Master Dynamique des Structures,

Materiaux et Systemes Couples

Space-time finite elements methodsin elastodynamics

Student:Jonathan Baptista

Supervisor:Dr. Mathias Legrand

Vibrations and Structural Dynamics LaboratoryFaculty of Engineering - McGill University

Ecole Centrale de Paris

Supmeca

2010 - 2011

Contents

Introduction 7McGill University and the Structural Dynamics and Vibration labora-

tory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Numerical methods in elastodynamics . . . . . . . . . . . . . . . . . 7

1 Wave propagation in elastic structures 91.1 Preliminaries: a simple spring-mass system . . . . . . . . . . . 91.2 Generic wave equation . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 Longitudinal wave equation for infinite rod . . . . . . . 101.2.2 Longitudinal wave equation for finite rod . . . . . . . . 11

1.3 Impact boundary condition . . . . . . . . . . . . . . . . . . . . 11

2 Newmark schemes for elastodynamics 152.1 Time integration schemes . . . . . . . . . . . . . . . . . . . . . . 15

2.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . 162.1.3 Numerical properties . . . . . . . . . . . . . . . . . . . . 16

2.2 Simple spring-mass system . . . . . . . . . . . . . . . . . . . . . 172.2.1 Central differences scheme . . . . . . . . . . . . . . . . . 172.2.2 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Rod with time dependent boundary conditions . . . . . . . . . 182.3.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 182.3.2 Numerical scheme and implementation . . . . . . . . . 192.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Continuous Galerkin Methods 253.1 Time finite elements method . . . . . . . . . . . . . . . . . . . . 25

3.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 253.1.2 Numerical scheme for linear elements . . . . . . . . . . 263.1.3 Numerical scheme for quadratic elements . . . . . . . . 273.1.4 Errors analysis . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Space-time finite element method . . . . . . . . . . . . . . . . . 293.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.2 Numerical scheme and implementation . . . . . . . . . 323.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Contents

4 Discontinuous Galerkin Methods 374.1 Derivation in time domain . . . . . . . . . . . . . . . . . . . . . 37

4.1.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 374.1.2 Numerical scheme and implementation . . . . . . . . . 394.1.3 Errors analysis . . . . . . . . . . . . . . . . . . . . . . . . 39

4.2 Space-time discontinuous finite element method . . . . . . . . 414.2.1 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 414.2.2 Numerical scheme and implementation . . . . . . . . . 444.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Conclusion 49

References 51

Resume

Ce rapport parcourt differentes methodes numeriques utiles au traitement dela propagation d’ondes dans un milieu homogene avec, en arriere plan, uneproblematique d’impact. Une attention particuliere est portee aux phenomenesnumeriques conduisant a une propagation d’ondes a vitesse infinie. Apres unbref rappel des solutions theoriques pour une barre encastree, les methodesd’integration temporelle directe de Newmark et les formulations en elementsfinis continus et discontinus de Galerkin exploitees sur des domaines espace-temps non separes sont comparees pour differents types de chargement endeplacement. Les perspectives d’utilisation de chacune de ces methodes con-cluent ce rapport. Entre autres, il est mis en evidence que l’utilisation deformulations faibles couplant espace et temps permet de rendre compte de lapropagation a vitesse finie de l’infomation dans la barre. Enfin, l’utilisationde formulations de Galerkin discontinues offre des perspectives interessantesdans le cas de systemes soumis a des chargements non reguliers.

Abstract

In this report, different numerical methods dedicated to the treatment of wavepropagation in homogeneous media are investigated, with an underlying inter-est to impact problems. A particular attention is paid to numerical phenomenaleading to wave propagation at infinite velocity. After a brief background ontheoretical solutions for a clamped homogeneous rod, direct time integrationNewmark methods and Galerkin continuous and discontinuous finite elementsmethods expressed in non-separated time-space domains are compared forvarious prescribed displacements. In the conclusion, perspectives for use ofeach of these methods is discussed. It is shown that coupled space-time fi-nite element formulations properly described the traveling of informationthrought the rod at finite velocity. It is highlighted that the use of discon-tinuous Galerkin formulations provides interesting properties in the case ofnon-smooth loading.

Introduction

McGill University and the Structural Dynamics andVibration laboratory

McGill University is based in Montreal, Canada, and is the oldest university ofthe city. Founded in 1821, it is nowadays one of the best-known institutionsof higher education in the country: ranked among the top 20 universitiesworldwide, it gathers over 36,000 students from more than 150 countries.

As for the research, McGill is ranked as one of the top research-intensiveuniversities of the country, working in both fundamental and applied sciences,it has made significant contributions not only in health sciences, but also inengineering, computer sciences and social sciences.

Within the Faculty of Engineering, the Structural Dynamics and VibrationLaboratory is directed by Prof. Christophe Pierre and Dr. Mathias Legrand.The research carried out in this laboratory mainly focuses on nonlinear vibra-tion, wave propagation, reduced-order models and linear and non-linear rotordynamics, and is closely related to industrial needs in aeronautics.

Numerical methods in elastodynamics

Traditional numerical techniques involving finite element procedures for time-dependent phenomena assume a solution separated in time and space. Forone-dimentional space problem, the sought solution takes the form u(x, t) =N (x)U (t) where u(x, t) stands for the displacement of a point x of the structureat time t. Such procedures are widely used in practice and well understoodnumerically. Through the derivation of finite elements in space, the originalpartial differential equations to be solved reduce to a system of ordinary differ-ential equations in time which are, in turn, discretized by finite difference-likenumerical schemes. The aforementioned technique leads to numerical diffi-culties because information of interest may propagate through the structuresat infinite velocity. It is thought that non-separated space-time formulationscould help circumventing this class of numerical difficulties. Such formulationsrequire space-time domains of four dimensions when full three-dimensionalstructures are involved together with classical spatial boundary conditions andinitial and final, if necessary, conditions.

Usually mechanical systems are macroscopically described by equations ofcontinuum mechanics while discontnuities or sharp gradients in the sought

8 Introduction

solution need a smaller scale to be accurately captured. In this line, severalnumerical approaches exist such as the multi-scale LArge Time INcrement(LATIN) method [4], the meshless Partition of Unity Finite Element Method [21]and the eXtented Finite Element Method [24].

This work focuses on the extension of classical finite element methods toperiodic time-dependent systems with a particular emphasis on the space-timecoupling and the treatment of potential jumps in the sought solution. FirstNewmark methods assuming separated space-time formulations are inves-tigated. Then, non-separated space-time formulations and dedicated finiteelements methods are introduced. Finally space-time discontinuous finiteelement methods are presented to address configurations involving disconti-nuities in the responses the mechanical systems of interest.

Chapter 1

Wave propagation in elasticstructures

This chapter introduces the generic governing equations of motion and theo-retical solutions for two physical systems of interest, solutions later employedfor comparison purposes with the developped numerical strategies:

— a 1 degree-of-freedom (dof) spring-mass system;— a 1 dimensional (1D) homogeneous elastic rod undergoing longitudinal

travelling waves.

The aim of this chapter is also to present the notations and quantites of interestthat will be later used throughout the document. It illustrates the difficultiesimplied by the propagation of information in elastic structures on two aspects:

— the smoothness of the travelling wave shape;— the finite velocity of the travelling wave.

1.1 Preliminaries: a simple spring-mass system

First let us consider a one dof system consisting of a mass m and a spring k asillustrated in figure 1.1. This system is governed by the following equation of

mk

u(t)

f (t)

Figure 1.1: Spring mass system

motion together with the prescribed initial conditions:

∀t ∈]0;+∞[, mu(t) + ku(t) = f (t)u(0) = u0

u(0) = v0

(1.1)

10 1. Wave propagation in elastic structures

When f (t) = 0, the exact solution is given by:

∀t ∈ [0;+∞[, u(t) = u0 cos(ωt) +v0

ωsin(ωt) with ω =

√km

(1.2)

Obviously, no wave can propagate in such a system. It nevertheless offers aquite simple analytical solution very useful for comparison with numericalresults.

