Highly Accurate Subcell Integration in the Context of The ... · The concept of FCM implies that...

Faculty of Civil, Geo and Environmental Engineering

Chair for Computation in Engineering

Prof. Dr. rer. nat. Ernst Rank

Highly Accurate Subcell Integration

in the Context of The Finite Cell Method

Laszlo Kudela

Master’s thesis

for the Master of Science program Computational Mechanics

Author: Laszlo Kudela

Matriculation number: 3627061

Supervisor: Prof. Dr.rer.nat. Ernst Rank

Nils Zander, M.Sc.

Date of issue: 01. April 2013

Date of submission: 20. September 2013

Involved Organisations

Chair for Computation in EngineeringFaculty of Civil, Geo and Environmental EngineeringTechnische Universitat MunchenArcisstrasse 21D-80333 Munchen

Declaration

With this statement I declare, that I have independently completed this Master’s thesis. Thethoughts taken directly or indirectly from external sources are properly marked as such. Thisthesis was not previously submitted to another academic institution and has also not yetbeen published.

Munchen, September 19, 2013

Laszlo Kudela

Laszlo KudelaEggenfeldener Straße 119D-81929 MunchenE-mail: [email protected]

ACKNOWLEDGEMENTS

I would like to offer my special thanks to Nils Zander, my main supervisor, for his continoussupport in the last six months. The completion of this thesis would have been not possiblewithout his help.

I would like to thank Prof. Ernst Rank and Dr. Stefan Kollmannsberger for offering me theopportunity of writing this thesis.

I thank Tino Bog for his valuable and constructive input regarding implementation questionsand for his help in the revision of this work.

Advice and assistance given by Felix Frischmann in the field of geometric modeling is greatlyappreciated.

I thank my colleague, Mohamed Elhaddad for the fruitful discussions we had about the theoryand implementation of FEM.

Thanks to all people from the Chair for Computation in Engineering for the support and thegreat atmosphere. It is a pleasure to work with them.

I am particularly grateful for the financial support of the German Academic Exchange Service(DAAD) in the last two years.

Last but not least, I would like to thank my family for supporting me during all the years ofmy studies. Especially, I want to thank my girlfriend Emese for her loving understanding inthe past times.

ABSTRACT

The Finite Cell Method is an immersed boundary method using higher order finite elements.In the FCM, the material parameters of the fictitious domain are scaled down, thus a dis-continuity is introduced in the cut finite cells. This introduces an error in the discretizedweak form of the governing physcial equations. Typically, the integrals are evaluated usingcomposed Gaussian quadrature, based on a space tree decomposition. The drawback of thespacetree decomposition is that it introduces a high number of quadrature points and yieldsonly a 0th order aproximation of the geometry. This thesis proposes a different apporachto deal with integration: the cut cells are decomposed into quadrilaterals and triangles withcurved sides and the quadrature points are distributed in their parameter spaces. To thisend, an algorithm is proposed that performs this decomposition in two dimensions. Thealgorithm is able to work on complex geometries. The results show that the blended subcellintegration is able to produce results with higher precision using less integration points thanthe spacetree approach. Thus, the overall computational effort of FCM analyses is consider-ably reduced in comparison to the spacetree integration. This makes the proposed approacha good fit into the design-through-analysis pipeline.

VII

Contents

1 Introduction and Motivation 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Theoretical background 3

2.1 The p-version of the Finite Element Method . . . . . . . . . . . . . . . . . . . 3

2.2 The blending function method . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Parametric curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Trimmed curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.3 NURBS curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.4 Blended mappings for quadrilaterals . . . . . . . . . . . . . . . . . . . 12

2.2.5 Blended mappings for triangles . . . . . . . . . . . . . . . . . . . . . . 14

2.2.6 Remark on the consistency of blended elements in p-FEM . . . . . . . 14

2.3 Numerical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Numerical integration of parametric curves . . . . . . . . . . . . . . . 19

2.3.2 Numerical integration of blended elements . . . . . . . . . . . . . . . . 21

2.4 The Formulation of the Finite Cell Method . . . . . . . . . . . . . . . . . . . 22

2.4.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4.2 Adaptive integration of cut cells . . . . . . . . . . . . . . . . . . . . . 24

3 Algorithmic partitioning of cut cells 27

3.1 Marching squares approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Triangulated cut cell approach . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.1 Numbering and naming conventions . . . . . . . . . . . . . . . . . . . 29

3.2.2 Determining whether the cell is cut . . . . . . . . . . . . . . . . . . . . 31

3.2.3 Determining which edges are cut . . . . . . . . . . . . . . . . . . . . . 32

3.2.4 Creating the auxiliary point and line, intersecting the edges . . . . . . 32

3.2.5 Create blended triangles and quadrilaterals . . . . . . . . . . . . . . . 33

3.2.6 Mapping to the parameter space of the finite cell . . . . . . . . . . . . 35

3.2.7 Subdivision for too complex geometries . . . . . . . . . . . . . . . . . 36

3.3 Implementation of the concept . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 Patch test of the integration cells 37

4.1 Curved domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2 Rectangular domain with moving circle . . . . . . . . . . . . . . . . . . . . . 39

4.3 Violin with f-holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

5 Finite Cell Method examples 455.1 Perforated plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2 Poisson problem on a curved domain . . . . . . . . . . . . . . . . . . . . . . . 485.3 Compression wave on a violin . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

6 Summary, conclusion and outlook 556.1 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

1

Chapter 1

Introduction and Motivation

The Finite Element Method (FEM) has been used by scientists and engineers for decades asa standard approach for solving partial differential equations (PDE) numerically. As a resultof the growth of the computational power in the second half of the 20th century, the methodallowed its users to solve problems of increasing complexity and size. However, the core ideaof FEM remains the same. The domain is subdivided by a suitable spatial discretizationmethod into a set of smaller subdomains, often called mesh, and the solution is representedby the linear combination of shape functions whose support is local to the these subdomains.

One major issue with the standard FEM approach is the generation of a proper mesh. Themethod requires the elements of the mesh to conform with geometric boundaries. While thisrequirement is easy to fulfill in case of simple geometries, it constitutes a severe bottleneckfor domains with higher geometric complexity. Translating the geometric data that is usu-ally available in the form of CAD files into a conforming FEM discretization is often timeconsuming and expensive. This translation is hard to fully-automate, and the resulting meshmight need manual adjustments in order to be able to perform numerical computations onit. Studies show that the time required for creating an analysis suitable geometry and itscomputational mesh accounts for about 80% of the overall analysis time [1].

The desire to overcome meshing problems and to bring CAD and FEA closer together resultedin the method of Isogeometric Analysis (IGA), in which the spline basis functions describingthe geometry are directly used as the shape functions of the solution field [1].

Another approach to avoid expensive mesh generation of complex geometries is the immersedboundary and fictitious domain methods [2]. This group of methods extend the domain ofinterest with complex boundaries to a larger embedding domain whose geometry is simplerand thus can be meshed easily with a structured grid. The influence of the fictitious domainextension is eliminated by scaling down its material parameters by a penalty factor such thatthe consistency with the original problem is preserved.

The Finite Cell Method (FCM) is an immersed boundary method combined with higherorder finite elements (p-FEM) [3]. The method shows exponential convergence characteristicsin the energy norm known from p-FEM, and allows accurate numerical calculations to beperformed without having to mesh complex geometries. The method is also suitable foranalyses on implicitly defined or voxel based geometries that are typical in e.g. biomechanicalapplications [4]. Its biggest advantage lies in fast convergence at almost no meshing costs.

2 1. Introduction and Motivation

1.1 Motivation

The concept of FCM implies that the effort of a structural analysis is shifted from meshgeneration to the level of integration. The penalty factor introduces a discontinuity at theboundary of the fictitious and physical domain that has to be taken into account at theconstruction time of the stiffness matrix. As the numerical quadrature is usually chosento be the Gaussian quadrature that does not generally perform well along discontinuities,the quadrature points have to be distributed in a better way. In the context of FCM thisdistribution is usually done by an adaptive spacetree approach, which recursively refines thefinite cell mesh in the cut cells until a pre-defined depth, thus the discontinuity is capturedby a higher density of integration points along the boundary.

Although the spacetree approach converges as the depth of the tree is increased, it hassome drawbacks. On the one hand, the number of integration points increases exponentiallywith every new level of spacetree decomposition, thus making the analysis computationallyexpensive. On the other hand, using a spacetree results in a 0th order approximation of theboundary which does not balance well with the higher order shape functions of p-FEM [5].

These drawbacks indicate the need of another quadrature point distribution approach thatshould take the geometric description of the boundary into account. Investigations havealready been made in this direction by approximating the cut cells with Lagrangian elements[5]. A similar approach has also been applied to XFEM applications, where the elements thatare cut by the interface are partitioned such that the edges of the resulting subcells align withthe interface [6]. However, the possibility of directly using the geometrical description of theboundary for the cut cell integration in the context of FCM is still an open question and shallbe addressed in this work. To this end, the blending function method as a possible tool forthe quadrature point distribution is going to be investigated.

1.2 Outline

Chapter 2 presents the theoretical background of the Finite Cell Method and the geometricrepresentation of curves, as well as the blending function method. Chapter 3 discusses thealgorithmic approach to subdivide cut cells into blended subcells. An integration test ofthe resulting integration mesh is performed in Chapter 4. The results of the new approachare presented and compared with the conventional spacetree method in Chapter 5. Thediscussion is closed by some concluding remarks and an outlook in Chapter 6 .

3

Chapter 2

Theoretical background

2.1 The p-version of the Finite Element Method

This section shortly introduces the Finite Element Method and its p-version for steady linearelastic problems. For a complete discussion, refer to [7][8].

Let Ωphy be a physcial domain with its boundary ∂Ω divided into Neumann ΓN and DirichletΓD parts (Figure 2.18). Considering linear elastic behavior, the following equations hold [9]:

∇ · σ + b = 0 ∀x ∈ Ωphy (2.1a)

σ = C : ε ∀x ∈ Ωphy (2.1b)

ε =1

2+[

∇u+ (∇u)⊺]

∀x ∈ Ωphy (2.1c)

σ · n = t ∀x ∈ ΓN (2.1d)

u = up ∀x ∈ ΓD, (2.1e)

where σ, b, C, ε, t, up, denote the stress tensor, the body forces, the constitutive tensor,the strain tensor, the prescribed traction vectors and the prescribed displacement vectors,respectively.

Introducing a test function δu and following the principle of virtual work, Equation 2.1 canbe transformed into the following integral equation [7]:

δW (u, δu) =

∫

Ωphy

σ : (∇symδu)dV −∫

Ωphy

δu · bdV −∫

ΓN

δu · tdA = 0, (2.2)

where ∇sym denotes the symmetric part of the gradient.

In the context of p-FEM, the domain Ωphy is decomposed into a set of elements whose bound-aries are conforming with the geometric boundary ∂Ω. This set of elements is called mesh.The unknown quantities u and δu are discretised as a sum of shape functions multiplied withunknown coefficients:

4 2. Theoretical background

u =n∑

i=1

Niui (2.3)

δu =n∑

i=1

Niδui. (2.4)

The Ni shape functions have local support on the elements of the mesh. The vectors ui andδui contain the unknown coefficients. According to the standard Bubnov-Galerkin approach,equations 2.3 and 2.4 are substituted back to 2.2, yielding a system of linear equations:

Ku = f . (2.5)

The stiffness matrix K from Equation 2.5 is the result of the proper assembling of the ke

element stiffness matrices that are calculated as [3]:

ke =

∫ 1

−1

∫ 1

−1(LN )TC(LN )‖J‖dξdη, (2.6)

where L is the standard strain-displacement operator, N is the matrix of shape functions, Cis the constitutive matrix, and ‖J‖ is the Jacobian determinant of the mapping Q(ξ, η) thatmaps the local coordinates (ξ, η) of the element to the global coordinate system (x, y).

