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Master Thesis

Finite Element Method simulation for flow problems with elastic particles based on Eulerian approach

Presented By

Maneesh Kumar Mishra

Matriculation Number – 108 008 237 013

Computational Engineering Department

Ruhr Universitaet, Bochum

November 2010

Master Thesis

Finite Element Method simulation for flow problems with elastic particles based on Eulerian approach

Presented By

Maneesh Kumar Mishra

Under the guidance of:

Prof. Dr. Stefan Turek

Department of Applied Mathematics

Faculty of Mathematics

Technische Universität, Dortmund

Thesis Supervisor:

Prof. Dr. Rüdiger Verfürth

Department of Numerical Mathematics

Faculty of Mathematics

Ruhr Universitaet, Bochum

Finite Element Method simulation for flow

problems with elastic particles based on

Eulerian approach

Maneesh Mishra

November 25, 2010

Contents

List of Figures 3

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Aim of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Organization of Thesis . . . . . . . . . . . . . . . . . . . . . . 5

2 Problem Description 6

2.1 Fluid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Solid Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Oldroyd-B Model . . . . . . . . . . . . . . . . . . . . . 72.2.2 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Fluid-Solid Model . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Discretization of the Navier-Stokes equations in time . . . . . 82.5 Discretization of the Navier-Stokes equations in space . . . . . 9

2.5.1 Conforming Stokes Element Q2/P1 . . . . . . . . . . . 102.6 FEM formulation . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . 122.8 Solution algorithm . . . . . . . . . . . . . . . . . . . . . . . . 122.9 UMFPACK Solver . . . . . . . . . . . . . . . . . . . . . . . . 142.10 FEATFLOW software and ingredients . . . . . . . . . . . . . 14

2.10.1 Grid Generation using DeViSoR . . . . . . . . . . . . . 142.10.2 FEATFLOW software . . . . . . . . . . . . . . . . . . 152.10.3 Post processing tool . . . . . . . . . . . . . . . . . . . 16

3 Interface Capturing 17

3.1 Eulerian interface tracking methods . . . . . . . . . . . . . . . 183.2 Numerical Treatment . . . . . . . . . . . . . . . . . . . . . . . 21

1

CONTENTS 2

3.3 Reinitialization . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3.1 Brute Force Re-distancing . . . . . . . . . . . . . . . . 233.3.2 Implicit Reinitialization . . . . . . . . . . . . . . . . . 24

4 Numerical Experiments and Results 25

4.1 Reversibility Test . . . . . . . . . . . . . . . . . . . . . . . . . 254.1.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . 254.1.2 Results and Discussion . . . . . . . . . . . . . . . . . . 294.1.3 Hyperelasticity . . . . . . . . . . . . . . . . . . . . . . 314.1.4 Mass Conservation . . . . . . . . . . . . . . . . . . . . 33

4.2 Interaction of Particles . . . . . . . . . . . . . . . . . . . . . . 374.2.1 Problem Setup 1 . . . . . . . . . . . . . . . . . . . . . 374.2.2 Problem Setup 2 . . . . . . . . . . . . . . . . . . . . . 43

5 Conclusion 52

5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

List of Figures

1 Flow around a Parachute . . . . . . . . . . . . . . . . . . . . . 22 Fluid-structure interaction of a three-lobe valve . . . . . . . . 23 Partitioned approach, Lagrangian and Eulerian coupled . . . . 3

4 Location of the degrees of freedom for the Q2, P1 element. . . 11

5 Lagrangian interface description . . . . . . . . . . . . . . . . . 186 Eulerian interface description . . . . . . . . . . . . . . . . . . 187 Level set description with Γ = 0 . . . . . . . . . . . . . . . . . 208 Brute force redistancing . . . . . . . . . . . . . . . . . . . . . 23

9 Problem setup for cylinder shear test . . . . . . . . . . . . . . 2610 Grid with increasing refinement . . . . . . . . . . . . . . . . . 2711 Level set function after initialization . . . . . . . . . . . . . . 2812 Contour of level set function representing the interface . . . . 2821 Interface at maximum stretch for different levels of refinement 3126 Computed area of solid at each time step for different levels . 3427 % loss of area of solid for different grid levels . . . . . . . . . . 3528 Solid at maximum stretch for different viscosity . . . . . . . . 3629 Solid at maximum stretch for G = 1, 10 . . . . . . . . . . . . . 3730 Problem set up for 2 cylinder test . . . . . . . . . . . . . . . . 3831 Computational Grid . . . . . . . . . . . . . . . . . . . . . . . 3948 Problem setup for 2 balls in shear flow . . . . . . . . . . . . . 4349 Computational Grid for Problem Setup 2 . . . . . . . . . . . . 4359 Idea of clipping . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3

Abstract

This thesis is concerned with the numerical solution for fluid-elastic parti-cle interaction for incompressible viscous fluid flow with hyperelastic solidparticles in different situations. A complete Eulerian description is used todescribe both fluid and solid medium. Finite Element Method is used in fullycoupled monolithic formulation using Crank Nicholson time stepping schemeand the interface is captured using the Level Set Method. The simulationswere performed to predict the behaviour of elastic particles for different flowfields.Keywords: Finite Element Method, Eulerian, Fluid - Elastic Particles Interaction,

Coupled problems, Level Set Method, Hyperelastic

Acknowledgements

I would like to sincerely thank Prof. Dr. Stefan Turek for providing me withthis opportunity to work under him and providing me the resources requiredfor completion of this work. The topic he suggested was indeed interestingand I learnt a lot during the time I worked for my thesis.

My heartiest thanks to Hogenrich Damanik who gave me a lot of supportand guidance without which this thesis work would not have been possible. Iwould also like to thank Dr. Abderrahim Ouazzi for his suggestions wheneverI stumbled with some problems.

I would like to show my gratitude to all the members who are involvedin the FEATFLOW group for developing such a robust open-source softwareand have contributed in someway or the other in this work.

Finally, I would like to thank Anusha for being there during the toughtimes and supporting me and boosting my confidence.

Chapter 1

Introduction

The numerical simulation of multiphysics problems is one of the growingchallenges in scientific computing. Typical examples of such problems are thethermal-structural interaction, electromagnetic-thermal interaction, magnetic-structural interaction and fluid-solid interaction.

