This book presents an introduction to Matlab for students and professionals working in the field of engineering and other scientific and technical sectors, who have an interest or need to apply Matlab as a tool for undertaking simulations and formulating solutions for the problems concerned. The presentation is highly accessible, employing a step-by-step approach in discussing selected problems: deduction of the mathematical model from the physical phenomenon, followed by analysis of the solutions with Matlab. Since a physical phenomenon takes place in space and time, the corresponding mathematical model involves partial differential equations. For this reason, the book is dedicated to numerically solving these equations with the Finite Element Method and Finite Difference Method. Throughout, the text presents numerous examples and exercises with detailed worked solutions. Matlab for Engineering is a useful desktop reference for undergraduates and scientists alike in real world problem solving.
Author(s): Berardino D'acunto
Publisher: World Scientific Publishing
Year: 2021
Language: English
Pages: 325
City: Singapore
Contents
Preface
Chapter 1. Function Files
1.1 Matrices
1.1.1 Creating Matrices
1.1.2 Matrix Indexing
1.1.3 Matrix Manipulation
1.1.4 Tridiagonal Matrices
1.1.5 Matrix Operations
1.1.6 Right and Left Divisions
1.2 Script Files
1.2.1 For Loop
1.2.2 Examples of Script Files
1.3 Introduction to Function Files
1.3.1 Structure of Function Files
1.3.2 Function with a Multiple Output Variable
1.3.3 Flow Control Structures
1.3.4 Local Functions, Anonymous Functions
1.3.5 Logical Operators and Logical Functions
Chapter 2. The Finite Difference Method
2.1 Finite Difference Approximations of Derivatives
2.1.1 Forward, Backward and Central Approximations
2.1.2 Approximation of Functions Depending on Two Variables
2.1.3 Approximation of Higher Order Derivatives
2.2 Diffusion
2.2.1 Fourier’s Law and Heat Equation
2.2.2 Fick’s Law and Diffusion
2.2.3 Free Boundary Value Problems
2.3 Finite Difference Method
2.3.1 Explicit Euler Method
2.3.2 Stability, Convergence, Consistence
2.3.3 Boundary Value Problems
2.3.4 Diffusion in a Multi-layer Medium
2.3.5 Implicit Euler Method
2.3.6 Crank–Nicolson Method
2.3.7 Von Neumann Stability Criterium
2.4 Exercises
Chapter 3. Diffusion and Convection
3.1 Convection-diffusion Equation
3.1.1 Upwind Method
3.1.2 Other Finite Difference Methods for the Convection-Diffusion Equation
3.1.3 Advection Equation
3.2 Method of Lines
3.2.1 Heat Equation
3.2.2 Nonlinear Equations
3.2.3 Variable Diffusivity Coefficient
3.2.4 Convection-Diffusion Equation
3.3 Saving Data and Figures
3.3.1 Save Function
3.3.2 Load Function
3.3.3 Saving Figures
3.4 Exercises
Chapter 4. Introduction to the Finite Element Method
4.1 Numerical Integration
4.2 Finite Element Method
4.2.1 Axial Motion of a Bar
4.2.2 Weak Solution
4.2.3 Shape Functions
4.2.4 Boundary Value Problems
4.2.5 Axial Displacement and Stress in a Bar
4.2.6 Concentrated Force and Dirac Function
4.3 Partial Differential Equations
4.3.1 Diffusion Equation
4.3.2 Wave Equation
4.4 Exercises
Chapter 5. Introduction to the Finite Element Method in Two Spatial Dimensions
5.1 Elliptic Partial Differential Equations
5.1.1 Green’s Identities
5.1.2 Boundary Value Problems
5.2 Finite Element Method in Two Spatial Dimensions
5.2.1 Shape Functions
5.2.2 Weak Form of the Poisson Equation
5.2.4 Applications to the Dam and Sheet Pile Wall
5.3 Finite Difference Method
5.3.1 Five-Point Method
5.3.2 Model of a Dam
5.4 Exercises
Chapter 6. The Euler–Bernoulli Beam
6.1 Finite Element Method
6.1.1 Euler–Bernoulli Beam Equation
6.1.2 Shape Functions
6.1.3 Weak Form
6.2 Statics
6.3 Beam Subjected to Concentrated Forces
6.4 Exercises
Bibliography
Index