1.2 Generic wave equation

1.2.1 Longitudinal wave equation for infinite rod

In the context of the theory of small perturbations theory and when the externalforcing g as well as the body force fv are neglected, the behaviour of the elasticrod is completely described by the following equations:

∀(x, t) ∈R×]0;+∞[,∂2u

∂t2(x, t)− c2∂

2u

∂x2 (x, t) = 0 with c =

√Eρ

∀x ∈R, u(x,0) = u0(x)

∀x ∈R, ∂u∂t

(x,0) = v0(x)

(1.3)

The explicit solution of this system of equations is known to be [6, p. 10]:

u(x, t) =12

(u0(x − ct) +u0(x+ ct)

)+

12c

∫ x+ct

x−ctv0(s)ds (1.4)

Assuming v0 = 0, the solution is the superimposition of two harmonic wavesmoving in opposite directions at velocity c. Figure 1.2 represents solution (1.4)for an initial window u0 by families of lines of equation x + ct = cst andx − ct = cst starting from the initial condition u0 in the space-time plane andcalled characteristic lines.

0

t

x1

1/2 1/2

Figure 1.2: Solution of the wave equation for an infinite rod

1.3 Impact boundary condition 11

1.2.2 Longitudinal wave equation for finite rod

The second investigated system is a one-dimensional homogeneous elastic roddepicted in figure 1.3 and undergoing an external forcing load g(t) at the freeend and a body force ~fv = fv · ~x. The rod is of lenght L and cross-section areaS, with a Young modulus E and a mass density ρ. It is clamped at x = 0. The

u(x, t)

g(t)

~x0 L

Figure 1.3: One-dimensional elastic rod

space domain is restricted to the interval [0;L] such that (1.3) becomes:

∀(x, t) ∈ [0,L]×]0;+∞[,∂2u

∂t2(x, t)− c2∂

2u

∂x2 (x, t) = 0

∀t ∈ [0;+∞[, u(0, t) = 0

∀t ∈ [0;+∞[,∂u∂x

(L,t) = 0

∀x ∈ [0,L], u(x,0) = u0(x)

∀x ∈ [0,L],∂u∂t

(x,0) = v0(x)

(1.5)

Assuming v0(x) = 0, and knowing that the boundary conditions yield anti-symetry of solution (1.4) in x = 0 and symetry in x = L, a periodic solution isfound: a portion of it is illustrated in figure 1.4 which shows the reflexionsundergone by the initial window u0(x).

t

x

0 L

td

ctd

−1/201/2

Figure 1.4: Solution of the wave equation for a finite rod

1.3 Impact boundary condition

In order to approximate a contact occurring at the free end of the rod, adisplacement impulse is introduced at this point. This is reflected by a ramp

12 1. Wave propagation in elastic structures

entry with a finite slope as depicted in figure 1.5(a). When the slope tends toinfinity as displayed in figure 1.5(b), very sharp gradients in the solution maybe expected. Of high interest is the development of numerical strategies ableto address such non-differentiable entries.

It is assumed that the solution has two phases:

1. A first phase where displacement is prescribed: u(L,t) = uc(t), corre-sponding to impact/contact conditions, in red.

2. A second phase where the prescribed displacement is released: displace-ment of the tip of the rod is then free, in blue.

u(L,t)

t

1 2

Tsts(a) with a finite slope

u(L,t)

t

1 2

Tsts(b) with a infinite slope

Figure 1.5: Ramp entry

Such conditions are not exactly a displacement window as in section 1.2.2.These conditions correspond to a ramp followed by a constant step of dis-placement which then suddenly vanish. This case was studied in [19] and therespective graphical solution is shown in [22, p. 39]. The numerical solutionfor a finite slope is plotted in 3D in figure 2.2.

1.3 Impact boundary condition 13

−1/201/2

t

x0 L

Figure 1.6: Solution of the wave equation to a ramp entry with an infinite slopefor a finite rod

Chapter 2

Newmark schemes forelastodynamics

The Newmark family of time-integration methods is mainly employed forsolving systems of, potentially nonlinear, ordinary differential equations. Theseschemes are easy to implement and sufficiently versatile to address a broadclass of problems by seeking for a solution separated in time and space. Thischapter is dedicated to their description in the context previously discussed.They will later be compared to more sophisticated solution methods.

2.1 Time integration schemes

2.1.1 Formulation

Consider the following system of second order differential equations alongwith two respective initial conditions in displacement and velocity:

∀t ∈]0;+∞[, Mu(t) + Cu(t) + Ku(t) = f(t)u(0) = u0

u(0) = v0

(2.1)

These are the general settings used throughout the present report. QuantitiesM, C and K are respectively the mass, the viscuous damping, and the stiff-ness matrices, obtained through the usual finite element approach detailedin section 2.3.1. Vector f stores the discretized external force and u is thedisplacement vector.

By assuming a time discretization such that un = u(tn) = u(n∆t) where ∆tstands as the constant time-step, equation (2.1) reads:

∀n ∈N+∗, Man + Cvn + Kun = fnu(0) = u0

u(0) = v0

(2.2)

where the notations an = un and vn = un are used. By exploiting Taylor expan-sion theorem, displacement un, velocity vn and acceleration an can be expressed

16 2. Newmark schemes for elastodynamics

with respect to their first time derivatives for each time step as follows:

un+1 = un +∆tvn +∆t2

2an+2β

vn+1 = vn +∆tan+γ

an+2β = (1− 2β)an + 2βan+1

an+γ = (1−γ)an +γan+1

(2.3)

where β and γ are user-defined constants as detailed in section 2.1.3.

2.1.2 Implementation

Equations (2.2) and (2.3) are combined so that time-integration algorithm 1 canbe implemented. It involves quantities upn+1 and vpn+1 which are the predictorterms depending on the previous time step n [12].

Algorithm 1 Newmark time integration scheme

Require: γ , βRequire: Initial conditions: u0, v0

a0← (M +γ∆tC + β∆t2K)−1(f0 −Cv0 −Ku0)for n = 0 to N do

upn+1← un +∆tvn + ∆t22 (1− 2β)an

vpn+1← vn +∆t(1−γ)anan+1← (M +γ∆tC + β∆t2K)−1(fn+1 −Cvpn+1 −Kupn+1)un+1← upn+1 +∆t2βan+1

vn+1← vpn+1 +γ∆tan+1end for

Any system of second-order differential equation can be recasted in a systemof first-order differential equation. When generalized to the nth time stepthrough an amplification matrix A, the displacement and velocity at time tnare related to the initial displacement and velocity as follows:

∀n ∈N+∗,(unvn

)= An

(u0v0

)(2.4)

An eigenvalue analysis of matrix A provides information about the stability ofthe method. In order to be stable, the magnitude of the eigenvalues of A mustbe smaller than unity.

2.1.3 Numerical properties

As already mentioned, Newmark schemes depend on coefficients γ and β.These two coefficients drive the numerical properties of the underlying solutionmethod: it can be either explicit or implicit and conditionally or unconditionallystable. The most common combinations are summarized in table 2.1. Stabilityconditions are well define in literature [7] for this type of numerical schemes.

2.2 Simple spring-mass system 17

Method γ β Stability

Central Differences 1/2 0 explicit and conditionally stable

Fox-Goodwin 1/2 1/12 implicit and conditionally stable

Linear acceleration 1/2 1/6 implicit and conditionally stable

Average acceleration 1/2 1/4 implicit and unconditionally stable

Table 2.1: Parameters γ and β in Newmark Schemes

Newmark schemes are unstable for γ ≤ 1/2, unconditionally stable for 1/2 ≤γ ≤ 2β and conditionally stable if 1/2 ≤ γ and 2β ≤ γ (under a conditioncombining time step and spatial mesh size, see 3.2.3). Moreover for γ = 1/2,Newmark method is second order accurate and this is the reason why thisvalue is often assigned to γ . When β = 0, the scheme is explicit.

