container h(X(`, t), t) is pulled-back to a fixed time-

independent function on label space h0(`) by the determinant of the Jacobian |J | =∣∣Xi

,a

∣∣ = h0(`)

h(X(`,t),t) , where the short-hand notation Xi,a =

∂Xi

∂`a .

The shallow water equations in a rotational frame are derived from the stationary

state of the action principle

S =

∫ t2

t1

dt

∫da h0{

1

2|X|2 +R(X) ∙ X−

g

2(h0|J |

−1 + 2b)}, (4.1)

where da = d`ad`b and the rotation vector R is given by curl R = 2ω(X), where ω is

the angular velocity of the rotational frame relative to an inertial frame.

The variational derivative of S,

δS =

∫ t2

t1

dt

∫da h0{(X

i +Ri)δXi +gh0

2|J |−2δ|J |} = 0, (4.2)

where the potential has been derived by integrating over the depth of fluid for some

bottom topography b(x) = b

135

∫dA

∫ h+b

b

gzdz =

∫dA

g

2((h+ b)2 − b2) =

∫dA

gh

2(h+ 2b)

=

∫da |J |

g

2h0|J |

−1(h+ 2b)

=

∫da h0

g

2(h+ 2b),

(4.3)

where dA = dX1dX2. Integrating the first term in the stationary action principle by

parts and expressing δ|J | as a function of δX i gives

δS =

∫ t2

t1

dt

∫da h0[−X

i +Rj,iXj − gh0m

i]δX i = 0, (4.4)

where m1 = (|J |−2X2,b),a − (|J |−2X2,a),b and m

2 = (|J |−2X1,a),b − (|J |−2X1,b),a and it is

assumed that δX i(`, t2) = δXi(`, t1) = 0 for some arbitrary times t1 and t2.

The Euler-Lagrange equations for flow on velocity phase space are, after use of the

chain rule h,a = h,iXi,a in the expressions for m

i,

U = −g∇Xh− f0k×U,

X = U,(4.5)

where we have made the f-plane approximation, ∇ × R = f0k, k is the unit vector

in the direction of gravity and f0 is the Coriolis parameter which is given by f0 =

2|ω|. Under the assumption of hyper-regularity of the Lagrangian, the Hamiltonian

may be defined by the Legendre transformation. One may then derive the canonical

Hamiltonian equations of motion which define the flow on the momentum phase space,

dual to the velocity phase space.

In Section 4.3, we semi-discretise the Hamilton’s action principle for shallow water

with a free-Lagrange data structure, derive the semi-discrete Euler-Lagrange equations

and show that these are equivalent to the canonical Hamiltonian particle equations.

We refer to these semi-discrete Euler-Lagrange equations as the VFL equations since

they preserve the variational structure of the flow field and use the data structure of

the free-Lagrange method.

Discrete variational approach in fluids The concept of discretising Hamilton’s

principle with Lagrangian particles in the context of idealised fluid dynamics can be

attributed to Salmon (1983) and Buneman (1982). Salmon (1983) points out that not

136

only is the approach succinct as a beginning point for approximations, but accommod-

ates moving disconnecting fluid boundaries and gives conservation laws corresponding

to symmetries of the Hamiltonian. The subsequent derivations of the discrete equa-

tions of motion parallel his presentation. Bonet & Rodriguez-Paz (2005) give a similar

presentation leading to variational SPH methods for hydrodynamics. The wider scope

of their presentation gives useful details for future generalisation of the approach sub-

sequently described.

We begin with a description of the VFL method and show how the mean layer

depth field is reconstructed at the end of each time step to ensure that the mass in

each moving cell remains constant.

4.2 The Variational Free-Lagrange Method for Shallow

Water

We present the VFL method in a similar notation to that used in the Hamiltonian

particle mesh method (Frank et al. 2002, Frank & Reich 2004), considered in Chapter

5. We follow the standard methodology for deriving particle methods. Firstly the label

space is discretised into N2 particles which are labelled by α. The particle coordinates

in velocity phase space TQ ⊆ R4 are

(XT ,UT ) = ([X1, . . . ,XN2 ]T , [U1, . . . ,UN2 ]

T ),

where Xα = (Xα, Yα)T .

Voronoi Diagram We refer to the set of particle positions X = {X1, . . . ,XN2} as

the set of sites.

Definition 4.2.0.1 (Voronoi Cell). A Voronoi cell Vα with site Xα is a polygon con-

taining the set of points Xβ closer to Xα than to any other site.

A hexagonal Voronoi cell is shown in Figure 4.1 together with the specification of

a local index for referring to neighbouring particles.

The Voronoi diagram is the set of all closed Voronoi cells indexed by α. In its con-

struction, we assume that particle positions are distinct. Note that this assumption

does not prohibit the formation of shocks - characteristics may still collide as they are

not the particle trajectories (see, for example, Whitham 1974, for an explanation of

shock formation). We now describe how the conservation of cell mass law gives the

shallow water layer depth field.

137

Aα

Xβ1

Xα

Xβ2

Xβ3

Xβ4

Xβ5

Xβ6

Δβ1Xα

nβ5

Figure 4.1: A hexagonal (ne = 6) Voronoi cell containing the particle with label indexα. By construction of the cell, each line connecting particle α with its neighbours isa perpendicular bi-sector of a cell edge. A local index i := 1 → ne is used to refer tothe neighbouring particles βi. Since the ne cells edges are in one to one correspondencewith the neighbouring particles, cell edges are also indexed by βi. Each cell edge is oflength Δlβiα .

138

4.2.1 Mass conservation

The statement of conservation of mass of cell α

Dmα

Dt=

D

Dt

∫

Vα

h(x, t)dA = 0, (4.6)

reduces to the form, DDtmα =DDt

(hα(t)Aα(t)

), if the layer depth h(x, t) is chosen to be

piecewise constant over each Voronoi cell, h(x, t) = hα(t), x ∈ Vα. This conservation

law gives an explicit expression to compute the cell averaged layer depth which, in finite

time and using finite difference notation, takes the form

hn+1α =AnαAn+1α

hnα. (4.7)

We now semi-discretise the shallow water variational principle with the ’free-Lagrange

data structure’. This data structure is defined by a particle representation of the ma-

terial velocity field and a Voronoi diagram for the layer depth. We first begin with the

1D non-rotational model.

4.3 The Variational Free-Lagrange Equations for 1D Shal-

low Water

We begin by considering the semi-discrete material description of Hamilton’s action

principle for 1D (non-rotating) shallow water, expressed using the free-Lagrange data

structure,

L =1

2

∑

α

mαU2α −

g

2

∑

α

mαhα, (4.8)

where the first term is the kinetic energy over the particles and the second term is

the potential energy over the Voronoi cells. Substituting the expression for the cell-

averaged layer thickness hα =mαΔXα, where Aα = ΔXα =

12(Xα+1 − Xα−1), into the

above action principle and taking variations gives

∂L

∂Xα=g

2

(m2α−1(ΔXα−1)

−2 −m2α+1(ΔXα+1)−2) , (4.9)

andD

Dt

∂L

∂Xα

= mαUα. (4.10)

The semi-discrete Euler-Lagrange equations are therefore

139

Uα = −g

2mα

(

mα+1hα+1ΔXα+1

−mα−1hα−1ΔXα−1

)

. (4.11)

The right hand side of the above equations defines the discrete gradient operator

grad(hα) :=1

2mα

(

mα+1hα+1ΔXα+1

−mα−1hα−1ΔXα−1

)

. (4.12)

The semi-discrete Euler-Lagrange equation and the semi-discrete continuity equation

D

Dthα = mα

D

Dt(ΔXα)

−1, (4.13)

are referred to as the VFL equations for 1D shallow water.

4.3.1 Energy Conservation

We now show that the semi-discrete shallow water equations conserve energy.

Definition 4.3.1.1 (Hamiltonian for the VFL method). The Hamiltonian for the VFL

method is given by

H =1

2

∑

α

mαU2α +

g

2

∑

α

mαhα, (4.14)

where the first term is the kinetic energy, formed from particle velocities, and the second

term is the potential energy formed from the cell averaged layer depth.

Lemma 4.3.1.2. The Hamiltonian for the VFL method is conserved along particle

trajectoriesD

DtH = 0. (4.15)

Proof 4.3.1.3.

D

DtH =

∑

α

mαUαD

DtUα +

g

2

∑

α

mαD

Dthα

=∑

α

[−gmαUαgrad(hα) +g

2m2α

D

Dt(ΔXα)

−1]

=∑

α

−g

2Uα[mα+1

hα+1

ΔXα+1−mα−1

hα−1

ΔXα−1+mα−1

hα−1

ΔXα−1−mα+1

hα+1

ΔXα+1] = 0.

(4.16)

We now consider the derivation of the 2D VFL rotating shallow water equations.

140

4.4 A Variational Free-Lagrange method for 2D Shallow

Water

The semi-discrete material description of the Hamilton’s action principle, is expressed

using the free-Lagrange data structure

Definition 4.4.0.4 (The Semi-Discrete Hamilton’s Action Principle for Shallow Wa-

ter).

Sd =1

2

∫ tb

ta

dt∑

α

mα

(|Uα(t)|

2 + 2Rα ∙Uα

)− g

∑

α

mα

(h(Xα, t) + 2bα). (4.17)

Following Augenbaum (1984), we express the potential energy V in terms of the cell

Jacobian ˉ|J |α(t) :=h(Xα,t0)h(x,t) , x ∈ Aα to give

V =g

2

∑

α

mα(h(Xα, t0) ˉ|J |−1α (t) + 2bα), (4.18)

Stationarity of the discrete action principle gives

∂L

∂Xα= mαXα =: Pα, (4.19)

and

1

Aα

∫

Aα

∂L∂Xα

dA =1

Aα

∫

Aα

∂L∂|J |α

∂|J |α∂Xα

dA =1

Aα

∑

i

[∂L∂|J ||J |

]βi

α

dnβiα . (4.20)

We formulate the weak form of the Euler-Lagrange particle equations, given by

1

Aα

∫

Aα

{d

dt

∂L

∂Xα−∑

i

[∂L∂|J ||J |

]βi

α

dnβiα }dA = 0, (4.21)

which is satisfied by

d

dt

∂L

∂Xα=∑

i

[∂L∂|J ||J |

]βi

α

dnβiα , (4.22)

where the operator

[∙]βiα =1

2(∙α + ∙βi), (4.23)

expresses the value of a scalar quantity at the βthi edge of cell α as the mean of that

quantity over cell α and cell βi. Evaluating the derivatives inside the brackets [∙]βiα gives

[∂L∂|J ||J |

]βi

α

= −g[mh]βiα, (4.24)

141

giving the final expression for the semi-discrete Euler-Lagrange particle equations

Uα = −ggrad(hα)− f0k×Uα,

Xα = Uα,(4.25)

where

grad(hα) := −1

mα

∑

i

[mh]βiαdnβiα . (4.26)

and dnβiα := nβiα Δl

βiα . The semi-discrete continuity equation takes the form

D

Dthα = −hα(t0)

Aα(t0)

A2α(t)

D

DtAα(t)

= −hα1

Aα

∫

Aα

∇ ∙UdA.(4.27)

From the divergence theorem, the semi-discrete continuity equation can be rewritten

as1

Aα

∫

Aα

∇ ∙UdA =1

Aα

∮

∂Aα

n ∙Udl =1

Aα

ne∑

i=1

nβiα ∙∮

∂Aβiα

Udl, (4.28)

where ∂Aα denotes the boundary of the polygonal Voronoi cell, and ∂Aβiα denotes the

face shared with the neighbouring cell βi with unit normal vector nβiα .

Following Ringler & Randall (2002) we introduce the approximation

1

Aα

ne∑

i=1

nβiα ∙∫

∂Aβiα

Udl ≈1

Aα

ne∑

i=1

dnβiα ∙Uβiα := div(Uα), (4.29)

which gives the definition of the discrete divergence operator div(∙α) over cell α where

dnβiα := nβiα Δl

βiα . (4.30)

Recall that Δlβiα is the length of the side indexed by βi of cell α. The semi-discrete

Euler-Lagrange equation and the continuity equation

D

Dthα = −hαdiv(Uα), (4.31)

formulate the VFL method for rotating shallow water on a f-plane with bottom topo-

graphy.

142

4.4.1 Energy Conservation

We now show that the VFL 2D rotating shallow water equations conserve energy.

Consider the material derivative of the Hamiltonian given by

D

DtH =

∑

α

mαUα ∙D

DtUα + gmα

D

Dthα

=∑

α

−gmαUα ∙ grad(hα)− f0k×Uα − gmαhαdiv(Uα)

=∑

α

gUα ∙∑

i

[mh]βiαdnβiα − f0k×Uα − gmαhα

∑

i

[U]βiα ∙ dnβiα

=∑

α

gUα ∙∑

i

mβi hβidnβiα − f0k×Uα − gmαhα

∑

i

Uβi ∙ dnβiα

=∑

α

Uα ∙

(

g∑

i

mβi hβidnβiα − f0k×Uα − g

∑

i

mβi hβidnβiα

)

= 0,

(4.32)

where the term in [∙]βiα is simplified by the property that∑

i dnβiα = 0 and the last line

is obtained by shifting the indices of the terms of the potential energy.

