Practical Finite Element Modeling in Earth Science using Matlab

Guy Simpson obtained his PhD in Geology from ETH Zurich. He is currently within the Department of Earth Science at the University of Geneva. Over the past decade, he has taught numerous courses at the Universities of Geneva, École Normale Supérieure in Paris, and ETH Zurich on numerical modeling in Earth science using Matlab. He also uses this method in his own research that includes investigation of earthquakes, tectonics and erosion of active mountain ranges, fluid flow, magmatism, and tsunamis.

Preface xiii

Symbols xv

About the Companion Website xvii

Part I The Finite Element Method with Matlab 1

1 Preliminaries 3

1.1 Mathematical Models 3

1.2 Boundary and Initial Conditions 4

1.3 Analytical Solutions 5

1.4 Numerical Solutions 5

1.5 Numerical Solution Methods 7

1.6 Matlab Script 8

1.7 Exercises 10

Suggested Reading 12

2 Beginning with the Finite Element Method 13

2.1 The Governing PDE 13

2.2 Approximating the Continuous Variable 14

2.3 Minimizing the Residual 15

2.4 Evaluating the Element Matrices 17

2.5 Time Discretization 18

2.6 Assembly 19

2.7 Boundary and Initial Conditions 21

2.8 Solution of the Algebraic Equations 21

2.9 Exercises 22

Suggested Reading 23

3 Programming the Finite Element Method in Matlab 25

3.1 Program Structure and Philosophy 25

3.2 Summary of the Problem 25

3.3 Discretized Equations 26

3.4 The Program 27

3.4.1 Preprocessor Stage 27

3.4.2 Solution Stage 29

3.4.3 Postprocessor Stage 30

3.5 Matlab Script 30

3.6 Exercises 33

Suggested Reading 34

4 Numerical Integration and Local Coordinates 35

4.1 Gauss–Legendre Quadrature 36

4.2 Local Coordinates 37

4.3 Evaluating the Integrals 39

4.4 Variable Material Properties 40

4.5 Programming Considerations 41

4.6 Matlab Script 43

4.7 Exercises 45

Suggested Reading 47

5 The Finite Element Method in Two Dimensions 49

5.1 Discretization 50

5.2 Geometry and Nodal Connectivity 52

5.3 Integration of Element Matrices 54

5.4 Multielement Assembly 57

5.5 Boundary Conditions and Solution 60

5.6 Matlab Script 61

5.7 Exercises 65

Suggested Reading 66

6 The Finite Element Method in Three Dimensions 67

6.1 Discretization 67

6.2 Element Integration 69

6.3 Assembly for Multielement Mesh 72

6.4 Boundary Conditions and Solution 73

6.5 Matlab Program 74

6.6 Exercises 79

Suggested Reading 80

7 Generalization of Finite Element Concepts 81

7.1 The FEM for an Elliptic Problem 84

7.2 The FEM for a Hyperbolic Problem 96

7.3 The FEM for Systems of Equations 102

7.4 Exercises 116

Suggested Reading 116

Part II Applications of the Finite Element Method in Earth Science 119

8 Heat Transfer 121

8.1 Conductive Cooling in an Eroding Crust 122

8.2 Conductive Cooling of an Intrusion 126

Suggested Reading 135

9 Landscape Evolution 137

9.1 Evolution of a 1D River Profile 138

9.2 Evolution of a Fluvially Dissected Landscape 143

Suggested Reading 150

10 Fluid Flow in Porous Media 151

10.1 Fluid Flow Around a Fault 152

10.2 Viscous Fingering 157

Suggested Reading 166

11 Lithospheric Flexure 167

11.1 Governing Equations 167

11.2 FEM Discretization 168

11.3 Matlab Implementation 171

Suggested Reading 181

12 Deformation of Earth’s Crust 183

12.1 Governing Equations 183

12.2 Rate Formulation 185

12.3 FEM Discretization 186

12.4 Viscoelastoplasticity 188

12.5 Matlab Implementation 190

Suggested Reading 205

13 Going Further 207

13.1 Optimization 207

13.2 Using Other FEMs 213

13.3 Use of Existing Finite Element Software 215

Appendix A Derivation of the Diffusion Equation 217

Appendix B Basics of Linear Algebra with Matlab 221

Appendix C Comparison between Different Numerical Methods 227

Appendix D Integration by Parts 237

Appendix E Time Discretization 239

References 241

Index 245