ENGLISH

Computational Framework for the Finite Element Method in MATLAB® and Python

By Pavel Sumets

Book information

Publisher
CRC Press
Year
2022
ISBN
1032209259, 9781032209258
Language
english
Format
PDF
Filesize
7 MB (7231309 bytes)
Pages
166\182
Time added
2022-07-13 20:06:54

Description

Computational Framework for the Finite Element Method in MATLAB and Python aims to provide a programming framework for coding linear FEM using matrix-based MATLAB language and Python scripting language. It describes FEM algorithm implementation in the most generic formulation so that it is possible to apply this algorithm to as many application problems as possible. Readers can follow the step-by-step process of developing algorithms with clear explanations of its underlying mathematics and how to put it into MATLAB and Python code. The content is focused on aspects of numerical methods and coding FEM rather than FEM mathematical analysis. However, basic mathematical formulations for numerical techniques which are needed to implement FEM are provided. Particular attention is paid to an efficient programming style using sparse matrices. Features Contains ready-to-use coding recipes allowing fast prototyping and solving of mathematical problems using FEM. Suitable for upper-level undergraduates and graduates in applied mathematics, science, or engineering. Both MATLAB and Python programming codes are provided to give readers more flexibility in the practical framework implementation. Cover Half Title Title Page Copyright Page Contents Introduction Background and Basic Concepts CHAPTER 1: Finite Element Method for the One-Dimensional Boundary Value Problem 1.1. FORMULATION OF THE PROBLEM 1.2. INTEGRAL EQUATION 1.3. LAGRANGE INTERPOLATING POLYNOMIALS 1.3.1. Global Interpolating Polynomial 1.3.2. Piecewise Polynomial Interpolation 1.3.3. Linear System of the FEM 1.4. ILLUSTRATIVE PROBLEM 1.5. ALGORITHMS OF THE FINITE ELEMENT METHOD 1.5.1. Composing Linear System of the FEM 1.5.2. Assembling Global Matrices 1.5.3. Calculating Elements of the Local Matrices 1.6. QUADRATURE RULES 1.7. DEFINING PARAMETERS OF THE FEM CHAPTER 2: Programming One-Dimensional Finite Element Method 2.1. SPARSE MATRICES IN MATLAB® 2.2. INPUT DATA STRUCTURES 2.3. CODING QUADRATURE RULES 2.4. INTERPOLATING AND DIFFERENTIATING MATRICES 2.5. CALCULATING AND ASSEMBLING FEM MATRICES 2.6. PYTHON IMPLEMENTATION CHAPTER 3: Finite Element Method for the Two-Dimensional Boundary Value Problem 3.1. MODEL PROBLEM 3.2. FINITE ELEMENTS DEFINITION 3.3. TRIANGULATION EXAMPLES 3.4. LINEAR SYSTEM OF THE FEM 3.5. STIFFNESS MATRIX AND FORCING VECTOR 3.6. ALGORITHM OF SOLVING PROBLEM CHAPTER 4: Building Two-Dimensional Meshes 4.1. DEFINING GEOMETRY 4.2. REPRESENTING MESHES IN MATRIX FORM FOR LINEAR INTERPOLATION FUNCTIONS 4.3. COMPLEMENTARY MESH 4.4. BUILDING MESHES IN MATLAB® 4.5. BUILDING MESHES IN PYTHON CHAPTER 5: Programming Two-Dimensional Finite Element Method 5.1. ASSEMBLING GLOBAL STIFFNESS MATRIX 5.2. ASSEMBLING GLOBAL FORCING VECTOR 5.3. CALCULATING LOCAL STIFFNESS MATRICES 5.4. CALCULATING EQUATION COEFFICIENTS 5.5. CALCULATING GLOBAL MATRICES 5.6. CALCULATING BOUNDARY CONDITIONS 5.7. ASSEMBLING BOUNDARY CONDITIONS 5.8. SOLVING EXAMPLE PROBLEM CHAPTER 6: Nonlinear Basis Functions 6.1. LINEAR TRIANGULAR ELEMENTS 6.2. CURVILINEAR TRIANGULAR ELEMENTS 6.3. STIFFNESS MATRIX WITH QUADRATIC BASIS Conclusion APPENDIX A: Variational Formulation of a BVP APPENDIX B: Discussion of Global Interpolation APPENDIX C: Interpolatory Quadrature Formulas APPENDIX D: Quadrature Rules and Orthogonal Polynomials APPENDIX E: Computational Framework in Python E.1. ASSEMBLING MATRICES E.2. ASSEMBLING BOUNDARY CONDITIONS E.3. SOLVING EXAMPLE PROBLEM Bibliography Index

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