Integration Method order Consider ψ, the numerical solution such thatψn = ψ(tn) and y, the exact solution known and given in 1.1 for a spring-masssystem. One can define δn the local error over one step of the integrationmethod by:

∀n ∈ ~0,N δn = ψn − y(tn) (2.5)

and δmax the global error by:

δmax = maxn∈~0,N

(δn) = maxn∈~0,N

(ψ(tn)− y(tn)

)(2.6)

The method is said to be of order p if δn = O(∆tp+1) or if δmax = O(∆tp) when∆t→ 0.

2.2 Simple spring-mass system

2.2.1 Central differences scheme

Equation (1.1) is now investigated through a finite central difference integrationapproach:

an +ω2un = 0 (2.7)

One can refer to equation (2.2) by considering scalars m and k as the 1-by-1matrices M and K, and C = 0. On a finite time interval [0,T ], Newmark methodwith the proper β and γ parameters yields:

an =un+1 − 2un +un−1

∆t2(2.8)

and the discretized equation to be solved reads:

∀n ∈ ~0,N, bun+1 + 2aun + bun−1 = 0 (2.9)

18 2. Newmark schemes for elastodynamics

with 2a = 1/∆t2 and b =ω2 − 2/∆t2, which is equivalent to the following linearsystem:

1 0 · · · · · · 0

−1 1 . . ....

b 2a b. . .

......

. . . . . . . . . 00 · · · b 2a b

u1

u2...

uN+1

=

u0

∆tv0...

fN+1

(2.10)

Solution method for linear system (2.10) is implemented in a new algorithmequivalent to algorithm 1 and detailed in section 3.1.2.

2.2.2 Error analysis

The spring-mass system used for the calculation over a period T = 10 s has thefollowing properties: m = 10 kg, k = 10 N/m, u0 = 0.5 m and v0 = −0.5 m/s.Respective convergence rate of various Newmark schemes is depicted in fig-ure 2.1. One can notice that implicite schemes provide the best results and

10−3 10−2 10−1 10010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

∆t

Errorδ m

ax

AACDLAFG

Figure 2.1: Convergence of the solution for Newmark schemes

particularly the Fox-Goodwin Newmark scheme (FG). Since methods are allsecond order accurate, their convergence rate features the same exponentiation:δmax ∼ ∆t2 when ∆t→ 0.

2.3 Rod with time dependent boundary conditions

2.3.1 Formulation

The formulation for a n-DoF system is a little bit different since boundaryconditions in space and time have to be dealt with.

2.3 Rod with time dependent boundary conditions 19

Consider the rod described in 1.2 and whose displacement at its end isprescribed during a period Tc and then released. The governing equation (1.5)are modified as follows:

∀(x, t) ∈ [0,L]×]0;+∞[, ES∂2u

∂x2 (x, t) + fv = ρS∂2u

∂t2(x, t)

∀t ∈ [0;+∞[, u(0, t) = 0∀t ∈ [0;Tc[, u(L,t) = uc(t)

∀t ∈ [Tc;+∞[, E∂u∂x

(L,t) = g(t)

∀x ∈ [0,L], u(x,0) = u0(x)

∀x ∈ [0,L],∂u∂t

(x,0) = v0(x)

(2.11)

Since displacement is released at the end of the contact period [0;Tc[, thepotential external forcing acting at x = L is zero, thus g = 0. In order to simplifycalculations u0, v0 and fv are all nill as well.

Then, the use of Newmark time integration scheme assumes a separtionbetween time and space in the sought solution 2.12. Space integration isrealized through the Finite Element Method (FEM) while integration over timeis performed through a numerical scheme detailed in section 2.1. Therefore,integration in space yields solutions of the form:

∀(x, t) ∈ [0,L]×]0;+∞[, u(x, t) =Ns∑i=1

ui(t)wi(x)

∀t ∈ [0;+∞[, u(0, t) = 0∀t ∈ [0;Tc[, u(L,t) = uc(t)

(2.12)

The corresponding weak formulation is obtained by reporting u(x, t) from (2.12)into the first equation of (2.11) and premultiplying by admissible trial func-tions ψi ∈ Ca = ψ ∈ H1([0,L]), ψ(0) = 0 such that: ∀j ∈ ~1,Ns,∀t ∈]0;+∞[,

ESNs∑i=1

∫ L

0

∂wi∂x

∂ψj∂x

dx

ui + ρSNs∑i=1

∫ L

0wiψjdx

∂2ui∂t2

= Sg(t)ψj(L) (2.13)

Remark 1 (Basis function in space). The condition u(L,t) = uc is not accountedfor in Ca because this condition is time dependent while basis functions are not.

2.3.2 Numerical scheme and implementation

Once again, one can refer to equation (2.2) by considering Ns ×Ns matrices Mand K define as follows: ∀(i, j) ∈ ~1,Ns2,

Kij = ES∫ L

0

∂wi∂x

∂ψj∂x

dx Mij = ρS∫ L

0wiψjdx (2.14)

20 2. Newmark schemes for elastodynamics

These matrices are symmetric if ψi = wi . Accordingly, linear trial functions inspace are chosen. This formulation leads to solve equation (2.2) taking intoaccount the clamped condition in x = 0 and the enforced displacement in x = Lby considering the following displacement vector un at time tn:

∀n ∈N+∗, un =

u1n = un(0) = 0

...uin...

uNsn = un(L) = uc

(2.15)

The first component of vectors un, vn and an such as the first line and the firstcolumn of matrices M, C and K have to be removed from system (2.2) which isnow of size Ns − 1.

Vector u is then partitioned in u1 and u2, so are v and a. Equation (2.2)writes ∀n ∈N+∗:[

M11 M12M21 M22

](a1na2n

)+[C11 C12C21 C22

](v1nv2n

)+[K11 K12K21 K22

](u1nu2n

)=

(f1nf2n

)(2.16)

The first block in equation (2.16) provides the system behaviour while thesecond one provides the resultant force f2 due to the enforced displacement.Since the displacement is only enforced over a period Tc, algorithm 2 has to beimplemented.

Algorithm 2 Newmark scheme with enforced/free displacementRequire: γ , β, uc and Initial conditions: u0, v0

u2← uc; v2← u2; a2← v2a10← (M11 +γ∆tC11 +β∆t2K11)−1(f10−C11v10−K11u10−M12a20−C12v20−K12u20)Enforced displacement phasefor n = 0 to NTc do

up1n+1← u1n +∆tv1n + ∆t22 (1− 2β)a1n

vp1n+1← v1n +∆t(1−γ)a1n

a1n+1← (M11+γ∆tC11+β∆t2K11)−1(f1n+1−C11vp1n+1−K11up1n+1−M12a2n−C12v2n−K12u2n)u1n+1← up1n+1 +∆t2βa1n+1

v1n+1← vp1n+1 +γ∆ta1n+1end forTransition: uNTc ← (u1NTc

;u2NTc); vNTc ← (v1NTc

;v2NTc); aNTc ← (a1NTc

;a2NTc)

Free displacement phase:for n =NTc to N do

upn+1← un +∆tvn + ∆t22 (1− 2β)an

vpn+1← vn +∆t(1−γ)anan+1← (M +γ∆tC + β∆t2K)−1(fn+1 −Cvpn+1 −Kupn+1)un+1← upn+1 +∆t2βan+1

vn+1← vpn+1 +γ∆tan+1end for

2.3 Rod with time dependent boundary conditions 21

2.3.3 Results

For the calculation, the structural damping C as well as g are neglected andv0 = 0 is prescribed. The enforced displacement is a ramp at the free end of therod x = L at ts = 0.3 s (see figure 1.5(a)). Then displacement and velocity areobserved at x = L/2 with ∆t = 10−3 s and ∆x = L/30 m. In figure 2.2, reponse

00.5

11.5

22.5

3

0

0.2

0.4

0.6

0.8

1

−1.5

−1

−0.5

0

0.5

1

time

space

Figure 2.2: Solution for a finite ramp entry

to a ramp entry with ∆t = 10−2 s and ∆x = L30 is plotted. Later in this report,

for more graphical convenience, results for the 1D case will be provided ascross-sections of this type of surfaces.