In the p-version of the FEM [10], the shape functions are chosen to be the integrated Legendrepolynomials, as introduced by Szabo and Babuska [8][11]. The definition of these polynomialsreads:

φj =

√

2j − 1

2

∫ ξ

−1Pj−1(t)dt =

1√4j − 2

(Pj(ξ)− Pj−2(ξ)) j = 2, 3, . . . , (2.7)

where Pj(ξ) are standard Legendre polynomials in local cell coordinates. The one dimensionalbasis is constructed by combining the standard linear shape functions with the integratedLegendre polynomials defined in Equation 2.7. Some of these shape functions are depictedon Figure 2.1.

-1 1

1

-0.5

-1 1

1

-0.5

-1 1

1

-0.5

N1(ξ)N2(ξ)

φ2(ξ) φ3(ξ)

Figure 2.1: 1D Shape functions

The basis functions for higher spatial dimensions are constructed by taking the tensor productof these functions. For a 2D case, one can distinguish between the following shape functions[8]:

2.2. The blending function method 5

• Nodal modes: The nodal modes defined as

NNi

1,1(ξ, η) =1

4(1 + ξiξ)(1 + ηiη) j = 1, . . . , 4 (2.8)

are the standard bilinear shape functions that are well known from the isoparametricbilinear elements. The value of a nodal mode is 1 at node i and vanishes for all othernodes. An example with NN1

1,1 (ξ, η) is shown in 2.2a.

• Edge modes: These modes are non-zero on one edge, and vanish at all other edges. Forexample, the definition for an edge mode for edge E1 reads:

NE1

i,1 (ξ, η) =1

2(1− η)φi(ξ). (2.9)

An example with NE1

2,1(ξ, η) is shown in 2.2b.

• Internal modes: The internal modes vanish at all edges and are non-zero only at theinternal parts of the element.

N inti,j (ξ, η) = φi(ξ)φj(η) (2.10)

An example for N int2,2 (ξ, η) is shown in 2.2c.

In Figure 2.2 an example for nodal modes, edge modes, and internal modes is depicted,respectively. The concept of modes can be extended to three dimensions, where also internalor volume modes have to be considered.

2.2 The blending function method

A mesh of finite elements with linear edges is not able to follow curved geometric boundariesexactly. Since p-FEM uses higher order polynomials to approximate the solution field, ele-ments that can capture higher order boundaries in an exact sense are needed. To this end, theblending function method was introduced in [12] as an approach for creating elements thatconform with curved boundaries. The following sections introduce the thoery of parametriccurves, which form the basis of blended elements. The introduction of parametric curves isfollowed by a short discussion of the blending function method, closely following [13] and [14].

2.2.1 Parametric curves

A parametric curve in space is described by a set of equations that map the parameter t froma one dimensional interval I to the three dimensional Euclidean space R

3.

xc = [x(t), y(t), z(t)]T , t ∈ I = [t0, t1]. (2.11)

This means that each coordinate of the curve is described by one equation an terms of t.


1

0

0.2

-10.6

0.4

-0.6

0.6

0.2

0.8

-0.2-0.2

1

0.2-0.6

0.6

-11

ξ η

(a) Nodal mode

1

-0.8

-

10.6

-0.6

-0.6

-0.4

0.2

-0.2

-0.2-0.2

0

0.2-0.6

0.6

-11

ξ η

(b) Edge mode

1

0

0.2

-1

0.6

0.4

-0.6

0.6

0.2

0.8

-0.2-0.2

1

0.2-0.60.6

-11ξ η

(c) Internal mode

Figure 2.2: Example for two dimensional nodal modes, edge modes and internal modes

The nth order derivative of a curve is a vector defined as:

∂nxc

∂tn=

[

∂nx(t)

∂tn,∂ny(t)

∂tn,∂nz(t)

∂tn

]T

n = 1, . . . ,m. (2.12)

A curve is said to be Cm continuous if it is m times continuously differentiable.

The tangent vector of a parametric curve is defined by substituting n = 1 into Equation 2.12.If the tangent vector is continuous along the whole parametric domain I of the curve, thecurve is said to be regular. If there exists a parameter t where the tangent vector vanishes,the curve is called singular at t. If every point (x, y, z) on the curve can be assigned toexactly one parameter in the interval t, the mapping is called bijective, and there exists aninverse mapping x−1

c (t) that maps the global coordinates (x, y, z) to the parametric spaceof the curve. Usually this inverse mapping cannot be given explicitly, but by means of anumerical iteration, for example the Newton-Raphson method [14]. In our discussion werestrict ourselves to curves that are regular with bijective mappings.

It is obvious that Equation 2.11 is not the only possible parametrization of a curve. If we


introduce a further mapping

γ = t(u) : [u0, u1] → [t0, t1] (2.13)

that maps the interval I = [u0, u1] to the interval I, we can compose a new mapping for thesame curve that reads:

xc(u) = xc(t(u)). (2.14)

The curve defined by Equation 2.11 and its reparametrized form in Equation 2.14 has thesame shape in the Euclidean space, although their parametrization is different. However,the reparametrization by introducing the second mapping t(u) influences the magnitude ofthe tangent vector of the curve. For example, the x component of the tangent vector fromEquation 2.12 with n = 1 of a reparametrized curve will read as:

∂x(u)

∂u=

∂x(t)

∂t

∂t

∂u. (2.15)

If we imagine t as the time it takes to travel along the curve, the tangent vector is analogousto the velocity vector, i.e. the direction of the travel. By looking at Equation 2.15, it isobvious that by introducing a suitable mapping γ, the direction of this travel can be flipped,i.e. it is possible to start from the point xc(t1) instead of xc(t0). This means that one canassign an orientation to every parametric curve.

2.2.2 Trimmed curves

It is sometimes desirable to work with parts of parametric curves only and not treat them asa whole. Moreover, the blending function method (Section 2.2.4) requires that curves whichconstitute the boundaries of curved triangles and quadrilaterals have to be defined such thattheir parameter space is [−1, 1] (Refer to Figure 2.6).

Given a curve xc defined by Equation 2.11, and the parameters tn, tm ∈ I, the curve is saidto be trimmed at parameters tn and tm if a new parametric space I ⊂ I is introduced forxc such that I = [tn, tm]. Thus the parametric equation of the trimmed curve remains thesame, only the parametric bounds change:

P : xc = [x(t), y(t), z(t)]T , t ∈ I = [tn, tm]. (2.16)

If the trimmed curve has to be reparametrized such that its parametric bounds will be [−1, 1],a similar mapping to Equation 2.13 can be introduced:

γ = t(u) : [−1, 1] → [tn, tm]. (2.17)

Thus the parametric equation of the curve in terms of u ∈ [−1, 1] is:

Q : xc(u) = xc(t(u)). (2.18)

If tm > tn, the original orientation of the curve is flipped. The concept of trimming andreparametrizing is depicted in Figure 2.3.


R3

R3

P

P

It0

tn

tn

tm

tm t1

xc(t0) xc(tn)

xc(tm)

xc(t1)

trimming

u

I

−1 1

γ

Q

xc(−1)

xc(1)

Figure 2.3: The method of curve trimming

2.2.3 NURBS curves

These days, non uniform rational B-Splines (NURBS) are predominant in the field of geo-metric modelling. A short introduction to the theory of these curves is given in the followingsection.

The discussion of NURBS starts from the theory of B-Splines. This type of curves are basedon piecewise defined polynomials that form a set of basis functions denoted by N , and a setof control points denoted as P i.

A B-Spline of polynomial order p is defined as [15]:

C(t) =

n∑

i=0

Ni,p(t)P i t0 < t < tm, (2.19)

where the shape functions are defined recursively:


For p = 0:

Ni,0(t) =

1 ti < t < ti + 1

0 otherwise, (2.20)

for p ≥ 1:

Ni,p(t) =t− ti

ti+p − tiNi,p−1(t) +

ti+p+1 − t

ti+p+1 − ti+1Ni+1,p−1(t),

0

0:= 1, (2.21)

where the set of non-decreasing numbers U = [t0, t1, . . . , tm] divides the parameter space ofthe curve into m subintervals. The set U is called knot vector, the ti are called knots. Thesubintervals are referred to as knot spans. A knot can appear multiple times in a knot span:in this case it is called as a multiple knot with multiplicity k.

Without going more into details (for a deeper discussion, refer to [15]), there are some prop-erties of B-Splines that have to be mentioned:

• The basis function Ni,p(t) is non zero on [ti, ti+p+1]. This is also called as the localsupport property of the B-Spline basis functions. This has the consequence that movingone control point influences only a subinterval of the overall curve shape.

• The sum of all non-zero basis functions on any knot span [ti, ti+1] is 1. This is calledthe partition of unity property.

• If the number of knots is m+ 1, the polynomial degree is p and the number of controlpoints is n, the following relation holds:

n = m− p− 1 (2.22)

• The basis function Ni,p(t) is a piecewise polynomial curve with joining points at knotsin the interval [ti, ti+p+1].

• At a knot of multiplicity k, the B-Spline curve is Cp−k continuous.

• The B-Spline curve is interpolatory at the end points only if the multiplicity of the firstand last knot is p+ 1.

• The B-Spline curve is always enclosed by the convex hull of its control polyline.

As an example, consider the basis functions of a curve with polynomial degree p = 2 anda knot vector U = [0, 0, 0, 0.5, 0.5, 1, 1, 1]. From Equation 2.22, we have n = 5 non zerobasis functions. Applying the recursive formula of Equation 2.21 yields the following basisfunctions:


N0,2(t) = 4

(

1

2− t

)2

0 ≤ t < 0.5 (2.23a)

N1,2(t) = 8t

(

1

2− t

)

0 ≤ t < 0.5 (2.23b)

N2,2(t) =

4t2 0 ≤ t < 0.5

4(1− t)2 0.5 ≤ t < 1(2.23c)

N3,2(t) = 8(1 − t)

(

t− 1

2

)

0.5 ≤ t < 1 (2.23d)

N4,2(t) = 4

(

t− 1

2

)2

0.5 ≤ t < 1 (2.23e)

These basis functions are depicted in Figure 2.4. Note that the multiplicity of the knot t = 0.5causes the basis function N2,2(t) to be C0 continuous at parameter t = 0.5. This results in a“kink” on the curve of the function.

0

0.2

0

0

0

1

0 0.2 0 0 0 1

N0,2(t)

N1,2(t)

N2,2(t)

N3,2(t)

N4,2(t)

t

Figure 2.4: An example for basis functions with p = 2

Let us now suppose that we have the following set of 2D control points for the B-Spline basisdefined in Equation 2.23:

P 0 = [1.0, 0]T

P 1 = [1.0, 1.0]T

P 2 = [0.0, 1.0]T

P 3 = [−1.0, 1.0]T

P 4 = [−1.0, 0]T .

