Finite Element Methods in Civil and Mechanical Engineering: A Mathematical Introduction

✍ Scribed by Ali Gerami Matin, Arzhang Angoshtari

• Publisher: CRC Press
• Year: 2020
• Tongue: English
• Leaves: 177
• Category: Library

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Chapter 1: Overview
Chapter 2: Mathematical Preliminaries
2.1. Real Numbers
2.2. Functions
2.3. Linear Spaces, Linear Mappings, and Bilinear Forms
2.4. Linear Independence, Hamel Bases, and Dimension
2.5. The Matrix Representation of Linear Mappings and Bilinear Forms
2.6. Normed Linear Spaces
2.7. Functionals and Dual Spaces
2.8. Green’s Formulas
Exercises
Comments and References
Chapter 3: Finite Element Interpolation
3.1. 1D Finite Element Interpolation
3.1.1. The Global Level
3.1.2. The Local Level
3.2 Finite Elements
3.2.1. Simplicial Lagrange Finite Elements of Type (k)
3.2.2. Simplicial Hermite Finite Elements of Type (3)
3.2.3. The Raviart-Thomas Finite Element
3.2.4. The Nedelec Finite Element
3.3. Meshes
3.4. Finite Element Spaces and Interpolations
3.4.1. H1-Conformal Finite Element Spaces
3.4.1.1. Lagrange Elements
3.4.1.2. Hermite Elements
3.4.2. H(div)-Conformal Finite Element Spaces
3.4.3. H(curl)-Conformal Finite Element Spaces
3.4.4. Affine Families of Finite Elements
3.5. Convergence of Interpolations
Exercises
Computer Exercises
Comments and References
Chapter 4: Conforming Finite Element Methods for PDEs
4.1. Second-Order Elliptic PDEs
4.2. Weak Formulations of Elliptic PDEs
4.2.1. Dirichlet Boundary Condition
4.2.2. Neumann Boundary Condition
4.2.3. Robin Boundary Condition
4.3. Well-posedness of Weak Formulations
4.4. Variational Structure
4.5. The Galerkin Method and Finite Element Methods
4.5.1. The Stiffness Matrix
4.5.2. Well-posedness of Coercive Discrete Problems
4.5.3. Convergence of Finite Element Solutions
4.6. Implementation: The Poisson Equation
4.6.1. Dirichlet Boundary Condition
4.6.2. Mixed Dirichlet-Neumann Boundary Condition
4.6.3. Robin Boundary Condition
4.7. Time-Dependent Problems: Parabolic Problems
4.7.1. Finite Element Approximations using the Method of Lines
4.7.2. Temporal Discretization
4.7.3. Implementation: A Diffusion Problem
4.8. Mixed Finite Element Methods
4.8.1. Mixed Formulations
4.8.2. Mixed Methods and inf-sup Conditions
4.8.3. Implementation
Exercises
Computer Exercises
Comments and References
Chapter 5: Applications
5.1. Elastic Bars
5.2. Euler-Bernoulli Beams
5.3. Elastic Membranes
5.4. The Wave Equation
5.5. Heat Transfer in a Turbine Blade
5.6. Seepage in Embankment
5.7. Soil Consolidation
5.8. The Stokes Equation for Incompressible Fluids
5.9. Linearized Elasticity
5.10. Linearized Elastodynamics: The Hamburg Wheel-Track Test
5.11. Nonlinear Elasticity
Exercises
Computer Exercises
Appendix A: FEniCS Installation
Appendix B: Introduction to Python
B.1. Running Python Programs
B.2. Lists
B.3. Branching and Loops
B.4. Functions
B.5. Classes and Objects
B.6. Reading and Writing Files
B.7. Numerical Python Arrays
B.8. Plotting with Matplotlib
References
Index