In section 2.1.3, implicit and explicit methods were introduced. Implicitmethods usually require more demanding computational effort since solutionat time tn+1 involves state of the sytem at time tn and tn+1. Nonetheless, theyare usefull when explicit methods require impractically small time steps.

In the case of an explicit Newmark scheme without structural damping(C = 0 and β = 0), the acceleration is computed as follows:

an+1←M−1(fn+1 −Kupn+1) (2.17)

However, the mass matrix M is not diagonal and propagates the informationwith an unbounded velocity into the discrete system. This inconsistency stemsfrom the assumed separation of variable and can be avoided through lump-ing only. This is most of the time not a problem but can lead to numericaldifficulties in very specific configurations.

In order to avoid this phenomena, one can choose to diagonalize the massmatrix. The effects of the so called lumped mass matrix [23] on the displace-ment are introduced in figure 2.3. It can be seen, for the first ramp at time

22 2. Newmark schemes for elastodynamics

t = ts + L2c that unlike consistant mass matrix, displacement at x = L/2 remains

nil until the propagating wave reaches this point. It means that no informationtravels along the rod with an infinite speed and the respective displacementis in agreement with the theoretical solution exposed in 1.3. The lumped

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

Consistent mass

Lumped mass

Figure 2.3: Displacement with central differences for a ramp entry at x = L/2

mass matrix also tends to soften the solution as illustrated on the velocity infigure 2.4. Also, high frequency oscillations around the theoretical solution

0 0.5 1 1.5 2 2.5 3

−10

−5

0

5

10

Time (s)

Velocity(m

/s)

Consistent mass

Lumped mass

Figure 2.4: Velocity with central differences for a ramp entry at x = L/2

can be noticed. A way to partially dissipate them is to introduce a structural

2.3 Rod with time dependent boundary conditions 23

damping which change the structural properties or to introduce a numericaldamping as proposed by the generalized-α method [8, 9]. Such formulationsmay have undesirable effects and should sometimes be avoided.

Chapter 3

Continuous Galerkin Methods

A popular time integration scheme is detailed in chapter 2. The use of finiteelements in space as well as in time is now of interest. This chapter is devotedto the extension of finite elements in space to the time domain by continuousspace-time fonctions. Few works have been reported on the Space Time Con-tinuous Finite Elements Method (STCFEM) also called Continuous GalerkinMethod (CGM). The reader will find further information in [1, 2, 16, 18].

3.1 Time finite elements method

3.1.1 Formulation

Consider the 1DoF spring-mass system described in 1.1 with a time intervalrestricted to [0,T ]:

∀t ∈]0;T ], mu(t) + ku(t) = f (t)u(0) = u0

u(0) = v0

(3.1)

From (3.1), the weak formulation for the unknown u is:

Find u ∈ Ca s.t.∫ T

0(ku +mu − f )ψdt = 0 ∀ψ ∈ C (3.2)

with Ca = w ∈H1([0;T ]), w(0) = u0 and C =H1([0;T ]).An integration by part of the acceleration term in equation (3.2) and assum-

ing no discontinuities at t = 0 and t = T yields:

Find u ∈ Ca∀ψ ∈ C,

∫ T

0(kuψ −muψ)dt +mu(T )ψ(T ) =

∫ T

0f ψdt +mv0ψ(0)

(3.3)

A regular time mesh of interval [0,T ] consisting in N subintervals [tn−1, tn]for n ∈ ~1,N is considered to interpolate the displacement with linear basisfunctions in time wi(t):

∀t ∈ [0;T ], u(t) =N∑i=1

uiwi(t) (3.4)

26 3. Continuous Galerkin Methods

The classical FEM is used to calculate the respective matrices and vectors:∀(i, j) ∈ ~1,N2,

Mij =m∫ T

0

dwjdt

dψidt

dt ; Kij = k∫ T

0wjψi dt ; Λij =mwj(T )ψi(T )

fi = k∫ T

0f ψi dt ; λ0i =mv0ψi(0)

(3.5)

Similarly to what was done in section 2.3.2, we choose ψi(t) = wi(t) in order tohave symmetric matrices.

Remark 2 (Assembled matrices). M and K are the assembled mass and springmatrices, respectively. They have been built from the elementary mass and springmatrices.

Other formulations are possible using the velocity v, strain or stress asunknown. According to [1], formulations limited to the displacement fieldgive the best results. A two-field formulation using u and v for the continuousGalerkin method is described in [15]. Since we are mainly interested in thedisplacement field, the single-field displacement formulation was selected.

3.1.2 Numerical scheme for linear elements

Combining equations (3.3) and (3.5) yields the following discrete system ofequations (3.6):

(K−M +Λ)u = f +λ0 ⇔ Ru = F (3.6)

where u is the discrete displacement vector consisting of displacement valuesat all time steps, R is the resultant matrix, and F stands as the generalized forcevector. The vector u is of size N + 1.

By construction, using linear functions and constant time steps, system (3.6)has the following structure:

a b 0 · · · 0

b 2a b...

0 . . . . . . . . . 0... b 2a b0 · · · 0 λ1 λ2

u1

...

uN+1

=

f1 +mv0

...

FN+1

(3.7)

Since the first coordinates u1 = u0 is known, the first line of system 3.6 leadsto:

u2 =f1 +mv0 − au1

b(3.8)

The band structure of R allows for a simple iterative scheme to calculatethe (n+1)th component of u. The last line of R is not useful when there is

3.1 Time finite elements method 27

no discontinuities at the final intant T . Accordingly, equation (3.7) can bereorganized as follows:

1 0 · · · · · · 0

0 b. . .

...

b 2a b. . .

....... . . . . . . . . 0

0 · · · b 2a b

u1

u2...

uN+1

=

u0

f1 +mv0 − au0...

FN+1

(3.9)

System (3.9) is an explicit scheme similar to the central differences scheme (2.10)because a constant time-step is used. Instead of inverting very the large matrixR which becomes even larger when spatial dimension increases, algorithm 3 ispreferred.

Algorithm 3 Continuous Galerkin with linear elementsRequire: a, b and fRequire: Initial conditions: u0, v0u1← u0

u2← f1+mv0−au0b

for n = 3 to N doun← Fn−2aun−1−bun−2

bend for

3.1.3 Numerical scheme for quadratic elements

For quadratic elements, assuming a constant time step ∆t, one can use thethree following basis functions:

w1 =(t − tn)(t − tn+1)

2∆t2; w2 = 1− (t − tn)2

∆t2; w3 =

(t − tn)(t − tn−1)2∆t2

(3.10)

The respective elementary mass and stiffness matrices are defined as follows:

Ke =k∆t15

4 2 −12 16 2−1 2 4

; Me =m

6∆t

−7 −8 −2−4 −8 4−2 4 −7

(3.11)

By plugging the elementary matrices (3.11) into equation (3.6), system (3.7)

28 3. Continuous Galerkin Methods

h8cm

ttn tn+1tn−1

1

∆t

Figure 3.1: Quadratic basis functions

becomes:

a b c 0 · · · · · · 0

b d b 0 0

c b 2a b c 0...

0 0 b d b 0 0...

0 c b 2a b c. . .

.... . . . . . . . . . . . . . . . . . 0

... 0 c b 2a b c

0 0 b d b

0 · · · · · · 0 c b a

u1

u2

...

...

uN+1

=

f1 +mv0

...

...

...

FN+1

(3.12)

Since u1 = u0, then one can implement algorithm 4, where:

p =(ab

), H =

(c b 2a0 0 b

)and A =

(b cd b

)

3.1.4 Errors analysis

The considered spring-mass system is the same as the one described in 2.2.2and the time step is constant. The number of degrees-of-freedom is the samefor quadratic and linear models. In figure 3.2, the error for the continuousGalerkin scheme is very closed to the one of the central differences schemedue to the similarities between systems (3.9) and (2.10). In the case of a nonconstant time step, the computation for the Galerkin scheme is much longerthan for the Newmark solution strategy.