(2.24)

Substituting the basis functions from Equation 2.23 and the control points from Equation 2.24into the definition of the B-Spline curve (Equation 2.19) yields the following parametric


equation for the curve:

C1(t) =

[

1− 4t2

−4t2 − 4t

]

0 ≤ t < 0.5 (2.25a)

C2(t) =

[

4t2 − 8t+ 3−4t2 − 4t

]

0.5 ≤ t < 1. (2.25b)

As Equation 2.25 suggests, the curve is composed of two quadratic subcurves that join atparameter t = 0.5, where the control point P 2 lies. The curve is depicted on Figure 2.5.

Obviously, not all curves can be represented by B-Splines. As B-Splines are piecewise poly-nomial curves, an extension of the B-Spline theory is needed to be able to represent rationalcurves, for example circles or ellipses. This lead to the development of NURBS (non-uniformrational B-Splines). The definition of a NURBS reads as [15]:

C(t) =

n∑

i=0Ni,p(t)wiP i

n∑

i=0Ni,p(t)wi

t0 ≤ t < tm (2.26)

where the weights denoted by wi are associated with the control points P i. The higher thevalue of wi is, the closer the curve approaches the associated control point P i and vice versa.

Consider the following vector of wi for the B-Spline curve defined by Equations 2.23 and 2.24:

W =

[

1,1√2, 1,

1√2, 1

]

. (2.27)

Substituting the basis functions, control points and the weight vector into the definition ofthe NURBS (Equation 2.26) yields the following parametric equation for the curve:

C1(t) =

−√32t2 + 4t2 +

√8t− 4t+ 1

−√32t2 + 8t2 +

√8t− 4t+ 1

−√32t2 + 4t2 +

√8t

−√32t2 + 8t2 +

√8t− 4t+ 1

0 ≤ t < 0.5 (2.28a)

C2(t) =

√32t2 − 4t2 − 3

√8t+ 4t+

√8− 1

−√32t2 + 8t2 + 3

√8t− 12t−

√8 + 5

−√32t2 + 4t2 + 3

√8t− 8t−

√8 + 4

−√32t2 + 8t2 + 3

√8t− 12t−

√8 + 5

0.5 ≤ t < 1. (2.28b)

The curve is depicted in 2.5. Note that by applying the weight vector W , the shape of thecurve is identical to the half of a circle.

This means that the well known sine and cosine parametrization is not the only representationthat one can choose for a circle.


−1.5 −1 −0.5 0 0.5 1 1.5

C(t)

C(t)

P 0

P 1

P 2P 3

P 4

Figure 2.5: An example of a B-Spline and a NURBS curve

2.2.4 Blended mappings for quadrilaterals

Given a quadrilateral with three linear edges and one curved edge in the global coordinatesystem, its corner vertices are denoted by X1,X2,X3,X4 (Figure 2.6). Let us denote theparametric equation of the curved edge as E2(t) and assume that t ∈ [−1, 1].

1

2

(−1,−1) (1,−1)

(1, 1)(−1, 1)

η

ξ

Q(ξ, η)

Q−1(ξ, η)X1

X2

X3X4

E2(t)

x

y

f2(η)

Figure 2.6: Blended quadrilateral with one curved side

The blended mapping that maps the local parameter space (ξ, η) of the quad to the global(x, y) space is defined as the standard bilinear mapping extended by an extra term f2(η)[12][13][14]:

Q(ξ, η) =1

4

(

(1− ξ)(1− η)X1 + (1 + ξ)(1 − η)X2+

(1 + ξ)(1 + η)X3 + (1− ξ)(1 + η)X4

)

+ f2(η). (2.29)

The extra term f2(η) is the difference between the curveE2(t) and the straight line connectingnodes X2 and X3 multiplied with a linear term that vanishes at the edge ξ = −1 (i.e. theedge opposite to E2(t)):


f2(η) =

(

E2(η)−1

2

(

(1− η)X2 + (1 + η)X3

)

)

1

2(1 + ξ). (2.30)

Substituting the extra term f(η) into Equation 2.29 yields:

Q(ξ, η) =1

4

(

(1− ξ)(1− η)X1 + (1− ξ)(1 + η)X4

)

+1

2E2(η)(1 + r). (2.31)

The blended function concept can be extended to quadrilaterals whose sides are all curved. Inthis case, Equation 2.29 is augmented by the extra terms f1(ξ), f3(ξ), f4(η) that are computedsimilarly to Equation 2.30. After substituting and refactoring, the complete blending for ageneric curved quadrilateral reads:

Q(ξ, η) =1

2

(

(1− η)E1(ξ) + (1 + ξ)E2(η)

+ (1 + η)E3(ξ) + (1− ξ)E4(η)

)

−4∑

i=1

N i(ξ, η)X i, (2.32)

where the last term denotes the bilinear mapping from Equation 2.29.

As it mentioned in Section 2.1, the Jacobian matrix of the mapping is required for integration.In 2D, the Jacobian matrix is defined as:

J =

∂Qx(ξ, η)

∂ξ

∂Qy(ξ, η)

∂ξ

∂Qx(ξ, η)

∂η

∂Qy(ξ, η)

∂η

. (2.33)

In the case of a blended quadrilateral, the partial derivatives are:

∂Q(ξ, η)

∂ξ=

1

2

(

(1− η)∂E1(ξ)

∂ξ+E2(η) + (1 + η)

∂E3(ξ)

∂ξ−E4(η)

)

− 1

4

(

(1− η)(X2 −X1) + (1 + η)(X3 −X4)

)

(2.34)

∂Q(ξ, η)

∂η=

1

2

(

−E1(ξ) + (1 + ξ)∂E2(η)

∂η+E3(ξ) + (1− ξ)

∂E4(η)

∂η

)

− 1

4

(

(1− ξ)(X4 −X1) + (1 + ξ)(X3 −X2)

)

. (2.35)

The inverse of the mapping Q(ξ, η), i.e. a mapping that maps the global (x, y) coordinatesto the local (ξ, η) space of the quadrilateral can usually not be given explicitly. As Q(ξ, η)is a nonlinear function, an iterative approach has to be applied to compute the inverse, forexample the Newton-Raphson method. A detailed description of how the inverse is computedcan be found in [13].


2.2.5 Blended mappings for triangles

It is possible to use blended mappings for triangles with curved edges. To this end, we startfrom the blended mapping of a quadrilateral and collapse one of its edges to one point [7].

1

2

(−1,−1) (1,−1)

(1, 1)(−1, 1)

η

ξ

Q(ξ, η)

Q−1(ξ, η)X1

X2

X3

E2(t)x

y

Figure 2.7: Blended triangle, edge 3 is collapsed

As edge 3 is now collapsed to a single point X3, the blended mapping from Equation 2.32takes the following form:

Q(ξ, η) =1

2

(

(1− η)E1(ξ) + (1 + ξ)E2(η) + (1 + η)X3 + (1− ξ)E4(η)

)

− 1

4

(

((1 − ξ)(1− η)X1 + (1ξ)(1 + η)X2 + 2(1 + η)X3)

)

. (2.36)

The partial derivatives for the Jacobian matrix are:

∂Q(ξ, η)

∂ξ=

1

2

(

(1− η)∂E1(ξ)

∂ξ+E2(η)−E4(η)

)

− 1

4

(

(1− η)(X2 −X1)

)

(2.37)

∂Q(ξ, η)

∂η=

1

2

(

−E1(ξ) + (1 + ξ)∂E2(η)

∂η+X3 + (1− ξ)

∂E4(η)

∂η

)

− 1

4

(

(1− ξ)X1 − (1 + ξ)X2 + 2X3

)

. (2.38)

2.2.6 Remark on the consistency of blended elements in p-FEM

One important feature of p-FEM applications is that the shape functions are defined in thelocal parameter space (ξ, η) of the finite elements that are conforming with the geometricboundaries. Since the solution of the problem is represented as the linear combination of theshape functions in the global space, the mapping Q(ξ, η) that maps the shape functions fromthe local to the global space has a major influence on the quality of the solution. For example,consider a linear shape function N(ξ, η) in the parameter space of the element and its mapin the global space N (ξ(x, y), η(x, y)). The mappings ξ(x, y) and η(x, y) are the inverses ofthe mapping Q(ξ, η). Since Q(x, y) is generally not linear, the shape function that is linearin the parameter space becomes nonlinear in the global space. Therefore, a shape function


of polynomial order p in the parameter space (ξ, η) is in general not reproduced to an orderp polynomial in the global (x, y) space. This loss of consistency can lead to cases where asolution of order p in (x, y) is not representable by shape functions of order p [16].

The opposite is also true: if a solution u(x, y) is not a polynomial in the global space, but itsmap u(x(ξ, η), y(ξ, η)) is a polynomial with order p in the parameter space of the elements,then u(x, y) can be represented exactly if the polynomial order of the shape functions ischosen to be p.

There are finite element techniques that avoid this loss of consistency: the so-called CartesianFEM uses shape functions defined in the global (x, y) space and integrate on their maps in theparameter space N(x(ξ, η), y(ξ, η)). Cartesian FEM approximates the geometry by isopara-metric mappings and therefore it is consistent but geometrically not exact. An extension ofthis idea is the NURBS-enhanced finite element method (NEFEM) that uses NURBS curvesdefined in the global space as shape functions (usually available from CAD geometry files)[16]. NEFEM captures the geometry exactly and remains consistent at the same time [16].

The influence of the mappings on the polynomial representability of the solution is addressedbriefly in the following two examples.

Poisson problem on a rectangular domain

Consider a rectangular domain, depicted in Figure 2.8. We are solving the steady 2D Poissonequation

∂2u

∂x2+

∂2u

∂y2+ s = 0 (2.39)

with the source term s = −2 and boundary conditions:

uΓ1= 2 + x2

uΓ2= 3

uΓ3= 2 + x2

uΓ4= 2.

(2.40)

The analytical solution of Equation 2.39 with the boundary conditions is

u(x, y) = 2 + x2. (2.41)

The internal energy for this problem is U = 23 . Using a mesh of 2× 2 elements with straight

edges and ansatz order p = 2 (Figure 2.9), the solution is represented exactly and the errorin the strain energy is in the order of numerical precision. Consider now the same mesh butreplace one straight edge of each element with a 2nd order B-Spline with one knot span (i.e.a Bezier curve) and use blended mappings for these new elements. For representing the exactsolution with this mesh of blended elements (Figure 2.9), choosing p = 2 is not satisfactory.Only p = 4 is sufficient.

The reason behind this phenomenon can be found by the following considerations: imaginean element in one dimension with the local parameter space ξ. The mapping Q(ξ) maps the


1

1x

y

Γ1

Γ2

Γ3

Γ4

u(x, y)

Figure 2.8: Poisson problem on a rectangular domain

(a) Linear elements (b) Blended elements

Figure 2.9: 2× 2 meshes of finite elements

parameter ξ to the global space x. For nonlinear mappings, we can say that

x ∼ ξq (2.42)

where q denotes the order of the mapping Q. If the solution u in the global space is of orders, we have

u ∼ xs ∼ (ξq)s ∼ ξqs. (2.43)

Let p denote the order of the shape functions defined on the parametric domain of the element:

N ∼ ξp. (2.44)

Bringing equations 2.43 and 2.44 together implies that the following inequality has to hold ifthe shape functions have to represent the solution exactly:

q · s ≤ p. (2.45)

Note that for isoparametric elements p = q, thus they cannot pass the higher order patchtest with s > 1 [16][17].

