Adaptive Finite Element Methods for Fluid StructureInteraction Problems with Applications to Human

Phonation

NİYAZİ CEM DEĞİRMENCİ

Doctoral ThesisStockholm, Sweden 2018

TRITA EECS-AVL-2018:38ISBN 978-91-7729-764-2

KTH School of Electrical Engineeringand Computer ScienceSE-100 44 Stockholm

SWEDEN

Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framläggestill offentlig granskning för avläggande av teknologie doktorsexamen i datalogi ons-dagen den 23 mai 2018 klockan 10.30 i Sing-Sing, våningsplan 2, Kungl Tekniskahögskolan, Lindstedtsvägen 26, Stockholm.

© NİYAZİ CEM DEĞİRMENCİ, May 2018

Tryck: Universitetsservice US AB

iii

Abstract

This work presents a unified framework for numerical solution of FluidStructure Interaction (FSI) and acoustics problems with focus on humanphonation.

The Finite Element Method is employed for numerical investigation of par-tial differential equations that model conservation of momentum and mass.Since the resulting system of equations is very large, an efficient open sourcehigh performance implementation is constructed and provided. In order togain accuracy for the numerical solutions, an adaptive mesh refinement strat-egy is employed which reduces the computational cost in comparison to auniform refinement. Adaptive refinement of the mesh relies on computableerror indicators which appear as a combination of a computable residual andthe solution of a so-called dual problem acting as weights on computed resid-uals. The first main achievement of this thesis is to apply this strategy tonumerical simulations of a benchmark problem for FSI. This FSI model isfurther extended for contact handling and applied to a realistic vocal foldsgeometry where the glottic wave formation was captured in the numericalsimulations. This is the second achievement in the presented work. The FSImodel is further coupled to an acoustics model through an acoustic analogy,for vocal folds with flow induced oscillations for a domain constructed to cre-ate the vowel /i/. The comparisons of the obtained pressure signal at specifiedpoints with respect to results from literature for the same vowel is reported,which is the final main result presented.

iv

Sammanfattning

Detta arbete presenterar en enhetlig ram för numerisk lösning av fluid-strukturinteraktion (FSI) och akustikproblem med fokus på det mänskligatalet. En finita elementmetod används för numerisk lösning av de partiella dif-ferentialekvationer som beskriver konserveringslagar för moment och massa.Eftersom det resulterande systemet av ekvationer är mycket stort, konstruerasen öppen källkod med hög prestanda. För att få hög noggrannhet i de nu-meriska lösningarna används en adaptiv nätförfiningsstrategi vilken minskarberäkningskostnaden jämfört med en uniform förfining.

Adaptiv förfining av nätet bygger på beräknade felindikatorer som byg-ger på en kombination av en beräkningsbar residual och lösningen av ett såkallat dualt problem. Den första huvudresultatet av denna avhandling är attutveckla en och validera denna strategi för en FSI-modell i ett benchmarkproblem.

Denna FSI-modell utvidgas vidare för att hantera kontaktmekanik, ochanvänds sedan för en realistisk modell av stämbandsstrukturerna där denglottiska vågformationen fångas i de numeriska simuleringarna. Detta är detandra huvudresultatet i det presenterade arbetet.

FSI-modellen kopplas också till en akustikmodell genom en akustisk analogi,för modell konstruerad för att skapa vokalen / i /. Den erhållna trycksignaleni ett antal punkter jämförs med resultat från litteraturen, vilket är det slutligahuvudresultatet som presenteras.

v

To Nurettin, Süreyya and Ceren Değirmenci.

I would like to thank my supervisors Johan Jansson and Johan Hoffman forproviding a liberal research atmosphere and fruitful discussions.Doghonay Arjmand, Jeannette Hiromi Spühler, Love Lindholm, Henrik Holstand Kaspar Müller thank you for the inspiration at KTH.Laura Saavedra, Margarida Moragues Ginard, Massimiliano Leoni thank youfor the inspiration at BCAM.Berat Beran Çamak desteğin için teşekkürler. Melih Temel, Arman Toplu,Türker Küçük ve Emre Atsan iyi ki varsınız.

Preface

This thesis consists of an introduction and six papers. The included papers are:

Paper I

Johan Hoffman, Johan Jansson, Rodrigo Vilela De Abreu, Niyazi Cem Degir-menci, Niclas Jansson, Kaspar Müller, Murtazo Nazarov, Jeannette Hiromi Spühler,"Unicorn: Parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry", Computers &Fluids, Vol. 80, pp. 310–419, 2013.The author of this thesis developed the topological mesh adaptation implementa-tion.

Paper II

Johan Jansson, Niyazi Cem Degirmenci, Johan Hoffman, "Framework for adaptivefluid-structure interaction with industrial applications", Int. J. Materials Engineer-ing Innovation, Vol. 4, No. 2, 2013.The author of this thesis has done the implementation and performed the numericalsimulations, and conributed to the writing of the manuscript.

Paper III

Johan Jansson, Niyazi Cem Degirmenci, Johan Hoffman, "Adaptive Unified Con-tinuum FEM Modeling of a 3D FSI Benchmark Problem". Int. J. Numer. Meth.Biomed. Engng., doi: 10.1002/cnm.2851, 2016.The author of this thesis has done the implementation of the primal solver, theintegration of an existing dual solver into the adaptive algorithm, performed thenumerical simulations and authored parts of the manuscript.

vii

viii

Paper IV

Jeannette Hiromi Spühler, Niyazi Cem Degirmenci, Johan Jansson, Johan Hoffman,"A 3D full-friction contact model for fluid-structure interaction problem", Submit-ted, 2017.The first and second author performed the implementations and the numerical ex-periments in close cooperation. The modeling and authoring of the manuscript wasdone in collaboration between all authors.

Paper V

Johan Hoffman, Johan Jansson, Niyazi Cem Degirmenci, Jeannette Hiromi Spühler,Rodrigo Vilela De Abreu, Niclas Jansson, Aurélien Larcher, "FEniCS-HPC coupledmulti-physics in computational fluid dynamics", High-Performance Scientific Com-puting: First JARA-HPC Symposium, JHPCS 2016, Aachen, Germany, October4–5, 2016, Revised Selected Papers, pp. 58–69, 2017.The author of this thesis has done the implementation and run the numerical sim-ulations for the setting of the vocal folds.

Paper VI

Niyazi Cem Degirmenci, Johan Jansson, Johan Hoffman, Marc Arnela, PatriciaSánchez-Martín, Oriol Guasch, Sten Ternström, "A unified numerical simulation ofvowel production that comprises phonation and the emitted sound". Proceedingsof INTERSPEECH 2017 (peer-reviewed), Stockholm, Sweden, pp. 3492–3496, Au-gust 20–24, 2017The author of this thesis has done the implementation and run the numerical sim-ulations as well as writing parts of the manuscript.

Contents

Contents ix

1 Introduction 11.1 Research Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Voice Production Physics . . . . . . . . . . . . . . . . . . . . 21.2.2 The Mathematical Model . . . . . . . . . . . . . . . . . . . . 4

Mathematical Modeling of the Incompressible Continuum . . 4Mathematical Modeling of the Acoustics . . . . . . . . . . . . 5Mathematical Modeling of the Contact Detection . . . . . . . 6

1.2.3 A First Introduction to Numerical Solution of PDE’s by theFinite Element Method . . . . . . . . . . . . . . . . . . . . . 6The Model Problem . . . . . . . . . . . . . . . . . . . . . . . 6Weak Formulation of The Model Problem . . . . . . . . . . . 7Ritz-Galerkin Approximation . . . . . . . . . . . . . . . . . . 7

1.2.4 A Second Introduction to Solving PDE’s with the Finite El-ement Method . . . . . . . . . . . . . . . . . . . . . . . . . . 8Approximation of Weak Solutions . . . . . . . . . . . . . . . 8The Weak Derivative . . . . . . . . . . . . . . . . . . . . . . . 8Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9The Galerkin Method . . . . . . . . . . . . . . . . . . . . . . 9The Finite Element Method . . . . . . . . . . . . . . . . . . . 10

1.2.5 Assessing the Scalability of the Implementations . . . . . . . 10Strong Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . 11Amdahl’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . 11Weak Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Gustafson’s Law . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1 Existence, uniqueness, regularity results for the solutions . . 121.3.2 Numerical results for the problem . . . . . . . . . . . . . . . 13

2 Methods 15

ix

x CONTENTS

2.1 Fluid-Structure Interaction Coupling . . . . . . . . . . . . . . . . . . 152.2 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Eulerian description of the motion . . . . . . . . . . . . . . . 172.2.2 Lagrangian description of the motion . . . . . . . . . . . . . . 172.2.3 Arbitrary Lagrangian Eulerian description of the motion . . . 17

