Practical Finite Element Modeling in Earth Science using Matlab

Guy Simpson

• Publisher: Wiley-Blackwell
• Year: 2017
• ISBN:
  9781119248668
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

Mathematical models have become a crucial way for the Earth scientist to understand and predict how our planet functions and evolves through time and space. The finite element method (FEM) is a remarkably flexible and powerful tool with enormous potential in the Earth Sciences. This pragmatic guide explores how a variety of different Earth science problems can be translated and solved with FEM, assuming only basic programming experience.

This book begins with a general introduction to numerical modeling and includes multiple sample Matlab codes to illustrate how FEM is implemented in practice. Textboxes have been included to provide additional detail, such as specialized Matlab usage or advanced topics. Covering all the key aspects, this is essential reading for those looking to master the technique, as well as those simply seeking to increase their basic level of understanding and appreciation of FEM.

Part I
The Finite Element Method with Matlab

Part I has two main purposes. The first purpose is to introduce readers to the Galerkin form of the finite element method (FEM), which is a numerical technique for discretizing partial differential equations (PDEs). The second purpose is to show practically how the resulting equations are programmed and solved on a computer using Matlab. Each chapter builds on the previous one and introduces one or more key concept. We will consider how the FEM is applied in one dimension (1D), two dimension (2D), and three dimension (3D), using a parabolic (diffusion) equation as an example. Chapter 7 generalizes the concepts and extends application of the FEM to systems of equations and to elliptic and hyperbolic problems.

1
Preliminaries

This chapter provides a short introduction to mathematical models consisting of systems of partial differential equations (PDEs) along with auxiliary (boundary and initial) conditions. We discuss how these equations can be solved, either exactly or using numerical methods. We also briefly consider the important issues of precision and stability of a numerical solution. A Matlab script is provided at the end of the chapter to enable readers to compare an analytical solution with its corresponding numerical approximation.

1.1 Mathematical Models

The application of the principles of conservation of mass, momentum, and energy combined with experimentally derived laws produces sets of PDEs that describe variations in velocity (or displacement), pressure, and temperature in space and time. When combined with boundary and initial conditions, these equations constitute mathematical models that can be solved and studied in a way somewhat similar to performing experiments in a laboratory. Whether a model is mathematical or analogue, both are simplified abstractions of reality. However, such models are useful because they can help isolate the influence of certain parameters or scenarios, study complex system interactions, and make predictions.

An example of a mathematical model that has important application in Eart...

Table of contents

• Cover
• Title Page
• Table of Contents
• Preface
• Symbols
• About the Companion Website
• Part I: The Finite Element Method with Matlab
• Part II: Applications of the Finite Element Method in Earth Science
• Appendix A: Derivation of the Diffusion Equation
• Appendix B: Basics of Linear Algebra with Matlab
• Appendix C: Comparison between Different Numerical Methods
• Appendix D: Integration by Parts
• Appendix E: Time Discretization
• References
• Index
• End User License Agreement