1.1 Motivation

Viscous fluid flow interacting with an elastic body has many real life appli-cations. General examples of such interaction problems include flow trans-porting elastic particles (particulate flow), flow around elastic structures (air-planes, marine applications, parachutes (Fig. 1)), material processing andflows in elastic structures (haemodynamics (Fig. 2)), transport of fluids inclosed containers) etc. In aeronautical applications such computations aremainly used for flutter analysis, where a loose coupling exists between thefluid and the structural problem. In the biomedical application typical com-pliant structures like vessels or the blood flow interaction with the heartwall, result in strongly coupled problems. The interest in these simulationsis growing since it is expected to predict the effects of the medical treat-ment and to provide operational instructions along with rapid progresses inimaging and computational technologies. Diagnostic imaging devices suchas a computed tomography and a magnetic resonance imaging can visualizethe inner side of the human body, and provide the multi-component geom-etry as voxel data for each patient, which would be incorporated into themulti-physics simulation.

1

CHAPTER 1. INTRODUCTION 2

Figure 1: Flow around a Parachute[14]

Figure 2: Fluid-structure interaction of a three-lobe valve [19]


Fluid

SolidSolid

Tn+1 Tn+2Tn

Fluid

Solid

Fluid

Figure 3: Partitioned approach, Lagrangian and Eulerian frameworks coupled

Modeling of such physical systems which contain both solid and fluidphase is challenging as there are conflicting situations one faces. Fluids arenaturally described using the Eulerian (spatial) description while Solids aredescribed using the Lagrangian (referential) description. From analysis pointof view, decoupling of this problem presents significant technical challenges.[2] Combining the Eulerian and Lagrangian setting involves conceptionaldifficulties. Here are some of the existing approaches one uses:

Partitioned approach

The fluid domain is time-dependent and it depends on the deformation ofthe solid particle. On the other hand, for the structure, the fluid boundaryvalues i.e. velocity and normal stresses are needed. In both the cases, valuesfrom one problem are used for the other, which can lead to a drastic lossof accuracy and is costly. A solution to such a problem is to separate thetwo models, solve each of them separately, and so converge iteratively to asolution which satisfies both together with the interface conditions. Solvingthe separated problems serially multiple times is referred to as ’partitionedapproach’ as can be seen in Figure 3. It has an advantage as there are manywell tested finite element based numerical methods for separate problems ofelastic deformation and fluid flow [2]. The treatment of the interface andthe interaction is problematic. The interface is explicitly tracked by meshadjustment and are generally referred to as ’interface tracking ’ methods.


Lagrangian Approach

In this approach, both the solid and fluid domain are represented by the La-grangian description. Elastic particles are typically described in Lagrangiancoordinates as it facilitates the calculation of deformation gradient and hencethe calculation of deformation tensor and the solid stresses.

Arbitrary Lagrangian Eulerian approach

The arbitrary Lagrangian-Eulerian (ALE) [15] approach combines the useof two reference frames. There is one flexible grid and another grid thatallows for material to flow through. In essence, it takes the best part of bothreference frames and assimilate them in one. This is helpful in problems withlarge deformations in solid mechanics and in fluid-structure interaction. Thismethod allows the grid to deform and track the material. The difficulty whenusing the ALE approach is deciding how much to allow a grid to deform. Thismethod allows to accurately account for continuity of velocities and stressesat the interface. They are difficult to implement and time-consuming inparticular for 3D systems undergoing large deformations.

Eulerian approach

In this work, the coupled problem of fluid and solid interaction is solved in acomplete Eulerian framework. A similar work has been done by Walkingtonet al. [8] and Sugiyama et al. [13]. There is no need to explicitly store thereference configuration of the solid for calculation of the deformation gradientand hence, the deformation tensor. Instead, an additional equation describ-ing the temporal rate of the Cauchy deformation tensor is used to describethe solid deformation. In such an Eulerian setting, a phase variable is em-ployed on the fixed mesh to distinguish between the different phases of liquidand solid. This approach of identifying the fluid-solid interface is referred toas ’interface capturing ’ which is commonly used in multiphase flows. Such aformulation makes the coupling easy. It requires only a single mesh throughout the simulation and has an easier computational implementation.


1.2 Aim of Thesis

A full Eulerian approach has not been explored much for fluid-solid interac-tion problems. The aim of the thesis is to explore the possibility of such aformulation and its capabilities to capture such interaction problems basedon Finite Element Method. This thesis describes how such an Eulerian de-scription could be realized for elastic particles in viscous fluid flow usinginterface capturing methods.

1.3 Organization of Thesis

The first chapter gives a brief introduction into the applications of fluid-solidinteraction problems and the existing methodologies for solving them. Thenext section gives a brief outline regarding the objective of the work.

The second chapter describes the theory, governing equations, finite ele-ment method and the solution methodology. An introduction to the FEAT-

FLOW software is discussed in the end.The third chapter mentions about the methods for determining the inter-

face and gives a brief overview about the Level Set Method and the numericalalgorithms for solving it.

The fourth chapter deals with numerical results for different test con-figurations and a brief discussion regarding the results is mentioned. Thesolutions are tested for grid convergent results and conservation of mass.

The fifth chapter is the conclusion part which overviews the results ob-tained from the numerical simulations and scope for future work.

Chapter 2

Problem Description

In many fluid applications, particularly in flow problems with fluid-solid in-teraction based on the incompressible Navier-Stokes equations, the mathe-matical description and the numerical schemes have to be designed in sucha way that quite complicated constitutive relations and interactions betweenfluid and solids can be incorporated. In the following sections, the problemand solution strategy in a complete Eulerian framework is described.

2.1 Fluid Model

Motion of fluid is governed by Navier Stokes equations which is a set ofPartial Differential Equations which describe the momentum and continuityconditions for a fluid. Consider Ωf (t) as the domain occupied by the fluid attime t, and Ωs(t) as the domain occupied by the solid. For an incompressibleNewtonian fluid, these equations can be written as:

∂u

∂t+ u · ∇u− 2ν∇ ·D+∇p = ~f, Divu = 0 . (2.1.1)

where u is the velocity vector, p is the pressure, ν is the kinematicviscosity, D is the deviatoric strain rate tensor which is defined as, D =12(∇u +∇uT ). Now consider Ω = Ωf (t) ∪ Ωs(t)as the entire computational

domain independent of t. Fluids are naturally defined in Eulerian frame ofreference which means fluid motion is observed on specific locations in thespace through which the fluid flows as time passes.