Mesh Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Stabilized Galerkin Formulation . . . . . . . . . . . . . . . . . . . . . 19

3 Contributions and Results 213.1 A Posteriori Error Estimation . . . . . . . . . . . . . . . . . . . . . . 213.2 Contact Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Acoustic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4 Implementation Details . . . . . . . . . . . . . . . . . . . . . . . . . 27

4 Discussion 294.1 Adaptive FEM for FSI problems . . . . . . . . . . . . . . . . . . . . 294.2 Contact Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Acoustic Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Bibliography 33

Chapter 1

Introduction

1.1 Research Question

The focus of the presented work is to develop an efficient adaptive finite elementmethod (FEM) for fluid-structure interaction (FSI) problems that can be appliedto simulate human phonation. The main objectives can be listed as:

• Adaptive FEM for FSI problems:The objective is to reduce the error in the quantity interest in an efficientway for the numerical approximation, where a particular challenge is theformulation of an estimate for a time dependent nonlinear problem and itsimplementation. Papers II and III focus on this objective and an introductionis provided in Section 3.1

• Contact Modeling:The objective is to be able to handle problems when the domain boundariescome in close proximity, with challenges to keep the condition number of theresulting linear system low as well as preventing inverted elements. Paper IVfocus on this objective and an introduction is provided in Section 3.2

• Acoustic Coupling:The objective is to be able to use incompressible FSI-contact computationresults as an input to a wave propagation solver for the simulation of humanphonation. The challenges involve testing different possible acoustic analo-gies, testing different boundary conditions to prevent numerical artefacts inthe form of wave reflections from the domain boundaries, implementation ofsolvers in the same framework, with particular challenges in mesh partitioningand data transfer for two solvers operating on two different domains and dis-cretizations. Paper VI focus on this objective and an introduction is providedn Section 3.3

1

2 CHAPTER 1. INTRODUCTION

• HPC implementation:An implementation with good scalability properties is necessary to be able toapproximate solution of 3D problems numerically with high accuracy. PapersI and V focus on this objective and some details of the open source finiteelement framework are provided in Section 3.4

1.2 Background

1.2.1 Voice Production Physics

Larynx is an organ which is considered to have a key role during the migration ofthe first vertebrates from an aquatic to a terrestrial environment nearly 350 millionyears ago in the early Carboniferous period of the Paleozoic era [52]. Similar tothat of bichir Polypterus, which possesses both external gills and paired ventrallungs, the ancient "larynx" is thought to be composed of a simple muscle sphincterto protect the lung from the water [60].

The larynx has evolved and changed during time and besides airway protection,and for humans it began to fulfill the additional role of being an acoustic source forsound and speech production with the help of the vocal fold structures containedinside. Figure 1.1 depicts the organ and inner structures as well as the importantmuscles used for controlling it.

The physics of voice production is rather intricate. In a nutshell, the air em-anating from the lungs impinges on the vocal folds (VF) and pushes them apartuntil their elasticity takes over and the VF close again. These vibrations resultin a pulsating glottal jet with maximum velocity ∼ 14 − 45ms−1 and a Reynoldsnumber in the range O(102) − O(104) [47]. Humans can adduct the VF as wellas manipulate the tension and change elastic properties actively by controlling thethyroarytenoid, posterior cricoarytenoid, cricothyroid, and lateral cricoarytenoidmuscles. The pressure at the glottis decreases due to the jet flow in the vocal tract(VT) by Bernoulli’s law and a self sustained oscillation is established. See e.g., [59]for a comprehensive introduction to the topic.

The flow dynamics and VF vibrations result in acoustic sources of mono polar,dipolar and quadrupolar character [66]. The Fundamental frequency is between80− 220 Hz. with lower frequencies for male voice due to longer length of the VF.Acoustic waves are generated, then propagate through the VT and finally becomeradiated outwards. In the case of vowels, the VT resonances (formants) get excited,which actually allows one to distinguish one vowel sound from another.

The process is described as “chaotic” in [32], referring to a pseudo randombehavior where the precise output is unpredictable and extremely sensitive to slightdifferences in initial conditions. The modeling and the numerical simulation of suchcomplex phenomena poses a big challenge.

1.2. BACKGROUND 3

(a) (b)

(c) (d)

Figure 1.1: The Larynx (a,b), and the muscles involved in adducting as well asmanipulating the tension (c,d) [54] (Images under public license in their country oforigin.)

4 CHAPTER 1. INTRODUCTION

1.2.2 The Mathematical ModelMathematical Modeling of the Incompressible Continuum

In order to describe the motion of the incompressible continuum consisting of afluid and solid structures, conservation of mass [49], conservation of momentum[48] and constitutive laws derived from the fundamental stress principle of Cauchy-Euler [62] are used as components to formulate a mathematical model, which iselucidated in [27].

The conservation laws can be represented as partial differential equations (PDE)of the form:

∂

∂tui(x, t) + ∂

∂xifi(u(x, t)) = si. (1.1)

where u is the vector valued function for the conserved quantity, with ui represent-ing its ith spatial component, f is the flux function and s is the source term. Herethe Einstein convention is used where repeated indices signify summation.

The corresponding Cauchy problem for unbounded domains is

∂

∂tui(x, t) + ∂

∂xifi(u(x, t)) = si, (x, t) ∈ Rn × T, (1.2)

u(x, t0) = u0(x), x ∈ Rn,

where T = [t0,∞), or an initial boundary value problem (IBVP) in a finitespatial domain Ω ∈ Rn with a boundary Γ:

∂

∂tui(x, t) + ∂

∂xifi(u(x, t)) = si, (x, t) ∈ Ω× T, (1.3)

u(x, t0) = u0(x), x ∈ Ω,

αu(x, t) + β∂u(x, t)∂xj

nj = uΓ(t) (x, t),∈ Γ× T,

where n represents the outward normal for the boundary Γ. Appropriate initialand boundary conditions must be considered for the well-posedness of the resultingproblem. The system of equations for the conservation of the momentum andthe mass for incompressible Newtonian fluid flow with constant density and withDirichlet boundary conditions are given as:

ρ(∂tu+ (u · ∇)u)− µ∇2u+∇p− s = 0, (x, t) ∈ Ω× T, (1.4)∇ · u = 0, (x, t) ∈ Ω× T,

u(·, t0) = u0, x ∈ Ω,u(x, t) = uΓ(x, t) (x, t),∈ Γ× T,

where u is the velocity, p is the pressure, ρ is the density, µ dynamic viscosity and sexternal forces, inside the spatial domain Ω with boundary Γ, and the time domainT .

1.2. BACKGROUND 5

The constitutive equations for the solid structure in the context of fluid-structureinteraction model corresponds to the relation between internal stresses and the de-formations of the structure. A diligent and step by step derivation of the con-stitutive equations can be found in [21]. We are in particular interested in theNeo-Hookean material model [61], which for the incompressible case reduces to:

σs = −pI + C10FFT , (1.5)

where σs is the Cauchy deviatoric, p is the pressure, C10 is a material constant(shear modulus) and F is the deformation gradient.

Mathematical Modeling of the Acoustics

We refer to [19] for the modeling of the emanating sound which is done by observingthese three principles during the motion of a sound wave:

• The gas moves and changes the density.

• The change in density corresponds to a change in pressure.

• Pressure inequalities generate gas motion.

The resulting equation for the sound assuming that the velocity is independentof the wavelength is given as:

ptt = c2pxx (1.6)

where c is the speed of the sound and p is the acoustic pressure. The problem canbe posed for example as an IVP in 1D such as:

ptt = c2pxx −∞ < x <∞, 0 < t <∞ (1.7)p(x, 0) = f(x) −∞ < x <∞pt(x, 0) = g(x) −∞ < x <∞

or can also be posed for example as an IBVP for a finite domain Ω together withinitial and boundary conditions as in:

ptt = c2pxx x ∈ Ω, 0 < t <∞ (1.8)p(x, 0) = f(x) x ∈ Ωpt(x, 0) = g(x) x ∈ Ωp(x, t) = h(x, t) x ∈ ∂Ω, 0 < t <∞

In this work we are interested in the system of first order PDEs that describesound propagation in a fluid moving with mean velocity U :

1ρ0c20

Dtp+∇.u = Q (1.9)

ρ0Dtu+∇p = f (1.10)

6 CHAPTER 1. INTRODUCTION

Where Dt := ∂t + U .∇ is the material derivative, p acoustic pressure, u isacoustic velocity, Q a volume source distribution and f external body force per unitvolume, ρ0 mean air density and c0 speed of the sound, together with appropriateboundary and initial conditions [23].

Mathematical Modeling of the Contact Detection

A problematic phenomenon to model is the contact, for which we defer address-ing the reason for the difficulty as well as the methodology followed to resolve itnumerically to later sections.