Remark 4.4.1.1 (Canonical VFL equations on momentum phase space). An alternat-

ive approach to proving energy conservation is outlined in Appendix C. This approach

constructs the canonical VFL equations on momentum phase space and verifies that

they are Hamiltonian. The approach relies on the existence of a smooth and invertible

Legendre transformation.

Remark 4.4.1.2 (Particle relabelling symmetry). The discrete Lagrangian is not in-

variant under continuous transformations of the labels - there is no particle relabelling

symmetry (see Ripa 1981, Salmon 1982). Padhye & Morrison (1996) succinctly sum-

marise the implications of the particle relabelling symmetry for hydrodynamics. The

symmetry gives Ertel’s theorem of conservation of potential vorticity which in turn

recovers the known connection with Kelvin’s circulation theorem along surfaces of con-

stant entropy. The relation between Ertel’s theorem and Hamilton’s action principle

was made earlier by Salmon (1982). Moreover, he pointed out that the Hamilton’s

principle provides a means of unifying all forms of Ertel’s theorem of hydrodynamics.

The discrete Lagrangian is, however, invariant under permutation of the indices.

This symmetry is deemed by Serrano, Espanol & Zuniga (2005) to give an associated

discrete Kelvin circulation theorem. We point out that this theorem is only approximate,

however, and is in fact exhibited by any convergent scheme.

143

4.5 The Shallow Water Vorticity Equation

By analogy with the continuum shallow water theory, we derive the semi-discrete po-

tential vorticity and divergence equations exhibited by the VFL method and show

that the latter quantity is conserved. The following derivation of the shallow water

vorticity equation requires the definition of the discrete curl, div and grad operators.

Semi-discretisation of Hamilton’s action principle with the free-Lagrange data structure

gives the form of the grad operator. Following Ringler & Randall (2002), we formulate

the semi-discrete continuity equation, given by equation (4.31), in terms of the div

operator which is restated below for convenience. Further following Ringler & Randall

(2002), we define the normal component of the discrete curl operator by analogy with

the derivation of the discrete divergence operator given by equation (4.34). The Voro-

noi diagram is assumed to rest in the X-Y plane so that the vertical unit vector k is

normal to the plane. From Stoke’s theorem

1

Aα

∫

Aα

k ∙ ∇ ×UdA =1

Aα

∮

∂Aα

τ ∙Udl =1

Aα

ne∑

i=1

τβiα ∙∮

∂Aβiα

Udl, (4.33)

which approximates to

1

Aα

ne∑

i=1

τβiα ∙∫

∂Aβiα

Udl ≈1

Aα

ne∑

i=1

dτβiα ∙ (U)βiα := k ∙ curl(Uα), (4.34)

where

dτβiα := τβiα Δl

βiα . (4.35)

Recall that Δlβiα is the length of the side indexed by βi of cell α. To summarise, the

discrete div and curl operators take the form

div (∙α) :=1

Aα

ne∑

i=1

dnβiα ∙(∙βiα),

k ∙ curl (∙α) :=1

Aα

ne∑

i=1

dτβiα ∙(∙βiα).

(4.36)

The vertical component of the discrete curl of the semi-discrete Euler-Lagrange

equation over cell α takes the form

k ∙ curl(Uα) = −gk ∙ curl(grad(h(Xα, t))

)− f0k ∙ curl(k×Uα). (4.37)

144

From the definition of curl, the left hand side of equation (4.37)

k ∙ curl(Uα)

=1

Aα

∑

i

Uβiα ∙ dτ

βiα

=D

Dt

(1

Aα

∑

i

Uβiα ∙ dτβiα

)

+1

A2α

D

Dt(Aα)

∑

i

Uβiα ∙ dτβiα −

1

Aα

∑

i

Uβiα ∙D

Dtdτβiα

=D

Dt

(1

Aα

∑

i

Uβiα ∙ dτβiα

)

+1

A2α

D

Dt(Aα)

∑

i

Uβiα ∙ dτβiα +

1

Aα

∑

i

Uβiα ∙ΔUβiα

︸︷︷︸=0

=

(D

Dt+1

Aα

D

DtAα

)1

Aα

∑

i

Uβiα ∙ dτ

βiα

=

(D

Dt+1

Aα

D

DtAα

)1

Aα

∑

i

Uβiα ∙ dτ

βiα

=

(D

Dt+ div(Uα)

)1

Aα

∑

i

Uβiα ∙ dτ

βiα

=D

Dt

(k ∙ curl(Uα)

)+ div(Uα)k ∙ curl(Uα),

(4.38)

where

D

Dtdτβiα =

D

Dt

(τβiα Δl

βiα

)= Δ

D

Dt(τ l)βiα = ΔU

βiα := U

βi+1α − Uβi−1α . (4.39)

Remark 4.5.0.3. The vanishing under-braced term in the fourth line follows from the

definition of ΔUβiα and the property that the terms cancel around a closed loop

∑

i

Uβiα

(Uβi+1α −Uβi−1α

)= 0. (4.40)

Note that in the continuum limit, the expression takes the form

1

2A

∮

A

d|U|2 = 0. (4.41)

Substituting the definition of the relative vorticity ζα = k ∙ curl(Uα) into the last

line of equation (4.38) simplifies it to

k ∙ curlDUα

Dt=

D

Dtζα + ζαdiv(Uα). (4.42)

Substituting this expression in the semi-discrete vorticity equation in equation (4.37)

145

givesD

Dtζα − ζαdiv(Uα) + f0k ∙ curl

(U⊥)= −gεα, (4.43)

where the term εα := k ∙ curl(grad(hα)

)6= 0 arises from the property that the discrete

curl of the discrete grad operator does not vanish

k ∙ curl(grad(hα)) =1

Aα

∑

i

(grad(hα)

)βiα∙ dτβiα

= −1

mαAα

∑

i

∑

j

[mh]βjαdn

βjα

βi

α

∙ dτβiα

= −1

mαAα

∑

i

∑

j

[mh]βjαnβjα Δl

βjα

βi

α

∙ τβiα Δlβiα = εα 6= 0.

(4.44)

The semi-discrete vorticity equation simplifies to

D

Dt(ζα + f0) + (ζα + f0) div(Uα) = −gεα, (4.45)

where we have used the property of the discrete curl and div operators

k ∙ curl(U⊥α ) =1

Aα

ne∑

i=1

dτβiα ∙(U⊥α

):=1

Aα

ne∑

i=1

dτβiα ∙(Uα × k

)

=1

Aα

ne∑

i=1

dnβiα ∙ (Uα) = div(Uα).

(4.46)

Remark 4.5.0.4 (Discrete curl operator). εα is an error term in the semi-discrete

vorticity equations which arises because the discrete curl of the discrete gradient operator

does not vanish. Certainly, the absence of a discrete potential vorticity law is consistent

with the remark made earlier on particle relabelling symmetry. For further insight, recall

that the discrete curl operator, unlike the discrete gradient operator is not implied from

the discrete Euler-Lagrange equations, but is chosen from a discrete approximation of

the integral identity for curl given by Stoke’s law. The discrete curl operator is therefore

formulated outside of the variational framework. To remedy the property of the discrete

curl operator, we suggest adding a constraint of the form

〈Jβiα ,1

mαAα

∑

i

∑

j

[mh]βjαdn

βjα

βi

α

∙ dτβiα 〉, (4.47)

146

where Jβiα is a Lagrange multiplier. The constrained discrete semi-discrete action prin-

ciple would then give a Mimetic differencing scheme (see Lipnikov et al. 2006, Margolin

et al. 2002). Mimetic differencing schemes are based on discrete operators which pre-

serve critical properties of the original continuous differential operators. Conservation

laws, solution symmetries and relationships between differential operators are just some

examples of such properties. They have been applied to a wide class of problems includ-

ing continuum mechanical models.

In these models, Mimetic differencing schemes have taken the form of arbitrary-

Lagrangian-Eulerian schemes defined on irregular unstructured meshes, rather than on

Voronoi diagrams. The approach is not variational by construction either. These reas-

ons render it more difficult to infer the form of the discrete operators required to make

the VFL method Mimetic. Crucially, we must establish whether existing discrete differ-

ential operators on unstructured meshes, which satisfy curl(grad(∙)) = 0, can be formu-

lated on a Voronoi diagram before attempting to cast this approach into a variational

framework.

Potential vorticity We now derive the semi-discrete potential vorticity equation

from the semi-discrete vorticity equation and the semi-discrete mass continuity equation

D

Dthα = −hαdiv(Uα), (4.48)

over the cell Vα.

Following Salmon (1998), we substitute the expression for the discrete divergence

of the particle velocity Uα into the semi-discrete vorticity equation, given by equation

(4.45) to give the statement of conservation of semi-discrete potential vorticity over cell

α

D

Dt(ζα + f0) +

(ζα + f0hα

)D

Dthα =

D

Dt

(ζα + f0hα

)

= −gεα. (4.49)

This equation states that potential vorticity is only conserved if the discrete curl of the

discrete gradient of the cell averaged layer depth vanishes. Enforcement of this property

would then put the VFL method on a par with the energy and potential vorticity

conserving Poisson-bracket method developed by Salmon (2004). This approach, based

on direct discretisation of the Poisson-bracket may give insight into the form and other

properties of the gradient and curl operators required for the VFL method to conserve

potential vorticity. Curiously, in his earlier work, Salmon (1983) suggests expressing

the rotational velocity in the discrete action principle as a functional of particle labels

and the invariant potential vorticity on particles.

147

4.6 The Semi-Discrete Divergence Form of the Shallow

Water Equations

We now repeat the previous steps for deriving the semi-discrete potential vorticity equa-

tion, only this time, taking the discrete divergence of the semi-discrete Euler-Lagrange

equations. The discrete divergence of the semi-discrete Euler-Lagrange equation over

cell α takes the form

div(Uα) = −gdiv(grad(h(Xα, t))

)− f0div(k×Uα). (4.50)

From the definition of div, the left hand side of equation (4.50)

div(Uα) =1

Aα

∑

i

Uβiα ∙ dn

βiα

=D

Dt

(1

Aα

∑

i

Uβiα ∙ dn

βiα

)

+1

A2α

D

Dt(Aα)

∑

i

Uβiα ∙ dnβiα −

1

Aα

∑

i

Uβiα ∙D

Dtdnβiα

=D

Dt

(1

Aα

∑

i

Uβiα ∙ dn

βiα

)

+1

A2α

D

Dt(Aα)

∑

i

Uβiα ∙ dnβiα −

1

Aα

∑

i

Uβiα ∙Δ(U⊥)βiα

︸︷︷︸:=Γα 6=0

=

(D

Dt+1

Aα

D

DtAα

)1

Aα

∑

i

Uβiα ∙ dn

βiα − Γα

=

(D

Dt+1

Aα

D

DtAα

)1

Aα

∑

i

Uβiα ∙ dn

βiα − Γα

=

(D

Dt+ div(Uα)

)1

Aα

∑

i

Uβiα ∙ dnβiα − Γα

=D

Dtdiv(Uα) + div(Uα)

2 − Γα.

(4.51)

Denoting the divergence of Uα as δα := div(Uα), the last line of the above equation

becomes

div(D

DtUα) =

D

Dtδα + δ

2α − Γα. (4.52)

So the semi-discrete divergence equation over cell α is

D

Dtδα + δ

2α − Γα + gdiv(grad(hα))− f0div(U

⊥α ) = 0. (4.53)

Remark 4.6.0.5. We compare the semi-discrete shallow water divergence equation

with the continuous form of this equation given by

148

div

(D

DtU

)

= ∂tδ + div(U ∙ ∇U)

= ∂tδ + ∂X(U∂XU + V ∂Y U) + ∂Y (U∂XV + V ∂Y V )

= ∂tδ + (∂XU)2 + (∂Y V )

2 + 2∂Y U∂XV + U∂2XU + V ∂

2Y V + V ∂XY U + U∂XY V

= ∂tδ + (∂XU)2 + (∂Y V )

2 + 2∂Y U∂XV +U ∙ ∇δ

=D

Dtδ + δ2 + 2(∂XV ∂Y U − ∂XU∂Y V )

=D

Dtδ + δ2 − 2det(∇U).

(4.54)

The first two terms correspond to those in the semi-discrete divergence equation

(4.52) and the last term corresponds to Γα which evaluates to1Aα

∮∂AαU ∙ d(U⊥) in the

continuum limit.