6

CHAPTER 2. PROBLEM DESCRIPTION 7

2.2 Solid Model

Solids are naturally defined in Lagrangian frame of reference. Instead of usinga transformation function to map the variables from Lagrangian to Euleriandomain for a solid, a stress rate equation is used to compute the solid stressesevolving over time. The reference configuration is not specifically stored forsuch a formulation. Oldroyd Stress rate model is used for computing thestress. In the present analysis the solid object for simulations are consideredto be hyperelastic.

2.2.1 Oldroyd-B Model

The Oldroyd stress rate model [7] is based on the idea to transform back(pullback) the stress to the reference configuration, execute there the differ-entiation with respect to time and then bring (push forward) the rate to theactual configuration. The following relation can be written based on this ratefrom [6]

∂B

∂t+ u · ∇B = ∇u ·B+B · ∇uT (2.2.1)

where B is the left Cauchy-Green deformation tensor.

2.2.2 Hyperelasticity

The solid in our problem is modeled based on a hyperelastic constitutivemodel based on a Neo-Hookean material model. Examples of such materialmodel would be human tissue (heart valves, lung tissue), red blood cells,rubber etc. When the work done by the stresses during a deformation processis dependent only on the initial state at time t0 and the final configurationat time t, the behavior of the material is said to be path independent andthe material is termed hyperelastic. A hyperelastic material is characterizedby reversibility in shape on removal of stretching force.

A Neo-Hookean solid [20] is a hyperelastic material model and is an ex-tension to Hooke’s law.This model was proposed by Ronald Rivlin in 1948. Itincorporates the stress-strain behavior for materials experiencing large defor-mations. The stress strain curve is linear for most part but after sometime,the stress-strain curve plateaus because of release of energy which dissipatesin the form of heat.


Based on the evaluation of the left Cauchy deformation tensor from Eq.2.2.1, the corresponding stress based on a hyperelastic material model canbe calculated. An incompressible Neo-Hookean solid model is chosen and thestress can be written as [6],

σ = GB− pI

where G is the modulus of transverse elasticity.

2.3 Fluid-Solid Model

The complete set of equations considering both solid and fluid phase cannow be written. For the fluid, an additional stress term comes from thesolid. This accounts for the coupling of forces between the solid and the fluiddomain and the final set of equations which looks like the following,

Momentum Equation:

∂u

∂t+ (u · ∇)u = −∇p+ 2ν∇ ·D+ G∇ ·B

︸︷︷︸

solid stress

Continuity Equation:∇ · u = 0

Oldroyd-Stress Rate:

∂B

∂t+ u · ∇B = ∇u ·B+B · ∇uT

2.4 Discretization of the Navier-Stokes equa-

tions in time

The numerical solution techniques for the incompressible Navier-Stokes equa-tions is considered, [18]

ut − ν∆u+ u · ∇u+∇p = f , Divu = 0, in Ω× (0, T ] (2.4.1)

for given force f , with prescribed boundary values on the boundary ∂Ω andan initial condition at t = 0. In order to solve this problem numerically athigh Reynolds number, is still considered to be a difficult task for long time


calculations. The case is even more difficult if, the dynamics over time iscomplex. A 3D computation makes it even more challenging however, only2D simulations are dealt with in this work.

The common approach is to have separate discretization in space andtime. The equations are first discretized in time by one of the usual meth-ods such as the Forward Euler or Backward Euler, the Crank-Nicolson-or Fractional-Step-θ–scheme, or others, and obtain generalized stationaryNavier-Stokes problems.

Basic θ-scheme:

Given un and k = tn+1 − tn, then solve for u = un+1 and p = pn+1

u− un

k+ θ[−ν∆u+ u · ∇u] +∇p = gn+1 , Divu = 0 , in Ω (2.4.2)

with right hand side gn+1 := θfn+1+(1−θ)fn−(1−θ)[−ν∆un+un ·∇un].

The parameter θ is chosen depending on the time-stepping scheme, fore.g., θ = 1 for the Backward Euler, or θ = 0.5 for the Crank-Nicolson-scheme.Here, in our case, Crank-Nicolson-scheme is used for all the simulations whichare of 2nd order. A nonlinear saddle point problem is solved at every timestep and then, discretized in space.

These methods belong to the group of one-step-θ-schemes. The CrankNicholson scheme sometimes suffers from numerical instabilities because ofits weak damping property. Another possibility which could be used is theFractional-Step-θ-scheme (FS). It uses three fractional time steps and threedifferent values for θ at each time level.

Here, only implicit time stepping schemes are used. Explicit schemescould have been implemented as well but they suffer from stability problemsand need very small time steps and prohibit long time flow simulations.Because of more efficient solvers, implicit schemes have become more feasible.

2.5 Discretization of the Navier-Stokes equa-

tions in space

A finite element approach for the spatial discretization is applied. A finiteelement model of the Navier-Stokes equations is based on variational for-


mulation. The governing equations are multiplied with test functions andthen integrated in space. On the finite mesh Th (triangles, quadrilaterals)covering the domain Ω with local mesh size h, polynomial trial functions forvelocity and pressure are defined. These spaces Hh and Lh should lead tonumerically stable approximations as h → 0, and should satisfy the LBB(Ladyzhenskaya, Babouska and Brezzi) stability condition which says, [3]

minqh∈Lh

maxvh∈Hh

(qh,Div vh)

||qh||0 ||∇vh||0≥ γ > 0 (2.5.1)

with a mesh–independent constant γ. To summarize, the LBB conditioncomes down to increasing the order (or the number of degrees of freedom)of the basis functions for the velocity compared to the basis function for thepressure. Here, a Q2/P1 element is utilized which satisfies the LBB condition.