Here however we present the Eikonal Equation, in order to model the distancefrom a point in the domain to the domain boundaries. In our application it is usedto check the distance of the vocal fold boundaries from each other.

The Eikonal equation used in our work for the domain Ω is of the form:

|∇u(x)| = 1, x ∈ Ω (1.11)u(x) = 0, x ∈ ∂Ω

together with its boundary conditions. The derivation of the Eikonal equation fromthe wave equation can be found in many textbooks including [8].

The meaning of the derivatives and solution spaces is the topic of the subsection1.2.4. Another point that should be clarified is that for the equations (1.4) and (1.5)the constitutive laws for the structure are presented for a material point x, whilefor the fluid flow part x is a fixed spatial coordinate. The coordinate systems arediscussed in the section 2.2

1.2.3 A First Introduction to Numerical Solution of PDE’s bythe Finite Element Method

In this section we are going to examine a model PDE and apply the classicalsteps to obtain a numerical solution using the FEM without the intention of beingcomplete. Some of the missing foundation for the ideas are explained in the secondintroduction part.

The Model Problem

Consider the boundary value problem in the interval I = [0, 1]

−∂2u

∂x2 = f x ∈ (0, 1) (1.12)

u(0) = u(1) = 0 (1.13)

which is in strong form and will be used for our demonstration purpose.

1.2. BACKGROUND 7

Weak Formulation of The Model Problem

If we multiply the strong form with any sufficiently smooth function v with v(0) =v(1) = 0 and perform integration by parts we get:

−∫I

∂2u

∂x2 v dx =∫I

fv dx (1.14)

−∂u∂xv|10 +

∫I

∂u

∂x

∂v

∂xdx =

∫I

fv dx (1.15)∫I

∂u

∂x

∂v

∂xdx =

∫I

fv dx (1.16)

where the boundary terms disappear due to v(0) = v(1) = 0.By using the space V = v ∈ L2(0, 1) : v(0) = v(1) = 0,

∫I(∂v∂x

)2 dx < ∞ andthe definitions:

a(u, v) :=∫I

∂u

∂x

∂v

∂xdx (1.17)

L(v) :=∫I

fv dx (1.18)

we can formulate the problem as find u ∈ V such that:

a(u, v) = L(v) ∀v ∈ V (1.19)

This is the weak formulation of the same problem and the equivalence of bothformulations under regularity assumptions on f and u can be shown [6].

Ritz-Galerkin Approximation

The space V defined in the previous subsection has infinite dimensions. Considera finite dimensional subspace S ⊂ V and the Ritz-Galerkin approximation of theweak solution in this space is formulated as follows:Find us ∈ S s.t:

a(us, vs) = L(vs) ∀vs ∈ S (1.20)

Using the information that S has a finite basis φi : 1 ≤ i ≤ n, we can constructa finite linear system of equations by testing 1.20 against each φi

uS :=n∑j=1

Ujφj (1.21)

Aij := a(φj , φi) (1.22)bi := L(φi) (1.23)

8 CHAPTER 1. INTRODUCTION

and finally getting the system

AU = b (1.24)

It can be shown that this system has a unique solution [6].In this work we use the Ritz-Galerkin method to obtain a numerical approxi-

mation of the weak solutions. This method is also a very useful tool to show theexistence of a weak solution for 1.19 by showing the existence of solutions for aseries of problems on finite dimensional spaces converging to the original problemin infinite dimensions as shown in section 1.2.4.

1.2.4 A Second Introduction to Solving PDE’s with the FiniteElement Method

Here we provide some more foundation to the ideas presented previously and applythem for the partial differential equations resulting from the conservation laws.

Approximation of Weak Solutions

Since finding solutions to PDEs in the classical sense is not always possible, a wayto proceed is to search for a solution in the distributional sense, by testing theequations against smooth functions with compact support (C∞c ) and thus relaxingthe requirements for the point-wise regularity.

Assuming φ ∈ C∞c (Rn×T ), the weak formulation of the problem (1.2) is statedas: ∫

Rn×T

∂

∂tφ(x, t)ui(x, t) + ∂

∂xiφ(x, t)fi(u(x, t)) + φ(x, t)si dxdt+ (1.25)∫

Rn

(u0i )φ(x, t0)dx = 0

which is constructed by multiplying the original PDE by φ and integrating in Rn×Tand then using integration by parts. A solution u is called a weak solution if itsatisfies the equation (1.25) for every φ ∈ C∞c (Rn × T )

However searching for solutions in special complete normed vector spaces (Ba-nach spaces) has the advantages of being able to use powerful functional analysistools.

The Weak Derivative

Assume U ⊂ Rn is an open and bounded domain, u, v ∈ L1loc(U) are locally inte-

grable functions on every compact subset of U , then v is called the αth weak partialderivative of u if: ∫

U

uDαφdU = (−1)|α|∫U

vφdU

1.2. BACKGROUND 9

for all φ ∈ C∞c (U). Here for the multi index α = (α1, ...αn) Dα is defined as :

Dαφ = ∂|α|

∂xα11 ...∂xαn

n

We also write v = Dαu.

Sobolev Spaces

The Sobolev space W k,p(U) is defined as

W k,p(U) := u : u ∈ L1loc(U), Dαu ∈ Lp(U) ∀|α| ≤ k

||u||Wk,p(U) :=

(∑|α|≤k

∫U|Dαu|pdU)1/p (1 ≤ p <∞)∑

|α|≤k ess supU |Dαu| (p =∞)

The Sobolev Spaces are complete [15] and the closure of C∞c (U) in W k,p(U) isdenoted by W k,p

0 (U).Now taking T ′ = (t0, t∗) with t0 < t∗ < ∞, the problem (1.4) can be stated as

find ui, p ∈W 1,20 (Ω× T ′) for i = 1, .., n such that:∫

Ω×T ′

n∑i=1

ρ(∂tuivi + u · ∇uivi) + µ∂ui∂xi

∂vi∂xi

+ ∂p

∂xivi − sivi + ∂ui

∂xiq dx dt = 0

(1.26)

is satisfied ∀vi, q ∈W 1,20 (Ω× T ′).

The Galerkin Method

One way to construct a weak solution to the PDE is by finding solutions on a seriesof finite dimensional subspaces. When these subspaces converge to the originalSobolev space, it is possible to pass to limits for the finite dimensional solutionsunder certain conditions.

The summary of the method in [15] after adapting the notation to the problem(1.3) is as follows:

Let f be a first order partial differential operator in (1.3) and wk∞k=1 be anorthogonal basis of W 1,2

0 (Ω) and orthonormal basis of L2(Ω). For each m, we seeka function um,i : [t0, t∗]→W 1,2

0 (Ω) of the form:

um,i(t) :=m∑k=1

dkm,i(t)wk (1.27)

10 CHAPTER 1. INTRODUCTION

where the coefficients dkm,i(t)(t0 ≤ t∗, k = 1, · · · ,m) satisfy

dkm,i(0) =∫

Ωu0iwkdx (1.28)∫

Ωu′iwk − fi(u)∂wk

∂xi− siwk dx = 0 (i = 1 · · ·n) (1.29)

.If it is possible to show the existence of each um,i and prove necessary energy

estimates that depend on f and s; one can show existence and uniqueness of weaksolutions for (1.3).

The Finite Element Method

Using piecewise polynomials as the basis wk is a natural option to build finitedimensional subspaces of the previously mentioned Sobolev spaces in a systematicmanner.

The following definition is taken from [6] which is following Ciarlet’s definitionof a finite element [9] .

Let:

• K ⊂ Rn be a bounded closed set with nonempty interior and piecewise smoothboundary (the element domain)

• PK be a finite-dimensional space of functions on K (the space of shape func-tions)

• NK = NK,1, NK,2 · · ·NK,m be a basis for the dual space of PK

Then (K,PK , NK) is called a finite element.A set th = Ki, where U = ∪thKi, for a finite number of element domains

Ki with the requirement that i 6= j ⇒ Ki ∩ Kj = ∅, is called a mesh. In thesimple case of U ⊂ R2, the element domains may be triangles, and if U ⊂ R3 theelement domains may be tetrahedrons. If the mesh further satisfies the geometricalrestriction that 2 neighbor elements share only one facet (no hanging nodes), thenit is called a geometrically admissible mesh.

Let th be a geometrically admissible mesh and Vh = vh : vh|K ∈ PK. If forevery K ∈ th, PK ⊂ W 1,2(K) and Vh ⊂ C0(Ω) then Vh ∈ W 1,2(Ω). If in additionx ∈ ∂Ω⇒ vh(x) = 0, ∀vh ∈ Vh, then Vh ⊂W 1,2

0 (Ω) [10].