4.7 A Symplectic Time Stepping Scheme

The Stormer-Verlet method is used to integrate the 1D VFL shallow water equations of

motion. Similar procedures are described in Dixon & Reich (2004), Frank, Gottwald &

Reich (2002), Frank & Reich (2004). This integrator is a 2nd order explicit partitioned

Runge-Kutta method of the form

Pn+ 1

2α = Pnα −

Δt

2VX(X

nα),

Xn+1α = Xn

α +ΔtPn+ 1

2α

mα,

Pn+1α = Pn+ 1

2α −

Δt

2VX(X

n+1α ),

(4.55)

which preserves the symplectic two-form

ω =N2∑

α=1

dXα ∧ dPα. (4.56)

Computational considerations The gradient of the potential in the above time

stepping scheme is computed on a Voronoi diagram which is newly generated at the

beginning of each time step. Each Voronoi cell may have a variable number of sides and

change shape over each time step. Keeping track of the cell vertices is computationally

149

complex and challenges the scalability of the approach to higher dimensions. Harlen,

Rallison & Szabo (1995) outline an alternative approach which retains the nodes as

material points and reconnects them in the way that optimally triangulates the mesh.

For this shallow water model, this takes the form of a Delauney triangulation which is

dual to the Voronoi diagram. This triangulation can be generated at a relatively low

computational cost.

150

4.8 1D Numerical Experiments

This Section describes two numerical experiments of the 1D VFL rotating shallow water

equations on a periodic domain. The purpose of the first experiment is to investigate

the conservative properties of the VFL method for rotating 1D shallow water. In the

second experiment, we consider the problem of geostrophic adjustment of shallow water

and show that the VFL method produces results which are consistent with the theory

of geostrophic adjustment established by Rossby (1938).

This theory describes the physical mechanism by which perturbed rotating shallow

water recovers to a geostrophically balanced state. Layer depth perturbations h′ of the

scale L′ << LD, where LD =√gH0/f0 is the Rossby deformation radius for shallow

water in a f-plane, result in gravity waves which propagate energy and momentum

away from the source leaving behind a geostrophically balanced flow. This mechanism

is consistent with the advection law oF potential vorticity conservation.

The experiments are initialised by perturbing the Voronoi cell masses with a Gaus-

sian perturbation and the velocity field is initially zero.

4.8.1 Experiment 1: conservative properties of VFL

Unless otherwise stated, we use 128 cells, where each cell is initially of the same size.

The parameters of the simulations are provided in Table 4.1 below.

Parameter Value

Number of cells N 128Time step 0.01Domain length 2πInitial conditions mα(t0) = 1 + 0.1exp(−800(Xα(t0)− L/2)2/L2)

Uα(t0) = 0Vα(t0) = 0

f0 2πg 4π2

Table 4.1: This Table lists the simulation parameters for experiment 1, which investig-ates the conservative properties of the VFL method for 1D rotating shallow water.

Results The following Figures show the energy, potential (and total) vorticity and

potential enstrophy (square of PV) errors. Note that thesse quantities are summed

over all particles. Figure 4.2 shows the scaling of the energy, potential vorticity and

total vorticity with the size of the time step. The error in the energy scales with the

151

order of integrator, O(Δt2), where as the potential and total vorticity do not scale

with this order. This suggests that the VFL method introduces error in the discrete

vorticity. Figure 4.3 shows (from top to bottom) the relative energy, potential vorticity

and potential enstrophy error over 106 time steps, with a time step of 0.01. Each error

does not drift which suggests that the VFL method is long-time stable and conservative.

1 0 -1 -2 -3 -4 -5 -6 -7 -8-25

-20

-15

-10

-5

0

log 2(e

rror

)

log2(Δ t)

Energy errorPotential vorticity error Total vorticity error

Figure 4.2: This Figure shows the scaling laws of the mean energy error, potentialvorticity and total vorticity with the size of the time step over 106 time steps. Theenergy error scales to an order Δt2 which is consistent with the order of the StormerVerlet integrator. Potential vorticity (PV) and total vorticity (TV) error do not scalewith the order of the integrator suggesting that the order of the error of the PV andTV are not governed by the integrator but by the VFL method itself. The simulationparameters are given in Table 4.1.

152

0 1 2 3 4 5 6 7 8 9 10

x 105

-6

-4

-2

0

2

4

6

8x 10

-4 Energy Error

t

E-E

0

0 1 2 3 4 5 6 7 8 9 10

x 105

-1.5

-1

-0.5

0

0.5

1

1.5x 10

-4

|| q

|| -||

q 0||

t

Potential Vorticity Error

0 1 2 3 4 5 6 7 8 9 10

x 105

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-4

|| q2 ||

-||q

02 ||

t

Potential Enstrophy Error

Figure 4.3: These graphs show (from top to bottom) the relative energy, potentialvorticity and potential enstrophy error over 106 time steps. Each error does not driftwhich suggests that VFL method is long-time stable and conservative. The simulationparameters are given in Table 4.1.

153

4.8.2 Experiment 2: geostrophic adjustment

This experiment uses the VFL method to simulate the mechanism of geostrophic adjust-

ment of shallow water in a plane in which the layer depth and velocity are independent

of the y-direction (meridional direction)

U = −g∇Xh+ f0V,

V = −f0U,

X = U,

(4.57)

Given a geostrophically balanced layer depth H0, meridional velocity V 0 = gf0∇Xh

and horizontal velocity U0 = 0, we perturb the layer depth h = H0 + h′ by a smooth,

domain centred, gaussian function h′ with support L′ less than the Rossby deformation

radius LD. The dynamics, governed by equations (4.57), are observed to exhibit gravity

waves which propagate energy and momentum away from the source leaving behind

geostrophically balanced flow.

We verify in this numerical experiment that this mechanism only occurs if the scale

of layer depth perturbation is smaller than the Rossby deformation radius by performing

two simulations, one in which the scale of perturbation is larger and one is which it is

smaller than the Rossby deformation radius. The parameters for this simulations are

provided in Table 4.2.

Parameter Value

Number of cells N 512Time step 0.01Domain length 2π

Initial conditions mα(t0) = 1 +∑2

i=1 0.01exp(−βi(Xα(t0)− L/2)2/L2)Uα(t0) = 0Vα(t0) = −

gf0grad(H0α)

f0 2πg 4π2

H0 1LD 1

Table 4.2: This Table lists the simulation parameters for experiment 2 which modelsgeostrophic adjustment when β1 = 10(L

′ << LD) and β2 = 1000(L′ >> LD). H

0α de-

notes the discrete steady state profile, from which the meridional velocity is initialised.

Computational issues

154

Results Figures 4.4 and 4.5 compare two different rotating shallow water regimes

distinguished by the scale of perturbation of the layer depth. In the first case, the layer

depth is perturbed on the scale of the Rossby radius LR = 1 and in the second, on a

scale smaller than the Rossby radius. In the latter case, the sequence of layer depth

graphs (shown at increasing simulation times) in Figure 4.5 show the gravity waves

which propagate from the source. The bottom profile of each of these graphs in this

Figure also shows that a geostrophically balanced region is recovered in the region of

the source.

0 1 2 3 4 5 6 7-0.01

0

0.01time: 0.27 days

x

h-h

0

0 1 2 3 4 5 6 71

1.005

x

h

0 1 2 3 4 5 6 7-0.02

0

0.02

x

u

ui

0 1 2 3 4 5 6 7-0.05

0

0.05

x

v

0 1 2 3 4 5 6 7-0.05

0

0.05

x

v+ g

/f 0 hx

Figure 4.4: These graphs show a snapshot of the shallow water taken at time t = 0.27”days” (1 ”day” is a rotational unit) for which the initial scale of perturbation L′ >>LD, where LD is the Rossby deformation radius. The top two graphs show the layerdepth perturbation and layer depth respectively. The third and fourth graphs showthe horizontal and meridional velocities. The bottom graph shows the difference of themeridional velocity with the layer depth gradient. Note that the source region is notrestored to a geostrophically balanced state because geostrophic adjustment does notoccur. The simulation parameters for this experiment are given in Table 4.2 in whichβ = 1000.

Figure 4.6 show the corresponding energy and potential vorticity errors over 2 ro-

tation units (i.e. 2 days). We firstly observe that the profiles exhibit high frequency

oscillations arising from gravity waves but do not exhibit drift. There are also jumps in

the profiles with an approximate period of 0.3 days. These jumps occur when the grav-

ity waves collide (recall that the domain is periodic). Figure 4.7 shows how the discrete

approximation of the layer depth, over the central region of the domain, converges to

155

0 1 2 3 4 5 6 71

h

0 1 2 3 4 5 6 7-0.05

0

0.05

u

0 1 2 3 4 5 6 7-0.05

0

0.05v

0 1 2 3 4 5 6 7-0.2

0

0.2

v- g

/f0 h

x

0 1 2 3 4 5 6 76

6.5

x

PV

hh

0

0 1 2 3 4 5 6 70.95

1

1.05

h

0 1 2 3 4 5 6 7-0.05

0

0.05

u

0 1 2 3 4 5 6 7-0.1

0

0.1

v

0 1 2 3 4 5 6 7-0.2

0

0.2

v- g

/f0 h

x

0 1 2 3 4 5 6 76

6.5

x

PV

hh

0

Figure 4.5: These graphs show snapshots of the shallow water taken at times t ={0.2, 0.5} ”days” (1 ”day” is a rotational unit) for which the initial scale of perturba-tion L′ << LD, where LD is the Rossby deformation radius. The top graph show thelayer depth perturbation and layer depth respectively. The third and fourth graphsshow the horizontal and meridional velocities. The bottom graph shows the differ-ence of the meridional velocity with the layer depth gradient. This difference is zerowhen the flow is geostrophically balanced. Note that the source region does restoreto a geostrophically balanced state because gravity waves propagate away energy andmomentum to restore the balance. The simulation parameters for this experiment aregiven in Table 4.1, in which β = 50.

the steady state after 0.4 days for various number of grid points. The outer regions of

the domain exhibit gravity waves which have propagated away from the source.

156

0 0.5 1 1.5 2-3

-2

-1

0

1

2

3x 10

-6

time (days)

(E-E

0)

0 0.5 1 1.5 2-1

-0.5

0

0.5

1

1.5x 10

-5

time (days)

(PV

-PV

0)/P

V0

Figure 4.6: These graphs show the energy (top) and potential vorticity (bottom) errorprofiles of shallow water over 2 ”days” (1 ”day” is a rotational unit) for which theinitial scale of perturbation is L′ << LD, where LD is the Rossby deformation radius.Both profiles exhibit high frequency oscillations arising from gravity waves but do notexhibit drift. There are also jumps in the profiles with an approximate period of 0.3days. These jumps occur when the gravity waves collide (recall that the domain isperiodic). The simulation parameters for this experiment are given in Table 4.2 inwhich β = 50.

157

0 1 2 3 4 5 6 70.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

x

h

1282565121024

Figure 4.7: This graph shows the layer depth at 0.4 days for various number of gridpoints. In the central region of the domain, the layer depth has reached a steadystate (geostrophic balance). The graph shows how the discrete approximation of thelayer depth converges to the steady state. The outer regions of the domain exhibitgravity waves which have propagated away from the source. This outer region is notin geostrophic balance after 0.4 days.

158

4.9 Summary

This Chapter derives and investigates a variational integrator for the rotating shallow

water equations which uses a Voronoi diagram to represent the layer-depth. This

method, referred to as the Variational free-Lagrange (VFL) method, locally conserves

mass and conserves energy to the order of the symplectic integrator used to approximate

the semi-discrete Euler-Lagrange equations. This Chapter culminates in three main

outcomes.

Firstly, we investigate the properties of the semi-discrete Euler-Lagrange shallow

water equations. We form discrete gradient, divergence and curl operators and verify

that the discrete divergence of the discrete gradient operator is zero. We verify that

the semi-discrete 2D Euler-Lagrange rotating shallow water equations conserve energy.

Secondly, we form the semi-discrete shallow water divergence and potential vorticity

equations which describe the evolution of the respective divergence and potential vorti-

city of the semi-discrete shallow water equations. We find that conservation of potential

vorticity depends upon the property that the discrete curl of the discrete gradient is

zero. This suggests that an additional constraint is needed in the discrete action prin-

ciple in order to derive a discrete curl operator with the correct property. The addition

of the constraint would, however, destroy the symplecticity of the VFL method.

Thirdly, we integrate the semi-discrete Euler-Lagrange equations using a Stormer-

Verlet symplectic integrator. Augenbaum did not appreciate the importance of pre-

serving symplectic integrator on long-time energy conservation properties and con-

sequently did not present numerical results, as we do, which demonstrate the energy

error scaling and energy and potential vorticity conservative properties of the VFL

method over long time intervals. We also verify that the VFL method exhibits the

mechanism of geostrophic adjustment. These results are only for 1D rotating shal-

low water and further numerical experiments are required to verify the conservative

properties of the VFL method in 2D with bottom topography.