2.5.1 Conforming Stokes Element Q2/P1

The domain Ω is approximated by a domain Ωh with piecewise linear bound-ary which is equipped with a quadrilateral mesh Th. It is assumed that Th

is regular which means that any two quadrilaterals are disjoint or have acommon vertex or a common edge. The finite dimensional spaces Uh, Ph

and Bh for the velocity, pressure and strain approximations are defined onthis mesh. The spaces U, P and B would be approximated in the case ofQ2, P1 element as [15]

Uh =uh ∈ [C(Ωh)]

2,uh|T ∈ [Q2(T )]2 ∀ T ∈ Th,uh = 0 on ∂Ωh

Ph =ph ∈ L2(Ωh), ph|T ∈ P1(T ) ∀ T ∈ Th

Bh =Bh ∈ [C(Ωh)]

2,Bh|T ∈ [Q2(T )]2 ∀ T ∈ Th,Bh = 0 on ∂Ωh

Q2(T ) denotes the standard bi-quadratic space on the quadrilateral Twhich, when transformed by bilinear transformation to the reference quadri-lateral Tref = (−1, 1)2, is defined by

Q2(Tref ) =q ψ−1

T : q ∈ span⟨1, ξ, η, ξη, ξ2, η2, ξ2η, ξη2, ξ2η2

⟩(2.5.2)


~vh, ~uh

ph,∂ph∂x, ∂ph

∂y

x

y

Figure 4: Location of the degrees of freedom for the Q2, P1 element.

consists of nine local degrees of freedom which are located at the vertices,midpoints of edges and in the center of the quadrilateral. The space P1(T )consists of linear functions defined on the reference element by

P1(T ) =q ψ−1

T : q ∈ 〈1, ξ, η〉

(2.5.3)

has three local degrees of freedom with the function value and both partialderivatives in the center of the quadrilateral.

The corresponding degrees of freedom are illustrated in figure 4

2.6 FEM formulation

The governing equations are multiplied with test functions (χ, ω, λ) ∈ (Uh, Ph,Bh)and solutions of this problem are sought by integrating them over the com-plete domain and over time interval [0, T ]

∫

T

∫

Ω

(

ρ∂u

∂t+ ρ(u · ∇)u+∇p− 2η∇ ·D−G∇ ·B

)

χdV dt = 0

∫

T

∫

Ω

(∇ · u)ω dV dt = 0


∫

T

∫

Ω

(∂B

∂t+ u · ∇B−∇u ·B−B · ∇uT

)

λ dV dt = 0

2.7 Numerical Treatment

The time discretization of the govering equations using the single step θ-scheme is performed:

The time discretization of the Continuity Equation and Navier StokesEquation

∇ · un+1 = 0

un+1 − un

∆t+ θ

[un+1 · ∇un+1 − (ν∆un+1 −G∇ ·Bn+1)

]+∇pn+1

+(1− θ) [un · ∇un − (ν∆un −G∇ ·Bn)] = 0

where un ∼ u(tn). The left Cauchy-Green deformation tensor equationgiven in the Oldroyd derivative is discretized in the same way so that

Bn+1 −Bn

∆t+ θ

[un+1 · ∇Bn+1 − (∇un+1 ·Bn+1 −∇Bn+1 · un+1)

]+

+(1− θ) [un · ∇Bn − (∇un ·Bn −∇Bn · un)] = 0

By choosing θ =1

2, the fully implicit Crank-Nicholson method with sec-

ond order accuracy is obtained.

2.8 Solution algorithm

The system of nonlinear algebraic equations is solved using iterative Newtonmethod. Newton iteration can be written as [15]

Xn+1 = Xn −

[∂F

∂X(Xn)

]−1

F(X n) (2.8.1)

If the initial guess is sufficiently close to the solution, the basic iterationcan exhibit quadratic convergence.


The damped Newton method with line search improves convergence byadaptively changing the length of the correction vector. In the Newtonmethod, the solution update step is replaced by

Xn+1 = Xn + ωδX (2.8.2)

where the parameter ω is determined such that a certain error measureddecreases.

• Let Xn be some starting guess.

• The residuum vector is set to Rn = F (Xn) and the tangent matrix is

set to A =∂F

∂X(Xn)

• The correlation ∂X is solved for AδX = Rn

• The optimal step length ω is found.

• The new solution is Xn+1 = Xn − ωδX

For non-linear convergence, an adaptive time-stepping scheme helps. Whenthere was quadratic convergence, only three Newton steps were needed to ob-tain the required precision and hence, an increased time step could also beused. If not, then the time step needs to be reduced.

The structure of the Jacobian matrix∂F

∂Xis

∂F

∂X(X) =

Suu Suv 0Svu Svv Bu + Bv

BTu BT

v 0

(2.8.3)

and it can be computed by finite differences from the residual vector F(X)

[∂F

∂X

]

ij

(Xn) ≈[F ]i (X

n + αjej)− [F ]i (Xn − αjej)

2αj

(2.8.4)

where ej are the unit basis vectors in Rn and the coefficients αj are

adaptively taken according to the change in the solution in the previous timestep. Because of the sparsity of the Jacobian matrix due to the finite elementmethod formulation, the solution of the linear problems is the dominant partin terms of the CPU time.


2.9 UMFPACK Solver

The Unsymmetric-Pattern MultiFrontal Package (UMFPACK) [1] is a set ofsubroutines designed to solve linear systems of the form Ax = b, where A isan n-by-n general unsymmetric sparse matrix, and x and b are n-by-1 vec-tors. It uses LU factorization, and performs pivoting for numerical purposesand to maintain sparsity. The sparse matrix A can be square or rectan-gular, singular or non-singular, and real or complex (or any combination).Only square matrices A can be used to solve Ax = b or related systems.Rectangular matrices can only be factorized.

UMFPACK is based on the unsymmetric-pattern multifrontal method.The method relies on dense matrix kernels to factorize frontal matrices, whichare dense sub-matrices of the sparse matrix being factorized [1]. UMFPACKis written in ANSI Fortran-77, with compiler directives for the Cray Fortrancompiler (which are simply seen as comments by other compilers). Bothdouble-precision and single-precision (REAL) versions are included.

2.10 FEATFLOW software and ingredients

2.10.1 Grid Generation using DeViSoR

DeViSoR [9] is abbreviated for Design and Visualization Software Resourceand is implemented in Java. The DeViSoR Grid application is used for thefollowing tasks:

• geometry generation

• manual coarse mesh definition

• grid visualization at all levels

The preprocessing part contains descriptions of 2D domains and coarselevel triangulation at coarse level. Programs such as trigen2d, tr2to3 andtrigen3D provide coarse triangulations and their graphical representationroutines. Using this, the computational domains are defined for conductingnumerical tests. Once the geometry is defined, the domain can be discretizedinto smaller pieces called elements. In general, an equidistant mesh has beenused in this mesh. However, there are some situations where the requirementwas to make a non-uniform mesh to reduce computation time.