Returning to the Poisson problem, the same sequence of reasoning can be applied. If theelement is blended by a 2nd order B-Spline of Figure 2.9, we have for the mapping:

x(ξ, η) ∼ ξ · η2

y(ξ, η) ∼ ξ · η.(2.46)


The solution is 2nd order in x and constant in y, thus

u(x, y) ∼ x2 · y0 ∼ (ξ · η2)2 · (ξ · η)0 ∼ ξ2 · η4 (2.47)

This means that choosing p = 2 as ansatz order is not sufficient to represent the exact solutionin the parametric η direction, which in our case coincides with the global y direction. Therequirement is to have p = 4, which is also confirmed by the mentioned numerical study.

Poisson problem on a curved domain

x

y

r

ϕ

Γ1

Γ2

Γ3

Γ4

0.5

1

Figure 2.10: Poisson problem on a circular domain

There are cases where solutions that are not representable by polynomials in the global spacecan still be represented exactly. To show this, let us take a Poisson problem on a curveddomain, depicted on Figure 2.10 The Poisson equation in polar coordinates reads:

1

r

∂

∂r

(

r∂u(r, ϕ)

∂r

)

+1

r2∂2u(r, ϕ)

∂ϕ2+ s = 0. (2.48)

If we take the source term s(r, ϕ) = −4 + 1rand the boundary conditions as

∂u

∂r

∣

∣

∣

∣

Γ1

= 0

u|Γ2

=1

4∂u

∂r

∣

∣

∣

∣

Γ3

= 0

∂u

∂r

∣

∣

∣

∣

Γ4

= 0

(2.49)

then the solution is:u(r, ϕ) = (r − 0.5)2. (2.50)


In Cartesian coordinates, the solution is not a polynomial expression:

u(x, y) =(

√

x2 + y2 − 0.5)2

. (2.51)

If the mesh is composed of one blended element, whose sides are circular arcs, the mappingQ(ξ, η) reads as:

Qx(ξ, η) =1

4r cos

(πs

4+

π

4

)

+3

4cos(πs

4+

π

4

)

Qy(ξ, η) =1

4r sin

(πs

4+

π

4

)

+3

4sin(πs

4+

π

4

)

.

(2.52)

Applying this mapping on Equation 2.51 the solution in the parameter space (ξ, η) is:

u(ξ, η) = −ξ2 + 2ξ − 1

16, (2.53)

which is a polynomial in ξ. This means that it is sufficient to choose p = 2 in order to representthe exact solution with one element. This is also confirmed by the numerical solution depictedin Figure 2.11.

Figure 2.11: Solution of the Poisson problem with p = 2, one blended element

2.3 Numerical considerations

As this discussion focuses on integration by utilizing the parametric description of geometricentities, in the following the numerical aspects of working with parametric curves and blendedelements are given.

2.3. Numerical considerations 19

2.3.1 Numerical integration of parametric curves

The length of a curve is given by the integral of the magnitude of the tangent vector betweenits parametric bounds [18]:

L =

∫ t1

t0

∣

∣

∣

∣

∂xc(t)

∂t

∣

∣

∣

∣

dt =

∫ t1

t0

√

(

∂x(t)

∂t

)2

+

(

∂y(t)

∂t

)2

+

(

∂z(t)

∂t

)2

dt. (2.54)

This integral can be evaluated numerically using Gaussian quadrature.

As an example, consider the example NURBS circle and compare it with its classical sine-cosine representation:

Cclassical(t) =

[

1 · cos(t)1 · sin(t)

]

0 ≤ t < π. (2.55)

In order to employ the Gaussian quadrature on interval u ∈ [−1, 1], we apply a mapping onparameter t:

t(u) =π

2+

π

2u u ∈ [−1, 1]. (2.56)

Thus the curve length integral of Equation 2.54 becomes:

L =

∫ 1

−1

√

(

∂xc(u)

∂u

)2

+

(

∂yc(u)

∂u

)2

du, (2.57)

where ∂x(u)∂u

is a term from Equation 2.15. Similarly, the parametric bounds of the NURBScircle are mapped to [−1, 1]. The magnitude of the tangent vectors for the two representationsis depicted in Figure 2.12.

0.2

1.4

1.45

1.5

1.55

1.6

1.65

1.7

-1 -0.8 -0.6 -0.4 -0.2 0-1 -0.8 -0.6 -0.4 -0.2 0 0.4 0.6 0.8 1

u

∂Cclassical(u)∂u

∂CNURBS (u)∂u

Figure 2.12: Comparison of tangent vector magnitudes for a classical circle and its NURBS repre-sentation

Figure 2.12 implies that the major difference between the sine-cosine and the NURBS rep-resentation of a circle lies in the derivatives: while the magnitude of the tangent vectoris constant throughout the whole parameter domain, the NURBS representation shows os-cillations in [−1, 1]. As the Gaussian quadrature can be exact only if the integrand is acontinuous polynomial, and the NURBS circle is a piecewise rational function, the precision


of the quadrature deteriorates.

Let us examine the error in the curve length in relation to the number of Gauss points. Theerror is defined by the following formula:

e =|Lexact − Lnumerical|

Lexact. (2.58)

t0 . . .. . . tn−1 tn+1 tm

−1 10

Figure 2.13: Mapping of quadrature points to knot spans

Since B-Splines are piecewise polynomials, the standard Gaussian quadrature is not ableto integrate the curve precisely enough. Thus a different approach has to be applied: thequadrature points are distributed on each knot span of the curve. Then they are mapped tothe parameter space I (refer to Figure 2.13). The performance of this approach is superiorto the standard Gaussian point distribution, which can be seen on the error plot as well(Figure 2.14).

Error in

cu

rve len

gth

1−10

1−05

1

Number of Gauss points

0 5 10 15 20

Quadr. points in parameter space

Quadr. points knotspan-wise

Figure 2.14: Error in curve length of a NURBS curve with Gaussian quadrature

Note that the integral cannot be exact even with a relatively high number of Gauss points,because on the one hand, the NURBS curve is a piecewise rational function, and on theother hand, the length of a tangent vector is a square root expression, while the Gaussianquadrature is only exact for polynomials.

2.3. Numerical considerations 21

2.3.2 Numerical integration of blended elements

Section 2.3.1 discussed some numerical considerations when dealing with parametric curves.As parametric curves are directly used in the formulation of the blended elements (Equa-tion 2.32 and 2.36), the numerical aspects of the integration of blended elements are investi-gated in the following.

(−1,−1) (1,−1)

(1, 0)

(0, 1)

(−1, 0)

E3(t)

Figure 2.15: A quad with one circular edge

As an example, consider the curves of Section 2.2.3 (Equation 2.55 and 2.28) and extendthem to form a quadrilateral with one curved edge (Figure 2.15).

The area of the quad can be computed by the Gaussian quadrature:

A =

∫ 1

−1

∫ 1

−1‖J‖dξdη =

ni∑

i=1

nj∑

j=1

‖J‖∣

∣

∣

∣

(ξi,ηi)

wξiwηi , (2.59)

where ni and nj are the number of quadrature points in the ξ and η direction, and wξi andwηi are their corresponding weights.

The arguments here are similar to the discussion in Section 2.3.1. The piecewise rationalnature of the NURBS curve invalidates the exactness of the Gaussian quadrature, thus asubdivision in the (ξ, η) space has to be introduced for the distribution of Gauss points, similarto the method that was discussed in Section 2.3.1. This is done such that the parameter space(ξ, η) of the quadrilateral is decomposed into a set of smaller quadrilaterals. The edges ofthe smaller quadrilaterals are determined by the knot locations of the bounding curves. Inour example, the position of the single knot on curve E3(ξ) is at ξ = 0.0, thus the parameterspace is cut in halves. The Gauss points are then distributed in the parametric space (ξ, η)of these smaller quadrilaterals. This is shown in Figure 2.16.

x

y

E2(ξ)

E2(0)

Q(ξ, η)ξ = 0

ξ

ηξ

η

Figure 2.16: Subdivision of parameter space along a knot


The exact area of the quadrilateral is A = 2+ π2 . If the curve E3(ξ) is replaced by a B-Spline

that was defined in Equation 2.25, the exact area is A = 2+ 106 . The relative error in the area

integration is given similarly to Equation 2.58. A plot of errors with different quadraturepoint distributions for the NURBS, B-Spline, and circular arc represented boundaries areshown in Figure 2.17.

Error in

area

−15

−10

−05

1


0 100 200 3

NS notspan- adr. points

NS adr. points in parameter space

-SB adr. points in parameter space

-SB notspan- adr. points

Ac . points in parameter space

Figure 2.17: Error in the area integral of a blended quadrilateral for different curves and quadraturepoint distributions

If the quadrature points are distributed in the parameter space of the quadrilateral, theintegral does not get exact even for a high number of quadrature points. However, if theintegration points are distributed in the knotspan-wise subdivided parameter space, the in-tegral becomes exact. For the quadrilateral with a NURBS edge, the error converges to thenumerical precision. In comparison, the area of the quadrilateral with a B-Spline edge isexact with 2 × 2 Gauss points in each knot span. The difference comes from the piecewiserational nature of the NURBS curves [16].

In general, for piecewise polynomials the integration is exact if p+12 or p+2

2 quadrature pointsare used per dimension per knot span, when p is odd or even, respectively [19]. A furtherdiscussion about the numerical treatment of integrals involving NURBS can be found in[19][20].

2.4 The Formulation of the Finite Cell Method

The following section presents the essential ideas of the Finite Cell Method. For a deeperdiscussion refer to [3][4][21][22].

2.4.1 Variational formulation

Considering the physical domain Ωphy of Section 2.1, we extend Ωphy with the fictitiousdomain Ωfict. The extension is done in such a way that Ωphys∪Ωfict = Ω∪ results in a simplegeometry that can be meshed easily (Figure 2.18).

2.4. The Formulation of the Finite Cell Method 23

Ωphy

tt = 0on ∂Ω∪

ΓN

ΓD

Ωfic

Ω∪=Ωphy ∪ Ωfic α = 1.0

α = 0.0

Figure 2.18: The core concept of the fictitious domain approach: the physical domain Ωphys isextended by the fictitious domain Ωfict such that their union forms the embedding domain Ω∪. Thisallows the creation of a structured grid. To extinguish the influence the fictitious domain, the materialparameters of Ωfict are penalized by α [21]

In the context of the FCM, the principle of virtual work (Equation 2.2) takes the followingform:

δW (u, δu) =

∫

Ω

σ : (∇symδu)dV −∫

Ωphy

δu · bdV −∫

ΓN

δu · tdA = 0. (2.60)

On the boundary of Ω∪ Neumann boundary conditions are applied. On ΓN of the physicaldomain Neumann boundary conditions are specified by the traction vector t. The constitu-tive tensor C is penalized by the factor α in Ωfict in order to extinguish the influence of thefictitious domain. The stress-strain relationship of Equation 2.1 thus becomes:

σ = αC : ε (2.61)

with α defined such that:

α(x) =

1 ∀x ∈ Ωphy

10−q ∀x ∈ Ωfict.(2.62)

For numerical computations, the value of α on Ωfict should be chosen as small as possible,but large enough to prevent extreme ill-conditioning of the stiffness matrices of elements thatare partly lying in Ωfict, also called as cut cells. Typically α is chosen in the range between10−4 to 10−15 [3][22].

The main difference in comparison to p-FEM is that a structured mesh of rectangular finiteelements is created on Ω instead of a conforming mesh on Ωphy. The elements of the structuredmesh are called finite cells that can be represented by the nodal modes alone:

X =nvert∑

i=1

Nni Xi, (2.63)

where nvert are the corner vertices of the finite cells, andNi andXi denote their correspondingnodal modes and locations in the global coordinate system, respectively. This means thatthe Finite Cell Method can be classified as a sub-parametric finite element scheme [13].