Further research Serrano et al. (2005) consider the application of the free-Lagrange

method, which they refer to as the Voronoi fluid-particle model, to the Euler equations.

They formulate a discrete gradient operator with the remarkable property that semi-

discrete free-Lagrange approximation of the Euler equations is exact when the pressure

field is linear. This property avoids artificial numerical instabilities in the limit of zero

viscosity and suggests statistical features that actually correspond to similar features

observed in experiments on Lagrangian tracers in homogeneous fully developed turbu-

lence. We propose putting the VFL method in the context of Serrano et al. (2005) by

applying the method to the Euler equations and compiling, as they did, the distribution

of accelerations in the dynamical equilibrium state of the Voronoi fluid particles.

159

In the next Chapter we describe how ghost particles can be introduced into a

Hamiltonian framework to implement velocity boundary conditions. This is most

simply demonstrated for the 2D rotating shallow water equations through implement-

ation in a more established numerical method, referred to as the Hamiltonian particle

mesh method. It remains an open question as to whether this approach can be adapted

for the VFL method presented here.

160

Chapter 5

A Hamiltonian Particle Mesh

Method for Shallow Water in a

Bounded Domain

Synopsis This Chapter formulates free-slip boundary conditions for the HPM ap-

proximation of 2D rotating shallow water. We introduce ghost or image particles into

the HPM method and show how to modify the basis of the mesh to implicitly rep-

resent them. We also extend the layer depth smoothing matrix for bounded domains

by imposing a Neumann boundary condition on the smoothed layer depth. We finally

present numerical results showing the conservative properties of the HPM method for

shallow water channels and the motion of a vortex pair in rotating shallow water as it

reaches the channel wall.

Overview This Chapter is structured as follows. We begin with a Eulerian de-

scription of rotating shallow water before revisiting the Lagrangian description of the

continuum shallow water equations, only this time on a bounded domain. Section 5.2

reviews the formulation of the Hamiltonian particle-mesh method for 2D shallow water

in a channel and presents the extension of this method to bounded domains. Section

5.3 describes how the layer depth is smoothed on a bounded domain and considers the

1D case first before extending the approach to shallow water in channels and basins.

We refer the reader to the previous Chapter for a description of the time-stepper used

to solve the HPM equations for non-rotating shallow water. The inclusion of a Coriolis

term, however, calls for a modified symplectic integrator which is presented in Sec-

tion 5.2.1. Section 5.4 presents numerical results from four experiments which validate

the stability and conservative properties of the HPM approximation of a bounded 1D

shallow water model, compare the HPM approximation of shallow water in a channel

161

(without rotation) with a spectral-Chebyshev method and show the motion of a pair of

vortices in a channel of rotating shallow water on a f-plane as they reach the channel

wall.

5.1 Introduction

We consider a bounded domain of shallow water rotating with angular velocity ω re-

lative to an inertial frame. We assume that the shallow water is in a rotating plane,

referred to as a f-plane. This plane is tangent to the surface of the earth, across which

the Coriolis force 2ω×u takes its value at the point of tangency and only the component

of the angular velocity vector corresponding to rotations about the unit gravity vector

k is considered. The associated constant Coriolis parameter in the f-plane is f0 = 2|ω|.

The Eulerian description of bounded rotating shallow water provides two equations

for the Eulerian velocity u and the layer depth h and a Dirichlet boundary condition

on the normal component of the velocity

∂u

∂t+ u ∙ ∇u+ f0k× u = −g∇h,

∂h

∂t+ u ∙ ∇h = −h∇ ∙ u,

n ∙ u = 0, x ∈ ∂Ω.

(5.1)

The boundary condition on the component of the Eulerian velocity, normal to the

boundary n ∙ u = 0, is commonly referred to in inviscid fluid dynamics as a free-slip

boundary condition, since the tangential component of the velocity is not constrained.

For the case when f0 = 0, the free-slip boundary condition is equivalent to the Neumann

boundary condition on the layer depth n ∙ ∇h = 0. For the purposes of this Chapter,

the Eulerian description serves little more than to introduce the bounded shallow water

model in the most intuitive form.

Lagrangian description This Chapter shall pursue the formulation and implement-

ation of bounded shallow water in the Lagrangian description instead. Recall from the

previous Chapter, that in this description, each fluid parcel is positioned labelled by

some ` ∈ L ⊂ R2 meaning that the value of the label is the initial position of the par-

cel. The position of each parcel in the fluid container R2 is given by a diffeomorphism

X = X(`, t) where

X : L × R+ → R2.

162

The independent variables in this description are the fluid parcel labels and time and

the shallow water equations are

DU

Dt= −g∇h− f0k×U,

Dh

Dt= −h∇ ∙U,

n ∙ ∇h = 0, x ∈ ∂Ω,

(5.2)

where DDt =

∂∂t + u ∙ ∇ is the material derivative. We impose the zero condition on

the normal component of the gradient of the layer depth. For the case when f0 = 0

(the non-rotating case), this boundary condition on the layer depth is equivalent to the

kinematic form the free-slip boundary condition on the velocity

n ∙D

Dt(U) = n ∙ ∇h = 0. (5.3)

For the rotating case (f0 6= 0) the Neumann boundary condition n ∙ ∇h takes the

form of a zero mass transport condition normal to the boundary

Dh

Dt=∂h

∂t+ (n ∙U) (n ∙ ∇h)

︸︷︷︸=0

+(τ ∙U)(τ ∙ ∇h). (5.4)

The boundary is regarded as a line of symmetry in an extended domain across which

fluid parcels are symmetric (see Figure 5.1). Parcels reflect off the boundary by moving

across the boundary and exchanging their positions and velocities with parcels on the

other side of the boundary. Because the boundary is a line of symmetry, the normal

component of the gradient of the layer depth vanishes and no mass is transferred across

the boundary.

5.2 The Hamiltonian Particle Mesh Approximation

This Section introduces the HPM approximation for 2D rotating shallow water in a

channel Ω := [0, Lx]× [0, Ly) which is bounded in x and periodic in y. Further details

of this approach, without consideration of the boundary conditions, can be found in

Frank et al. (2002), Frank & Reich (2004).

We follow the standard methodology for deriving particle-mesh methods by adopt-

ing two stages. Firstly, the label space is discretised into N particles which are labelled

by greek indices. The particle coordinates in momentum phase space are

Z := (XT ,PT ) = ([X1, . . . ,Xα, . . . ,XN ]T , [P1, . . . ,Pα, . . . ,PN ]

T ). (5.5)

163

i=n−1i=0i=−1 i=n

x=-L

Ω ΩRΩL

x=0 x=L x=2L

Figure 5.1: This diagram provides the conceptual view of the boundary in the Lag-rangian description of shallow water in a channel. Each boundary is a line of symmetrypartitioning the domains Ω with ΩL and ΩR. In the HPM implementation, it is suffi-cient to extend the basis of cubic B-splines to the shaded region by positioning splinesat i = −1 and i = n. This ensures that the basis forms a partition of unity over Ω.

We introduce one non-standard feature at this stage, namely ghost or image particles.

We will shortly show that these particles are needed to impose the zero Neumann

boundary condition on the layer depth in HPM.

Remark 5.2.0.1 (Terminology). We shall use both names because ghost particles are

frequently referred to in the literature but in the opinion of the author, image better con-

veys that these particles are simply images of particles about each of the four boundaries,

moving with opposite velocities in the x-direction to their pre-images.

The definition of these particles is now given.

Definition 5.2.0.2 (Image Particles). Particles whose phase space coordinates are

given by the total relation Zα 7→ [Z−α,Z+α] where Z−α := ([−Xα, Yα], [−P xα , Pyα ]) and

Z+α := ([2L−Xα, Yα], [−P xα , Pyα ]) and whose mass m±α = mα are referred to as image

or ghost particles.

Gridded layer depth In the second stage of the HPM construction, the layer depth

H is discretised over a fixed mesh whose nodes are denoted by roman indices Ωh =

{xi,j : xi,j = (iΔx, jΔy) , i = 0, . . . , nx − 1, j = 0, . . . , ny − 1}, Ωh ⊆ Ω. We denote

the element of H at xi,j as hi,j and subsequently refer to the set of hi,j on Ωh as the

gridded layer depth.

The gridded layer depth is determined by interpolating particle masses mα at

position Xα using a basis of (cubic) B-splines functions {φi,j , i = −1, . . . , nx, j =

164

−1, . . . , ny}, where

φi,j(Xα) := φ(|xi −Xα|)⊗ φ(|yj − Yα|), (5.6)

is the tensor product of 1D cubic B-splines centred at xi,j . A horizontal Section of this

basis, in the region of the boundary at x = 0, is shown in Figure 5.2. The basis is

extended by k = sx4Δx number of basis functions (where sx denotes the support of the

basis function in the x direction), centred at grid cells beyond the boundary. When

the basis function are cubic B-splines, with a support width of sx = 4Δx, the basis

is extended by k = 1 splines (shown with the dotted line in Figure 5.2) which are

positioned at (−iΔx, jΔy) and (L + iΔx, jΔy), i = 1, . . . , k so that a partition of

unity is formed over Ω

nx−1+k,ny−1∑

i=−k,j=0

φi,j(x) = 1, x ∈ Ω. (5.7)

The partition of unity ensures global conservation of mass when interpolating particle

masses (and their images) to the gridded layer depth using the equation

hi,j =N∑

α=1

mαγ (φi,j(Xα) + φi,j(−Xα) + φi,j(+Xα)) , (5.8)

where mα := mαΔ`aΔ`b, the weighting γ :=∫Ω φ(x)dxdy. From the definiton of the

ghost particles and the symmetry of the basis functions, we observe that the subset of

basis functions φi,j implicity represent the image particles

φi,j(Xα) + φi,j(X−α) + φi,j(X+α) =(φi,j + φ−i,j + φ2(nx−1)−i,j

)(Xα) =: φi,j(Xα).

(5.9)

Note that if any of the basis functions in this subset are not in the basis, defined

above, then they are neglected.

Definition 5.2.0.3 (Gradient of the layer depth). The gradient of the layer depth at

position Xα is defined as

∇h(Xα) =nx−1+k,ny−1∑

i=−k,j=0

hi,j ∇φi,j(Xα). (5.10)

Lemma 5.2.0.4. HPM satisfies the boundary condition n ∙ ∇h = 0 through the sym-

metry properties of the image particles.

Proof 5.2.0.5. We shall consider the boundary xL = [0, y] along which the normal

165

component of the gradient of the layer depth is given by

n ∙ ∇h(xL) =nx−1+k,ny−1∑

i=−k,j=0

hi,j n ∙ ∇φi,j(xL). (5.11)

Given that the basis functions have a support of sx, this expression simplifies to

n ∙ ∇h(xL) =k,ny−1∑

i=−k,j=0

hi,j n ∙ ∇φi,j(xL). (5.12)

Using the property that n ∙∇φi,j(xL) = −n ∙∇φ−i,j(xL) and hi,j = h−i,j, the expression

for the gradient is

n ∙ ∇h(xL) = n ∙ ∇φ0,j(xL) = 0, (5.13)

which is the required boundary condition.

particle

Xα-Xα x=0

Additional basis function

Ghost particle

Rigid wall

ΩL Ω

Basis functions

Figure 5.2: This diagram shows a horizontal Section of the basis of cubic B-splines inthe boundary region around x = 0. There are two distinguishing features of the HPMimplementation to note. Firstly, when the basis function are cubic B-splines, the basisis extended by one spline (shown with the dotted line). This extended basis forms apartition of unity over Ω. This property is needed for HPM to globally conserve mass.Secondly, ghost or image particles are implicitly introduced to impose the boundarycondition n ∙ ∇h = 0. These are images of the particles reflected about the boundary.The image particles have the same mass, but opposite velocities as their pre-images.HPM does not explicitly use image particles but instead uses an appropriate set of basisfunctions to represent them.

166

Canonical HPM equations The canonical Hamiltonian equations of motion for

each particle in rotating shallow water are are

Pα = −c20mα

∑

i,j

hi,j(t)∇Xαφi,j(Xα)− f0k×Pα,

Xα =Pαmα

,

(5.14)

where hi,j denotes a smoothed gridded layer depth and is defined in the next Section.

These equations of motion preserve the discrete Hamiltonian

H =1

2

∑

α

||Pα(t)||2

mα+c202

∑

i,j

hi,j(t)hi,j(t)ΔxΔy, (5.15)

where the first term is the kinetic energy of the particles and the second term is the

potential energy over the gridded layer depth.

5.2.1 Symplectic time stepping

No rotation In the absence of a Coriolis term, the symplectic Stormer-Verlet method

is used as the time-stepper for the HPM shallow water equations as described in Section

4.7 for the 1D VFL shallow water equations.