2.10.2 FEATFLOW software

FEATFLOW [17] is a general purpose subroutine system for the numericalsolution of partial differential equations by the finite element method. Asolver package, CC2d is a direct, coupled approach for solving the discreteversion of the incompressible Navier Stokes equations at low Reynolds num-bers for stationary and instationary flows .

CC2d solver

After time and spatial discretization, the discrete NS equations can be writtenin the following form from the FEATFLOW Manual, [10]:

Given un and k = tn+1 − tn, then solve for u = un+1 and p = pn+1 by

solving

Su+ kBp = g BTu = 0,

with matrix S and g on right hand side such that

Su = [αM + θkN(u)],

N(u)u := −ν∆u+ u · u := −νL+K(u),

where

• α = 1 for time dependent problems,

• α = 0, k = 1 for steady state formulation,

• M is the lumped mass matrix,

• B is the discrete gradient operator,

• BT is the divergence operator,

• N is the advection matrix where L is discrete Laplacian and convectiveK(u) is the nonlinear transport operator


2.10.3 Post processing tool

General Mesh Viewer (GMV) [4] tool is used for visualization. GMV is afree visualization tool for viewing 2D and 3D simulation data on structuredor unstructured meshes.

Paraview [12] has been used for generating some of the plots for compar-ative studies and generating movies for the simulations. It is an open-sourcevisualization software which can run on multiple platforms and is very pop-ular for data analysis. It can build visualizations very fast to allow the usersto have a quick analysis of data. It also allows data to be analyzed quantita-tively and qualitatively. The plugin GMVReader for Paraview developed atLS3, TU Dortmund has been used for importing GMV files into Paraview.

Chapter 3

Interface Capturing

Interface is defined as the point of interaction between two phases. Here inthis case, the two phases are fluid and solid. Interfaces can be determinedby the following two generalized methods:

• Lagrangian Methods : Using this method, the interfaces are explic-itly tracked. The cell edges are aligned with the interfaces as shownin Fig. 5. A sharp interface can be represented using this approachbut it adds additional algorithmic complexities to handle complicatedsituations such as breaking up and merging of the interface segments.As there is movement of grid in this case, the governing equations arealso modified accordingly. And as a result, this method is plagued bythe fact that it requires periodic meshing if there is large deformationof the interfaces.

• Eulerian Methods : In this method the interfaces are allowed tointersect the cells arbitrarily as shown in 6. Instead of tracking theinterface through the computational domain explicitly, it is now calcu-lated implicitly on a fixed grid. This allows for fixed grids to be usedwhich is more desirable from both computational and implementationpoints of view. The drawback of using Eulerian methods is that oneneeds a more refined grid to obtain the interfaces with higher resolutionwhen compared to the Lagrangian methods.

17

CHAPTER 3. INTERFACE CAPTURING 18

Figure 5: Lagrangian interface description

[5]Figure 6: Eulerian interface description

[5]

3.1 Eulerian interface tracking methods

The two most popular Eulerian methods are Volume of Fluid(VOF) methodand Level Set Method.

Volume of Fluid Methods

In this method, motion of the interior region is tracked by assigning to eachcell on the computational grid a “volume fraction” of the 2 phases. Hence,for each cell the size of cell’s portion belonging to the two phases underconsideration is known as a fraction. Each cell is assigned a volume fractionbetween 0 and 1 to indicate the averaged relative amount of solid and fluid.When a cell is having a volume fraction of 0 or 1, it is completely consistingof one of the phases i.e. solid or fluid. An interface cell would be having afraction of both the phases and its easy to keep track of the cells in whichthe interface lies. Consequently, it has the following advantages [11]:

• There is no need of any additional computational elements as the samegrid can be utilized which was used for computing the velocities.

• Complicated boundaries can be easily handled.


However, its very difficult to handle curvature dependent problems.

Level Set Method

Another possibility to model the moving front, avoiding the problems of La-grangian formulation and taking curvature dependencies into account con-sists in describing the interface as zero level set of a level set function. Thelevel set method, first introduced by Osher and Sethian [11], can be usedin many diverse fields such as multiphase flow, medical image processing,crystal growth, and inverse problems. The essence of the level set methodis to represent the interface Γ(t) (represented by a curve in 2D) in a higherdimensional function ϕ, that is,

Γ(t) = x ∈ Rd|ϕ(x, t) = vls

where vls is the contour level value which represents the interface implic-itly. The choice of vls is arbitrary. In this case, the interface is representedwith zero level set.

ϕ is initialized as a signed distance function,

ϕ(x, 0) = d(Γ,x) = dist(Γ,x) < 0 for solid

dist(Γ,x) = 0 for interface

dist(Γ,x) > 0 for fluid

A distance function is smooth and it simplifies the implementation of aregularized Heaviside function. The normal and curvature are defined as

n =∇ϕ

|∇ϕ|, κ = −∇ · n (3.1.1)

An example for the level set function initialized as a signed distancefunction, where Γ is a circle can be seen in Figure: 7


−2−1

01

2

−2

0

2−1

−0.5

0

0.5

1

1.5

2

Figure 7: Level set description with Γ = 0 representing the interface shown withdotted black line


In order to derive the evolution of the level set function ϕ, it is importantthat the following must hold true for the interface [5]

ϕ(x(t), t) = 0

On differentiating with chain rule,

∂ϕ

∂t+∇ϕ ·

∂x(t)

∂t= 0

The speed with which the front propagates in the normal direction is

given by F = n ·∂x(t)

∂twhere n =

∇ϕ

|∇ϕ|This results in the evolution equation for ϕ as

∂ϕ

∂t+ F · |∇ϕ| = 0

which must hold globally for all values of ϕ. The speed function F de-pends on variables such as external forces, mean curvature, but in this case,it will depend only on the fluid velocity, that is F = n · u, which gives

∂ϕ

∂t+ (u.∇)ϕ = 0 (3.1.2)

With this, an implicit formulation for the interface is defined. However,there is a cost involved for computing its evolution. Interface curvatureand normals can be reconstructed globally. The location of the interface isknown explicitly, as opposed to the Volume-of-fluids method and it can bereconstructed when required.