2.4.2 Adaptive integration of cut cells

In general, the integral of the stiffness matrix (Equation 2.6) cannot be evaluated analytically.Therefore, the Gaussian quadrature [7] is applied as a numerical integration scheme. For theelement stiffness matrix in equation 2.6, the integration performed by Gaussian quadratureis:

ke ≈nj∑

j

ni∑

i

(LN )TC(LN )‖J‖, |ξi,ηiwiwj (2.64)

where ξi and ηi are the location of the quadrature points in local coordinates of the cell, wi

and wj are their weights.

Due to the discontinuity that is introduced by penalizing the constitutive matrix C (Equa-tion 2.62), the Gaussian quadrature loses its accuracy in cut cells [3][22]. In order to improvethe accuracy of the numerical integration, FCM uses composed Gaussian quadrature that isbased on hierarchical decomposition of the cells that are cut by the domain boundaries. Intwo dimensions, the structure of the decomposed cells is generated in a sense of a quadtree.Starting from the initial finite cell mesh with a sub-cell level of k = 0, each cell with sub-celllevel k = i is checked whether it is cut by the domain boundary or not. If it is cut, thecell is subdivided into 4 cells of equal size whose sub-cell level will be k = i + 1 . If thecell is not cut, (p+ 1)× (p + 1) quadrature points are distributed on its domain, where p isthe polynomial order of the shape functions. The partitioning is repeated until a predefineddepth of k = m is reached (refer to Figure 2.19). This approach can be analogously extendedto three dimensions, in which case the resulting sub-cell structure is represented by an octree.The distributed Gauss points are mapped to the local parameter space of the finite cell, sincethe integration is performed in this space. This means that the ‖J‖ term in Equation 2.64 isthe product of the Jacobian determinants of the mappings Q(ξ, η) and Q(ξi, ηi), where theterms with ∼ denote the local coordinates of the integration cell and the mapping from thelocal space of the integration cell to the local space of the finite cell. The concept of thesetwo mappings is depicted in Figure 2.20.

finite cell mesh

k=0 k=1 k=2

k=3 k=4 k=5

with geometric

boundary

Figure 2.19: The method of quadtree partitioning. Each cut cell is recursively subdivided until apredefined subdivision depth is reached [21].

2.4. The Formulation of the Finite Cell Method 25

x

y

ξ

η

ξ

η

Q(ξ, η)

Q(ξ, η)

Figure 2.20: The concept of mappings for k=1 quadtree depth

The spacetree decomposition has already proven to be a viable approach for performingsimulations on geometries where the geometric information is available implicitly or in aform of a voxel based model [4]. The two main advantages of the spacetree decompositionare that it is easy to implement and it can operate on any geometry.

The drawback of subdividing the cut cells using the quadtree approach is that increasingthe maximum level of subdivision m can lead to exponential growth in the number of Gausspoints. This results in a dramatic increase in the computational effort that is needed forcomputing the stiffness matrices of cut elements. Furthermore, the quadtree approach, beinga 0th order approximation of the geometry, influences the good convergence characteristicsthat could be achieved by higher order shape functions [5].

There have been investigations to find a better method for distributing the integration pointsin the cut cells. One possible approach is to use Lagrangian elements whose boundaries areapproximating the geometric boundary in the cut cells and distribute the Gauss points inthe parameter space of these Lagrangian elements [5]. A similar approach has been appliedto XFEM applications, where elements that are cut by the interface are decomposed suchthat the interface conforms with the boundaries of the resulting subcells [6]. For XFEM,the use of NURBS-enhanced finite elements as possible subcells in the cut elements was alsoinvestigated [23].

The present thesis discusses another possible remedy for the challenge of integration: similarlyto the approximation with Lagrangian elements, the cut cells are going to be decomposedinto triangles and quadrilaterals with curved boundaries. The quadrature points distributedin the local parameter space (ξ, η) of these elements are then to be mapped to the parameterspace of the finite cell (ξ, η) using blended mappings as Q(ξ, η) (Chapter 2.2). This conceptis depicted on Figure 2.21.

x

y

ξ

η

ξ

η

Q(ξ, η)

Q(ξ, η)

Figure 2.21: The concept of cut cell integration with blended mapping of a curved quadrilateral

27

Chapter 3

Algorithmic partitioning of cut cells

As stated in the previous chapter, the aim of this work is to decompose the cut finite cellsinto quadrilaterals and triangles with curved edges. The quadrature points distributed intheir local parametric space are mapped to the parametric space of the cut finite cells byblended mappings. In comparison to the quadtree decomposition, this method will use theparametric definition of the boundary directly, thus giving a high accurate approximationof the geometry with possibly less integration cells. On the computational side this meansless quadrature points for the same precision, and because of that, less overall runtime. Theproposed approach for partitioning is depicted on Figure 3.1

x

y

Partitioning algorithm

Figure 3.1: The concept of the proposed partitioning algorithm

The spacetree decomposition relies on simple inside-outside tests, and this is its main advan-tage: it is robust enough to work on geometries with any complexity. Its disadvantage comesfrom the fact that it leaves the rest of the geometric information (if availableq) unexploited.Because of that, it requires a high level of subdivision depth to capture fine details of thegeometry.

The algorithm that performs the decomposition to blended cells has to be as robust as thespacetree method, but should exploit the available geometric information on a higher level.If the requirements of robustness and accuracy are fulfilled, the algorithm would be a betterfit to the design-through-analysis pipeline, bringing CAD and FEA closer together.

Working with higher level geometry information poses a number of challenges, such as:

• When non-convex boundaries are cutting the cell, what configuration of blended quadri-laterals and triangles can be created, if there is any?

28 3. Algorithmic partitioning of cut cells

• How to deal with boundaries that cut a cell more than once?

• How to treat cases, where there are more boundaries cutting one cell?

The first two of these points are depicted in Figure 3.11.

Dealing with cut rectangles is a well-known problem in computer graphics. The standardapproach to deal with them is the marching squares method [24] which is briefly presentedin the following.

3.1 Marching squares approach

Ω

Ω

ΩΩ

Ω

Ω

Ω

ΩΩ

ΩΩ

ΩΩ

ΩΩ

ΩΩ

1

1

1

1

1 1

11

1

1

1 11

111

1

11

1

11

1 1

1

1 1

1

11

11

0

0

0

0

0 0

00

0

000

0

0

0 0

00

00

0

00

0

00

0 0

0

0 0

0

Adjacentcu

tcases

Oppositecu

tcases

Figure 3.2: The possible cases of cuts in 2D

Given a quadrilateral (e.g. a finite cell) in two dimensions and a closed domain Ω with theboundary ∂Ω, the four corner vertices of the quadrilateral are evaluated whether they lieinside or outside Ω. Let ai denote the inside-outside state of the ith corner vertex of the

3.2. Triangulated cut cell approach 29

quadrilateral, and Xi its (x, y) coordinates:

ai =

1 Xi ∈ Ω

0 Xi 6∈ Ω(3.1)

As the quadrilateral has 4 corners, and each corner can have 2 states, the number of possiblecombinations is 24 = 16. Figure 3.2 shows these 16 cases with the corresponding ai values atthe cell corners. The first two rows show cases where adjacent edges are cut. The third rowdepicts cases where opposite edges are cut. The first two elements in the fourth row showquadrilaterals that lie completely outside or completely inside Ω. The last two elements inthe fourth row are cases where two pairs of adjacent edges are cut.

One possible approach to decompose the cut cell into triangles and quadrilaterals is to assigneach case an integer number and introduce an if-else or a switch-case structure that han-dles these cases separately. Although this method seems to be straightforward and easy toimplement, it has two major disadvantages:

1. It results in a massive algorithmic overhead. Its extension in 3D is even more complex:finite cells in 3D are hexahedra with 8 corner vertices, thus the number of possible casesrise up to 28 = 256.

2. It does not take into account all the problematic cases mentioned in the beginning ofthis chapter. This is no surprise as the marching squares method was designed for fieldvalue interpolation and not for cell decomposition.

These drawbacks indicate that the marching squares approach is not able to fulfill the cri-teria of robustness. Therefore, Section 3.2 will introduce a method with a higher level ofabstraction, such that the task can be handled in a more generic way.

3.2 Triangulated cut cell approach

The idea is that the cut cases can be reduced to one general case if the quadrilateral is firstintersected along its diagonal. The resulting pair of non-overlapping triangles is always cutby the boundary into a quadrilateral and a triangle. This approach is depicted in Figure 3.3.Obviously there are cases where this idea does not apply. For example, if the geometry is toocomplex, or there are multiple intersections between the boundary and the edges of the cell.These cases can be dealt with by subdividing the cell into four equal subcells, and performingthe partitioning again (Section 3.2.7). By subdivision, the algorithm is always able to reducecomplicated cell cuts to the general case.

3.2.1 Numbering and naming conventions

For the description of the partitioning algorithm, the following naming and numbering con-ventions are going to be used (refer to Figure 3.4):

• Edges of the cell are denoted by ei, where i = 0, 1, 2, 3.


∂Ω

∂Ω

TQ

Figure 3.3: Examples of triangular decomposition. T=triangle, Q=quadrilateral

• Corner vertices of the cell are denoted by vi, where i = 0, 1, 2, 3.

• In order to end up with triangles and quadrilaterals, an auxiliary point is introduced(denoted by v4). The diagonal edge that connects two adjacent corners through theauxiliary point is called auxiliary line (denoted by ea).

• The boundary that is cutting the cell is denoted by eb, the intersection pointsbetween the boundary and the cell edges are denoted by v5 and v6. If there are moreboundaries, their notation is eb0 , eb1 , . . . , ebn , where n is the number of boundaries inthe cell.

The task of the partitioning algorithm is to find for each cell the vi vertices using the availablegeometric information, then connect them in such a way that from the resulting edges a setof non-overlapping blended triangles and quadrilaterals can be formed. These triangles and


v0 v1

v2v3

v4

v5

v6

e0

e1

e2

e3

ebea

Figure 3.4: Numbering and naming conventions for the partitioning algorithm

quadrilaterals are then mapped to the parameter space of the finite cell, and quadraturepoints are distributed on their parametric space (refer to Figure 2.21).

Therefore, the algorithm has to perform the following operations:

1. Determine if a cell is cut by the boundary or not. If the cell is not cut, return the cellitself. If it is cut, proceed.

2. Find the edges ek and el that are cut by the boundary and compute the intersectionsek ∩ eb = v5 and el ∩ eb = v6.

3. Find the auxiliary point v4.

4. Connect the resulting set of points.

5. Based on the result of step 4, form blended triangles and quadrilaterals.

6. Map the triangles and quadrilaterals to the parameter space of the finite cell.

Obviously, there are a lot of cases when this algorithm can fail. Usually problems can occurif the geometry is too complex and there is no subdivision that can result in a set of wellconditioned triangles and quadrilaterals (e.g. there are more than one intersection betweenthe boundary and an edge). Therefore, every step has to include a self-test, to check whetherthe overall flow of the algorithm can proceed. If not, a subdivision is performed on the cellsimilarly to the quadtree decomposition and the steps are repeated on the decomposed cells.

3.2.2 Determining whether the cell is cut

The first step is to determine whether the cell is cut by the domain boundary ∂Ω. To this end,the algorithm distributes n× n seed points on the domain of the cell, where n is the numberof points in one direction. Starting from the first point, it iterates through the distributedseed points and checks whether the point is inside or outside Ω, similarly to Equation 3.1.If there is a change in the inside-outside state between two points, the domain is cut by theboundary, and the algorithm proceeds to the next step. If all the points are inside or outside,the cell is not cut, and the algorithm returns the unpartitioned cell.