1.2.5 Assessing the Scalability of the ImplementationsAn important property of the parallel algorithms is how well they scale when thenumber of processes increase. In order to quantify this, mostly strong and weakscaling concepts are used which are defined below. Paper I and V contains theseevaluations.

1.2. BACKGROUND 11

Strong Scaling

Strong scaling evaluates the algorithms by relating the total execution time to thenumber of processes for a constant problem size. The strong scaling of the idealalgorithm is the linear function Ss(N) = N . It uses the Speedup in Latency conceptS(N):

S(N) := t1tN

(1.30)

where N is the total number of processes, t1 is the time needed for one process toaccomplish the task, and tN is the time needed for N processes to accomplish thesame task.

Amdahl’s Law

Amdahl has used 1.30 as an argument against parallel architectures in [2]. With0 < α < 1 representing the proportion of the total time spent in the portion of thealgorithm which is not parallelizable, he rewrites the speedup in latency as:

S(N) := t1tN

= t1αtN + (1− α)tN

(1.31)

= t1

αt1 + (1− α)t1N

= 1

α+ (1− α)N

(1.32)

limN→∞

S(N) = 1α

(1.33)

.The Speedup in latency is therefore bounded by 1/α. The speedup graph for

various α with respect to N is given in Figure 1.2.

Weak Scaling

A more optimistic perspective on the evaluation is proposed by Gustafson in [24]where the author argues that the problem size will likely grow with the numberof processes for many industrial applications instead of being constant as in theprevious perspective.

Weak scaling evaluates the algorithms by relating the total execution time tothe number of processes for a constant problem size per process. The weak scalingof the ideal algorithm is the constant function Sw(N) = 1.

Gustafson’s Law

Rather than using the Speedup in Latency, Gustafson uses the definition of scaledspeedup SS(N) which is the time a single process would need to crunch the sameamount of data that N processes can process in unit time.

12 CHAPTER 1. INTRODUCTION

10 0 10 1 10 2 10 3 10 4 10 5

N

0

2

4

6

8

10

12

14

16

18

20

S(N

)

α=0.5

α=0.2

α=0.05

Figure 1.2: Speedup in latency versus number of processes for different levels of α.

SS(N) := αtN +N(1− α)tNtN

= N + (1−N)α (1.34)

limN→∞

SS(N) =∞ (1.35)

.Assuming unbounded problem sizes available, the scaled speedup is not bounded.

This perspective is applicable to most of the industrial applications where FEM isapplied, as bigger solution subspaces are necessary to reduce the error as much aspossible.

1.3 State of the Art

1.3.1 Existence, uniqueness, regularity results for the solutions

The global existence, uniqueness and regularity of solutions of the incompressibleNavier-Stokes equations for dimensions of three is an open problem [1].

Leray introduced a global weak solution [44] that is shown to be unique andregular for the case of dimension 2 [57, 41]. Strong solutions with more regularityhave been shown either locally or when the initial data and force term is small, areview of important advances on this problem can be found in [20].

For the Fluid-Structure Interaction (FSI) problem an existence result of weaksolutions for a 2 dimensional fluid domain interacting with a 1 dimensional structurewithout contact is shown in [5], as well as an overview of recent advances focusingon existence, uniqueness and regularity results.

1.3. STATE OF THE ART 13

Existence of strong solutions for a similar setting is shown in [42, 43]. As statedin these studies, the contact phenomenon for the incompressible continuum is asource of difficulty for theoretical as well numerical studies, which is relevant forthe mathematical modeling of vocal folds.

Existence, uniqueness and regularity results for wave equations can be found in[15] among other textbooks.

1.3.2 Numerical results for the problemAs for numerical test results for the FSI problem, a full-scale wind turbine applica-tion is presented in [4], which is one of the most advanced industrial FSI applicationswith turbulence as a key phenomenon.

Three dimensional numerical modeling of human vocal folds is studied in [39, 53],with prescribed vocal folds motion.

Mittal and co-workers in [65, 67] carry out Direct Numerical Simulation (DNS),with the Reynolds number artificially reduced by one order of magnitude, andwith a partitioned FSI approach where the structure is represented as an immersedboundary in a fluid simulation. Contact is enforced by a kinematic constraint,which is not described in detail.

Jiang and co-workers in [34] simulate full coupling with acoustics for the problemusing a compressible continuum model as well as a multi layered structure model forthe self oscillating vocal folds. For these simulations however the Reynolds numberis again reduced and contact is prevented by always ensuring some gap between thevocal folds by limiting the movement of the vocal folds.

Chapter 2

Methods

2.1 Fluid-Structure Interaction Coupling

In order to model the coupling of the fluid and the structure, employing a weakcoupling strategy [16, 30] is the first possibility. This strategy is characterizedby independent discretization of the fluid and the solid domains together with anexplicit time stepping scheme and exchanging information over the fluid-structureinterface. This strategy makes it possible to reuse existing solvers focused on solvingeach problem separately. This however comes with the cost of introducing extrainstability sources, and these methods are not suitable for the numerical solutionof every problem.

In order to provide stability, implicit time stepping methods (also known asstrong coupling) [56], or semi-implicit time stepping methods [18] using a combina-tion of implicit and explicit time stepping for different terms is possible.

Another direction is employing a monolithic strategy [63] where the fluid-structurecontinuum is discretized as one, allowing straightforward description of error con-trol techniques and providing stability at the expense of solving a bigger system ofequations.

A monolithic description based on a fully Eulerian coordinate formulation of afluid-structure model is described in [14]. Here the continuum model consists of theconservation equations with a general stress variable, with an additional structuredisplacement variable and an additional equation for interface capturing. This lastequation is necessary to prevent interface smearing on a fixed computational mesh.This way an initial position˝ set(IP) is explicitly tracked. An adaptive methodis presented with 2D numerical test results.

The theoretical foundation for a goal-oriented adaptive method in an FSI-ALEsetting on a fixed reference domain is presented in [50]. The method is verified andvalidated for the 2D FSI-I Hron and Turek benchmark in [64] where the deformationis very small, and also for a 3D test problem, again with very small deformations.

15

16 CHAPTER 2. METHODS

(a) (b)

(c) (d)

Figure 2.1: Two choices for the description of the motion of a deforming solid in astationary fluid flow: initial configuration (a), a bent structure in an Eulerian coordinatesystem together with a mesh (b), a bent structure on a Lagrangian coordinate systemtogether with a mesh (c), a bent structure on an ALE coordinate system with a a smoothedmesh (d)

2.2 Coordinate Systems

It is possible to present the same PDE in different coordinate systems assumingexistence of a one to one mapping between the two coordinate systems at each timeinstant. In the discretized problem we let the mesh track the deformation of thecoordinate system so that the velocity of the coordinate system β is related to themesh velocity m.

2.2. COORDINATE SYSTEMS 17

2.2.1 Eulerian description of the motion

The Eulerian description of motion is defined with respect to a fixed coordinatesystem, which is often preferred in fluid mechanics.

The main problem with this setting in the case of describing motion of a solidbody is that since the spatial resolution is finite, smearing of the interface of thebody occurs. This is of course not a problem in the case of a homogeneous fluidflow problem, where the exact location of a certain particle is uninteresting. Aproblematic case in fluid-structure interaction however is illustrated in Figure 2.1b,where the motion of a bending bar is examined and the shape in bent state is ofteninteresting.

2.2.2 Lagrangian description of the motion

The Lagrangian description presents the equations for a particle and the coordinatesystem follows the particles in motion. This setting is preferred in solid mechanicsand often the convective terms –that appear in the Eulerian description of motion–disappear leaving a simpler system of equations to solve. Since the mesh tracks theindividual particles, the tracking of interfaces becomes trivial.

The main problem with this setting in the case of large deformations is thatthe quality of the mesh decreases, often leading to a bad condition number forthe system of the resulting linear equations. This problem is illustrated in Figure2.1c. In the extreme cases inverted elements may appear where the conditioni 6= j ⇒ Ki ∩ Kj = ∅ is not satisfied. In such cases an expensive re-meshingoperation becomes necessary, and the projection of the variables from the previoustime step introduces an additional source of error.

2.2.3 Arbitrary Lagrangian Eulerian description of the motion

The Arbitrary Lagrangian Eulerian (ALE) description is designed to overcome theshortcomings of the previously described perspectives. The mesh can be movedwith an arbitrary velocity such that some phenomenon is captured in some region(like the interface of a bending bar) and in other parts the velocity may be chosento optimize the quality of the mesh. In the case of conservation laws, the ALEdescription adds convective terms to the partial differential equations with thevelocity of the coordinate system [13]. In Figure 2.1d the coordinate velocity isequal to the structure velocity in the solid part to capture the interface, and in thefluid part the coordinate velocity is decided using a mesh smoothing algorithm.