3.2 Numerical Treatment

The level set equation (3.1.2), is a pure hyperbolic transport equation. Timediscretization follows by applying the single step θ-scheme, which results inthe problem formulation [5]

Given ϕn and the time step k = tn+1 − tn, then solve for ϕ = ϕn+1

ϕ− ϕn

k+ θ(u · ∇)ϕ = b


with the right hand side

b = (θ − 1)(u · ∇)ϕn

Same discretization scheme is chosen as for the Navier-Stokes equationand hence, the same approximation order is applied as for the velocities anda Q2 element is utilized with the standard finite element method.

The final form of level set equation looks like,

ϕn+1s − ϕn+1

s

∆t+ θ

[un+1 · ∇ϕn+1

s

]+ (1− θ) [u · ∇ϕn

s ] (3.2.1)

[M(l) + kθA

]ϕn+1 = bn,

bn =[M(l) − k(1− θ)A

]ϕn

where M(l) =∫

Ωv1v2 dΩ is the mass matrix which can be lumped. A

is the transport matrix which is responsible for convection of the level setfunction and hence, the interface is also implicitly convected, and is given by,

A =

∫

Ω

(u · ∇)v1v2 dΩ

Here, v1 and v2 denote the Q2 basis functions and test functions respec-tively.

3.3 Reinitialization

After the level set has been initialized with distance function, it normallydistorts significantly with time. This smears the zero level set with timeand affects the interface being captured. The level set equation is a pureconvection equation and there are no diffusion terms and so it results intoconvergence difficulties. The distance function property can be preserved if,the velocity with which the level set field is transported satisfies,

∇ϕ · ∇F = 0 (3.3.1)

is fulfilled.


A common approach to solve such a problem is to apply re-distancingor reinitialization to the smeared level set field, and thus, retain the dis-tance function property. In order find its usage in practical applications, thereinitialization methods must fulfill a number of requirements.

• The chosen method should ideally not move the interface or zero levelset, which would cause loss of area and hence, the conservation of massis lost.

• It should be as accurate as possible, since an accurate level set fieldwill result in better normal and curvature fields.

• The computational overhead due to reinitialization should not be sig-nificant that it dominates the time taken by other components.

3.3.1 Brute Force Re-distancing

The simple brute force method consists of tracking the interface intersectionwith the element and determining which edges have a change of sign of thelevel set function. After determining the intersecting edges, the distancefield for nodes outside (ϕ > 0) is reconstructed. The distance for each gridpoint to all approximated zero distance segments is computed, from whichthe minimum is taken as the new distance. The algorithm can be seen fromFig. 8 . This algorithm is obviously of order O(NM) with N being thenumber of grid points and M the number of zero distance segments. As thenumber of interface segments increases, the algorithm approaches quadraticcosts but if there are small number of interface segments, then the cost islinear.

1. Mark the interface elements and corresponding edges which undergo achange of sign for level set function ϕ

2. Store the location of dofs which fall on the interface in a array

3. Compute distance for nodes where ϕ > 0 i.e. fluid medium, with allthe interface points and store the minimum, min

4. Reset the level set function ϕ with min and loop over all time steps

Figure 8: Brute force redistancing


3.3.2 Implicit Reinitialization

In addition to equation 3.1.2, the level set function also satisfies

‖ϕ‖ = 1 (3.3.2)

or equivalently, complete set of equation takes the following form:

∂ϕ

∂t+ (u.∇)ϕ = 0

n · ∇ϕ = 1

(3.3.3)

Then, ϕ is a solution of the following variational formulation [16]

∫

Ω

(∂ϕ

∂t+ (u.∇)ϕ

)

ψ dx+λ

∫

Ω

(n ·∇ϕ)(n ·∇ψ) dx = λ

∫

Ω

n ·∇ψ dx (3.3.4)

where λ is a penalization parameter.The original convection equation now includes additional second order

diffusion terms. This method reduces the computational costs significantly.But it suffers from a drawback that when the n changes significantly, thenthis approach fails.

Chapter 4

Numerical Experiments and

Results

Based on the complete Eulerian formulation as discussed in Chapters 2 and3, some test configurations are defined and then examined in different situa-tions.

4.1 Reversibility Test

A hyperelastic material is characterized by reversibility in shape when thestretching force is released. In the full Lagrangian formulation using finiteelement mesh, which memorizes the reference configuration even under defor-mation, the reversal can be captured with no difficulty. On the other hand,the Eulerian approach does not explicitly keep the reference configuration inthe equation set and smears out the interface. Hence, one may expect itsshortcomings in capturing the reversal in shape for a hyperelastic material.Therefore, a reversibility examination is performed for the present simulationmethod.

4.1.1 Problem Setup

A cylindrical solid in a viscous, shear flow between two plane walls is defined.The physical dimensions of the problem can be seen in Fig. 9.

25

CHAPTER 4. NUMERICAL EXPERIMENTS AND RESULTS 26

Solid Cylinder

Fluid Domain

u = 0.5

u = −0.5

1.02.0

2.0

Figure 9: Problem setup for cylinder shear test

Initial Conditions

The initial solution is found by considering everywhere fluid region and solv-ing a direct steady set of the original instationary equation set as discussedin Section 2.3:

∇ · u = 0

ρ(u · ∇)u = −∇p+ 2ν∇ ·D

B = I

with B propotional to identity so that the deviatoric stress is zero at rest,and r is the radius of the initial solid.


Hence, the initial conditions can be summarized as follows:

• B = I for solid domain, Ωs and B = 0 for the fluid domain, Ωf

• Initially, ϕ is set as distance function, i.e. ϕ =√

x2 + y2 − r where ris the radius of the cylinder ϕ > 0 for Ωs and ϕ < 0 for Ωf . ϕ = 0represents the interface, as shown in Figure 11 and 12

• Both fluid and solid cylinder are at rest initially.

• Density of fluid and solid, ρ = 1

• The viscosity of the fluid is set to µ = 1.0

• Constant modulus of transverse elasticity, G = 4.0.

• Time step, ∆t = 0.1.

Boundary Conditions

• Dirichlet Boundary Conditions For a time period of 0 ≤ t ≤ 2,both top and bottom plates are moved in the horizontal direction atspeed of 0.5 and -0.5, respectively to impose shearing force on the solid.After t > 2, the shear velocity on the walls is released.