One would note that a quadratic approximation of the displacement leadsto a piecewise constant acceleration. The linear acceleration scheme (LA) uses

3.2 Space-time finite element method 29

Algorithm 4 Continuous Galerkin for quadratic elementsRequire: A, H, and pRequire: Initial conditions: u0, v0u1← u0y1← u1 ·p(u2u3

)←A−1

(f1 +mv0f2

)− y1

for n = 3 to N − 2 do

yn←H ·un−2un−1un

(un+1un+2

)←A−1

( FnFn+1

)− yn

end for

10−3 10−2 10−1 10010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

∆t

Errorδ m

ax

CD Newmark

CG

CG quad

FG Newmark

Figure 3.2: Convergence for a continuous Garlerkin scheme

a linear approximation of the acceleration. Nonetheless, while second orderaccurate, the CGM with quadratic elements is about ten times more accuratethan the LA method.

3.2 Space-time finite element method

3.2.1 Formulation

In the previous sections, it was shown for a single degree-of-freedom systemcontinuous Galerkin formulations and Newmar integration schemes sharethe same properties. Consider the one-dimensional rod described by equa-tion (2.11) for a finite time interval [0;T ] where 0 ≤ Tc ≤ T . Boundary andinitial conditions are now seen as boundary conditions over the space-time

30 3. Continuous Galerkin Methods

t

x0 L

T

Ω0

ΩT

Γ0

Γ1

Γ2

∆t

t = Tc

1

Ns2 3 5

Ns + 1

64

23

2Ns − 1

1 S1

S2

S3

Figure 3.3: Meshed space time domain

domain S = Ω× Γ = [0,L]× [0,T ]. Using the displacement field as unknown,the following weak formulation is derived:

Find u ∈ Ca∀ψ ∈ C,

∫ T

0

∫ L

0

(ES∂2u

∂x2 − ρS∂2u

∂t2+ fv

)ψdxdt = 0

(3.13)

Through integrations by parts, second-order derivatives in time or space willbecome first-order derivatives so that linear basis functions are used (cf. re-mark 3). Since functions are chosen to be continuous over the space-timedomain S , integration by part has the following form:∫ T

0

∫ L

0ES∂2u

∂x2ψdxdt = −∫ T

0

∫ L

0ES∂u∂x

∂ψ

∂xdxdt +

∫ T

0ES

[∂u∂xψ]L0

dt∫ T

0

∫ L

0ρS∂2u

∂t2ψdxdt = −

∫ T

0

∫ L

0ρS∂u∂t

∂ψ

∂tdxdt +

∫ L

0ρS

[∂u∂tψ]T0

dx

(3.14)

which leads to the following weak formulation for the rod:

Find u ∈ Ca, ∀ψ ∈ C,"S

(ES∂u∂x

∂ψ

∂x− ρS ∂u

∂t

∂ψ

∂t

)dxdt−

∫ T

0Sgψ(L)dt ="Sfvψdxdt +

∫ L

0ρSv0ψ(x,0)dx

(3.15)

with

Ca = w ∈H1(S), w = u0 on Ω0, w = 0 on Γ0 and w = uc on Γ1C = ψ ∈H1(S),ψ = 0 on ΩT and ψ = 0 on Γ0

3.2 Space-time finite element method 31

Remark 3 (Accuracy of the solution). Since linear functions on S have beenchosen to approximate the displacement, velocity is constant over an finite elementand acceleration is zero.

Due to the space time mesh depicted in figure 3.3, an approximated solutionof the form:

∀(x, t) ∈ S , u(x, t) =Ndof∑i=1

uiwi(x, t) (3.16)

is employed. Finite elements are triangles on the 2D space-time map. Mesh isregular in order to facilitate the computation and the time length of a slab ∆t isthus constant. It is also worth to notice that remeshing is easier with simplicesthan with multiplices [1, 18].

Once again ψi(x, t) = wi(x, t) so that one gets symmetric matrices. Sinceg = 0 during the free displacement phase, one can consider the followingglobal vectors and global matrices over the full space time domain for theimplementation: ∀(i, j) ∈ ~1,Ndof

2,

Mij ="SρS∂wj∂t

∂ψi∂t

dxdt ; Kij ="SES∂wj∂x

∂ψi∂x

dxdt

fi ="Sfvψi dxdt ; λ0i =

∫ L

0ρSv0(x)ψi(x,0)dx

(3.17)

These global matrices and vectors are large and should not be numericallyimplemented as such. The solution method processes iteratively for each slabinstead.

Remark 4 (Line of integration). In the case of an non laminated and regular meshas the one illustrated in figure 3.3, slabs are not defined in the same way. One canuse a space-time integration line as proposed in [1, p.77].

Triangle based formulation Considering regular and laminated meshes aswell as the numbering of nodes in figure 3.3 and a constant ∆t (see remark ??),elementary matrices are built from triangular elements (3.18) and then assem-bled for each time slabs Sn. Such that ∀k ∈ ~1,Ne,∀(i, j, ) ∈ ~1,32,

Mkij =∫ 1−xr

0

∫ 1

0ρS

(∂wri∂xr

∂xr∂t

+∂wri∂tr

∂tr∂t

)(∂wrj∂xr

∂xr∂t

+∂wrj∂tr

∂tr∂t

)Jφk dxr dtr

Kkij =∫ 1−xr

0

∫ 1

0ES

(∂wri∂xr

∂xr∂x

+∂wri∂tr

∂tr∂x

)(∂wrj∂xr

∂xr∂x

+∂wrj∂tr

∂tr∂x

)Jφk dxr dtr

with wr1 = 1− xr − tr , wr2 = xr , and wr3 = tr (3.18)

whereNe is the number of elements per slab and Jφk is the Jacobian determinantof the transformation from Tr (the triangular reference element) to Tk (the realelement):

φk :(

Tr → Tk(xr , tr) 7→ (x, t)

)(3.19)

32 3. Continuous Galerkin Methods

Quadrangle based formulation Using the same regular and laminated meshas previously but with rectangular elements, the basis functions (4.28) shouldbe considered and reorganized as: ∀i ∈ ~1,4, wi(x, t) = fi(x)gi(t). Accordingly,mass and stiffness elementary matrices from (3.17) read:

Mij = ρS∫ L

0fifj dx

∫ T

0

∂gi∂t

∂gj∂t

dt

Kij = ES∫ T

0gigj dt

∫ L

0

∂fi∂x

∂fj∂x

dx

(3.20)

and the analytical expressions for the elementary matrices are:

Ke =∆t

6∆x

2 −2 −1 1−2 2 1 −1−1 1 1 −11 −1 −1 1

; Me =∆x6∆t

2 1 −1 −21 2 −2 −1−1 −2 2 1−1 −1 1 2

(3.21)

3.2.2 Numerical scheme and implementation

Assembling matrices M and K and assuming fv = 0, the following linear systemto be solved is obtained:

(K−M)u = f +λ0 ⇔ Ru = F (3.22)

which is the matrix counterpart of (3.7):

A B 0 · · · 0

Bt 2A B...

0 . . . . . . . . . 0... Bt 2A B0 · · · 0 Bt 2A

u1

...

uN+1

=

f1 +mv0

...

FN+1

(3.23)

where Ns denotes the number of nodes at the interface between Sn−1 and Sn,assumed to be constant, andN denotes the number of time slabs. IfND =N×Nsthe size of u and F is ND , the size of R is ND ×ND , the size of A and B is Ns×Ns,and the size un is Ns.

Similarly as in section 3.1.2, the very large matrix R is replaced smallermatrices A and B, and an iterative numerical scheme is implemented, whichwould lead to the following procedure ∀n ∈ ~3,N:

∀n ∈ ~3,N, Bun+1 = B−1(Fn −Btun−1 − 2Aun) (3.24)

For each time step n ∈ ~1,N, the first and the last components of un are known(un1 = 0 and unNs = u2

n = uc, see equation (2.15)). Accordingly, matrices A andB can be reduced to size (Ns − 1)× (Ns − 1), by discarding the first row and thefirst column corresponding to the clamped DoF. Then, the method exposedin 2.3.2 can be used with matrices (3.25) where B22, A22 and u2

n are scalars.