With Ω ⊂ R3 the spatial domain, and Q = Ω × I is a space-time domain withI = (t0, t∗] a time interval and ∂Ω = Γ1 ∪ Γ2, Γ1 ∩ Γ2 = ∅, the FSI equationsfor the unified continuum model with unknowns u, p, σs, θ can be given in an ALE

18 CHAPTER 2. METHODS

coordinate system with β velocity as:

ρ(u′ + ((u− β) · ∇)u)−∇ · σ − ρg = 0 (x, t) ∈ Q (2.1)∇ · u = 0 (x, t) ∈ Q

u(x, t0) = u0(x) x ∈ Ωu(x, t) = uΓ1(t) (x, t) ∈ Γ1 × [t0, t∗]

σ = σ − pI (x, t) ∈ Q (2.2)σ = θσf + (1− θ)σs (x, t) ∈ Qσf = µf (∇u+∇u>) (x, t) ∈ Q

p(x, t) = pΓ2(t) (x, t) ∈ Γ1 × [t0, t∗]

σ′s = 2µsε+∇uσs + σs∇u> (x, t) ∈ Q (2.3)θ′ + ((u− β) · ∇)θ = 0 (x, t) ∈ Q

σs(x, t0) = σ0(x) x ∈ Ωθ(x, t0) = θ0(x) x ∈ Ω

where µs is the shear modulus (solid) and µf is the dynamic viscosity (fluid) andθ marks the continuum either as fluid if θ = 1 or solid if θ = 0.

Mesh Smoothing

In order to capture the structure interface, β is defined to be equal to the continuumvelocity u when θ = 0. For the fluid part, the velocity of the mesh is obtained bysolving two additional equations, one Laplace type equation and one nonlinearequation to improve the mesh quality as explained in Paper IV.

Linear Smoother The Linear Smoother solves a linear elastic equation for themesh velocity m where the vertices are diffusively relocated over the domain,

m = ∇ · τm (2.4)τm = 2µmε(m). (2.5)

This is a simple and fast method similar to the FSI-GST in [58]. We measure thatadjusting the mesh quality by advancing 1 time step with the Non-Linear Smoothertypically takes 10 times longer than 1 time step with the Linear Smoother. How-ever, there is no certainty that the mesh quality is enhanced since no informationwith respect to shape of the cells is taken into account.

2.3. STABILIZED GALERKIN FORMULATION 19

Non-Linear Smoother In the Non-Linear Smoother the deformation of themesh is formulated as a time-dependent non-linear elasticity problem where weapply the incompressible Neo-Hookean constitutive model to describe the elasticproperty of the mesh. The stiffness of the model is weighted by the quality Q(K)of the cell element K ∈ Tn:

Q(K) := ||F ||2Fd · det(F )2/d , (2.6)

where d stands for the dimension of the mesh. F is the deformation gradientbetween a tetrahedral elementK ∈ Tn and an ideal element K with optimal qualityand ||F ||F indicates its Frobenius norm.

To calculate the deformation velocity m of the elastic model, the followingpartial differential equations are solved:

F0 = deformation gradient between K ∈ T 0 and K.F−1 = −F−1∇m,

ε = 12(1− FFT ), strain tensor

τm = 2µmε+ λtr(ε)I, stress tensorm = ∇ · (qτm), (2.7)

where q(x) = Q(K) forx ∈ K, I is the identity matrix and λ the Lamé’s firstparameter. The quality of the mesh is improved towards its goal of optimal shapeas the elasticity problem is approaching a stationary solution.

2.3 Stabilized Galerkin Formulation

It is known that instabilities may arise when seeking numerical solutions to (1.27,1.28)for finite dimensional discretizations. A typical example for hyperbolic systems isthe oscillations caused by convection dominated solutions [36]. Another examplefor mixed problems is violation of the Ladyshenskaya, Babuska, Brezzi (LBB) con-dition [6] if a non suitable pair of finite elements are used. Stabilization terms ofstreamline diffusion type are developed e.g. in [7], [35], [36], [37], [38].

The stabilized finite element formulation of the Unified Continuum Model takenfrom Paper IV is stated in the rest of this section.

Let Ωtn ⊂ R3 be a polygonal domain. We introduce a sequence of discrete timesteps 0 := t0 < t1 < · · · < tN := t where In := [tn−1, tn) is a time interval of lengthkn := tn − tn−1.

Tn = K specifies the spatial discretization of Ωtn and hn identifies the maxi-mal diameter of the cell elements K ∈ Tn. We also introduce the space-time slabSn := Ωtn × In.

20 CHAPTER 2. METHODS

The Sobolev-space H1(Ωtn) and Wn the finite dimensional function space ofpolynomials of continuous piecewise linear functions on Sn are defined as follows:

H1(Ωtn) : = v ∈ L2(Ωtn)| ∂v∂xk

∈ L2(Ωtn) k = 1, 2, 3 (2.8)

Wn : = v ∈ H1(Sn)| v ∈ C0(Sn), v ∈ P 1(K )× P 1(In),∀K ∈ Tn (2.9)Wn

0 : = v ∈Wn| v = 0 on ∂Ω (2.10)Wn

0 : = [Wn0 ]3. (2.11)

The space of piecewise constant in space functions Dn is defined as

Dn : = v ∈ L2(Sn)| v ∈ P 0(K )× P 1(In),∀K ∈ Tn (2.12)Dn

0 : = [Dn0 ]3×3. (2.13)

We identify the discrete solution for velocity, pressure and solid stress as U =(U, P, τs), the stress for both fluid and solid as T = −P I + θ2µfε(U) + (1− θ)τs,the discrete mesh velocity as M and the test function as v = (v, q, s).

We now formulate the spatially and temporally discretized variational formula-tion of the continuum model (2.1), (2.2), (2.3) based on these definition by usingthe midpoint quadrature rule in time, we obtain a Crank-Nicholson time steppingscheme: for each tn, find (Un, Pn, τn

s ) := (U(tn), P (tn), τs(tn)) with Un ∈ Wn0 ,

Pn ∈Wn and τs ∈ Dn and T n = T (tn), such that:

(ρk−1n (Un −Un−1) + (ρ(Un −Mn) · ∇)Un,v) + (T n : ∇v) + (∇ · U , q) (2.14)

+ SDδ(Un,Mn, Pn,v, q, ρ) = 0,(k−1n (τn

s − τn−1s : s)) = (2µsε(un−1) +∇un−1τn−1

s + τn−1s (∇un−1)T : s)(2.15)

for ∀(v, q, s) ∈Wn0 ×Wn ×Dn where Un = 1

2 (Un + Un−1) and

(v, w) =∑K∈Tn

∫K

v · w dx. (2.16)

We use a simplified Galerkin/least-square method, to stabilize the convection dom-inated problem, where the time derivative term is dropped since the test functionsare piecewise constant in time:

SDδ(Un,Mn, Pn,v, q, ρ) = (δ2∇ · Un,∇ · v)+ (2.17)(δ1ρ(((Un −Mn) · ∇)Un +∇Pn), ρ((Un −Mn) · ∇)v +∇q)

The stabilization parameters are chosen as δ2 = κ2ρh|Un−1| and δ1 = κ1ρ−1(k−2

n +|Un−1 −Mn−1|2h−2

n )−1/2, where κ1, κ1 are problem independent positive constantsof order O(1).

Chapter 3

Contributions and Results

3.1 A Posteriori Error Estimation

As the dimension of the finite element subspace increases, the system of equationsthat should be solved numerically also grows. Combined with the restrictions onthe time step size of the type of a Courant, Friedrichs, Lewy (CFL) condition [12], itbecomes essential to employ a strategic method to optimize the mesh for controllingthe error of a quantity of interest in particular.

For simplicity let A be a linear operator i the problem A(u) = f , for u the exactsolution in a Hilbert space V . We are interested in the error in the form of thebounded functional M(u − Uh), for the computed solution Uh on a discretizationwith cell diameter h such that:

M(u− Uh) = (u− Uh, ζ)

A dual problem A∗(φ) = ζ with initial and boundary conditions is constructedsuch that the bilinear equality (Au, v) = (u,A∗v) is satisfied ∀u, v ∈ V . Then onecan get:

|M(u− Uh)| = |(u− Uh, ζ)|= |(u− Uh, A∗(φ))|= |(A(u− Uh), φ)|= |(R(Uh), φ)|= |h(R(Uh), h−1(φ− πhφ))|≤ ‖h(R(Uh)‖‖CDφ‖

where πhφ is the linear interpolant of the dual solution in the finite dimensionalsolution subspace Vh ⊂ V and R(Uh) = f − A(Uh) is the residual. Due to thebilinear equality the dual equation has to be solved backwards in time if the primal

21

22 CHAPTER 3. CONTRIBUTIONS AND RESULTS

equation is time dependent. In the case of a non-linear operator, the adjoint (dual)operator is linearized at the u and Uh.