• Neumann Boundary Conditions The Neumann boundary condi-tion “DO NOTHING” (NBC) is applied to the left and the right bound-ary walls.

The computational mesh used for the simulation is a simple cartesianmesh as shown in Figure 10

Figure 10: Grid with increasing refinement starting from LEVEL 3 till LEVEL5


Figure 11: Level set function after initialization

Figure 12: Contour of level set function representing the interface


4.1.2 Results and Discussion

The following section deals with the results for the simulation at differentgrid levels. At t = 2, the solid experiences the maximum stretching.

Figure 13: Interface for LEVEL 3 attime, t = 0





Grid convergence

Grid convergence is the term used to describe the improvement of results byusing successively smaller cell sizes for the calculations. A calculation shouldapproach the correct answer as the mesh becomes finer (h → 0), hence theterm grid convergence. In Figure 21, it is obesrved that as the grid is refined,grid independency is achieved. The results for LEVEL 5 and LEVEL 6 areseen to be co-incident.

Figure 21: Interface at maximum stretch for different levels of refinement

4.1.3 Hyperelasticity

As mentioned earlier in Section 4.1, hyperelastic materials are characterizedby reversibility in shape after the release of stretching force. In this case, theDirichlet boundary condition is released on the walls for velocity for t > 2.After sufficiently long time, the interface is observed. Figure 22 , 23, 24 and25 show the results for different grids.


The cylinder returns to its original position at t = 12. And hence, thephenomenon of hyperelasticity using the full Eulerian method is captured.

4.1.4 Mass Conservation

Level set method suffers from the drawback of loss of area in the process ofthe solution. Even if ϕ is initialized as the distance function, the evolutionequation, eq. 3.1.2 destroys this property while convecting the solution withvelocity u. Therefore, the conservation of area of the cylinder is computedover all the time steps to check for mass/area loss during the process ofsimulation.

In Table 4.1, the computed area for different levels and the correspond-ing error are presented. The analytical area for the cylinder is given byA = πr2 = π0.52 = 0.7853. The computed area (C.A.) is calculated us-ing Paraview with ’Integrate Variables’ filter. The error is computed as∣∣∣∣

C.A.− πr2

πr2

∣∣∣∣. It is observed from Table 4.1, that with refinement the error

reduces ∼ h2 where h is the mesh width.

LEVEL NDOF NEL Area at t=0 Area at t=2 Area at t=12

Comp.

Area

% age

error

Comp.

Area

% age

error

Comp.

Area

% age

error

3 1926 64 0.7222 8.04 0.7041 10.34 0.7264 7.51

4 7302 256 0.7700 1.96 0.7644 2.67 0.7706 1.88

5 28422 1024 0.7815 0.49 0.7791 0.78 0.7796 0.73

6 112134 4096 0.7841 0.15 0.7842 0.15 0.7840 0.17

Table 4.1: Loss of Area

In Figure 26, it can be noted that there is a sudden loss of area after t = 2which is because when the stretching force is released, the solver experiencesdifficulty. However, with time, the lost area is regained. Also, as one movesto a finer computational grid, one reaches closer to the analytical solutionand there is less oscillation in the computed area value. Figure 27 shows the% age loss of area to get a comparitive idea of the loss of area during thesimulation.

CHAPTER

4.

NUMERIC

ALEXPERIM

ENTSAND

RESULTS

34Figure 26: Computed area of the solid at each time step for different levels

CHAPTER

4.

NUMERIC

ALEXPERIM

ENTSAND

RESULTS

35

Figure 27: % loss of area of solid for different grid levels


Next, the effect of changing the viscosity of fluid on the shape of interfaceis determined. Figure 28 shows the interface at maximum stretch i.e. t =2. It can be observed that with increase in viscosity, there is proportional

increase in the shearing force, τ = µdu

dy. Also, as µ is increased, the solver

becomes more stable because Reynolds number Re ∼1

µand as Re decreases

with increasing µ, the solver experiences less difficulty.

Figure 28: Solid at maximum stretch for different viscosity

The effect of change of modulus of transverse elasticity is tested by ini-tializing G to a higher value and behavior of the solver is noted. Physically,it is expected that as G is increased, the solid should undergo less deforma-


tion as it has more stiffness now. The same behavior is predicted with thisformulation and the results can be seen in Figure 29. However, the solverbegins to experience problems for very high values of G.

Figure 29: Solid at maximum stretch for G = 1, 10

4.2 Interaction of Particles

Two particles in different situations are now defined and tested to check theirbehavior on interaction.

4.2.1 Problem Setup 1

The problem setup can be seen in Fig. 30. Two cylinders are placed in be-tween concentric circles and the outer wall is rotated with some wall velocity,


vwall and the inner wall is kept static. The behavior of particles in such asituation is observed.

2 elastic particles

v=0.1

D=1.0

d=0.6

rigid static wall

d1 = 0.1

Fluid

Figure 30: Problem set up for 2 cylinder test

Figure 32: vwall = v1 Figure 33: vwall = 2v1


Figure 31: Computational Grid

Figure 34: vwall = 0.1, t = 0 Figure 35: vwall = 0.2, t = 0


Comments

It can be seen that in the case where vwall = 2v1, the crossover of the twoballs took place at t = 3 shown in Fig. 41 whereas for vwall = v1, crossovertakes place at t = 6 shown in Fig. 46. The Eulerian method was able tocapture such “racing situation” of the two particles.

4.2.2 Problem Setup 2

The following configuration for 2 balls in shear flow placed asymmetricallyis defined, as shown in Fig. 48

Figure 48: Problem setup for 2 balls in shear flow

Figure 49: Computational Grid for Problem Setup 2


On using the level set method without reinitialization, the solution suf-fered from intrusions as seen in Fig. 51,

Figure 50: t=0

Figure 51: Intrusion at t=4.5


Implicit Reinitialization

Implicit Reinitialization as discussed in Section 3.3.2 is applied as a remedy.The case with a cylinder placed asymmetrically in a viscous, shear flow ischecked. The balls become smaller with time. The solid undergoes largedeformation and as a result, the normals change very quickly. The methodseems not to work for interfaces with fast change of normals.

Figure 52: t = 0 Figure 53: t = 1


With this method, the solution till t = 1 were computed without anyproblems. But after that, the solver experienced problems and the cylinderlost area over time.