B =[B11 B12B21 B22

], A =

[A11 A12A21 A22

](3.25)

3.2 Space-time finite element method 33

and

∀n ∈ ~3,N, un =(u1n

u2n

), Fn =

(F1n

F2n

)(3.26)

Therefore, algorithm 5 can be implemented.

Algorithm 5 CG time integration scheme for enforced/free displacementRequire: A, B, ucRequire: Initial conditions: u0, v0

u1← u0u1

2← B−111(F1

1 −A11u11 −A12u2

1 −B12u22)

u22← uc(t2)

Enforced displacement phase:for n = 3 to NTc do

u2n← uc(tn)

u1n← B−1

11(F1n−1 − 2A11u1

n−1 − 2A12u2n−1 −B12u2

n −Bt11u1n−2 −Bt12u2

n−2)end forTransition:uNTc ← (u1

NTc;u2NTc

)Free displacement phase:for n =NTc to N do

un← B−1(Fn−1 − 2Aun−1 −Btun−2)end for

3.2.3 Results

The investigated rod has the mechanical properties defined in section 2.3.3with fv = 0 and v0 = 0. Are compared in figure 3.4 and 3.5 the central dif-ference Newmark scheme with lumped and consistent mass, respectively, tocontinuous Galerkin scheme with triangular or rectangular elements. It is wor-thy to note that Garlerkin and central difference scheme are equivalent whenthe lumped mass option is selected. Also the results given by the Newmarkscheme with consistent mass are identical to those of a continuous Galerkinscheme with rectangular elements. Convergence properties look identical fora continuous Galerkin approach with triangular or rectangular elements andcentral difference technique with a lumped mass. Also, Newmark schemeswith lumped mass (see 2.1.3) and Galerkin scheme converge only if c∆t < ∆x(CFL condition).

Figure 3.5 shows that with rectangular elements, information progates inthe rod with unbounded velocity. This yields spurious oscillations before theprescribed displacement command. This phenomena is not observable whentriangular elements are chosen. Displacement at a point of a rod is rigorouslyzero before the initial travelling wave reaches this point.

High-frequency oscillations keep existing as opposed to the theoreticalsolution. These oscillations only depend on the mesh size in space ∆x. Thenumber of oscillations increases and their amplitude decreases with ∆x asillustrated in figure 3.6.

34 3. Continuous Galerkin Methods

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

CG triangle

Lumped CDNM

CG rectangle

Figure 3.4: Response at x = L2 to a ramp entry for ∆t = 10−2 s and ∆x = L

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

Rectangular CG

Consistent mass CDN

Triangular CG

Figure 3.5: Response at x = L/2 to a ramp entry for ∆t = 10−3 s and ∆x = L/30

Finally, when linear functions in time are used, a continuous Galerkinscheme with triangular elements is equivalent to a central differences Newmarkscheme with lumped mass. A continuous Galerkin scheme with rectangularelements is equivalent to a central differences Newmark scheme with consistentmass matrix.

Ramp with an infinite slope A discontinuity due to the displacement win-dow has been introduced. All spurious phenomena described before are ampli-

3.2 Space-time finite element method 35

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

∆x = L/10

∆x = L/20

∆x = L/30

Figure 3.6: Convergence of CG methods for a ramp entry at x = L for ∆t = 10−2 s

fied. This discontinuity is shown in figure 3.7: the prescribed displacement atthe end of the rod causes much larger amplitude spurious high-frequency os-cillations than with finite slope displacement command, even when triangularelements are used.

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

TCG

RCG

Figure 3.7: Response at x = L/2 to a ramp entry with an infinite slope for∆t = 10−3 s and ∆x = L/30

Chapter 4

Discontinuous Galerkin Methods

In previous chapters, it was shown that time integration schemes may generatespurious high-frequency oscillations in the response when the external loadingtends to be nonsmooth. In order to mitigate these undesirable phenomena,discontinuous Galerkin methods were introduced for elastodynamics [11, 14].The main idea of this class of tehcniques is to assume space-time discontinuousbasis functions in the discretization. The method was first introduced in fluidmechanic to capture sharp gradients in the solution. It was implementedwith a h-adaptivity1 [26] and the stability and the convergence of the methodis analysed in [10, 17]. In [13], Hulbert compares single-field and two-fieldformulations for discontinuous Galerkin methods: both formulations are stableand convergent.

4.1 Derivation in time domain

The method is introduced in the time domain only.

4.1.1 Formulation

Equation (3.1) is written within a two-field formulation, diplacement andvelocity:

mv(t) + ku(t) = f (t) (4.1)v(t)− u(t) = 0 (4.2)u(0) = u0 v(0) = v0 (4.3)

As illustrated in figure 4.1, the discontinuous Galerkin method is character-ized by the presence of discontinuities in the solution at different intants tn.This greatly helpds capturing potential sharp gradients. In other words, thecontinuity of the unknown fields is weakly enforced at time tn:

[u(t)] = u(t+)−u(t−) = 0 (4.4)[v(t)] = v(t+)− v(t−) = 0 (4.5)

1hp-FEM are FEM using mesh size h and polynomial degree p as elements of variables.

38 4. Discontinuous Galerkin Methods

tt−n+1t+nt−nt+n−1

Figure 4.1: Discontinuous linear approximation

This conditions may be enforced through penalties [3] which yields the follow-ing weak two-field formulation:

Find (u,v) ∈ Cu ×Cv , ∀(ψ1,ψ2) ∈ C1 ×C2, (4.6)∫ T

0(ku +mv − f )ψ1dt +

∑n

c1[v(tn)]ψ1(tn) = 0 (4.7)∫ T

0(c3(u − v))ψ2dt +

∑n

c2[u(tn)]ψ2(tn) = 0 (4.8)

where c1, c2 and c3 are constants to be chosen. In this formulation, piecewiselinear functions are going to be employed for the approximation of u and v.Since the same approximation basis is chosen for u and v, Cu = Cv = C1 = C2 =C0p ([0,T ]). By discretizing the time domain [0,T ] as pictured in figure 4.1 and

looking for u and v under the form: ∀t ∈ [0;T ],

u(t) =N∑i=0

u+i w1i(t) +u−i w2i(t) (4.9)

v(t) =N∑i=0

v+i w1i(t) + v−i w2i(t) (4.10)

one obtain the following discrete two-field weak formulation:

Find (u,v) ∈ Cu ×Cv , ∀(ψ1,ψ2) ∈ C1 ×C2, (4.11)N∑n=0

(∫ t−n+1

t+n

(ku +mv)ψ1dt + c1[v(tn)]ψ1(tn))

=N∑n=0

∫ t−n+1

t+n

f ψ1dt (4.12)

N∑n=0

(∫ t−n+1

t+n

c3(u − v)ψ2dt + c2[u(tn)]ψ2(tn))

= 0 (4.13)

Remark 5 (Numerical fluxes). The continuity enforcement through penalties isa way to connect elements to each other. Indeed since discontinuities are allowedbetween elements, each element is initially independent. Another way to connectelements together is to enforce numerical fluxes between elements. This methodis mainly used for convection-diffusion problems in fluid dynamics [20] but canbe adapted to elastodynamics. Nonetheless the choice of the numerical flux is notobvious [5] since it may affect the stabitlity and the accuracy of the DGFEM [3].