The algorithm for adaptive mesh refinement based on a posteriori estimatesbecomes:

Algorithm 1 Adaptive Mesh Refinement Algorithm

1. solve the primal problem until final time.

2. solve the dual problem backwards in time.

3. calculate the error indicators using computed residuals and dual problemgradient.

4. if the sum of the indicators is below a given tolerance terminate.

5. refine a specified percentage of the cells with highest error indicator value.

6. return to step 1.

Paper II focuses on challenges in the derivation of the dual problem for the timedependent non-linear FSI problem, and solving it backwards in time with exactlythe same time stepping and data of the primal problem, obtaining successful resultsfor a serial computation.

Paper III shows an application of the method for a parallel implementationobtaining good results, even though with simplifications in the dual problem anddata of the primal problem.

3.2. CONTACT MODELING 23

3.2 Contact Modeling

θ(t0) θ(t1)

U(t0) U(t1)

Figure 3.1: Visualizations of the vocal folds and quantities computed for the contactmethod, at two time points: t0 - opened folds and t1 - closed folds. Phase functionθ (first row) and velocity function U at the mid plane (second row)

One difficulty for the simulation of human vocal folds is to sustain mesh qualitywhen vocal folds get close to each other. A way to solve this problem is usingre-meshing. In the Unified Continuum model the approach is solving an additionalEikonal equation to decide if the solids are close enough and to change the type ofcontinuum from fluid to solid to sustain mesh quality. This approach is detailed inPaper IV. In the same paper it is also shown that the contact model is in particularsuitable for describing the motion of the continuum for the self oscillating vocalfolds problem as it qualitatively forms the glottal waves as shown in Figure 3.3.The contact model is presented here as Algorithm 2 and its application to vocalfolds simulations is given in Figures 3.1, 3.2.

24 CHAPTER 3. CONTRIBUTIONS AND RESULTS

Algorithm 2 Unified Continuum Contact Algorithm

1. Mark all cells K as non-contact.

2. Solve the Eikonal equation |∇Dh| = 1 for the discrete distance Dh = Dh(x),using an artificial viscosity stabilized cG(1) method with Dh = 0 on theboundary of the fluid sub domain Ωf .

3. Compute |∇Dh|, and define the medial axis M as: M =x∣∣∣|∇Dh(x)| ≤ γ

,

with the threshold parameter γ < 1.

4. Define the contact medial axis: M =x∣∣∣x ∈ M,x 6∈ Ωs, Dh(x) < αh

, with

h the minimum cell size in the mesh.

5. Solve the Eikonal equation |∇DM | = 1 for the discrete distance DM , usingan artificial viscosity stabilized cG(1) method with DM = 0 for the distancefrom M .

6. Mark all fluid cells as contact which fulfill: C =x∣∣∣DM ≤ βh

.

3.2. CONTACT MODELING 25

θ(t0) θ(t1)

D(t0) D(t1)

|∇D(t0)| |∇D(t1)|

DM (t0) DM (t1)

Figure 3.2: Visualizations of the vocal folds and quantities computed for the contactmethod, at two time points: t0 - opened folds and t1 - closed folds.

26 CHAPTER 3. CONTRIBUTIONS AND RESULTS

a b c d

e f g h

a b c d

e f g h

a b c d

e f g h

Figure 3.3: Comparison of the glottal wave between a schematic figure (top row)of the expected contact pattern in the oscillatory cycle and simulations from [55],with a slice through the center of the vocal folds (second row), a clip through thesolid phase (third row) and a clip through the solid phase together with volumerendering of the magnitude of the velocity (bottom row).

3.3 Acoustic Coupling

The computed flow field u and pressure field p from the FSI solver is used to con-struct the acoustic source terms for a wave operator, following an acoustic analogy

3.4. IMPLEMENTATION DETAILS 27

approach. In an ALE setting, the standard wave equation becomes inappropriatefor that purpose [22] and one has to resort to a mixed formulation [11] for theacoustic pressure, pa, and acoustic particle velocity, ua, to account for the movingboundaries (in our case the VF). Let Qa = Ωf ×I stand for the space-time acousticdomain with β denoting again the ALE velocity. The equations to be solved aregiven by [22],

1ρ0c20

∂tpa −1

ρ0c20β · ∇pa +∇ · ua + αapa = Qs in Qa, (3.1)

ρ0∂tua − ρ0 β · ∇ua +∇pa + α∗aua = fs in Qa. (3.2)

Qs(x, t) represents a volume source distribution and fs(x, t) an external bodyforce per unit volume. To perform the one-way coupling between FSI and theacoustics equations, the terms Qs = (ρ0c

20)−1∂tp and fs = 0 have been chosen.

That corresponds to the mixed form of the analogy in [51], which as said, can beviewed as a simplification of the acoustic perturbation equations in [31, 23].

Equation (18) is to be supplemented with the following boundary and initialconditions

ua(x, t) · n = γapa x ∈ Γt,ua(x, t) · n = 0 x ∈ Γo,ua(x, 0) = 0, (3.3)

corresponding to absorbing boundary conditions on walls Γt and homogeneousDirichlet boundary conditions on Γo where we have taken γa = 0.05/c0ρ0, c0 =350, ρ0 = 1.225 for the numerical example in Paper VI.

The boundary conditions are imposed partly strongly and partly weakly asexplained in variational form III of [3] The coupling methodology was successful forcorrectly reproducing the first 3 formants of vowel /i/ as presented in Paper VI

3.4 Implementation Details

The implementation is part of FEniCS-HPC, an open source framework for auto-mated solution of PDE on massively parallel architectures, providing automatedevaluation of variational forms given a high-level description in mathematical no-tation, duality-based adaptive error control, implicit turbulence modeling by use ofstabilized FEM and strong linear scaling up to thousands of cores as shown in PaperI, Paper V and [33, 40, 46, 28, 29]. FEniCS-HPC is a branch of the FEniCS [45, 17]framework focusing on high performance on massively parallel architectures.

The framework is based on components with clearly defined responsibilities.The main components are the following, with their dependencies shown in thedependency diagram in figure 3.4:

• The Finite Element Automated Tabulator (FIAT): Automated generation offinite elements/basis functions

28 CHAPTER 3. CONTRIBUTIONS AND RESULTS

• Fenics Form Compiler (FFC): Automated evaluation of variational forms onone cell based on code generation

• Dolfin-HPC: Library for automated high performance assembly of discretesystems, parallel finite element mesh refinement, input-output, etc.

• Unicorn: Solvers, methods, algorithms using DOLFIN-HPC library for real-istic Unified Continuum modeling.

Figure 3.4: FEniCS-HPC component dependency diagram.

The implementation tasks in this thesis includes work on FFC (extending theinterface of version 1.0.0 to be compatible with DOLFIN-HPC), DOLFIN-HPC(implementation of parallel assembly for forms with jump terms, bug fixing) andUnicorn (implementation of primal and dual solvers). More information about theimplementation is provided in Paper I and Paper V.

Chapter 4

Discussion

The contributions of this work can be grouped in 3 parts:

4.1 Adaptive FEM for FSI problems

The development of an efficient adaptive algorithm for the FSI problem which re-duces the error in the functional of interest is done. The adaptive algorithm is basedon the solution of the dual problem whose gradient is used to compute weightedresiduals to have an error estimate. By computing the individual contribution ofdifferent cells to the error estimate, it is possible to mark the cells with the highestcontribution and refine a specified percentage of “worst” cells at each iteration. Thechallenges and the chosen methods for these challenges are:

• The nonlinearity of the primal problem is a difficulty for the formulation ofthe dual problem and the error estimates. This is handled by linearizingaround the primal solution which also means that data in the form of theprimal problem is needed for constructing and solving the dual problem.

• For the differential operator A corresponding to the operator of the primalproblem Au = f , the dual problem A∗(φ) = ζ with initial and boundaryconditions is constructed such that (Au, v) = (u,A∗v) for ∀u, v ∈ V . Fora time dependent initial value problem this actually means that the dualproblem should be solved backwards in time. Combined with the previouschallenge, this requires handling of the data in the form of the primal solutionfor the whole computation time, and in the context of a moving mesh approachalso involves mesh movement information for each time step. For the workpresented in Paper II, data for the complete time step series is saved andrestored. In this case the changes to the mesh at each time step to maintainits quality while capturing a moving interface was kept minimal by the use oflocal refinement, coarsening and swapping operations through the MADLiblibrary. Thus it was possible to save only the local changes of the mesh at

29

30 CHAPTER 4. DISCUSSION

each time step reducing the amount of data to be saved and enabling thisproblem to be solved for a 2D setting in serial.