Rotation The time-stepper for the HPM approximation of rotating shallow water

is adapted from the above symplectic second order explicit integrator to include an

implicit mid-point approximation of the Coriolis terms

Pn+1/2α = Pnα −Δt

2(∇V (Xnα)− f0k× (P

nα +P

n+1/2α )/2),

Xn+1α = Xnα +ΔtPn+1/2α

mα,

Pn+1α = Pn+1/2α −Δt

2(∇V (Xn+1α )− f0k× (P

n+1α +Pn+1/2α )/2).

(5.16)

Other approaches This is a convenient point to comment on the connection between

semi-Lagrangian schemes (Staniforth 1997) which are widely used in meterological ap-

plications and Lagrange-Galerkin finite element methods which have been developed

for a more general class of problems but also includes shallow water (Giraldo 2000).

Essentially, semi-Lagrangian schemes can be regarded as an extension of the above

167

time-stepper by interpolating the particle velocities to the mesh and then resetting

the particle positions to the mesh at the end of the time step. Interpolating back to

the grid renders the method as convenient for computations as Eulerian methods, but

with the added benefit of enhanced stability through computation of the shallow water

equations in the Lagrangian frame.

Semi-Lagrangian methods, however, integrate the particle shallow water equations

along the backward trajectory rather than, as above, the forward trajectory. This has

led to the development of ”remapped” particle-mesh methods for shallow water by

Cotter et al. (2007) which use a leap-frog scheme, similar to the above, but remap the

particles to the grid at the end of the time step.

Of course, the interpolation of velocities and resetting of particle positions destroys

the Hamiltonian structure of the flow field. This, however, may not be a primary

property for resolving significant geophysical dynamics when the equations of motion

are appropriately spatially discretised. Indeed, Giraldo (2000) combines a spectral

element approximation with a semi-Lagrangian scheme to resolve the motion of a pair

of equatorially trapped Rossby soliton waves. The scheme is able to model these waves

without an observed change in their profile or distance apart. The author points out

that the only disadvantage of using an explicit time-stepper is the more stringent CFL

constraint determined by the fastest modes. For this reason, we relax this constraint by

slowing down the gravity waves by introducing a dispersively regularising operator. The

following Section describes how the gridded layer depth is regularised over a bounded

domain.

5.3 Layer Depth Smoothing on a Bounded Domain

The HPM method is constructed using a regularisation operator. In the continuum

limit, the HPM equations for rotating shallow water correspond, not to the Lagrangian

description of the rotating shallow water given by equation (5.2) but the regularised

rotating shallow water equations

DU

Dt= −g∇h− f0k×U,

Dh

Dt= −h∇ ∙U,

n ∙ ∇h = 0, x ∈ ∂Ω,

(5.17)

in which the boundary condition n ∙ ∇h = 0 is imposed on the regularised layer depth.

Following Frank & Reich (2003), the layer depth is regularised by the inverse Helm-

holtz equation h = Sh, where S is a symmetric inverse Helmholtz operator of the form

168

S = (1− α2∇2)−1 with fixed smoothing length α typically of size 2Δx. In this Section,

we shall begin by using the finite element method to approximate the variational (weak)

form of the 1D Helmholtz equation. Neumann boundary conditions on the smoothed

gridded layer depth are imposed naturally, thus preserving the symmetry of the mat-

rix approximation of the inverse Helmholtz operator Sx, referred to as the smoothing

matrix. We will then consider the smoothing of 2D layer depths in a channel and a

basin.

5.3.1 A smoothing operator for 1D shallow water

Definition 5.3.1.1 (Variational Form). The variational form of the 1D Helmholtz

equation, paired with a smooth test function v(x) ∈ V, V := {v :∫Ω[v(x)

2+v′(x)2]dx <

∞}, x ∈ R is given by

∫

Ωv(x)h(x)dx =

∫

Ωv(x)(1− α2∂2x)h(x)dx, ∀v, (5.18)

with boundary condition

∂xh(x) = 0, x ∈ ∂Ω. (5.19)

Integrating equation (5.18) by parts yields

∫

Ωv(x)h(x)dx =

∫

Ωvh(x) + α2∂xv(x)∂xh(x)dx−

=0︷︸︸︷∫

∂Ωv(x)∂xh(x)dS, ∀v. (5.20)

where the boundary integral is formed over the set ∂Ω := {0, L} and vanishes to

impose the natural boundary condition. The 1D continuous piecewise linear finite

element solution to the variational form is now formulated on a uniform mesh, defined

as before, Ωh := {iΔx, i = 0, . . . , n− 1}.

Definition 5.3.1.2. The space of linear functions V h := {v : v ∈ C0(Ω), v is a linear

function on (xk−1, xk), k = 1, . . . , n} ⊆ V . Define a basis of piecewise linear functions

Nk(xj) = δkj , ∀k, j = 1, . . . , n.

{Nk} forms a nodal basis for V h (see Brenner & Scott 2002). hI(x) ∈ V h is the

interpolant of h(x), where hI(x) =∑

l hlNl(x) using the notation hl := h(xl) are the

nodal values of h(x). Similarly hI(x) =∑

l hlNl(x).

Choose, by construction of the (Ritz-Galerkin) finite-element method (Brenner &

Scott 2002), a test function Nk(x) ∈ V h is introduced so that equation (5.20) becomes

∫ xk+1

xk−1

Nk(x)hI(x)dx =

∫ xk+1

xk−1

Nk(x)hI(x) + α2∂xNk(x)∂xhI(x)dx, ∀k. (5.21)

169

Substituting these expressions into the previous equations yields

∑

l

Mklhl =∑

l

Aklhl, ∀k, (5.22)

where

Mkl =

∫ xk+1

xk−1

Nk(x)Nl(x)dx, Akl =

∫ xk+1

xk−1

Nk(x)Nl(x)+α2∂xNk(x)∂xNl(x)dx. (5.23)

Alternatively, equation (5.22) can be expressed in the matrix form

Mh = Ah. (5.24)

It follows from the form of the elements of A and M , that A and M are symmetric

tridiagonal matrices. The smoothing matrix Sx = A−1M requires the inversion of a

tridiagonal symmetric matrix, which is relatively efficient to compute.

5.3.2 A smoothing operator for 2D shallow water

We now consider the smoothing of the 2D gridded layer depth H, whose elements

are denoted hi,j and are defined on a regular nx by ny grid. We consider two types of

bounded domains: (i) a channel of the form Ω := [0, Lx]× [0, Ly), where the x-direction

is bounded and the y-direction is periodic and (ii) a (rectangular) basin of the form

Ω := [0, Lx]× [0, Ly], where the x and y directions are bounded.

In each case, we introduce a splitting of the smoothing operator into x and y

components, so that the inverse Helmholtz equation becomes

H = SyHSx, (5.25)

where H denotes the smoothed gridded layer depth, Sx and Sy are matrix approxim-

ations of symmetric inverse Helmholtz smoothing operators which each smooth inde-

pendently in the x and y directions respectively with smoothing parameters αx and

αy.

We smooth in two steps. Firstly, the finite element approximation of the x compon-

ent of the smoothing operator is extended to act on H. Substituting a new definition

of the linear interpolant (hI)m =∑

l hmlNl(x) into equation (5.21) gives

170

∫ xk+1

xk−1

Nk(x)(hI)m(x)dx =

∫ xk+1

xk−1

Nk(x)(hI)m(x) + α2∂xNk(x)∂x(hI)m(x)dx, ∀k,m,

(5.26)

or in matrix form

MxHT = AxH

T , (5.27)

where Mx is the mass matrix, Ax is the stiffness matrix and H := HSx = S−1y H is the

layer depth, smoothed in the x-direction.

The second step, for our purposes, is dependent on whether the domain is a channel

or a basin.

Channel The spectral approximation of the y component of the inverse Helmholtz

operator is the diagonal matrix

(Sy)i,i =(1 + α2(kiy)

2)−1

, kiy = −ny

2+ i, i = 1, . . . , ny. (5.28)

{kiy} is the finite set of all wave numbers of the streamwise Fourier modes permissable

over Ωh. Taking the product of this matrix with the discrete Fourier transform of the

finite element smoothed layer depth H gives an expression for the combined smoothing

in each direction

ˆH = Sy

H, (5.29)

or equivalently, using the finite convolution theorem,

H = Sy ∗ H. (5.30)

Basin In the case when the domain is a basin, the smoothing operator in the y

direction is formulated as a finite element approximation of the form

MyHT = AyH

T , (5.31)

which is solved for HT .

Remark 5.3.2.1 (Smoothing operator splitting). The splitting of the smoothing matrix

works well in practice because the boundary conditions are imposed naturally without the

computational overhead of a two dimensional finite element method for the Helmholtz

equation. Note also that Ax and Ay are constant nx × ny matrices and need only be

inverted once. The splitting, however, breaks the rotational symmetry of the Helmholtz

171

operator and therefore violates angular momentum conservation. In order to preserve

the isotropy of the operator, one can formulate a 2D Galerkin finite element approxim-

ation of the smoothing operator. The resulting stiffness matrix is nx×ny in length and

is more computationally expensive to invert. Cotter (2005) addresses this problem by

inverting the matrix using a preconditioned conjugate gradient approach. Alternatively,

one could use the split operator defined above as an ADI preconditioner, a description

of which is given in Miellou & Spiteri (2002).

Numerical simulations of this model are provided in the next Section.


In this Section, we measure the conservative properties of 1D shallow water and show

the effect on the shallow water dynamics of preserving the symmetry of the smoothing

matrix under a finite element approximation of the Helmholtz equation.

The purpose of the numerical experiments that will now be described is to firstly

measure the long-time energy error, compare a 1D implementation of the HPM method

with a spectral-Chebyshev method and secondly simulate the motion of a vortex pair

in a channel of rotating shallow water as it reaches the channel wall.

The long-time energy properties of the HPM method are most conveniently assessed

through a 1D experiment.

5.4.1 HPM for 1D (non-rotating) shallow water

Experiment 1 This experiment measures the long-time energy error over 3 × 105

time steps using HPM with image particles and the finite element smoother in 1D. The

simulation parameters for this experiment are given in Table 5.1.

Parameter Value

N 256n 64Δt 5× 10−3

α 2ΔxL 2π

Initial conditions mα = 1 + a0e−50((X−L/2)2)/L2

Uα = 0

Table 5.1: Experiment 1: the set of simulation parameters and initial conditions for ex-periment 1 and 2, which measures the energy error and the effectiveness of the smooth-ing operator in the HPM simulation of 1D shallow water on a bounded domain.

172

Parameter Value

Time elapsed 3× 105 time stepsRelative energy error 5× 10−8

Table 5.2: Experiment 1 results: the relative energy error (E − E0)/E0 of the HPMapproximation of 1D shallow water over 3 × 105 time steps.

5.4.2 The 2D (non-rotating) shallow water equations in a channel

Experiment 2 We perform a simulation of shallow water in a channel and com-

pare the HPM method with the spectral-Chebyshev method. Figure 5.3 shows contour

plots at successive simulation times of the layer depth, starting from a Gaussian per-

turbation and Figure 5.4 shows a comparison of the HPM approximation (fixing the

particle-to-grid ratio at 4) of the layer depth and velocity to a spectral-Chebyshev ap-

proximation and compares energy and mass conservation. The parameters for channel

flow simulations, unless otherwise stated, are given in Table 5.3.

Parameter Value

Number of particles N 256Number of grid points n 64Time step 5× 10−4

Smoothing length α 2ΔxDomain length 2π × 2πInitial conditions mα = 1 + a0exp(−800[(Xα(t0)− L/2)2 + (Yα(t0)− L/2)2]/L2)

Uα = 0, Vα = 1Amplitude a0 0.5Coriolis parameter f0 0

Table 5.3: Experiment 2: this Table lists the simulation parameters for 2D shallowwater channel flow together with their values.

173

Figure 5.3: Experiment 3: contour plots of the smoothed layer depth in a 2D channelwith non-zero mean streamwise velocity shown starting from the top left and increasingin reading order at intervals of 20 iterations up to 160. The reason why the layer depthis not rotationally symmetric is because the split smoothing operator is not isotropic.

174

0 1000 2000 3000 4000 50000

0.5

1

1.5

2

2.5x 10

-5

time steps

||hhp

m-h

cheb

|| L2 (Ω)

16x1632x3264x64

0 1000 2000 3000 4000 50000

1

2

3

4

5

6x 10

-5

||uhp

m-u

cheb

|| L2 (Ω)

time steps

16x1632x3264x64

0 200 400 600 800 100010

-12

10-10

10-8

10-6

10-4

10-2

time

Rel

ativ

e en

ergy

err

or

HPMSpectral-Chebyshev

0 200 400 600 800 100010

-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Rel

ativ

e m

ass

erro

r

time

HPMSpectral-Chebyshev

Figure 5.4: Experiment 3: L2 error measures of the HPM approximation of the (topleft) layer depth and (top right) velocity field to the spectral-chebyshev solution, wherethe norm ||h− h′||2L2(Ω) is defined as ||h− h

′||2L2(Ω) :=1

nxny

∑i,j |hi,j − h

′i,j |2. The error

profiles do not exhibit a definitive convergence trend, especially in the layer depthprofile. (bottom left) Relative energy error and (bottom right) relative mass erroragainst the number of time steps.