4.1 Derivation in time domain 39

4.1.2 Numerical scheme and implementation

Considering the following unknown vector :

qtn =(u+n u−n+1 v+

n v−n+1

)(4.14)

the system to be solved for each time step tn is :

∀n ∈ [0,N ], Anqn = fn (4.15)

where An = A′n + A′′n with

A′n =∫ t−n+1

t+n

kw1nψ1n kw2nψ1n mw1nψ1n mw1nψ1nkw1nψ2n kw2nψ2n mw1nψ2n mw2nψ2nc3w1nψ1n c3w1nψ1n −c3w1nψ1n −c3w2nψ1nc3w1nψ2n c3w2nψ2n −c3w1nψ2n −c3w2nψ2n

dt (4.16)

and

A′′n =

0 0 c1ψ1n(tn) 00 0 0 00 c2ψ1n(tn) 0 00 0 0 0

(4.17)

The external forcing is expressed as follows:

fn =

∫ t−n+1t+n

f ψ1ndt + c1v(t−n )∫ t−n+1t+n

f ψ2ndt

c2u(t−n )0

(4.18)

Matrix An has interesting numerical properties for the following three penaltyparamters: c1 = m, c2 = k and c3 = k [3, 26]. Linear time dicontinuous trialfunctions are assumed so that ψ1i = w1i and ψ2i = w2i . Also, any external bodyforce is neglected, i.e. f = 0. Accordingly, expression (4.15) becomes:

13∆tk

16∆tk −1

2m+ c112m

16∆tk

13∆tk −1

2m12m

−12c3 + c2

12c3 −1

3∆tc3 −16∆tc3

−12c3

12c3 −1

6∆tc3 −13∆tc3

u+n

u−n+1

v+n

v−n+1

=

c1v(t−n )

0

c2u(t−n )

0

(4.19)

For a constant time step ∆t, matrix An is also constant and named A. Then thetime integration scheme is quite simple as described by algorithm 6.

4.1.3 Errors analysis

Spring-mass system described in section 3.1.4 is used. The DGFEM impliestwo values at each time step tn and the global error is estimated at times t+n .

40 4. Discontinuous Galerkin Methods

Algorithm 6 Discontinuous Galerkin time integration schemeRequire: c1, c2 and A, Initial conditions: u0, v0u−0 ← u0v−0 ← v0for n = 0 to N dou(t−n )← u−nv(t−n )← v−n(u+n u−n+1 v+

n v−n+1

)T ←A−1(c1v(t−n ) 0 c2u(t−n ) 0

)Tend for

10−3 10−2 10−1 10010−11

10−9

10−7

10−5

10−3

10−1

∆t

Errorδ m

ax

Linear CG

FG Newmark

DG same number of DoF

DG same number of elts

Figure 4.2: Convergence rate for discontinuous Garlerkin scheme

In figure 4.2, DGFEM error is compared to Newmark schemes. One cansee that a DG scheme is much more accurate and converge faster than allschemes described and use so far. For ∆t = 10−3, DGFEM is 100 to 1000times more accurate than the Fox-Goodwin scheme which is 10 times moreaccurate than CGFEM. However one should note that DGM uses two valuesfor each approximated field at each time step. Therefore, for the same numberof elements, the number of DoF is twice larger than for a linear Newmarkmethod or a linear CGM (if Ne is the number of elements, in the linear caseN = Ne + 1 for Newmark or CG while N = 2Ne for DG). One can see that DGmethod is third order accurate, ie, δmax ∼ ∆t3 when ∆t→ 0. Even with twiceless elements, the dicontinuous Galerkin method is still more accurate thanother methods.

4.2 Space-time discontinuous finite element method 41

4.2 Space-time discontinuous finite element method

4.2.1 Formulation

The same way as the 1DoF system, a two-field formulation is now derived.Consider equations (2.11) for a finite time interval [0 ;T ] to which the velocitydefinition is added : ∀(x, t) ∈ [0 ;L]× ]0 ;T [,

ES∂2u

∂x2 (x, t) + fv = ρS∂v∂t

(x, t)

∂u∂t

(x, t) = v(x, t)(4.20)

with boundary conditions (0 ≤ Tc ≤ T ) :

∀t ∈ [0;T [, u(0, t) = 0∀t ∈ [0;Tc[, u(L,t) = uc(t)

∀t ∈ [Tc;T [, E∂u∂x

(L,t) = g(t)

(4.21)

and initial conditions :

∀x ∈ [0,L], u(x,0) = u0(x)

∀x ∈ [0,L],∂u∂t

(x,0) = v0(x)(4.22)

Similarly to what was done in section 4.1.1, displacement continuity has to beenforced at each time t. Integration over slabs Sn = Ω× [t+n−1 ; t−n ] for n ∈ [1,N ],domain displayed in figure 4.3, is considered. With both displacement andvelocity as unknown fields, two trial functions ψ1 and ψ2 are required. Then,the following weak formulation is derived:

Find (u,v) ∈ Cu ×Cv , ∀(ψ1,ψ2) ∈ C1 ×C2, ∀n ∈ [1,N ],"Sn

(ES∂2u

∂x2 (x, t)− ρS ∂v∂t

(x, t)) + fv)ψ1dxdt

+∫ΩC1[v(x, tn−1)]ψ1(x, t+n−1)dx = 0"

SnC3

(∂u∂t

(x, t)− v(x, t))ψ2dxdt +

∫ΩC2[u(x, tn−1)]ψ2(x, t+n−1)dx = 0

(4.23)

Integration by part of the term with second order derivatives in space yields"SnES∂2u

∂x2ψ1dxdt = −"SnES∂u∂x

∂ψ1

∂xdxdt +

∫ t−n

t+n−1

ES[∂u∂xψ1

]L0dt (4.24)

42 4. Discontinuous Galerkin Methods

Accordingly, the following weak two-field formulation is obtained for thesystem of equations (4.20), (4.21) and (4.22):

Find (u,v) ∈ Cu ×Cv , ∀(ψ1,ψ2) ∈ C1 ×C2, ∀n ∈ [1,N ],"SnES∂u∂x

(x, t)∂ψ1

∂xdxdt +

"SnρS∂v∂t

(x, t)ψ1dxdt −∫ t−n

t+n−1

Sgψ1(L,t)dt

+∫ΩC1[v(x, tn−1)]ψ1(x, t+n−1)dx =

"Snfvψ1dxdt"

SnC3

(∂u∂t

(x, t)− v(x, t))ψ2dxdt +

∫ΩC2[u(x, tn−1)]ψ2(x, t+n−1)dx = 0

(4.25)

within the following functional spaces :

Cu = w ∈H1p (S), w = u0 on Ω0, w = 0 on Γ0 and w = uc on Γ1

Cv = w ∈H1p (S), w = v0 on Ω0, w = 0 on Γ0 and w = uc on Γ1

C1 = ψ ∈H1p (S),ψ = 0 on Γ0

C2 =H1p (S)

For the sake of simplicity, g = 0, fv = 0, u0 = 0 and v0 = 0. Also, the sameapproximation has been chosen for u and v such that ψ1 = ψ2 and C2 = C1 =ψ ∈H1

p (S),ψ = 0 on Γ0. The solution is sought in the form:

∀(x, t) ∈ Sn, u(x, t) =Nd∑i=1

uiwi(x, t) and v(x, t) =Nd∑i=1

viwi(x, t) (4.26)

where Nd is the number of DoF per slab and ψi = wi .Several formulations are possible depending on the type of elements chosen

for the FEM. ∀(i, j,n) ∈ ~1,Nd2 × ~1,N,

Muijn ="SnρS∂wj∂t

ψidxdt ; Mvijn ="SnC3∂wj∂t

ψidxdt

Kuijn ="SnES∂wj∂x

∂ψi∂x

dxdt ; Qijn ="SnC3wjψidxdt

Λvijn =∫ΩC1wj(x, t

+n−1)ψi(x, tn−1)dx

Λuijn =∫ΩC1wj(x, t

+n−1)ψi(x, tn−1)dx

λvin =Nd∑j=1

∫ΩC1ujwj(x, t

−n−1)ψi(x, tn−1)dx

λuin =Nd∑j=1

∫ΩC2vjwj(x, t

−n−1)ψi(x, tn−1)dx

(4.27)

4.2 Space-time discontinuous finite element method 43

t

x

t−n

t+n

t−n+1

t+n−1

Sn−1

Sn

Sn+1

L0

Figure 4.3: Slabs over the space-time discontinuous domain

Quadrangle based formulation In that formulation rectangular elementsare chosen with the four basis functions: ∀(ξ,ζ) ∈ [−1;1]2,

w1(ξ,ζ) =14

(1− ξ)(1− ζ) ; w2(ξ,ζ) =14

(1 + ξ)(1− ζ)

w3(ξ,ζ) =14

(1 + ξ)(1 + ζ) ; w2(ξ,ζ) =14

(1 + ξ)(1 + ζ)(4.28)