Brute force re-distancing

As discussed in Section 3.3.1, the algorithm for brute force re-distancing wasapplied and tested for a 1 ball placed asymmetrically in a shear flow.


Figure 58: t = 2

Here again, it was observed that at t = 2, the solid seemed to get smeared.


Clipping

The idea of clipping is to create a sort of shield (refer Fig. 59) for thelevel set function and to protect it from changes/intrusions occurring at theboundaries. So, the level set function is set to a constant value outside somebuffer zone and then its impact on the solution is noted.

Figure 59: Idea of clipping





With clipping, the solution could be computed till t = 6 but after thatthe interface began to get smeared.


Cyclic BC

A modification in the velocity boundary condition as shown in Fig. 66 wastested to prevent intrusions from outside. And then, tests were done tosee how the simulations run for this case. After undergoing an initial sheardeformation, Fig. 72 at t = 4, the deformed particle experiences a “roll over”after their first interaction and at t = 9, Fig. 76 they regain their shape asone would expect from a hyperelastic material.

Figure 66: Changed problem set up with cyclic velocity BC




Figure 71: t = 3.5 Figure 72: t = 4


Chapter 5

Conclusion

In the following section, the results obtained from numerical simulation forthe full Eulerian approach for coupled fluid-solid problems are summarized.

5.1 Summary

Using FEATFLOW software, a fully coupled Eulerian framework for solvingcoupled problems was developed and interface was captured using the LevelSet Method. Such a method is characterized by simplified grid generationand plain extension of the standard algorithms for incompressible fluid flowsolvers. There is not need to explicitly couple the interface, as a fully coupledsolver is used.

First, a single cylinder in a shear flow is tested and the case for maximumdeformation is checked. Grid independent results were achieved for the sheartest. It was observed that, as the wall velocity was increased, there wereproblems experienced by the solver because of the large velocity gradients.The characteristic property of shape recovery by hyperelastic particles wasthen validated. After sufficiently long time, it was observed that the cylinderwas able to regain its original shape on removal of wall shear velocity. Thearea of the cylinder was computed and checked for mass loss. Mass losswas < 0.2% for finest grid at LEVEL 6 which can be considered sufficientlyaccurate. The property of hyperelasticity was captured with this formulation.The solver was then tested for different material and fluid parameters andthe results were in accordance to the physical situations.

Then, interaction of 2 particles in the system were tested. First, two

52

CHAPTER 5. CONCLUSION 53

particles in a concentric cylinder, with outer wall rotating at some velocitywere defined and examined for the ’racing’ condition. The problem waschecked for different wall velocities and verified. Secondly, 2 balls in a shearflow placed asymmetrically were simulated. Here, the solver experiencedproblems with the level set, as there were intrusions from the outside forthe level set function. Some strategies based on reinitialization techniqueswere attempted. The brute force re-distancing, implicit reinitialization andclipping were attempted as solution strategies, but all of them began to showproblems after some certain time steps. On changing the boundary conditionsfor the problem to a cyclic velocity boundary condition, the roll-over of thetwo particles could be captured.

5.2 Future Work

There is a lot of scope for improvement in the existing code. Perhaps, amore robust level set reinitialization method could be tested and then, betterresults for the two ball in viscous, shear flow problem could be expected.Also, numerical analysis of the components need to be done, to check howthey affect the solver. The numerics for the solver could also be improved,as currently, there are convergence problems when there is large velocitygradient or higher rigidity of solid. The formulation could be examined formore complex real life applications.

Bibliography

[1] Timothy A. Davis. Unsymmetric multifrontal pack (umfpack) solver.http://www.cise.ufl.edu/research/sparse/umfpack/.

[2] Th. Dunne. An eulerian approach to fluidstructure interaction and goal-oriented mesh adaptation. Int. J. Numer. Meth. Fluids, 51(9-10):1017–1039, 2006.

[3] Vivette Girault and Pierre-Arnaud Raviart. Finite element approxi-

mation of the Navier-Stokes equations. Lecture Notes in Mathematics.Springer, Berlin, 1979.

[4] GMV. http://www.generalmeshviewer.com/.

[5] S. Hysing. Numerical simulation of immiscible fluids with FEM level set

techniques. PhD thesis, Technische Universitat Dortmund, December2007.

[6] Richard D. Wood Javier Bonet. Nonlinear Continuum Mechanics for

Finite Element Analysis. Cambridge, 2008.

[7] D. Kolymbas and I. Herle. Shear and objective stress rates in hypoplas-ticity. Int. J. Numer. Anal. Meth. Geomech.,, 27:733744, 2003.

[8] Chun Liu and Noel J. Walkington. An eulerian description of fluidscontaining visco-elastic particles. Archive for Rational Mechanics and

Analysis, 159:229–252, 2001. 10.1007/s002050100158.

[9] TU Dortmund LS3. Devisor 2.1.

[10] TU Dortmund LS3. Featflow User Manual Release 1.1.

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[11] Stanley Osher and James A Sethian. Fronts propagating with curvature-dependent speed: Algorithms based on hamilton-jacobi formulations.Journal of Computational Physics, 79(1):12 – 49, 1988.

[12] Paraview. Paraview open source visualization software.

[13] Kazuyasu Sugiyama, Satoshi Ii, Shintaro Takeuchi, Shu Takagi, andYoichiro Matsumoto. Full eulerian simulations of biconcave neo-hookeanparticles in a poiseuille flow. Computational Mechanics, 46:147–157,2010. 10.1007/s00466-010-0484-2.

[14] T.E. Tezduyar. http://www.tafsm.org/proj/as/orion08a/.

[15] S. Turek and J. Hron. A monolithic FEM solver for an ALE formulationof fluid–structure interaction with configuration for numerical bench-marking, 2006. Eccomas CFD 2006.

[16] S. Turek, O. Mierka, S. Hysing, and D. Kuzmin. A high order 3D FEM–level set ap–proach for multiphase flow with application to monodispersedroplet generation. International Journal for Numerical Methods in Flu-

ids, 2010. submitted.

[17] Stefan Turek. Featflow software. http://www.featflow.de/.

[18] Stefan Turek. Efficient Solvers for Incompressible Flow Problems: An

Algorithmic and Computational Approach. Springer, 1999.

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[20] Wikipedia. Neo-hookean solid.

http://www.featflow.de/

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