175

5.4.3 Rotating shallow water in a channel

Experiment 4 This experiment shows the motion of a vortex pair in a channel of

rotating shallow water as simulated by HPM. The vortex pair moves to the boundary

and then separates, each vortex moving apart along the wall. The vorticity field is

plotted at successive times in Figure 5.5 starting from a geostrophically balanced flow

with a layer depth described by a double Gaussian function. The parameters for this

experiment and the conservative properties of the HPM method are shown in Tables

5.4 and 5.5 respectively.

Parameter value

Nt 6000 (6 days)c0 48L2

f0 4√2π

dt 5× 10−3

L 2πH 1N 252×252nx 64ny 64

α√120(c0Hdt2)/(4 + f20dt

2)

Table 5.4: Experiment 4: the simulation parameters for the rotating shallow waterequations in a channel initialised from a geostrophically balanced flow with the layerdepth described by a double Gaussian function.

Parameter value

Time elapsed 6 daysMean energy 32277.698

Relative energy error 9.929× 10−5

Total mass 16384Total mass error 0

PV(0) 6534.525PV(Nt) 6543.632

Relative PV error 1.393× 10−3

Table 5.5: Experiment 4 results: the conservative properties of the HPM approximationof 2D rotating shallow water in a channel after 8 days.

176

Figure 5.5: Experiment 4: this graph shows the vorticity field of rotating shallowwater in a channel as computed by HPM. The vorticity field is shown at times t ={0, 2, 3, 4, 5, 6, 7, 7.4, 8.2} days, increasing in reading order. We observe that undergeostrophically balanced initial flow conditions, the vortex pair moves to the wall andthen separates. Each vortex then moves further apart along the wall.

177

5.5 Summary and Further Research

The HPM method is an efficient geometric numerical method for modelling shallow wa-

ter without adding numerical dissipation to ensure long-time stability. In this Chapter

we describe the use of image particles to extend the HPM method to rotating shal-

low water in a bounded domain. The Neumann boundary condition on the layer depth

restricts smoothed layer depth solutions to those which are symmetric about the bound-

aries. The HPMmethod imposes this boundary condition by introducing ghost or image

particles as reflections about the boundaries. This can be implemented very efficiently

by modifying the basis of cubic-B-splines to implicitly represent the image particles.

This Chapter only considers planar boundaries which coincide with a uniform mesh

and it is the subject of future research to consider more general boundary geometries

and meshes.

Another necessary step for long-time stability of the method is to approximate the

dispersively regularising Helmholtz operator with a finite element method. Numerical

results in 1D demonstrate that the implementation of the Neumann boundary condition

on the smoothed gridded layer depth must result in a symmetric Helmholtz matrix. The

finite element method imposes Neumann boundary conditions naturally and therefore

satisfies this property. This approach can be easily extended to a channel, by con-

volving the finite element regularisation of spanwise layer depth Fourier modes with

a spectral regularisation of the streamwise layer depth Fourier modes, and to a basin,

by successively smoothing with the 1D finite element method in the x and y direc-

tions. A study of the splitting error introduced by using this computational convenient

approach, should be assessed as further research.

Numerical experiments show that the HPM approximation of (non-rotating) shallow

water in a channel does not distinctly convergence to a spectral-Chebyshev approxima-

tion as the mesh is refined. These experiments should be repeated to study convergence

of the HPM approximation to a spectral-Chebyshev approximation as the particle to

grid ratio is increased, keeping instead the mesh fixed. The numerical results also in-

clude a simulation of a vortex pair in rotating shallow water as it reaches a channel

wall.

Further experiments should compare HPM with other (dissipative numerical meth-

ods) for shallow water in a beta plane with the intent of studying the effect of numerical

dissipation in models of western boundary current intensification (see Stommel 1948).

178

Chapter 6

Summary

6.1 Contribution of this Thesis

Geometric integrators transfer powerful concepts in geometric mechanics to computa-

tional continuum dynamics by preserving properties of the continuous system such as

the geometric structure, symmetries and phase space volume. Holm et al. (1986) presen-

ted a unified geometric approach for the study of idealised Hamiltonian continuum

models (fluids, plasmas, elasticity, etc.) in the material, inverse material, spatial and

convective representations. This unified approach is based on momentum maps, which

carry the Poisson brackets in one representation into another.

6.1.1 Development of a unified framework

In this thesis, we pursued the development of a unified computational framework for de-

riving geometric integrators for the convective and spatial representation of continuum

dynamics. This computational framework transfers the following powerful concepts

given by the unified framework of Holm et al. (1986) , for the convective and spatial

Hamiltonian continuum dynamics, to computational continuum dynamics:

1 Holm et al. (1986) show that the group action for passing between the represent-

ations generates an infinitesimally equivariant momentum map which carry the

Poisson brackets in one representation to those of the other. Using the discrete

Clebsch approach (Cotter & Holm 2006), we give the corresponding (diagonal)

group actions for passing between the representations and their momentum maps

from the cotangent bundle to the dual of the Lie algebra of the group.

2 Holm et al. (1986) show that the equations of continuum motion with advected

quantities are coadjoint orbits for the action of a semi-direct product Lie-algebra

on the dual of a semi-direct product Lie algebra. These orbits are symplectic

179

foliations of the Poisson manifold P defined by the augmented cotangent bundle.

We show that the discrete Clebsch approach gives discrete equations of motion

with advected quantities which define co-adjoint orbits for the action of the semi-

direct product group on the dual of its Lie-algebra. These orbits and are also

symplectic foliations of P . The co-adjoint actions preserves the ± Lie-Poisson

brackets on the dual of this semi-direct product Lie algebra.

3 Holm et al. (1986) show through various examples, that these momentum maps

encode fundamental conservation laws of Hamiltonian continuum dynamics. These

conserved momentummaps are generated from the Noether symmetries for passing

to the spatial and convective representations, the latter of which is referred to

as a particle relabelling symmetry. We show that the discrete Clebsch approach

gives

I conserved momentum maps for the polar decomposed pseudo-rigid body

which are the conserved spatial angular momentum and the discrete Kelvin

circulation theorem generated by the respective rotational and material re-

labelling symmetries.

II conserved momentum maps for the inextensible and geometrically exact

elastic rods which are the total spatial angular momentum generated by

the rotational symmetries.

For the latter case, there is a fundamental difference between the form of the

momentum maps derived from the continuum framework and our computational

framework, however. Our computational framework represents the continuum as

a finite dimensional system of particles. The conserved momentum map then

takes the form of a discrete sum over all particle labels and is only the form of

the conserved momentum map for the continuum, in the continuum limit of the

particle system.

4 Our computational framework gives a prototype MV integrator for the convective

and spatial representations of compressible fluids.

Metric tensors are central to the theory of continuum mechanics. Holm et al.

(1986) consider a compressible fluid flow, in which the passage to the convective

and spatial representation is by reduction under the group of diffeomorphisms,

and show how the metric tensor and densities respectively transform in the differ-

ent representations. Analogously, we consider ellipsoidal motion and demonstrate

how our framework transforms the metric tensor (the Cauchy-Green matrix) and

180

shape matrices under reduction by the group GL(n)+ to the convective and spa-

tial representations. The discrete Clebsch approach gives MV integrators for

these representations in which the metric and shape matrix are respectively the

advected quantities.

6.1.2 Development of new DMV algorithms

This thesis shows that, under a forward in-time finite difference approximation of the

continuous Clebsch constrained action principle for the body representation of the rigid

body, that the discrete Clebsch approach recovers the Moser-Veselov integrator. MV

integrators are computed using the explicit DMV algorithm developed by McLachlan

& Zanna (2005). In parallel with the development of our computational framework,

as described above, we give new DMV algorithms to solve for the MV integrators and

verify their conservative properties by numerical experiment.

• Rigid body motions in the spatial representation We modify the DMV algorithm,

for the body representation of rigid bodies, to solve the spatial MV integrators

derived in Chapter 2. We then provided several numerical experiments to study

the comparative conservative properties of the spatial DMV algorithm. The res-

ults are conclusive and show that there is a negligible difference between the

conservative properties of each algorithm.

• Rigidly Coupled motions We develop a DMV algorithm for the coupled matrix

Ricatti equation which arises from coupled rigid body motion between the free

rigid body motions of the coupled rigid body and the circulatory and rotational

rigid body motions of the polar decomposed pseudo-rigid body. We implement a

model of a Mooney-Rivlin type pseudo-rigid body to describe the stretching and

rotational components of the motion and show that the DMV algorithm conserves

angular momentum and vorticity (relative to the Lagrangian frame) and exhibits

no secular drift in the energy.

• Elastically coupled motions We solve a system of elastically coupled MV integrat-

ors for the elastically coupled director motions of the geometrically exact elastic

rod. The DMV algorithm for an elastically coupled rigid body differs from that

of the rigidly coupled rigid body. In the former case, the coupling is only through

the source term and not, as in the latter case, through the Coriolis term and re-

quires minor modification. The discrete Clebsch approach also gives a variational

integrator for the material representation of the rod centroid positions which is

equivalent to a Stormer-Verlet symplectic integrator. The DMV algorithm for

a rod of 50 sections conserves total spatial angular momentum, total linear mo-

mentum and exhibits no secular drift in the energy.

181

6.1.3 Geometric integrators for shallow water

This thesis pursues geometric integrators for shallow water in a variational framework.

These integrators preserve the canonical symplectic structure of a finite dimensional

system of fluid particles and are partially expressed in terms of the Eulerian quantities

(such as the layer depth).

• Variational free-Lagrange method A variational free-Lagrange method for rotating

shallow water with bottom topography is presented in Chapter 4. We establish

the conservative properties of the semi-discrete shallow water equations and de-

rive a semi-discrete shallow water divergence conservation law and an evolution

equation for the potential vorticity equation with a non-zero right hand side.

The semi-discrete divergence equation is given by the extrema of the discrete

action principle. The potential vorticity equations, however, are formed through

a choice of a discrete curl operator which is not resolved from the extrema of

the action principle. This suggests the need for an additional constraint in the

action principle to constrain the form of the discrete curl operator so that the

semi-discrete potential vorticity equation is satisified too. Numerical results are

also presented which show that the VFL method for rotating shallow water in 1D

exhibits long-time energy and potential vorticity conservative properties in 1D

and exhibits the geostrophic adjustment mechanism of rotating shallow water.

• Boundary conditions in the HPM method We extend the HPMmethod to bounded

rotating shallow water flows by implicity introducing ghost or image particles.

We demonstrate that the HPM approximation of rotating shallow water in a

bounded domain conserves mass, exhibits no secular drift in the energy and re-

mains stable over long-time simulations. We compare the HPM approximation of

(non-rotating) shallow water in a channel to a spectral-Chebyshev approximation

and also use HPM to simulate the motion of a vortex pair as it approaches a

shallow water channel wall.

6.2 Conclusions

In this thesis, we have pursued the development of a unified computational framework

for deriving geometric integrators for the convective and spatial representation of con-

tinuum dynamics. We consider the application of this framework to ellipsoidal motions

and rod models and derive MV integrators for the reduced motions. We solve these in-

tegrators using DMV algorithms and assess their conservative properties by numerical

experiment. The extension of MV integrators to fluids is a challenging open problem

182

and requires the use of a finite dimensional representation of the group of diffeomorph-

isms which has been pursued by Zhong & Scovel (1994), Zeitlin (2004). We turn to

the formulation of a variational free-Lagrange method to represent the Hamiltonian

structure of shallow water in terms of the (material) particle velocities and the spatial

scalar quantity, referred to as the layer depth. We study the conservative properties of

the semi-discrete VFL method and present the semi-discrete divergence and potential

vorticity equations, the latter of which takes the form of a conservation law. We then

assess the conservative properites of the fully discrete VFL method by simple numerical

experiments. We finally consider the formulation of velocity boundary conditions in a

Hamiltonian framework.

Throughout the thesis, we identified limitations in our methodology and results

and proposed future research possibilities to address these limitations, which we now

conclude.

6.2.1 Future directions

Compressible fluid dynamics The unified computational framework seeks applic-

ation to the convective and spatial representations of fluid dynamics. The convective

and spatial representations of (compressible) ellipsoidal motion provide a basic proto-

type for ideal compressible fluid dynamics for reasons which we now state.