Time and space are uncoupled in the expression of the basis functions (4.28)and one can write:

wi(x, t) = fi(x)gi(t) (4.29)

and accordingly ∀(i, j,n) ∈ ~1,Nd2 × ~1,N,"Sn

∂wj∂t

widxdt =∫ L

0fifjdx

∫ t−n

t+n−1

∂gj∂tgidt"

Sn

∂wj∂x

∂wi∂x

dxdt =∫ t−n

t+n−1

gigjdt

∫ L

0

∂fj∂x

∂fi∂xdx"

Snwjwidxdt =

∫ L

0fifjdx

∫ t−n

t+n−1

gigjdt∫Ωwjwidx = gigj

∫Ωfjfidx

(4.30)

The number of spatial nodes at time t+n is Ns = Nd/2. Mass matrix M andstiffness matrix K are defined ∀(i, j) ∈ ~1,Ns2 as :

Mij =∫ L

0ρSfifjdx ; Kij =

∫ L

0ES∂fi∂x

∂fj∂xdx (4.31)

44 4. Discontinuous Galerkin Methods

Choosing C1 = C2 = C3 = ρS, the matricial system (4.32) is derived. It is thesame system as (4.19), in the case where c1 = c2 = c3 =m, generalized to higherdimension.

13∆tK

16∆tK

12M 1

2M16∆tK

13∆tK −1

2M 12M

12M 1

2M −13∆tM −1

6∆tM−1

2M 12M −1

6∆tM −13∆tM

u+n

u−n+1v+n

v−n+1

=

Mv−n

0Mu−n

0

(4.32)

In [26], Wiberg and Li define the velocity as follows :

∀(x, t) ∈ [0,L]×]0;T [,∂∂x

(∂u∂t

(x, t)− v(x, t))

= 0 (4.33)

which leads to a weak formulation as follows:

Find (u,v) ∈ Cu ×Cv , ∀(ψ1,ψ2) ∈ ×C2, ∀n ∈ [1,N ],"SnES∂u∂x

(x, t)∂ψ1

∂xdxdt +

"SnρS∂v∂t

(x, t)ψ1dxdt

+∫ΩρS[v(x, tn−1)]ψ1(x, t+n−1)dx = 0 (4.34)"

SnES

∂∂x

(∂u∂t

(x, t)− v(x, t))∂ψ2

∂xdxdt+

∫ΩES[u(x, tn−1)]ψ2(x, t+n−1)dx = 0

together with the respective system (4.35) assuming the same approximationas previously:

13∆tK

16∆tK

12M 1

2M16∆tK

13∆tK −1

2M 12M

12K 1

2K −13∆tK −1

6∆tK−1

2K 12K −1

6∆tK −13∆tK

u+n

u−n+1v+n

v−n+1

=

Mv−n

0Ku−n

0

(4.35)

The two definitions of the velocity field yield identical results. Neverthelessthe classical definition is considered because it is suitable for both quadrangleand triangle based formulations.

Triangle based formulation For triangles, velocity as defined in (4.33) wouldnot be appropriate since polynomial basis functions would linearly coupletime and space such that:

∀i ∈ ~1,Nd, ∂2wi∂x∂t

= 0 (4.36)

Formulation (4.25) is thus preferred.

4.2.2 Numerical scheme and implementation

In order to simplify the implementation, a regular laminated mesh with con-stant time step ∆t is assumed. At each time step n, the following linear systemshould be solved:[

Ku Mu +ΛvMv +Λu −Q

](unvn

)=

(λvnλun

)⇔ Rqn = In (4.37)

4.2 Space-time discontinuous finite element method 45

where matrix R is constant adn vector q is of size 4Np :

qt =(u0 . . .uNp ,uNp+1 . . .u2Np ,v0 . . .vNp ,vNp+1 . . .v2Np

)(4.38)

Similarly, the clamped DoF should be removed from (4.37) that is to say indexes(for q and λ) and rows and columns (for R) number 1, Np + 1, 2Np + 2 and3Np + 3. For the phase where displacement is enforced, one refers to 3.2.2:∀n ∈ ~1,NTc,[

R∗11 R∗12R∗21 R∗22

](q1n

q2n

)=

(l1n

l2n

)(4.39)

where q1n corresponds to the constrained displacement and velocity :

q1n =

(uc(t

+n−1) uc(t−n ) vc(t

+n−1) vc(t−n )

)t(4.40)

and

q2n = R∗22

−1(l2n −R∗21q1

n) (4.41)

Finally, the space time discontinuous integration scheme is implemented suchdiscribed in algorithm 7.

Algorithm 7 DG time integration scheme for enforced/free displacementRequire: R, P, uc, Initial conditions: u0, v0vc← ucu−0 ← u0v−0 ← v0R∗← PRtPEnforced displacement phase :for n = 1 to NTc do

ln← ln(u−n−1,v−n−1)

l∗n← Pln

q1n←

(uc(t

+n−1) uc(t−n ) vc(t

+n−1) vc(t−n )

)Tq2n← R∗−1

22(l2n −R∗21q1

n)

q∗n←(q1n q2

n

)Tqn← Ptq∗n =

(u+n−1 u−n v+

n−1 v−n)T

end forFree displacement phase :for n =NTc to N do

ln← ln(u−n−1,v−n−1)

qn← R−1lnend for

4.2.3 Results

We consider the rod model already used in 3.2.3. Results are plotted at time t+nfor ∆t = 10−2 s and for ∆x = L/30 m. One can notice in figure 4.4 that DGM pro-

46 4. Discontinuous Galerkin Methods

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

CG triangle

CG rectangle

DG rectangle EC

DG rectangle NC

Figure 4.4: Response at x = L2 to a ramp entry

vides a solution where high frequency oscillations are removed comparated toCGM and equivalent Newmark methods. This effect is even more visible whendiscontinuity is introduced in the enforced displacement as shown in figure 4.5.Also, the DG formulation provides access to both velocity and displacement

0 0.5 1 1.5 2 2.5 3−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Displacemen

t(m

)

CG rectangle

CG triangle

DG

Figure 4.5: Response at x = L/2 to a ramp entry with an infinite slope

with the same linear approximation and is third order accurate. It is thenmore accurate than CGM mainly when discontinuous enforced displacementis used as depicted in figure 4.6. It is also worth noticing the large unexpected

4.2 Space-time discontinuous finite element method 47

0 0.5 1 1.5 2 2.5 3

−40

−20

0

20

40

Time (s)

Velocity(m

/s)

TCG

DG

Figure 4.6: Response at x = L/2 to a ramp entry with an infinite slope

pertubation in the displacement in the DGM while the initial wave has not yetreach the middle point of the rod. These perturbations may be due to an errorin the code or more likely to the use of rectangular elements in the DGM.

Conclusion

In this report, time integration using finite elements has been investigated.First, it has been shown that continuous Galerkin finite element methodsusing linear basis functions could be related to central differences Newmarkmethods. Moreover the use of quadratic basis functions provides a great gainof accuracy. However, these formulations are less efficient when nonlinearitiesare introduced. That is why attention was paid to discontinuous Galerkin finiteelement methods.

As it has been foreseen, the type of elements used in space-time finiteelements method seems to be a sensitive choice. Indeed, muliplices alwayslead to a numerical model where information propagates at infinite velocitythrough the system while use of simplices tends to avoid this phenomenon.

Few investigations remain to be conducted. It would be of high interest tosearch and develop an error estimate for both discontinuous and continuousGalerkin methods in order to quantitatively compare them. Effects of higherorder basis functions on the accuracy of the method and formulations withseveral fields are also possible future roads of investigation. Ultimately, purelyperiodic motions as proposed in [25] could be explored by weakly enforcingthe periodicity conditions in time.

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