3.2.3 Determining which edges are cut

If the cell is cut by the boundary, the next step is to determine which edges of the cell areintersected. Let us assign a pair of integer numbers (o, p) to every corner vertex vi of thecell, such that o and p are the numbers of the two edges meeting at vi. This is depicted onFigure 3.5.

v0 → (3, 0) v1 → (0, 1)

v2 → (1, 2)v3 → (2, 3)

Figure 3.5: (o, p) coding of corner vertices

Starting from vertex v0 and an empty vector of integer numbers O, evaluate each cornervertex vi, whether it lies in Ω or not. If vi is in Ω, add its corresponding (o, p) values tothe end of the vector O. After the last corner v3 is evaluated, remove the entries that areduplicated in O. Depending on the remaining elements of O, there are four scenarios:

• If there are no entries in O, all the corner vertices are inside or outside the domain,although the cell was detected to be cut by the boundary. In this case, the cell has tobe refined (Figure 3.6a).

• If there are four entries in O, we are dealing with a case where two pairs of adjacentedges are cut. (These are the last two cases in the fourth row in Figure 3.2). In thiscase, the cell has to be refined (Figure 3.6b).

• If there are two entries in O, they denote the number of the edges that are cut by theboundary, the algorithm can proceed (Figure 3.6c).

After filtering the duplicates, the remaining entries in O are reordered such that they followeach other in a counterclockwise order.

3.2.4 Creating the auxiliary point and line, intersecting the edges

After reordering, the vertices of the cell are renumbered, such that v0 is the vertex whosenumber is the first element in the vector O. This vertex is the point where the auxiliary lineea(Figure 3.4) starts. The end point of the auxiliary line is v2. The location of the auxiliarypoint v4 depends on the number of boundaries intersecting the cell:

• If there is only one boundary in the cell, v4 is the intersection between the auxiliaryline and the boundary vb (Figure 3.7a).


Ω

O = ∅

O = ∅

Remove duplicate entries

(a) Boundary cuts cell, but all corners are outside

Ω

Ω

O = [3, 0, 1, 2]

O = [3, 0, 1, 2]


(3, 0)

(1, 2)

(b) All edges are cut

Ω

O = [3, 0, 1, 2, 2, 3]

O = [0, 1]


(3, 0)

(1, 2)(2, 3)

(c) Two edges are cut

Figure 3.6: Examples of collecting the intersected cell edges

• If there are two boundaries, v4 is the point where the two boundaries meet. (Figure 3.7b)If they do not meet at the same point, the cell is refined.

• If there are more than two boundaries, the cell is refined.

After vb is found, the algorithm checks whether the auxiliary point is the only intersectionbetween the auxiliary line and the boundary (or boundaries). If there are more intersections,the cell is refined.

The last two remaining vertices to find are v5 and v6 (Figure 3.4). To this end, the algorithmcomputes the intersections ek ∩ eb and el ∩ eb where k and l is the first and second elementin the vector O, respectively. If a boundary intersects an edge more than once, the cell isrefined.

3.2.5 Create blended triangles and quadrilaterals

When all the vi vertices are found, everything is set up for creating the set of triangles andquadrilaterals. To determine which vertices will form a triangle and a quad, the algorithm


Ω

O = [1, 0]

v3 v0

v1v2

v5

v6v4

ea

(a) One boundary cuts the cell

Ω

O = [1, 0]

v3 v0

v1v2

v5

v6

v4

ea

eb1

eb2

(b) Two boundaries cut the cell

Figure 3.7: Creating the auxiliary line and auxiliary point

checks if the cut edges are adjacent or not.

• For adjacent cut edges, the vertices [v0, v4, v5] and [v0, v6, v4] form two blended triangles,and vertices [v2, v3, v5, v4] and [v1, v2, v4, v6] form two quadrilaterals (Figure 3.9a).

• For opposite cut edges, vertices [v0, v4, v5] and [v2, v4, v6] form two blended triangles,and vertices [v2, v3, v5, v4] and [v0, v1, v6, v4] form two quadrilaterals (Figure 3.9b).

The last two vertices of these vectors are the locations where the boundary eb (or eb1 and eb2)is trimmed.

eb(tn)

v4

v0

v1v6

η =tn

η

ξQ1

Q2

Figure 3.8: Example of subdivision along knot span. The quadrilateral is subdivided along ξ = tn,where tn is the location of a knot on the boundary eb. The resulting quadrilaterals are Q1 and Q2


According to Section 2.3.2, the precision of the integration is maintained if the integrationis performed knotspan-wise on the blended elements. To this end, the created quadrilateralsand triangles have to be decomposed along the knot locations (if there are any) of the blendedcurves. An example of this decomposition for the blended quad [v0, v1, v6, v4] of Figure 3.9bis depicted on Figure 3.8, assuming that the boundary has one knot that is located at tn.

v3 v0

v1v2

v5

v6v4

ea

eb

(a) Adjacent edges are cut

v3 v0

v1v2

v5

v6

v4

(b) Opposite edges are cut

Figure 3.9: Numbering of different cuts for creating blended elements

3.2.6 Mapping to the parameter space of the finite cell

The blended integration cells are created in the global (x, y) space, but the aim is to find themapping that maps the parameter space (ξ, η) of the integration cell to the parameter space(ξ, η) of the finite cell (Figure 2.21). This is relatively easy to do. The mapping P ((ξ, η)) isprovided by the blending function method. Since the finite cell is rectangular, the mappingQ−1(ξ, η) that maps the global (x, y) coordinates to the local (ξ, η) coordinates of the cellcan be given in a closed form. The composition Q = Q−1 P yields the sought mapping.This is depicted in Figure 3.10.

x

y

ξ

ηξ

η

P (ξ, η)Q−1(ξ, η)

Q(ξ, η)

Figure 3.10: Mapping the blended subcell to the parameter space (ξ, η)


Non-convex boundarycuts the cell

The cell is subdividedinto four subcells

General case is constructedin the subcells

(a) Non-convex boundary cuts the cell

Boundary has two intersectionswith a cell edge

The cell is subdividedinto four subcells

General case is constructedin the subcells

(b) A boundary intersects a cell edge twice

Figure 3.11: Examples of subdivision for too complex geometries

3.2.7 Subdivision for too complex geometries

As it was mentioned a couple of times through this chapter, whenever the algorithm is notable to proceed because of the geometry is too complex, a subdivision takes place. This issimilar to the subdivision in the quadtree sense, i.e. the quadrilateral is cut into four equalpieces, then the algorithm performs the same operations described in Sections 3.2.2-3.2.5 onthese sub-quadrilaterals. This is done recursively, until either the triangles and quadrilateralscan be created or a predefined subdivision depth is reached.

3.3 Implementation of the concept

The algorithm described in Section 3.2 was implemented and tested in AdhoC++, which isa high order FEM code developed at the Chair for Computation in Engineering (CiE). Orig-inally, the geometry kernel of AdhoC++ was restricted to linear geometric entities. Thusthe code was extended by the geometry features provided by TUM.GeoFrame, a geometryand meshing framework developed also at CiE. The geometry kernel of TUM.GeoFrame isbased on OpenCASCADE, an open source CAD library for C++. OpenCASCADE pro-vides parametric curves including B-Splines and NURBS, as well as methods for computingintersections of curves.

37

Chapter 4

Patch test of the integration cells

In order to assess the quality of the blended decomposition of the geometry, a patch test isintroduced. The idea of the test is to solve a Poisson equation in the context of FCM, butchoose α = 1 on both Ωphy and Ωfic (Equation 2.61). Thus, the domain of the problemreduces to a rectangle whose boundaries are the boundaries of Ωfict. However, on the inte-gration level, the elements of the stiffness matrix (Equation 2.64) are calculated using theblended cells that are created by the blended decomposition of Section 3.2. The boundaryconditions are chosen such that:

u∣∣

Γ1

= 0

∂u

∂x

∣

∣

∣

∣

Γ2

= 0

u∣∣

Γ3

= h

∂u

∂x

∣

∣

∣

∣

Γ4

= 0,

(4.1)

where h denotes the height of the fictitious domain. The solution is:

u(x, y) = y, (4.2)

and the strain energy is:

Uex =wh

2(4.3)

where w is the width of the domain. (Figure 4.1)

The error of the integration will be assessed by measuring the error in the strain energy by:

e =|Uex − Unum|

Uex, (4.4)

where Uex is the exact strain energy (Equation 4.3) and Unum is the numerically calculatedstrain energy.

38 4. Patch test of the integration cells

Γ1

Γ2

Γ3

Γ4

α = 1

α = 1

Ωphy

Ωfic

w

h

Figure 4.1: Schematic setup of the integration tests. α = 1 on Ωphy and Ωfic.

Ωphy

Ωfic

α = 1

α = 1

1

1

partitioning

Figure 4.2: Example of a partitioned circular domain

4.1 Curved domain

As a first test, the partitioning of the curved domain from Section 2.10 is investigated. TheFinite Cell Mesh is composed of 4 × 4 cells, the polynomial order is chosen to be p = 1,since the solution is linear in y and constant in x. The curved domain is set up in twodifferent configurations: in the first one the domain is bounded by circular arcs with sine-cosine representation (similarly to Equation 2.55). In the second case, the curved edges arerepresented by NURBS (similar to Equation 2.28). As the NURBS boundaries are quartercircles, their parameter space is composed of only one knot span, thus no knotspan-wisesubdivision of the cells is required. The geometry setup and the resulting integration meshof blended triangles and quadrilaterals is depicted in Figure 4.2

The error in strain energy as a function of number of Gauss points in one direction is depicted

4.2. Rectangular domain with moving circle 39

in Figure 4.3. Note that the configuration with the conventional sine-cosine arc parametriza-tion converges faster to numerical precision than the NURBS parametrization. This is similarto what was found in Section 2.3.2.

Error in

str

ain

en

erg

y

−15

−10

−05

1

Number of Gauss points in one direction

1 2 5 6

ar

Figure 4.3: Error in strain energy for the circular domain integration test

4.2 Rectangular domain with moving circle

2

2

Ωphy

Ωfic

1

2

3

8

β

Figure 4.4: Geometry setup of the moving circle integration test

In the second test, Ωfic is defined as a circular hole moving on a circular path inside therectangular Ωphy. As in every time step, a different geometrical configuration has to bepartitioned, it can be assessed, how the algorithm copes with non-regular geometries. A


(a) β = 0 (b) β = 45

(c) β = 202.5 (d) β = 270

Figure 4.5: Results of partitioning for selected values of β

schematic overview of the setup is depicted in Figure 4.4. The moving circle is composed offour sine-cosine represented arcs. The locations, where the arcs join are marked by dots onthe figure.

The center point of the circle is moving on a path defined by:

xc = 1 + cos(β)

yc = 1 + sin(β)(4.5)

The angle β is:

β =2π

Tt, (4.6)

where T is the period of the rotation.

In accordance with the integration test of the previous section, the number of Gauss pointsare chosen to be 5 × 5 on each element, so that partitioned subcells can be integrated with

4.3. Violin with f-holes 41

maximum numerical precision. The results of the partitioning for a few β values are shownon Figure 4.5. The error in the strain energy stays under numerical precision in every timestep.

