A Space-Time Meshfree Collocation Method for CoupledProblems on Irregularly-Shaped Domains

Von der

Fakultät Architektur, Bauingenieurwesen und Umweltwissenschaften

der Technischen Universität Carolo-Wilhelmina

zu Braunschweig

zur Erlangung des Grades eines

Doktor-Ingenieurs (Dr.-Ing.)

genehmigte

Dissertation

von

Hennadiy Netuzhylov, MSc

geboren am 20.11.1978

aus Lviv, Ukraine

Eingereicht am

Disputation am

Berichterstatter

12. Juni 2008

10.~ovember2008

Prof. Dr.-Ing. Andreas Zilian

Prof. Dr. rer. nato Thomas Sonar

2008

Contents

Kurzfassung

Abstract

Preface

U

lU

lX

1 Introduction 1

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

1.2 Historical development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Current state of research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Classification of meshfree methods .. . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Basic concepts and Methodology 11

2.1 Deduction of the method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11

2.2 (Interpolating) Moving Least Squares . . . . . . . . . . . . . . . . . . . . . . . . .. 12

2.3 Comparison with other Least Squares methods . . . . . . . . . . . . . . . . . .. 15

2.4 Basis functions 16

2.5 Weight functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 17

2.5.1 Approximation case :: MLS . . . . . . . . . . . . . . . . . . . . . . . . . .. 17

2.5.2 Interpolation case :: IMLS . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18

2.6 Singularity-by-Construction................................ 19

CONTENTS

2.7

2.8

2.6.1 Singular Value Decomposition .

2.6.2 Regularization technique .

2.6.3 Derivatives and nodal values of the kerne! functions .

Boundary and initial conditions .. . . . . . . . . . . . . . . . . . . . . . . . .

Numerical properties of IMLS functions .

2.8.1 Patch tests :: A linear elasticity example .

2.8.2 Positivity conditions in meshfree collocations .

20

21

23

28

29

31

34

3 A Space-Time Meshfree Collocation Method 35

3.1 Splitting into space-time slabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35

3.2 Discretization of a coupled problem 37

3.3 Strategy for nonlinearities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38

3.4 Point distribution techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39

3.4.1 Halton points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39

3.4.2 Data management .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40

3.5 Numerical stability and trade-off principles . . . . . . . . . . . . . . . . . . . . .. 43

3.5.1 Choice of parameters " 43

3.6 Workflow............................................ 46

4 Discretization of Irregularly-Shaped Domains 47

4.1 Scattered data interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47

4.2 A multilevel surface reconstruction " 48

4.3 Interpolation of point cloud data by implicit surfaces 50

4.3.1 Demonstration of the approach in 2d. . . . . . . . . . . . . . . . . . . .. 54

4.3.2 Examples in 3d " 55

5 Interpolating Functions and Derivatives 63

5.1 Error indicators and convergence 63

5.2 Highly-oscillating functions :: Franke's function on a quadratic domain '" 66

5.3 Highly-oscillating functions :: A "blupp" function on a circular domain ... 69

CONTENTS

6 Numerical Study of STMCM for Differential Equations 73

6.1 Introduction.......................................... 73

6.2 Classification of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74

6.2.1 Problem types. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 74

6.2.2 Well-Posedness................................... 75

6.2.3 Equation types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 76

6.3 Linear Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . . . .. 77

6.3.1 A spring-mass oscillator 77

6.3.2 Foucault pendulum .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82

6.3.3 Stability and filter properties . . . . . . . . . . . . . . . . . . . . . . . . .. 86

6.4 Nonlinear Ordinary Differential Equations . . . . . . . . . . . . . . . . . . . . .. 89

6.4.1 Lorenz attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 90

6.5 Stationary Problems :: Elliptic Partial Differential Equations 92

6.5.1 Theoretical issues 95

6.5.2 Discretization and boundary conditions . . . . . . . . . . . . . . . . . .. 97

6.5.3 Laplace equation on a unit square . . . . . . . . . . . . . . . . . . . . . .. 100

6.5.4 Laplace equation on a circle . . . . . . . . . . . . . . . . . . . . . . . . . .. 100

6.5.5 A boundary layer problem 101

6.5.6 HeImholtz equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106

6.6 Time-Dependent Partial Differential Equations , 110

6.6.1 Diffusions :: Heat equation . . . . . . . . . . . . . . . . . . . . . . . . . .. 110

6.6.2 Vibrations :: Wave equation , 112

6.6.3 Convection :: Transport equation . . . . . . . . . . . . . . . . . . . . . .. 118

6.6.4 Hamilton-Jacobi equations :: Level-set equation . . . . . . . . . . . . .. 122

6.6.5 Curvature driven evolving interfaces . . . . . . . . . . . . . . . . . . . .. 124

6.7 Nonlinear Partial Differential Equations , 126

6.7.1 Shock waves :: Burgers' equation. . . . . . . . . . . . . . . . . . . . . . .. 126

6.7.2 Metastability :: Allen-Cahn equation . . . . . . . . . . . . . . . . . . . .. 128

6.8 Discussion........................................... 129

CONTENTS

7 Application to Biofilm and Tumour Growth Modelling 131

7.1 Biofilms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 131

7.2 State of research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 132

7.3 A mathematical model of biofilm growth . . . . . . . . . . . . . . . . . . . . . .. 133

7.3.1 Material description , 134

7.3.2 Nutrient description 135

7.3.3 Interface description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 135

7.3.4 Nutrient consumption and biomass formation 136

7.3.5 Validation...................................... 137

7.4 Tumours............................................ 143

7.4.1 A mathematical model of tumour growth . . . . . . . . . . . . . . . . .. 145

7.4.2 Validation :: a spherical avascular tumour . . . . . . . . . . . . . . . . .. 146

8 Summary and Outlook 149

8.1 Summary............................................ 149

8.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 150

A Analytical expressions 151

B Invariance of the Laplacian 155

Bibliography 157

Index 167