• Spatial : Holm et al. (1986) consider a compressible fluid and show that the pas-

sage to the spatial representation is by reduction under the group of diffeomorph-

isms. This group acts on the density by pull-back, but acts trivially on the

metric-tensor on Eulerian space, forming the kinetic energy. Analogously, the

(compressible) ellipsoid reduces to the spatial representation by the right action

of GL(n)+. This group acts on the shape matrix by conjugation and trivially on

the right Cauchy-Green Matrix (the metric-tensor on Eulerian space).

• Convective: Conversely, Holm et al. (1986) show that the passage to the con-

vective representation of compressible fluids is by reduction under the group of

spatial diffeomorphisms. This group acts trivially on the density and on the

metric-tensor on Eulerian space by pull-back, forming the kinetic energy. Ana-

logously, the (compressible) ellipsoid reduces to the convective representation by

the left action of GL(n)+. This group acts trivially on the shape matrix and on

the right Cauchy-Green Matrix by conjugation.

In order to extend the MV integrators for ellipsoidal motion to compressible fluids,

two problems must be addressed, formulation of the finite dimensional representation

of the group action of diffeomorphisms on (i) G, where G is a finite dimensional repres-

entation of the group of diffeomorphisms, and (ii) on V ∗ by pull-back. Use of a finite

183

dimensional representation of the group of diffeomorphisms in computational fluid mod-

els is a challenging open area of research which has made little progress over the last

decade. We refer the reader to Zhong & Scovel (1994), Zeitlin (2004) for further details.

MV integrators for the pseudo-rigid body The stretching component of the dis-

crete pseudo-rigid body motion is currently updated using a discrete Euler-Lagrange

equation. This update does not constrain the stretching matrices to the space of di-

agonal matrices with positive determinant. Further research should consider the for-

mulation of a multiplicative update procedure for the stretching motion. The MV

integrator should also be extended for an anisotropic polar decomposed pseudo-rigid

body in which the shape matrix is advected by both rotations and stretching. In order

to solve this integrator, the DMV algorithm must be extended to solve for stretching

motion.

MV integrators for the geometrically exact rod Future research should address

the extent to which the MV integrator for the geometrically exact rod model is Poisson

with respect to the Lie-Poisson bracket for the continuous SO(3) reduced rod motion

given by Krishnaprasad et al. (1988). The MV integrator for this rod model should

also be adapted to model the director orientations using quaternions, rather than Euler

angles, using the approach described in Chapter 2. The latter is motivated by the

need to model supercoiling and twisting motions of elastic materials such as DNA and

other polymer chains, for which Euler angle parameterisation is not suitable. Boundary

conditions, such as clamped end conditions, should also be investigated.

VFL methods for shallow water In Chapter 4, we find that the form of the

discrete curl operator does not satisfy the property that the discrete curl of the discrete

gradient is zero and results in a violation of the conservation of potential vorticity.

Further research should address how to constrain the semi-discrete curl operator in the

discrete action principle to give a semi-discrete PV conservation law. The question of

whether the VFL method can be formulated for the spatial or convective representation

of shallow water remains an open problem. The existence of a unified computational

approach for modelling the Hamiltonian structure of shallow water using spatial and

convective variables appears to rest on addressing this question.

Boundary conditions In Chapter 5, we show how a Hamiltonian particle mesh

method for rotating shallow water can be extended using ghost particles to impose

velocity boundary conditions. It remains an open problem as to whether this approach

can be integrated into our unified computational framework giving, for example, MV

integrators for rod models with clamped ends. By treating planar boundaries as a

184

particle symmetry, we show how image or ghost particles may be implicitly introduced

into the HPM method. We also derive a finite element approximation of the Helm-

holtz operator which is used to dispersively regularise the layer depth and thus relaxes

the CFL stability constraint. A Neumann boundary condition on the regularised layer

depth is imposed naturally and the resulting Helmholtz matrix is symmetric. Numer-

ical experiments demonstrate that symmetry of this matrix is essential for numerical

stability and further demonstrate the effects of preserving symplectic structure on a

vortex pair in a rotating shallow water basin.

185

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192

Appendix A

Properties of MV Integrators

A.1 Body and Spatial Representations in Continuous

and Discrete Time

Dixon Moser and Veselov Description

Λk Xk Attitude matrix at time tkI0 J Inertia matrixΩk = Λ

Tk−1Λk ωk = X

Tk Xk−1 body angular ’velocity’ at time tk

Mk = I0ΩTk − ΩkI0 Mk = ω

Tk J − Jωk body angular momentum at time tk

mk = Λk−1MkΛTk−1 mk = Xk−1MkX

Tk−1 spatial angular momentum at time tk

Table A.1: This Table provides a comparison of our notation with that of Moser &Veselov (1991).

Property Continuous Discrete

Body attitude Λ(t) ∈ SO(N) Λk ∈ SO(N)Angular velocity Ω = ΛT Λ = −ΩT Ωk+1 = Λ

TkΛk+1

Inertia Matrix I0Angular momentum M = I0Ω− ΩT I0 Mk = I0Ω

Tk − ΩkI0

Equations of motion M = ad∗ΩM Mk+1 = Ad∗ΩkMk

Right momentum map JR = skew(ΛP T ) Jk+1R = skew(ΛkPTk )

Table A.2: A comparison of the terms required to describe the body representation ofthe rigid body in continuous and discrete time as derived using the Clebsch approach.Blank items in the right-hand column indicate that they are identical to their discretetime descriptions.

193


Body attitude Λ(t) ∈ SO(N) Λk ∈ SO(N)Angular velocity ω = ΛΛT = −ωT ωk+1 = Λk+1Λ

Tk

Inertia Matrix I = ΛI0ΛT Ik = ΛkI0Λ

Tk

Angular momentum m = Iω − ωT I mk = IkωTk − ωkIk

Equations of motion m = ad∗ωm−∇IL � I = 0, mk+1 = Ad∗ωTk+1

mk + 2∇Ik lIk � Ik+1,

I = [ω, I] Ik+1 = ωk+1IkωTk+1

Left momentum map JL = P � Λ + J � I Jk+1L = Pk � Λk +Gk � Ik

Table A.3: Comparison of the terms required to describe the spatial representation ofthe rigid body in continuous and discrete time.


Body attitude Λ(t) ∈ SO(N) Λk ∈ SO(N)Angular velocity Ω = ΛT Λ = −ΩT Ωk+1 = Λ

TkΛk+1

Inertia Matrix I0Angular momentum M = I0Ω− ΩT I0 Mk = I0Ω

Tk − ΩkI0

Orientation of the z-axis Γ = ΛT z Γk = ΛTk z

Equations of motion M = ad∗ΩM +mgΓ � χ, Mk+1 = Ad∗ΩTkMk +mgΓk � χ,

Γ = −ΩΓ Γk+1 = ΩTk+1Γk

Right momentum map JR = P � Λ + Γ � JΓ JRk+1 = Pk � Λk + Γk �Gk

Table A.4: A summary of the terms required to describe the body representation of theheavy top in continuous and discrete time.

194


Body i attitude Λi(t) ∈ SO(N) Λki ∈ SO(N)Body i angular velocity Ω = ΛTi Λi = −Ω

Ti Ωk+1i = Λk

T

i Λk+1i

Position of c.o.m. of body i diOrientatn matrix (rel. to b. 1) Λ = ΛT1 Λ2

Mod. inertia matrix of body i Ii = Ii −m2im Dii

C1 C1 = I1 − εAΩ2 Ck+11 = I1 + εAk(Ωk+12 − Id)

C2 C2 = I2 − εATΩ1 Ck+12 = I2 + εAkT (Ωk+11 − Id)

Body 1 angular momentum M1 = C1Ω1 − ΩT1 CT1 Mk

1 = Ck1Ω

kT

1 − Ωk1C

kT

1

Body 2 angular momentum M2 = C2Ω2 − ΩT2 CT2 Mk

2 = Ck2Ω

kT

2 − Ωk2C

kT

2

Equations of motion Mi = ad∗ΩiMi +

δlδΛ � Λ, Mk+1

i = Ad∗ΩkiMki +∇Λk lΛk � Λ

k,

Λ = ΛΩ2 − Ω1Λ Λk+1 = Ωk+1T

1 ΛkΩk+12Right momentum map J iR = Pi � Λi + J � Λ J i

k

R = Pki � Λ

ki +G

k � Λk

Table A.5: A summary of the terms required to describe the body representation of thecoupled rigid body in continuous and discrete time, where i ∈ {1, 2}.

A.2 MV Integrators for the Cayley-Klein Parameterisa-

tion of Rigid Body Motion

Property SU(2) = h(Q) Q

Body attitude Λk ∈ SU(2) qk ∈ QAngular ’velocity’ Ωk+1 = Λ

†kΛk+1 Ωk+1 = qk ∗ qk+1

Moments of inertia Ij , j := 1→ 3

Inertia matrix Ak = Id − IiTr(Ω†kEi

)Ei Ak = [1,Av(tk)]

Av = ıjIjΩ(tk)jAngular momentum Mk+1 = Ak+1Ω

†k+1 − Ωk+1A

†k+1 Mk+1 =

12Ak+1 ∗ Ωk −

12Ωk ∗ Ak+1

Equations of motion Mk+1 = Ad∗ΩTkMk Mk+1 = Ωk ∗Mk ∗ Ωk

Right mom. map Jk+1R = Pk � Λk Jk+1R = pk � qk

Table A.6: A comparison of the terms required to describe the body representation ofdiscrete time rigid body motion as elements of SU(2) and quaternions.

195

A.3 The Spatial DMV Algorithm for the Rigid Body

For completeness, we include the specification of the (explicit) DMV algorithm for

the spatial representation of the rigid body. Following the approach of McLachlan &

Zanna (2005), the algorithm uses the Schur decomposition of the Hamiltonian for the

matrix Ricatti equation to construct a symmetric matrix Sk. Numerical experiments in

Section 2.10 show that there is little difference between the conservative properties of

the spatial and body versions of this algorithm. We observe that the numerical round-

off error differs between the two different versions of the DMV algorithm depending

upon the principal moments of inertia. Numerical experiments, not presented in this

thesis, find negligible difference in the stability and computational performance between

the two versions.

1. For k = 1 to NT

2. mk = ωk−1mk−1ωTk−1 + [Ik−1, ωk−1 + ω

Tk−1]

3. Hk =

(mk2 , Id

(mk2 )2, I2k−1 −

mTk2

)

4. [Rk, Uk] = Schur(Hk)

5. Sk = (Rk)21(Rk)−111

6. ωk = (Sk +mk2 )I

−1k

7. Ik = ωkIk−1ωTk

8. k = k + 1

A.4 A DMV Algorithm for Coupled Rigid Body Motion

Consider a coupled matrix Ricatti equation for discrete coupled rigid body motions of

the form

196

M1 =M′1 + J(Ω2)Ω

T1 − Ω1J(Ω2)

T ,

M2 =M′2 + J(Ω1)Ω

T2 − Ω2J(Ω1)

T ,(A.1)

whereMi and Ω denote the momenta and discrete velocity of each body, each multiplied

by a factor of Δt - the DMV algorithm of McLachlan & Zanna (2005) requires that

these variables are initially rescaled. M ′i denotes an uncoupled matrix Ricatti equation

of the form M ′i = IiΩTi − ΩiIi, for some symmetric positive definite matrix Ii.

[Step 1] Introduce a splitting of the coupled matrix Ricatti equations into two

separate matrix Ricatti equations in Ωi and formulate an iterative procedure for their

solution. The split matrix Ricatti equations are

M1 =M′1 + J(Ω2)Ω

T1 − Ω1J(Ω2)

T ,

M2 =M′1 + J(Ω1)Ω

T2 − Ω2J(Ω1)

T ,(A.2)

where J(Ωi) is a function of the previous value of Ωi and denoted as Ωi. Note that the

modified Coriolis terms are both O(Δt2), an O(Δt) smaller than the other terms.

[Step 2] The DMV algorithm given by McLachlan & Zanna (2005) is then applied

to compute the iterative solutions of equation A.2 which take the form

Ωi = I−1i (−M

′i + S

i), (A.3)

where Si is a symmetric matrix with expressions determined from each of the split

matrix Ricatti equations given by equation A.2.

[Step 3] The iterative solutions give updated values of J(Ωi), and using equation

A.2, updated values of M ′i . [Step 4] Repeat steps 2 and 3 until Ωi both converge to a

specified tolerance in their matrix two-norm and then rescale Mi by 1/Δt.

197

Appendix B

The Variational Description of

Elastic Body and Rod Models

B.1 The Anisotropic Pseudo-Rigid Body

We generalise the Euler-Poincare description of the polar decomposed isotropic pseudo-

rigid body by Sousa-Dias (2002) to the anisotropic free rigid body. This is the case

when the reference configuration is not spherical. In this case, the shape matrix is

advected by rotational and stretching motions.