It should be noted here that the resulting mesh of integration cells should not be looked atas a conventional FEM mesh. Regardless of how distorted the quadrilaterals and trianglesare, the integration is always exact (or converges to an exact value in the case of NURBS) aslong as sufficient number of quadrature points are distributed (Section 2.3.2).

4.3 Violin with f-holes

To demonstrate that the algorithm is capable of working with complex geometries, the coverplate model of a violin is partitioned. The model includes both big and small scales. Boththe outer boundary and the inner f-holes are defined by B-Splines of 3rd order.

80

120

Ωfic

Ωfic

Ωphy

Figure 4.6: Geometric setup for the violin partitioning example

The result of the partitioning is shown in Figures 4.8, 4.9 and 4.10. A comparison of theerror in the strain energy with and without knotspan subdivision is shown on Figure 4.7.

The partitioned violin without knotspan subdivison is depicted on Figure 4.9. Accordingto Figure 4.7, the error in integration becomes exact if subdivision along the knot locationstakes place. As all the bounding curves are B-Splines of 3rd order, it is sufficient to distribute3× 3 Gauss points in every integration cell to be exact. It is important to note here that thetotal number of integration points is influenced by the parametric definition of the curves toa great extent. This means that if there are many knots present in the boundary curves, a lotof knotspan-wise subdivision is performed, therefore a high number of quadrature points isdistributed which results in high overall runtime. Some commercial CAD software solutions(e.g. Rhinoceros 3D) are capable of remodelling curves such that their overall shape remainsapproximately the same, but the parameter space gets simplified. This way, the total numberof quadrature points can be reduced. Since the simplified curve is only an approximation ofthe original one, some precision is lost in exchange for less runtime.


Error in

str

ain

en

erg

y

−15

−10

−05

1

number of Gauss points

0

otspan

notspan

Figure 4.7: A comparison of integration error with and without knotspan subdivision

Figure 4.8: A detailed view of the F-hole after partitioning

4.3. Violin with f-holes 43

Figure 4.9: Partitioned violin without knotspan subdivision


Figure 4.10: Partitioned violin with knotspan subdivision

45

Chapter 5

Finite Cell Method examples

As it was implied in Chapter 3, using blended triangles and quadrilaterals for integrationpurposes reduces the total number of Gauss points distributed on the domain, resulting inless runtime but possibly higher accuracy. In the following, these properties are examined andthe results of the new method are compared with the conventional spacetree decompositionfor 2D problems.

Throughout this chapter, the relative error in the strain energy is computed as:

e =|Uref − Unum|

Uex(5.1)

where Uref is the reference strain energy and Unum is the strain energy obtained by thenumerical simulations.

5.1 Perforated plate

Consider a plane stress problem of linear elasticity, depicted in Figure 5.1. The material ofthe perforated plate is steel, with the properties E = 2.069·105 [MPa], ν = 0.29[−]. The plateis loaded by a vertical load of 100[MPa]. Symmetry conditions are applied on Γ1 and Γ4.The boundaries of the hole and Γ2 are treated as free boundaries. The domain is discretisedinto 2× 2 finite cells, the polynomial degree p of the shape functions is increased from p = 1to p = 12. The reference strain energy of the problem is U = 0.7021812127, obtained by an“overkill” p-FEM solution from [3].

The integration is performed on blended integration cells and on quadtree cells with a depthof k = 4 and k = 5. For comparison, the same problem is solved by means of linear finiteelements (h-FEM) with increasing number of degrees of freedom. Based on the observationsof Section 4.1, (p+4)2 integration points are distributed in the blended integration cells. Forthe quadtree integration cells and the linear finite elements the number of quadrature pointsis chosen as (p + 1)2. The integration cell meshes and a mesh of linear finite elements aredepicted on Figure 5.2. The error in the strain energy is plotted on Figure 5.3.

Both the quadtree and the blended integration shows exponential convergence, similarly top-FEM. However, depending on the number of refinements k, the quadtree integration curve

46 5. Finite Cell Method examples

Γ1

Γ2

Γ3

Γ4

100[MPa]

4

4

x

y

Figure 5.1: Perforated plate problem

levels off after a certain number of degrees of freedom. This is due to the fact that thequadtree refinement is only a 0th order approximation of the geometry and it never followsthe boundaries exactly. In comparison, the blended integration that uses the parametricdescription of the boundaries shows exponential convergence for higher numbers of degreesof freedom withouting leveling off.

It has already been mentioned that the number of quadrature points has a major influenceon the overall computational costs of a numerical simulation. Therefore the relationshipbetween the number of integration points and the relative error in strain energy is also animportant point when it comes to assessing the performance of the different approaches. Thisrelationship is depicted on Figure 5.4. Apart from the better convergence characteristics,another advantage of the blended subcell integration lies in the total number of Gauss pointsdistributed on the domain.

For the same error in the strain energy, the blended integration cells require approximatelyone order of magnitude less Gauss points than a quadtree integration mesh with a depth ofk = 4. If the depth of the quadtree is increased to k = 5, the point where the convergencecurve levels off is slightly shifted to a lower value. However, this gain in precision comes withthe cost of high computational overhead. As an example, consider a relative error of ∼ 10−4

in the strain energy, where the k = 5 quadtree integration mesh needs approximately 40000quadrature points. In comparison, for the same error the blended mesh needs approximately2500 integration points.

Both the blended and quadtree methods show better convergence characteristics in compar-ison to the standard h-FEM on the basis of the number of degrees of freedom (Figure 5.3).Comparing the number of integration points of the different approaches reveals that up toapproximately 1000 integration points the error in the strain energy of the h-FEM solutionis smaller than the error with blended integration. This point is located where the curveof the h-FEM error intersects the curve of the blended integration error on Figure 5.3. Forthe quadtree methods, this intersection with the h-FEM error curve lies in regions of highernumber of Gauss points.

It should be noted here that although the h-FEM delivers the same or better precision up

5.1. Perforated plate 47

(a) Quadtree integration cell mesh, k = 4 (b) Blended integration cell mesh

(c) Mesh of 246 linear quadrilateral finite el-ements

Figure 5.2: Comparison of integration cell meshes and a linear finite element mesh

to this intersection point as the blended subcell integration, it requires considerably moredegrees of freedom than the FCM approach.


Rela

tive e

rror in

str

ain

en

erg

y

−

−06

−05

0.0001

0.001

0.01

0.1

1

Degrees of Freedom

10 100

Quadtr

Quadtrtion

Figure 5.3: Error in strain energy for the perforated plate, p = 1, .., 12

Rela

tive e

rror in

str

ain

en

erg

y

−

−06

−05

0.0001

0.001

0.01

0.1

1

number of Gauss

100 0

Quadtr

Quadtrtion

Figure 5.4: Error in strain energy as a function of number of quadrature points

5.2 Poisson problem on a curved domain

In the following, the Poisson problem of Section 2.2.6 is solved by means of the FiniteCell Method. The results with blended subcell integration are compared to the results ob-tained with quadtree decomposition. The domain is decomposed into 2 × 2 finite cells, and(p+ 4)× (p+ 4) quadrature points are distributed over the domain of each integration cell,where p = 1, . . . , 8. The error in the strain energy against the number of degrees of freedom

5.3. Compression wave on a violin 49

is plotted on Figure 5.5. Note that for this problem the error curve of the blended integrationlevels off as well. This is due to the fact that the the constrained boundaries do not coincidewith the boundaries of the finite cells. Thus, the error due to imposing Dirichlet boundaryconditions in a weak sense is also included in this diagram. A discussion of how to imposeessential boundary conditions in a weak sense for FCM applications can be found in [25].

Error in

str

ain

en

erg

y

06

0.0001

0.01

1

100

!"#tion

Quadtr !"#a$ %=&

Figure 5.5: Error in strain energy as a function of number of degrees of freedom for the Poissonproblem

To compare the total number of Gauss points with the error in strain energy, the sameproblem with a mesh of 2 × 2 finite cells is taken with a polynomial order p = 4. For theblended integration, the number of distributed quadrature points is increased from 2 × 2 to8 × 8. For the quadtree integration, 8 × 8 quadrature points are chosen and the maximumdepth of the tree is increased from k = 1 to k = 6.

Figure 5.6 shows that for the same relative error in the strain energy, there is at least oneorder of magnitude difference in the required number of Gauss points between the blendedand quadtree integration. This difference gets higher as the relative error in the strain energydecreases. The high error of the blended integration up to ∼ 400 Gauss points is due to theinsufficient number of integration points distributed for integrating the blended cell (compareto Figure 4.3).

The solution field u is plotted with both the blended integration cells and the quadtree cellson Figure 5.7.

5.3 Compression wave on a violin

Considering a homogeneous isotropic material, the dynamic equilibrium equation in Cartesiancoordinates is [9]:

ρ∂2u

∂t2= µ∇2u+ (λ+ µ)∇(∇ · u) + s, (5.2)


Error in

str

ain

en

erg

y

'(−10

'(−05

1


100 ')*** '*)***

+,(./(/ 2.5(79:tion

Quadtr(( 2.5(79ation, k;')<<)>

Figure 5.6: Error in strain energy as a function of number of quadrature points for the Poissonproblem, p = 4

where ρ is the density, λ and µ are the Lame parameters, u is the vector of displacements, ands is the vector of source terms. Using the Helmholtz decomposition theorem, the displacementfield can be rewritten as a sum of the gradient of a scalar potential and the curl of a vectorpotential [9]:

u = ∇φ+∇×ϕ. (5.3)

Taking the divergence and the curl of Equation 5.2, respectively, the following two equationshold:

∂2φ

∂t2− α2∇2φ = s (5.4a)

∂2ϕ

∂t2− β2∇2ϕ = s (5.4b)

where

α =

√

λ+ 2µ

ρ(5.5a)

β =

√

µ

ρ. (5.5b)

The solution of Equation 5.4a characterizes a compressional wave (P-wave) advancing in thedomain with a velocity α, and the solution of Equation 5.4b represents a shear wave (S-wave)moving with a velocity β [26].

Considering the violin cover plate model of Section 4.3, the motion of a compressional waveon the domain is studied by solving the PDE of Equation 5.4a using the Finite Cell Method


(a) Quadtree integration mesh

(b) Blended integration mesh

Figure 5.7: FCM solution of the Poisson problem with different integration meshes

and blended subcell integration. The material parameters are chosen as:

ρ = 5460

[

kg

m3

]

λ = 6.52[GPa]

µ = 4.35[GPa]

(5.6)

and the source term is:

s(x, y, t) = 104 · e− x2

2·0.5·0.5 · e−(y−76.33333)2

2·0.5·0.5 ·A(t), (5.7)

which represents a Gaussian bell curve whose maximum is at x = 0, y = 7613 multiplied by a

time dependent scaling term A(t).

A(t) = −108 · (t− 0.005) · e(−5·105·(t−0.005)2). (5.8)


Thus the source term acts on the system as a “spike” excitation. Figure 5.8 shows thelocation where the source has its maximum. Both the outer boundaries and the F-holes arefixed Dirichlet boundaries, where φ = 0. The time integration scheme used for solving theequation of motion is chosen to be the Newmark-beta method [7] with parameters:

γ =1

2

β =1

4.

(5.9)

The duration of the simulation is chosen as tmax = 0.12[s]. The time interval is discretisedinto 120 time steps with constant time step size. The finite cell mesh is composed of 8× 14cells, with polynomial order p = 6.

