Inside Finite Elements
Description
All relevant implementation aspects of finite element methods are discussed in this book. The focus is on algorithms and data structures as well as on their concrete implementation. Theory is covered only as far as it gives insight into the construction of algorithms. In the exercises, a complete FE-solver for stationary 2D problems is implemented in Matlab/Octave.
Contents:
Finite Element Fundamentals
Grids and Finite Elements
Assembly
Solvers
Error Estimation
Mesh Refinement
Multigrid
Elastomechanics
Fluid Mechanics
Grid Data Structure
Function Reference
Reviews / Votes
"This book is a "must have" for mathematicians and engineers interested in scientific computing and applied mathematics."
Razvan Raducanu in: Zentralblatt MATH 1343
"Die Sachverhalte werden kurz und bündig beschrieben. Das macht das Buch für [.] Studenten, die schon etwas Erfahrung mit der Finiten Element Methode haben, zum idealen Nachschlagewerk bei praktischen Übungen.
Die Theorie wird kurz und knapp erläutert. Die praktischen Code Beispiele erleichtern den Studenten das Schreiben eigener Programme."
Dipl.-Ing. Martin Epperlein, Brandenburgische TU Cottbus Senftenberg
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Content
- Intro
- Contents
- Preface
- 1. Finite Element Fundamentals
- 1.1 Stationary Heat Equation
- 1.2 Boundary Conditions
- 1.3 Weak Formulation
- 1.4 Galerkin Discretization
- 1.5 Finite Elements
- Exercises
- 2. Grids and Finite Elements
- 2.1 Cells and Vertices
- 2.2 Shape and Ansatz Functions
- 2.3 Grid Generation and Visualization
- 2.3.1 Preprocessing
- 2.3.2 Postprocessing
- 2.4 Edges and Boundaries
- Exercises
- 3. Assembly
- 3.1 System Matrices
- 3.1.1 Elemental Stiffness Matrices
- 3.1.2 Scattering
- 3.1.3 Elemental Mass Matrices
- 3.2 Boundary Conditions
- 3.2.1 Robin Boundary Conditions
- 3.2.2 Dirichlet Boundary Conditions
- 3.3 Right Hand Sides
- 3.4 Examples
- 3.5 Sparse Matrix Storage Formats
- 3.5.1 Triplet Format
- 3.5.2 Compressed Row Storage
- 3.5.3 Computational Efficiency
- Exercises
- 4. Solvers
- 4.1 Direct Solvers
- 4.2 Iterative Solvers
- 4.2.1 Stationary Iterations
- 4.2.2 Gradient Descent
- 4.2.3 Conjugate Gradient Method
- 4.3 Examples
- Exercises
- 5. Error Estimation
- 5.1 Hierarchical Error Estimation
- 5.2 DLY simplification
- 5.3 Assembly
- 5.3.1 Elemental Matrices
- 5.3.2 Elemental Right Hand Sides
- 5.3.3 Quadrature Rules
- 5.4 Examples
- Exercises
- 6. Mesh Refinement
- 6.1 Marking Strategies
- 6.2 Refinement
- 6.2.1 Bisection
- 6.2.2 Triangle splitting
- 6.3 Grid Hierarchy
- 6.4 Examples
- Exercises
- 7. Multigrid
- 7.1 Smoothers
- 7.2 Coarse Grid Correction
- 7.3 Multigrid Solver
- 7.4 Additive Multilevel Preconditioners
- 7.5 Examples
- Exercises
- 8. Elastomechanics
- 8.1 Linear Elastomechanics
- 8.2 Vectorial Finite Elements
- 8.3 Time-harmonic Dynamics
- 8.3.1 Eigenfunctions
- 8.4 Examples
- 8.4.1 Stationary Problems
- 8.4.2 Oscillatory Problems
- Exercises
- 9. Fluid Mechanics
- 9.1 Stokes Equation
- 9.2 Uzawa's Method
- 9.3 Examples
- 9.3.1 Discretization
- 9.3.2 Iterative Solution
- Exercises
- A. Grid Data Structure
- B. Function Reference
- Bibliography
- Index