Following Holm et al. (1986), we consider the Lagrangian defined on TGL(3)+×V ∗

for the anisotropic pseudo-rigid body

LI0 =Tr

2(AI0A

T ), (B.1)

where I0 denotes that I becomes an active parameter under the right actions φ :

TGL(3)+ × V ∗ × GD : (A, I0) ∙ φ := (AD−1, DI0DT ) and φ′ : TGL(3)+ × V ∗ ×

SO(3) | (A, I0)∙φ := (AST , SI0ST ) where GD is the group of positive diagonal matrices.

Polar decomposition of RASTD−1 gives

L =Tr

2(AI0A

T )

=Tr

2

((RTDS +RT DS +RTDS)I0(R

TDS +RT DS +RTDS)T)

=Tr

2

(RT (ΩT + Γ +DωD−1)It(Ω + Γ

T +D−1ωTD)R)

=Tr

2

(ΣItΣ

T),

(B.2)

where It = DSI0STD, ω = SS−1 ∈ so(3), Γ = DD−1 belongs to the Lie algebra of all

positive diagonal matrices gD(3) ' R3+, Ω = RR−1 ∈ so(3) and Σ = ΩT + Γ +m∗ω,

198

for Σ ∈ gl(3) where

g = so(3)⊕ R3+ ⊕m∗so(3). (B.3)

Lemma B.1.0.1. m∗ : g→ h , m∗(ω) = DωD−1 is a Lie algebra isomorphism.

Proof. m∗ is a Lie algebra homomorphism since

m∗[Λ,Γ] = D[Λ,Γ]D−1

= DΛD−1DΓD−1 −DΓD−1DΛD−1

= [m∗Λ,m∗Γ],

(B.4)

for Λ,Γ ∈ gl(3).

The variations of each of the Lie algebras and the inertia matrix

δΓ = ΞD + [ΞD,Γ],

δω∗ = [ΞD, ω∗] +D(Ξs + [ΞS , ω])D

−1,

δΩ = ΞR + [ΞR,Ω],

δIt = ΞDIt + ItΞTD +D[Ξs, Is]D,

(B.5)

are substituted into the Hamilton’s action principle

∫dt〈

δL

δΓ, δΓ〉+ 〈

δL

δω∗, δω∗〉+ 〈

δL

δIt, δIt〉 = 0, (B.6)

to give, after rearrangement of terms,

〈πD(ΣIt), ΞD + [ΞD,Γ]〉+ 〈ΣIt, [ΞD, ω∗] +D(ΞS + [ΞS , ω])D

−1〉

+ 〈skew(ItΣT ), ΞR + [ΞR,Ω]〉+ 〈

DΣTΣD

2,ΞDIt + ItΞ

TD +D[Σs, Is]D〉 = 0.

(B.7)

where mD :=δLδΓ = πD(ΣIt), mR :=

δLδΩ = skew(ItΣ

T ) and m′S := D−1 δLδω∗D. πD de-

notes the diagonal component of the matrix. This gives three Euler-Poincare equations

for evolution of the projected momentum

m′s = ad∗ωm′s +D

δL

δItD � Is,

mR = ad∗ΩmR,

mD = 2ItδL

δIt+ ad∗ω∗

δL

δω∗+ ad∗ΓmD,

(B.8)

together with the auxillary equation for the Lie advected inertia tensor

199

It = ΞDIt + ItΞTD +D[ω, Is]D. (B.9)

These equations describe how the components of the momentum of the pseudo-rigid

body evolve.

This decomposition together with the defined left and right symmetry reductions

gives complex expressions for mD and m′s associated with pure stretching and internal

rotational-stretching deformations.

B.2 The Geometrically Exact Elastic Rod Model

This Appendix provides a variational description of the geometrically exact rod model,

using the Clebsch approach (Holm & Kupershmidt 1983). The derive equations of

motion correspond to the Lie-Poisson equations given by Krishnaprasad et al. (1988).

The application of the Clebsch approach, described here, to the geometrically exact

rod also provides a useful reference point for the application of the discrete Clebsch

approach to the geometrically exact rod model.

The Clebsch constrained Lagrangian density is defined as

ˆ= `+ 〈P , Λ− ΛΩ〉+ 〈P, φ− ΛV〉+ 〈J ,Λ′ − ΛΩ〉+ 〈J, φ′ − Λ(Γ + E3)〉. (B.10)

The pairing 〈., .〉 denotes the product for identifying elements in the tangent space

of a group with elements in the dual space for all labels S. For elements of Diff(C)

describing the rod centroid position and B → SO(3) describing the orientation of a

director, these pairing take the from

〈p, q〉 =∫

Bp(S)T q(S) dS, q ∈ TqDiff(C), p ∈ T

∗qDiff(C), and

〈P, Q〉 =∫

BTr(P (S)T Q(S)

)dS, Q ∈ TQSO(3), P ∈ T ∗QSO(3).

(B.11)

The stationary action principle gives the following Clebsch relations

P = PΩ− J ′ + J ΩT − P ⊗ V − J ⊗ (Γ + E3),

δ`

δV= ΛTP,

δ`

δΓ= ΛTJ,

δ`

δΩ= skew(ΛPT ),

δ`

δΩ= skew(ΛJ T ).

(B.12)

200

Lemma B.2.0.2. The left infinitesimally equivariant momentum map for the cotangent

lifted left action ΨL on T ∗C is the total angular momentum

JL =

∫

Bskew(ΛTP)dS. (B.13)

Proof. The infinitesimal generator ζC = ΨLζ of Ψ

L on C acts on T ∗C by the canonical

action ζT ∗C = ζ ′C . This canonical action is the cotangent lift of ζC to T∗C. By the

theorem on momentum maps for lifted actions (Marsden & Ratiu 1999), it follows that

the infinitesimal generator is Hamiltonian with an associated infinitesimally equivariant

momentum map JL : T ∗C → so(3)∗. We recall that the infinitesimal equivariance

property of JL is

〈JL(αQ(S)), ζ〉 = 〈P(S), ζM (Λ(S))〉 = 〈Λ � P, ζ〉, (B.14)

where Λ(S) belongs to M = SO(3) ⊂ C, P(S) belongs to T ∗ΛM and αQ(S) belongs

to T ∗Q(S)C. This property implies that the momentum map can be expressed in the

standard form

JL = Λ � P (B.15)

which is the statement that JL is a left infinitesimally equivariant momentum map

for the cotangent lifted left action ΨL on T ∗C. We now verify that the computed

expression for the momentum map is equivalent to the standard form, by pairing the

Clebsch relation for the body angular momentum δ`δΩ with its dual ζ ∈ so(3)

∗ to give

〈skew(Λ(S)PT (S)

), ζ〉 =

∫

B−Tr

(skew(Λ(S)PT (S))ζ

)dS

=

∫

B−Tr(PT (S)ζΛ(S)) dS

=

∫

BTr(PT (S)LζΛ(S)

)dS

= 〈Λ � P, ζ〉,

(B.16)

which shows that the computed and standard form for JL are equivalent.

201

The convective representation of the Lagrange-Poincare equations

The final step, given the form of the momentum map, is to derive the SO(3) reduced

equations of motion which are expressed relative to the rotating frames. Taking deriv-

atives with respect to time of δ`δV

∂

∂t(δ`

δV) = −Ω

δ`

δV− ΛTJ ′, (B.17)

and substituting J ′ = Λ′ δ`δΓ + Λ(δ`δΓ

)′into the above equation gives

∂

∂t(δ`

δV) = −Ω

δ`

δV− Ω

δ`

δΓ−

(δ`

δΓ

)′. (B.18)

Repeating this for δ`δΩ gives

∂

∂t

δ`

δΩ= skew(−ΩΛTP)︸︷︷︸

−ad∗Ωδ`δΩ

−skew(ΛT (J ′ + J Ω + P ⊗ V + J ⊗ (Γ + E3)

). (B.19)

Substituting(δ`

δΩ

)′= skew(ΩΛTJ ) + skew(ΛTJ ′) into the above equation then gives

∂

∂t

δ`

δΩ= −ad∗Ω

δ`

δΩ−

(δ`

δΩ

)′+ skew(ΩΛTJ − ΛTJ Ω)︸︷︷︸

ad∗Ω

δ`

δΩ

− skew

(δ`

δV⊗ V

)

︸︷︷︸=0

−skew

(δ`

δΓ⊗ (Γ + E3)

)

.

(B.20)

To summarise, the Euler-Lagrange equation for the material linear momentum in

the rotating frame is

∂

∂t

(δ`

δV

)

= −(Ω)ˆ ×δ`

δV− (Ω)ˆ ×N −N. (B.21)

Note that this equation is just a convenient form of the canonical Euler-Lagrange

equation

δ`

δφ=

d

dt

δ`

δφ+

d

dS

δ`

δφ′. (B.22)

The Lagrange-Poincare equation for the convective angular momentum is

∂

∂t

(δ`

δΩ

)

= −ad∗Ωδ`

δΩ−M ′ + ad∗

ΩM − skew (N ⊗ (Γ + E3)) , (B.23)

202

where M := δ`

δΩand N := δ`

δΓ . These equations are consistent with those presented

in (Krishnaprasad et al. 1988) and (Simo & Vu-Quoc 1988). The last term in the

convective angular momentum equation couples the director motion with the position

of the rod centroid and represents the shearing component of the motion. Indeed, when

the tangent vector of the rod centroid is aligned with the unit vector d3, Γ is zero and

the last term vanishes.

203

Appendix C

Additional aspects of the

Variational Free-Lagrange

Method

We outline two aspects of the VFL method which, though not essential to the material

presented in Chapter 4, respectively enhance the geometric description of the method

and applicability to computational shallow water models.

C.1 Canonical Formulation of the Variational Free-Lagrange

Method

This Section provides an alternative proof that the VFL method conserves energy by

formulating the corresponding semi-discrete canonical Hamiltonian equations on mo-

mentum phase space. Consider the smooth and invertible Legendre transformation from

the Lagrangian defined on the tangent bundle(velocity phase space) to the Hamiltonian

H defined on the cotangent bundle (momentum phase space)

H = 〈Pα, Xα〉 − L(Xα, Xα), (C.1)

where the pairing between elements of the tangent space and its dual 〈, 〉 is given by

〈Pα, Xα〉 =∑

α

Pα ∙ Xα, (C.2)

and Pα =∂L∂Xα. The Hamiltonian takes the form

H =∑

α

1

mα|Pα(t)|

2 + 2Rα ∙Pα − g∑

α

mα

(D(Xα, t)− 2bα

), (C.3)

204

and the canonical Hamiltonian particle equations are

Pα = −∂H∂Xα

,

Xα =∂H∂Pα

.

(C.4)

Evaluating the weak form of the derivatives of the Hamiltonian, analogously to the

derivation of the semi-discrete Euler-Lagrange equations given in Section 4.4, gives the

canonical Hamiltonian free-Lagrange equations

Pα = −gmαgrad(h(Xα, t)

)+ f0k ×Pα,

Xα =Pαmα= Uα.

(C.5)

By construction, these canonical Hamiltonian particle equations are dual to the Euler-

Lagrange equations defined in equation 4.25. Further, conservation of mass in each

cell surrounding a particle implies that the canonical particle equations, on momentum

phase space, are simply a constant scalar multiple of the semi-discrete Euler-Lagrange

equations on velocity phase space as given by equation 4.25.

Expressing the change in the Hamiltonian in terms of the change in momentum

space coordinatesDHDt=∑

α

∂H∂Xα

∙DXαDt

+∂H∂Pα

∙DPαDt

, (C.6)

it follows that particle trajectories on momentum phase space , described by the dual

of the Euler-Lagrange particle equations conserve the Hamiltonian

DHDt=∑

α

∂H∂Xα

∙∂H∂Pα

−∂H∂Pα

∙∂H∂Xα

= 0. (C.7)

This approach assumes that the Legendre transformation is smooth and invertible.

C.2 Representation on a Fixed Grid

It is often convenient to compute the velocity and layer depth fields on a fixed grid.

This Section briefly describes how this is performed in a locally mass conservative way

via a process referred to as ’rezoning’. The Eulerian (spatial) velocity field and layer

depth field are respectively computed over a fixed grid by interpolating the particle

velocities to a velocity grid and taking the average of the layer-depth in a fixed grid

cell from the weighted contribution of each Lagrangian cell intersecting the fixed cell.

The fixed grid cell-average layer depth is

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Di,j(t) =1

Ai,j

∑

α

∫

Vα(t)⋂Vi,j

D(x, t)dA, (C.8)

where Vi,j denotes the Eulerian grid cell with index i, j and Ai,j denotes its area.

The particle velocities at time tn are interpolated to the velocity grid according to

uni,j =∑

α

Unαφi,j(Xnα). (C.9)

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