• In order to attack challenging problems in 3D, such as the benchmark studypresented in [26] and [25], a parallel implementation with simplifications isinvestigated in Paper III. In this setting, the dual problem is solved only on thefluid domain together with uniform refinement in the structure domain. Thedata of the primal problem is saved at bigger time intervals and simple linearinterpolation is used for approximating the data at individual time steps. Ina parallel setting the data and the mesh is distributed to different processingelements and special implementation for handling the information between theprimal and dual solvers and refinement steps was necessary. The approachwas successfully validated showing fast convergence to the experimental dataas shown in Figure 4.1. A uniform refinement strategy which is also testedon the same problem resulted in much lower level of accuracy.

The possible future research directions in this aspect can be solving the dualproblem in the whole domain in parallel, extending the dual problem formulationwith also the contact model, investigating higher order finite elements or employingdifferent meshes for the dual problem, p and hp refinement techniques [68], etc.

4.2 Contact Modeling

For problems where the domain boundaries come in close proximity such as theopposing walls of the vocal folds for the simulation of human phonation, maintainingthe quality of the discretization is necessary if the boundaries are tracked in themethod. If the quality of the discretization is not satisfied, the condition numberof the resulting system will increase, and even inverted elements may appear. Themain challenge in the vocal folds application is that the mesh smoothing algorithmsare not adequate to sustain the mesh quality and re-meshing during the simulationin a parallel setting is inefficient.

As a remedy to this challenge, the FSI model is redesigned to employ a contactmedium that is enabled or disabled between the vocal folds when they come inclose contact and prevent penetration. The solution of the Eikonal equation andits gradient is used in the model for the decision of these states, and the approachis validated with the studies in Paper IV. In particular the glottal waves for thesimulation on a realistic domain was reported which can be seen in Figure 3.3. Themodel when coupled with an acoustic solver through an acoustic analogy couldrecover the first three formants of the vowel /i/ on a specific geometry which isdescribed in Paper VI.

The possible future research directions in this aspect can be investigation of dif-ferent material properties for the contact medium, employing Nitsche type unfittedmesh approaches with the model, etc.

4.3. ACOUSTIC COUPLING 31

Figure 4.1: Convergence plot for the filament deflection versus experimental datafor 3D FSI benchmark problem presented in Paper III

4.3 Acoustic Coupling

In order to simulate human phonation, the incompressible FSI-contact model shouldfurther be coupled to a wave propagation model. The choice of a suitable acousticanalogy, modeling of boundary conditions to prevent wave reflections from domainboundaries, and numerical stabilization choices could be listed as the challengesin the modeling level. The problem is also complicated in terms of implementa-tion since the acoustic equations are defined on the fluid sub domain of the wholeproblem domain and efficient data transfers between the two solvers is necessary.

In Paper VI the choice of the coupling strategy from [51] is validated and asnapshot of the problem and the results is shown in Figure 4.2. Artificial viscosityon parts of the computational domain is used to prevent reflections. The imple-mentation challenge of arranging the computational domains for fluid and acousticsolvers has been tackled by creating the computational mesh corresponding to theacoustic solver on the fly in the memory of each processing element by removingthe solid cells. Note that this implies the necessity of rearranging the ghosted andshared entity relationships in a parallel setting.

32 CHAPTER 4. DISCUSSION

Figure 4.2: Computational domain and captured formants for the 3D FSI-ContactAcoustic problem discussed in Paper VI

The possible future research directions in this aspect can be investigation ofmore advanced coupling strategies such as acoustic perturbation equations [31, 23]or using a monolithic formulation for the compressible flow, investigation of differentstabilization techniques, employing different structure models involving vocal foldswith multiple layers etc.

Bibliography

[1] Clay mathematical institute millenium problems.www.claymath.org/millennium/Navier-Stokes_Equations/.

[2] Gene M. Amdahl. Validity of the single processor approach to achieving largescale computing capabilities. In Proceedings of the April 18-20, 1967, SpringJoint Computer Conference, AFIPS ’67 (Spring), pages 483–485, New York,NY, USA, 1967. ACM.

[3] Santiago Badia, Ramon Codina, and Hector Espinoza. Stability, convergence,and accuracy of stabilized finite element methods for the wave equation inmixed form. SIAM Journal on Numerical Analysis, 52(4):1729–1752, 2014.

[4] Y Bazilevs, M-C Hsu, J Kiendl, R Wüchner, and K-U Bletzinger. 3d simu-lation of wind turbine rotors at full scale. part ii: Fluid–structure interactionmodeling with composite blades. International Journal for Numerical Methodsin Fluids, 65(1-3):236–253, 2011.

[5] Tomáš Bodnár, Giovanni P Galdi, and Šárka Nečasová. Fluid-Structure Inter-action and Biomedical Applications. Springer, 2014.

[6] A. C. Brenner and L. R. Scott. The Mathematical Theory of Finite ElementMethods. Springer-Verlag, 1994.

[7] A.N. Brooks and T.J.R Hughes. Streamline upwind/petrov-galerkin formula-tions for convection dominated flows with particular emphasis on the incom-pressible navier-stokes equations. Computer methods in applied mechanics andengineering, 32:199–259, 1982.

[8] V. Cerveny. Seismic Ray Theory. Cambridge University Press, 2005.

[9] P. Ciarlet. The Finite Element Method for Elliptic Problems. Society forIndustrial and Applied Mathematics, 2002.

[10] Philippe G Ciarlet, Srinivasan Kesavan, Akhil Ranjan, and Muthusamy Van-ninathan. Lectures on the finite element method, volume 49. Tata institute offundamental research Bombay, India, 1975.

33

34 BIBLIOGRAPHY

[11] R. Codina. Finite element approximation of the hyperbolic wave equationin mixed form. Computer Methods in Applied Mechanics and Engineering,197(13):1305–1322, 2008.

[12] Richard Courant, Kurt Friedrichs, and Hans Lewy. Über die partiellen dif-ferenzengleichungen der mathematischen physik. Mathematische annalen,100(1):32–74, 1928.

[13] J. Donea, A. Huerta, J.-Ph. Ponthot, and A. RodrÃguez-Ferran. ArbitraryLagrangian-Eulerian Methods, pages 413–437. Wiley, 2004.

[14] T. Dunne and R. Rannacher. Adaptive finite element simulation of fluidstructure interaction based on an Eulerian variational formulation. Fluid-structure interaction: Modelling, Simulation, Springer (H.-J. Bunartz et al.,eds), Springer, to appear.

[15] L. C. Evans. Partial Differential Equations. American Mathematical Society,1998.

[16] Charbel Farhat, Kristoffer G Van der Zee, and Philippe Geuzaine. Provablysecond-order time-accurate loosely-coupled solution algorithms for transientnonlinear computational aeroelasticity. Computer methods in applied mechan-ics and engineering, 195(17):1973–2001, 2006.

[17] FEniCS. Fenics project. http://www.fenics.org, 2003.

[18] Miguel A Fernández. Incremental displacement-correction schemes for incom-pressible fluid-structure interaction. Numerische Mathematik, 123(1):21–65,2013.

[19] R.P. Feynman, R.B. Leighton, and M. Sands. The Feynman Lectures onPhysics, Vol. I: The New Millennium Edition: Mainly Mechanics, Radiation,and Heat. Basic Books, 2015.

[20] Yoshikazu Giga. Weak and strong solutions of the navier-stokes initial valueproblem. Publications of the Research Institute for Mathematical Sciences,19(3):887–910, 1983.

[21] F. Grund. Ciarlet, P. G.; Lions, J. L. (eds): Handbook of Numerical Analysis.Vol. iii: Techniques of Scientific Computing (Part 1). Numerical Methods forSolids (Part 1). Solution of Equations in IRn (part 2). Amsterdam etc., North-Holland 1994. x, 778 pp., Dfl. 270.-. ISBN 0-444-89928-6. ZAMM - Journal ofApplied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematikund Mechanik, 77(2):159–159, 1997.

[22] O. Guasch, M. Arnela, R. Codina, and H. Espinoza. A stabilized finite elementmethod for the mixed wave equation in an ALE framework with applicationto diphthong production. Acta Acustica united with Acustica, 102(1):94–106,2016.

BIBLIOGRAPHY 35

[23] Oriol Guasch, Patricia Sánchez-Martín, Arnau Pont, Joan Baiges, and RamonCodina. Residual-based stabilization of the finite element approximation tothe acoustic perturbation equations for low mach number aeroacoustics. In-ternational Journal for Numerical Methods in Fluids, 82(12):839–857, 2016.

[24] John L. Gustafson. Reevaluating Amdahl’s law. Commun. ACM, 31(5):532–533, May 1988.