It is important to note here that an integration mesh with blended elements includingknotspan subdivision (Figure 4.10) and 7 × 7 quadrature points in every integration cellresults in a total of 85860 Gauss points distributed in the whole domain. In comparison, aquadtree mesh with a depth of k = 4 for the violin results in 311542 quadrature points. Con-sidering that in case of a transient simulation, the stiffness matrix might have to be evaluatedat every time step, this could mean a serious difference in the simulation times.

s(x, y) = max.

x

y

Figure 5.8: Location of excitation for the violin example

The following figures show the evolution of φ on the domain over time.


(a) t = 0.005 (b) t = 0.012

(c) t = 0.019 (d) t = 0.026

(e) t = 0.033 (f) t = 0.040

(g) t = 0.047 (h) t = 0.054

Figure 5.9: Values of φ plotted for selected t values


(a) t = 0.061 (b) t = 0.068

(c) t = 0.075 (d) t = 0.082

(e) t = 0.089 (f) t = 0.096

(g) t = 0.103 (h) t = 0.120

Figure 5.10: Values of φ plotted for selected t values (continued)

55

Chapter 6

Summary, conclusion and outlook

The Finite Cell Method is classified as an immersed boundary method: the physical domainof interest is extended by a fictitious domain to a larger embedding domain. This embeddingdomain has a simple geometry that can be meshed easily. The contribution of the fictitiousdomain is extinguished by scaling down its material parameters. In the p-Version of FCM theexact solution is approximated by high order, integrated Legendre polynomials, spanned onthe elements of the finite cell mesh. FCM shows similarly good convergence characteristicsas p-FEM applications.

Since the geometry is not taken into account on the mesh generation level, the simulationeffort is shifted to the level of integration. As the material parameters are scaled down,a discontinuity is introduced in the cut cells. Typically, composed Gaussian quadraturefor evaluating integrals in these cells is used, based on a spacetree decomposition. For thespacetree, the exponential convergence curve known from p-FEM levels off after a certainnumber of degrees of freedom. Increasing the spacetree depth shifts the point where thelevel-off happens, but every new level of spacetree decomposition introduces a vast amountof quadrature points, resulting in high simulation times.

The present thesis addressed a new method to overcome these issues by decomposing the cutcells into triangles and quadrilaterals with curved sides. Knowing the parametric descrip-tion of the boundary, these triangles and quadrilaterals use blended mappings to map thequadrature points distributed in their parent space to the parameter space of the cut cell.

To this end, the theory of blended mappings was introduced as well as the concept of paramet-ric curves, including the theory of non-uniform rational B-splines (NURBS). As this thesisfocuses on integration, the numerical aspects of integrating blended elements bounded byparametric curves were investigated.

Chapter 3 introduced an algorithmic approach for decomposing cut cells into quadrilateralsand triangles. The algorithm aims to be generic, such that its implementation does notrequire many if-else or switch-case branches, resulting in less code which is easier to maintain.Similarly to the quadtree decomposition, the algorithm is robust enough to work with complexgeometries.

The error in integration with blended integration cells was investigated on selected geometriesby setting up a “pseudo-FCM” problem that has an exact solution but computes on the

56 6. Summary, conclusion and outlook

blended cells at the construction time of the stiffness matrix. It was found that in case ofB-Splines curves, the integration can be exact only if the blended cells are subdivided alongthe knot locations of the bounding curves. For NURBS, the integration can converge tonumerical accuracy if subdivision along knot spans takes place.

In Chapter 5 some FCM simulations were performed and the results of the blended subcellintegration were compared to the conventional spacetree approach and a linear finite ele-ment solution. The blended subcell integration shows better convergence characteristics incomparison to the classical spacetree decomposition with at least one order of magnitudeless quadrature points for the same error. To demonstrate that the algorithm is capable ofworking with complex geometries, a cover plate model of a violin was partitioned, and themovement of a compressional wave was simulated on its domain. Thanks to its robustnessand high accuracy, the algorithm is a good fit into the design-through-analysis pipeline.

6.1 Outlook

Throughout this thesis only problems in 2D were considered. Obviously, most of the intro-duced concepts can be extended to 3D as well:

• The rectangular finite cells with 4 vertices and 4 edges become hexahedra with 8 vertices,12 edges and 6 faces.

• The geometric boundary of the physical domain evolves from parametric curves toparametric surfaces.

• The concept of blended quadrilaterals (triangles) can be extended to blended hexahe-dra (tetrahedra and wedges) [13][14]. In this case, the blended mapping includes theparametric description of the faces.

• The number of possible cell cut cases goes up from 24 = 16 to 28 = 256.

Due to the cases where multiple surfaces meet in one cell and the high number of cell cutcases, an extension of the algorithm of Section 3 to 3D is desirable. A starting point forfuture investigations in this direction is shortly presented in the following.

As it was discussed in Chapter 3, a cut cell in two dimensions can be decomposed intotwo linear triangles that are cut by the boundary to a triangle and a quadrilateral. Theorientation of these linear triangles are determined by performing inside-outside tests at the4 corner vertices: if adjacent edges are cut, the triangles are aligned, if opposite edges arecut, the triangles are flipped.

The concept in three dimensions shall be the following: a hexahedral finite cell can be dividedinto five tetrahedra that are cut further by the boundary to a tetrahedra with a curved faceand a wedge with a curved face (Figure 6.1), or two wedges with curved faces (Figure 6.2).Therefore the algorithm in 3D would have to perform the following steps (refer to Figures 6.1and 6.2):

1. Find which cells are cut by the boundary.

6.1. Outlook 57

2. Subdivide the cell into 5 tetrahedra.

3. Depending on the inside-outside state information of the corner vertices, determine thecut case for each tetrahedron (wedge-wedge, or wedge-tetrahedron).

4. Compute the intersection lines on the faces of the finite cell and the tetrahedra.

5. Create blended elements and map them to the parameter space of the finite cell.

Future investigations in this direction should address the following questions:

• Depending on the inside-outside state of the corner vertices, how can it be determinedwhich tetrahedron is cut into two wedges, and which one into a tetrahedron and awedge?

• If there is more than one boundary cutting the cell, is it feasible to reconstruct themto one face using e.g. approximating surfaces, or is there a way to keep the parametricdescription of the boundary?

• Trimming a curve is relatively easy by specifying the parametric bounds where thetrimming happens. How can this concept be extended to parametric surfaces, wherethe parametric bounds do not necessarily form a rectangle in the parameter space?

In three dimensions, increasing the level of the octree decomposition leads to an even steeperincrease in the number of quadrature points than in two dimensions. The investigationsin this thesis proved that the blended subcell integration delivers higher accuracy with lessintegration points in 2D. Therefore, these questions regarding the 3D implementation aredefinitely worth looking at in the future.


∂Ω

Internal tetrahedron

Intersection line

Wedge

Tetrahedron

1. 2.

3. 4.

Figure 6.1: Cut case where a tetrahedron and a wedge is created.

6.1. Outlook 59

∂Ω

Internal tetrahedron

Intersection line

Wedge

1. 2.

3. 4.

Figure 6.2: Cut case where two wedges are created.

LIST OF FIGURES 61

List of Figures

2.1 1D Shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Example for two dimensional nodal modes, edge modes and internal modes . 6

2.3 The method of curve trimming . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 An example for basis functions with p = 2 . . . . . . . . . . . . . . . . . . . . 10

2.5 An example of a B-Spline and a NURBS curve . . . . . . . . . . . . . . . . . 12

2.6 Blended quadrilateral with one curved side . . . . . . . . . . . . . . . . . . . 12

2.7 Blended triangle, edge 3 is collapsed . . . . . . . . . . . . . . . . . . . . . . . 14

2.8 Poisson problem on a rectangular domain . . . . . . . . . . . . . . . . . . . . 16

2.9 2× 2 meshes of finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.10 Poisson problem on a circular domain . . . . . . . . . . . . . . . . . . . . . . 17

2.11 Solution of the Poisson problem with p = 2, one blended element . . . . . . . 18

2.12 Comparison of tangent vector magnitudes . . . . . . . . . . . . . . . . . . . . 19

2.13 Mapping of quadrature points to knot spans . . . . . . . . . . . . . . . . . . . 20

2.14 Error in curve length with Gaussian quadrature . . . . . . . . . . . . . . . . . 20

2.15 A quad with one circular edge . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.16 Subdivision of parameter space along a knot . . . . . . . . . . . . . . . . . . . 21

2.17 Error in the area integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.18 The core concept of the fictitious domain approach . . . . . . . . . . . . . . . 23

2.19 The method of quadtree partitioning . . . . . . . . . . . . . . . . . . . . . . . 24

2.20 The concept of mappings for k=1 quadtree depth . . . . . . . . . . . . . . . . 25

2.21 The concept of cut cell integration with blended mapping . . . . . . . . . . . 25

3.1 The concept of the proposed partitioning algorithm . . . . . . . . . . . . . . . 27

3.2 The possible cases of cuts in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 28

62 LIST OF FIGURES

3.3 Examples of triangular decomposition. . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Numbering and naming conventions for the partitioning algorithm . . . . . . 31

3.5 (o, p) coding of corner vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.6 Examples of collecting the intersected cell edges . . . . . . . . . . . . . . . . . 33

3.7 Creating the auxiliary line and auxiliary point . . . . . . . . . . . . . . . . . . 34

3.8 Example of subdivision along knot span. . . . . . . . . . . . . . . . . . . . . . 34

3.9 Numbering of different cuts for creating blended elements . . . . . . . . . . . 35

3.10 Mapping the blended subcell to the parameter space (ξ, η) . . . . . . . . . . . 35

3.11 Examples of subdivision for too complex geometries . . . . . . . . . . . . . . 36

4.1 Schematic setup of the integration tests . . . . . . . . . . . . . . . . . . . . . 38

4.2 Example of a partitioned circular domain . . . . . . . . . . . . . . . . . . . . 38

4.3 Error in strain energy for the circular domain integration test . . . . . . . . . 39

4.4 Geometry setup of the moving circle integration test . . . . . . . . . . . . . . 39

4.5 Results of partitioning for selected values of β . . . . . . . . . . . . . . . . . . 40

4.6 Geometric setup for the violin partitioning example . . . . . . . . . . . . . . . 41

4.7 A comparison of integration error with and without knotspan subdivision . . 42

4.8 A detailed view of the F-hole after partitioning . . . . . . . . . . . . . . . . . 42

4.9 Partitioned violin without knotspan subdivision . . . . . . . . . . . . . . . . . 43

4.10 Partitioned violin with knotspan subdivision . . . . . . . . . . . . . . . . . . . 44

5.1 Perforated plate problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.2 Comparison of integration cell meshes and a linear finite element mesh . . . . 47

5.3 Error in strain energy for the perforated plate, p = 1, .., 12 . . . . . . . . . . . 48

5.4 Error in strain energy as a function of number of quadrature points . . . . . . 48

5.5 Error in strain energy for the curved domain problem . . . . . . . . . . . . . 49

5.6 Error in strain energy as a function of number of quadrature points . . . . . . 50

5.7 FCM solution of the Poisson problem with different integration meshes . . . . 51

5.8 Location of excitation for the violin example . . . . . . . . . . . . . . . . . . . 52

5.9 Values of φ plotted for selected t values . . . . . . . . . . . . . . . . . . . . . 53

5.10 Values of φ plotted for selected t values (continued) . . . . . . . . . . . . . . . 54

LIST OF FIGURES 63

6.1 Cut case where a tetrahedron and a wedge is created . . . . . . . . . . . . . . 58

6.2 Cut case where two wedges are created . . . . . . . . . . . . . . . . . . . . . . 59

64 LIST OF FIGURES

BIBLIOGRAPHY 65

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