The book presents the fundamentals of the Galerkin Finite Element Method for linear boundary value problems from an engineering perspective. Emphasis is given to the theoretical foundation of the method rooted in Functional Analysis using a language accessible to engineers. The book discusses standard procedures for applying the method to time-dependent and nonlinear problems and addresses essential aspects of applying the method to non-linear dynamics and multi-physics problems. It also provides several hand-calculation exercises as well as specific computer exercises with didactic character. About one fourth of the exercises reveals common pitfalls and sources of errors when applying the method. Carefully selected literature recommendations for further studies are provided at the end of each chapter.
The reader is expected to have prior knowledge in engineering mathematics, in particular real analysis and linear algebra. The elements of algebra and analysis required in the main part of the book are presented in corresponding sections of the appendix. Students should already have an education in strength of materials or another engineering field, such as heat or mass transport, which discusses boundary value problems for simple geometries and boundary conditions.
Author(s): Uwe Mühlich
Series: Solid Mechanics and Its Applications, 268
Publisher: Springer
Year: 2023
Language: English
Pages: 204
City: Cham
Preface
Contents
Symbols
1 Introduction
1.1 Boundary Value Problems and Their Strong Forms
1.2 The Structure of the Book
1.3 Challenges in Learning and Teaching the Method in an Engineering Context
2 Linear Boundary Value Problems
2.1 The Poisson Equation in mathbbR
2.1.1 Strong Form
2.1.2 The Weak Derivative
2.1.3 Variational and Weak Form
2.1.4 A Precursor of FEM
2.1.5 Comments on Functional Analysis Background
2.1.6 Galerkin FEM with a Piecewise Linear Global Basis
2.1.7 Galerkin FEM Using a Linear Local Basis
2.1.8 Non-homogeneous Dirichlet Conditions
2.1.9 Hand-Calculation Exercises
2.2 Poisson Equation in mathbbRN, N 2
2.2.1 Preliminary Remarks
2.2.2 Strong Form
2.2.3 First Order Weak Partial Derivatives
2.2.4 Variational and Weak Form
2.2.5 Galerkin FEM with Linear Quadrilateral Elements
2.3 Systematic Construction of Lagrange Elements
2.3.1 Overview
2.3.2 Lagrange Interpolation in mathbbR
2.3.3 Q-type Interpolation Bases in mathbbRN , N2
2.3.4 P-type Interpolation Bases in mathbbRN , N 2
2.3.5 Remarks on Integration Using Reference Domains
2.4 Linear Elastostatics
2.4.1 Preliminary Remarks
2.4.2 Strong Form
2.4.3 Strong Forms for Plane Stress and Plane Strain Problems
2.4.4 Variational and Weak Forms
2.4.5 Galerkin FEM for Plane Strain/Stress Problems
2.5 Boundary Value Problems of Fourth Order in mathbbR
2.5.1 Strong Form
2.5.2 Variational and Weak Form
2.5.3 Galerkin FEM Using C1 Continuity
2.6 Network Models
2.7 Hand Calculation Exercises
2.8 Computer Exercises
2.9 Bibliographical Remarks
References
3 Linear Initial Boundary Value Problems
3.1 Introductory Example
3.2 Integration of Initial Value Problems
3.2.1 General Aspects
3.2.2 Finite Difference Methods and Simple Euler Methods
3.2.3 Runge-Kutta Methods
3.3 Stability of Time Integration Schemes
3.4 Non-stationary Linear Transport
3.4.1 Time-Continuous Variational Form
3.4.2 Galerkin FEM
3.4.3 Time Integration
3.5 Linear Structural Dynamics
3.5.1 Strong Form and Time-Continuous Variational Form
3.5.2 Galerkin FEM and Time Integration with Central Differences
3.5.3 Newmark's Method
3.5.4 Stability of Time Integration Methods
3.6 Bibliographical Remarks
References
4 Non-linear Boundary Value Problems
4.1 Non-linear Poisson Equation in mathbbR
4.1.1 Strong Form
4.1.2 Variational and Weak Form
4.1.3 Galerkin FEM
4.2 Newton's Method for Solving Non-linear Equations
4.3 Solving the System of Non-linear FEM Equations
4.4 Gauss-Legendre Integration
4.5 Implementation Aspects
4.6 Bibliographical Remarks
References
5 A Primer on Non-linear Dynamics and Multiphysics
5.1 Non-linear Dynamics
5.1.1 Strong and Time Continuous Variational Form
5.1.2 Galerkin FEM and Time Integration
5.2 Thermo-Mechanical Coupling
5.2.1 Strong Form
5.2.2 Time Continuous Variational Form and FEM Discretisation
5.3 Bibliographical Remarks
References
Appendix A Elements of Linear Algebra
A.1 Preliminaries
A.2 Elementary Algebraic Structures
A.3 The Real Vector Space
A.4 Inner Product, Norm and Metric
A.5 The Vector Space mathbbRN
A.6 Linear Mappings and Tensors
A.7 Linear Mappings and Matrices
A.8 Systems of Linear Equations and Matrix Properties
A.9 Bibliographical Remarks
Appendix B Elements of Real Analysis
B.1 Basic Topological Aspects
B.2 Limits and Convergence of Sequences and Series
B.3 Real Functions: Continuity and Boundedness
B.4 Sequences and Series of Functions
B.5 Gradient and Differential in mathbbR
B.6 Gradient and Differential in mathbbRN
B.7 Notation Using Differential Operators in mathbbRN
B.8 Notes on Riemann Integral and Lebesgue Integral
B.9 Comments on Notation for Integrals in mathbbRN
B.10 Integration Theorems in mathbbRN
B.11 Mappings and Their Jacobians
B.12 Bibliographical Remarks
Appendix C Elements of Linear Functional Analysis
C.1 Motivation
C.2 Introduction to Function Spaces
C.3 Linear Mappings and Linear Forms
C.4 Weak Derivative
C.5 Sobolev Spaces
C.6 Variational Formulation of Boundary Value Problems
C.7 The Lax-Milgram Lemma
C.8 Weak Solutions of Boundary Value Problems
C.9 Generalisations
C.10 Bibliographical Remarks
Appendix D Solutions of Selected Problems
Index