[25] A. Hessenthaler, N. R. Gaddum, O. Holub, R. Sinkus, O. Röhrle, and D. Nord-sletten. Experiment for validation of fluid–structure interaction models andalgorithms. International Journal for Numerical Methods in Biomedical Engi-neering, 33(9):e2848.

[26] Andreas Hessenthaler, Stephanie Friedhoff, Oliver Röhrle, and David A. Nord-sletten. 3d fluid–structure interaction experiment and benchmark results.PAMM, 16(1):451–452.

[27] J. Hoffman, J. Jansson, and M. Stöckli. Unified continuum modeling of fluid-structure interaction. Mathematical Models and Methods in Applied Sciences,2011.

[28] Johan Hoffman, Johan Jansson, Cem Degirmenci, Niclas Jansson, and Mur-tazo Nazarov. Unicorn: a Unified Continuum Mechanics Solver, chapter 18.Springer, 2012.

[29] Johan Hoffman, Johan Jansson, Niclas Jansson, C. Johnson, and Rodrigo V.de Abreu. Turbulent Flow and Fluid-structure Interaction, chapter 28.Springer, 2012.

[30] Norbert Hosters, Jan Helmig, Atanas Stavrev, Marek Behr, and Stefanie El-geti. Fluidâstructure interaction with nurbs-based coupling. Computer Meth-ods in Applied Mechanics and Engineering, 332:520 – 539, 2018.

[31] A. Hüppe and M. Kaltenbacher. Spectral finite elements for computationalaeroacoustics using acoustic perturbation equations. Journal of ComputationalAcoustics, 20(02):1240005, 2012.

[32] Jack J. Jiang, Yu Zhang, and Clancy McGilligan. Chaos in Voice, FromModeling to Measurement. Journal of Voice, 20(1):2 – 17, 2006.

[33] Niclas Jansson, Johan Hoffman, and Johan Jansson. Framework for MassivelyParallel Adaptive Finite Element Computational Fluid Dynamics on Tetrahe-dral Meshes. SIAM J. Sci. Comput., 34(1):C24–C41, 2012.

[34] Weili Jiang, Xudong Zheng, and Qian Xue. Computational modeling of fluid-structure-acoustics interaction during voice production. Frontiers in bioengi-neering and biotechnology, 5:7, 2017.

36 BIBLIOGRAPHY

[35] C. Johnson, U. Nävert, and J. Pitkäranta. Finite element methods for linearhyperbolic equations. Comput. Methods Appl. Mech. Eng., 45:285–312, 1984.

[36] C. Johnson and J. Saranen. Streamline diffusion methods for the incompress-ible euler and navier-stokes equations. Math. Comp., 47:1–18, 1986.

[37] C. Johnson and A. Szepessy. Shock-capturing streamline diffusion finite ele-ment methods for nonlinear conservation laws. Recent Developments in Com-putatioonal Fluid Dynamics, edited by T. J. R. Hughes and T. Tezduyar,AMD-Vol 95, 1988.

[38] C. Johnson, A. Szepessy, and P. Hansbo. On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conser-vation laws. Mathematics of Computation, 54(189):107–129, 1990.

[39] Manfred Kaltenbacher, Stefan Zörner, Andreas Hüppe, and Petr Sidlof. 3dnumerical simulations of human phonation. In Proceedings of the 11th worldcongress on computational mechanics-WCCM XI, 2014.

[40] Robert C. Kirby. FIAT: Numerical Construction of Finite Element BasisFunctions,, chapter 13. Springer, 2012.

[41] Olga A Ladyzhenskaya and Richard A Silverman. The mathematical theory ofviscous incompressible flow, volume 76. Gordon and Breach New York, 1969.

[42] Julien Lequeurre. Existence of strong solutions to a fluid-structure system.SIAM Journal on Mathematical Analysis, 43(1):389–410, 2011.

[43] Julien Lequeurre. Existence of strong solutions for a system coupling thenavier–stokes equations and a damped wave equation. Journal of MathematicalFluid Mechanics, 15(2):249–271, 2013.

[44] J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. ActaMathematica, 63:193–248, 1934.

[45] Anders Logg, Kent-Andre Mardal, Garth N. Wells, et al. Automated Solutionof Differential Equations by the Finite Element Method. Springer, 2012.

[46] Anders Logg, Kristian B. Ølgaard, Marie E. Rognes, and Garth N. Wells. FFC:the FEniCS Form Compiler, chapter 11. Springer, 2012.

[47] Rajat Mittal, Byron D Erath, and Michael W Plesniak. Fluid dynamics ofhuman phonation and speech. Annual Review of Fluid Mechanics, 45:437–467,2013.

[48] Isaac Newton. The Principia: mathematical principles of natural philosophy.Univ of California Press, 1999.

BIBLIOGRAPHY 37

[49] Philip Pomper. Lomonosov and the discovery of the law of the conservationof matter in chemical transformations. Ambix, 10(3):119–127, 1962.

[50] Th Richter. Goal-oriented error estimation for fluid–structure interaction prob-lems. Computer Methods in Applied Mechanics and Engineering, 223:28–42,2012.

[51] M Roger. Aeroacoustics: Some theoretical background â the acoustic analogy.Computational Aeroacoustics, pages 2006–05, 2006.

[52] A.S. Romer. Vertebrate Paleontology. University of Chicago Press, 1966.

[53] Petr Šidlof, Jaromír Horáček, and Václav Řidky. Parallel cfd simulation of flowin a 3d model of vibrating human vocal folds. Computers & Fluids, 80:290–300,2013.

[54] S. Standring and N.R. Borley. Gray’s Anatomy: The Anatomical Basis ofClinical Practice. ClinicalKey 2012. Churchill Livingstone/Elsevier, 2008.

[55] Brad H. Story. An overview of the physiology, physics and modeling of thesound source for vowels. Acoustical Science and Technology, 23(4):195–206,2002.

[56] S. Sathe T. E. Tezduyar and K. Stein. Solution techniques for the fully-discretized equations in computation of fluid-structure interactions with thespace–time formulations. Computer Methods in Applied Mechanics and Engi-neering, 195:5743–5753, 2006.

[57] R. Temam. Navier-Stokes Equation: Theory and Numerical Analysis. Studiesin Mathematics and Its Applications. North-Holland, 1977.

[58] Tayfun E. Tezduyar, Sunil Sathe, Jason Pausewang, Matthew Schwaab, JasonChristopher, and Jason Crabtree. Interface projection techniques for fluid–structure interaction modeling with moving-mesh methods. ComputationalMechanics, 43(1):39–49, Dec 2008.

[59] Ingo R Titze and Fariborz Alipour. The myoelastic aerodynamic theory ofphonation. National Center for Voice and Speech, 2006.

[60] I.R. Titze and R.C. Scherer. Vocal fold physiology: biomechanics, acoustics,and phonatory control. [Vocal fold physiology. Denver Center for the Perform-ing Arts, 1983.

[61] Leslie Ronald George Treloar. The physics of rubber elasticity. Oxford Uni-versity Press, USA, 1975.

[62] Clifford A Truesdell. Cauchy and the modern mechanics of continua. Revued’histoire des sciences, pages 5–24, 1992.

38 BIBLIOGRAPHY

[63] S. Turek and J. Hron. A monolithic FEM solver for an ALE formulation offluid–structure interaction with configuration for numerical benchmarking. InP. Wesseling, E. Onate, and J. Periaux, editors, Books of Abstracts EuropeanConference on Computational Fluid Dynamics, page 176. nn, 2006. EccomasCFD 2006.

[64] Stefan Turek, J Hron, M Madlik, M Razzaq, H Wobker, and JF Acker. Numer-ical simulation and benchmarking of a monolithic multigrid solver for fluid-structure interaction problems with application to hemodynamics. In FluidStructure Interaction II, pages 193–220. Springer, 2011.

[65] Qian Xue, Xudong Zheng, Rajat Mittal, and Steven Bielamowicz. Subject-specific computational modeling of human phonation. The Journal of theAcoustical Society of America, 135(3):1445–1456, 2014.

[66] Wei Zhao, Cheng Zhang, Steven H Frankel, and Luc Mongeau. Computationalaeroacoustics of phonation, Part I: Computational methods and sound gener-ation mechanisms. Journal of the Acoustical Society of America, 112(5):2134–2146, 2002.

[67] X Zheng, R Mittal, Q Xue, and S Bielamowicz. Direct-numerical simulation ofthe glottal jet and vocal-fold dynamics in a three-dimensional laryngeal model.The Journal of the Acoustical Society of America, 130(1):404–415, 2011.

[68] O. C. Zienkiewicz and J. Z. Zhu. Adaptivity and mesh generation. Interna-tional Journal for Numerical Methods in Engineering, 32(4):783–810.

AppendixIncluded Papers