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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The purpose of this book is to introduce and study numerical methods basic and advanced ones for scientific computing. This last refers to the implementation of appropriate approaches to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or of engineering (mechanics of structures, mechanics of fluids, treatment signal, etc.). Each chapter of this book recalls the essence of the different methods resolution and presents several applications in the field of engineering as well as programs developed under Matlab software.
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Seitenzahl: 190
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Preface
Part 1: Solving Equations
1 Solving Nonlinear Equations
1.1. Introduction
1.2. Separating the roots
1.3. Approximating a separated root
1.4. Order of an iterative process
1.5. Using Matlab
2 Numerically Solving Differential Equations
2.1. Introduction
2.2. Cauchy problem and discretization
2.3. Euler’s method
2.4. One-step Runge–Kutta method
2.5. Multi-step Adams methods
2.6. Predictor–Corrector method
2.7. Using Matlab
Part 2: Solving PDEs
3 Finite Difference Methods
3.1. Introduction
3.2. Presentation of the finite difference method
3.3. Hyperbolic equations
3.4. Elliptic equations
3.5. Parabolic equations
3.6. Using Matlab
4 Finite Element Method
4.1. Introduction
4.2. One-dimensional finite element methods
4.3. Two-dimensional finite element methods
4.4. General procedure of the method
4.5. Finite element method for computing elastic structures
4.6. Using Matlab
5 Finite Volume Methods
5.1. Introduction
5.2. Finite volume method (FVM)
5.3. Advection schemes
5.4. Using Matlab
6 Meshless Methods
6.1. Introduction
6.2. Limitations of the FEM and motivation of meshless methods
6.3. Examples of meshless methods
6.4. Basis of meshless methods
6.5. Meshless method (EFG)
6.6. Application of the meshless method to elasticity
6.7. Numerical examples
6.8. Using Matlab
Part 3: Appendices
Appendix 1: Introduction to
Matlab
A1.1. Introduction
A1.2. Starting up Matlab
A1.3. Mathematical functions
A1.4. Operators and programming with Matlab
A1.5. Writing a Matlab script
A1.6. Generating figures with Matlab
Appendix 2: General Approximation Principles
A2.1. Standard results
A2.2. Linear variational problems
A2.3. Variational approximation
A2.4. General result on an upper bound for the error
A2.5. Speed of convergence
A2.6. Galerkin method
Bibliography
Index
End User License Agreement
List of Tables
2 Numerically Solving Differential Equations
Table 2.1. Approximate values computed by RK4
Table 2.2. The values of β
nk
Table 2.3. The values of γ
nk
6 Meshless Methods
Table 6.1. Basis functions
List of Illustrations
1 Solving Nonlinear Equations
Figure 1.1. Fixed-point method
Figure 1.2. Newton’s method
Figure 1.3. Regula falsi method
2 Numerically Solving Differential Equations
Figure 2.1. Graphical solution
3 Finite Difference Methods
Figure 3.1. Example of a grid
Figure 3.2. A finite difference grid
Figure 3.3. The 5-point Laplacian
Figure 3.4. The contours of the solution. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 3.5. Visualization of the solution. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
4 Finite Element Method
Figure 4.1. The function ϕ
Figure 4.2. The domain Ω
Figure 4.3. Triangulation
Figure 4.4. Graph
Figure 4.5. Elastic body in equilibrium under external loads: a) initial configuration of S; b) deformed configuration of S
Figure 4.6. Punctured plate
Figure 4.7. Triangle with three nodes
Figure 4.8. P2 triangle
Figure 4.9. P3 triangle
Figure 4.10. Axisymmetric problem
Figure 4.11. 3D elements
Figure 4.12. Window of the PDE Toolbox
Figure 4.13. Discretization of the domain. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 4.14. Mesh of the domain. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 4.15. Computed solution. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 4.16. Comparison of the computed solution and the exact solution. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 4.17. Structure of the problem
5 Finite Volume Methods
Figure 5.1. Control volume
Figure 5.2. Positioning of nodes and control volumes
Figure 5.3. The function ϕ
Figure 5.4. Discretization of the function T
Figure 5.5. Three-dimensional case
Figure 5.6. Control volume
Figure 5.7. Upwind scheme (positive direction)
Figure 5.8. Upwind scheme (negative direction)
Figure 5.9. Quadratic smoothing
Figure 5.10. Solution of the shallow water equation computed by finite volumes. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
6 Meshless Methods
Figure 6.1. Comparison of the finite element method and the meshless method
Figure 6.2. Discretization using a meshless method: nodes, domain of influence (circle)
Figure 6.3. Set of 5 × 5 regularly distributed nodes
Figure 6.4. Weight function and shape functions
Figure 6.5. Implicit meshes
Figure 6.6. General algorithm of the EFG method
Figure 6.7. Fixed-free beam subject to a concentrated force
Figure 6.8. Distribution of the nodes
Figure 6.9. Visualization of the deformation
Figure 6.10. Metal block
Figure 6.11. Geometry and boundary conditions
Figure 6.12. Distribution of nodes
Figure 6.13. Visualization of the deformation
Figure 6.14. The n
i
random points in the domain and n
b
points on the boundary. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 6.15. RMSE as a function of the shape parameter c for two sets of values. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Figure 6.16. Profile of the solution of the problem. For a color version of this figure, see www.iste.co.uk/radi/advanced2.zip
Appendix 1: Introduction to
Matlab
Figure A1.1. Matlab window
Guide
Cover
Table of Contents
Begin Reading
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Mathematical and Mechanical Engineering Set
coordinated byAbdelkhalak El Hami
Volume 7
Advanced Numerical Methods with Matlab® 2
Resolution of Nonlinear, Differential and Partial Differential Equations
Bouchaib Radi
Abdelkhalak El Hami
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2018
The rights of Bouchaib Radi and Abdelkhalak El Hami to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2018934991
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-293-9
Preface
Most physical problems can be expressed in the form of mathematical equations (e.g. differential equations, integral equations). Historically, mathematicians had to find analytic solutions to the equations encountered in engineering and related fields (e.g. mechanics, physics, biology). These equations are sometimes highly complex, requiring significant work to be simplified. However, in the mid-20th Century, the introduction of the first computers gave rise to new methods for solving equations: numerical methods. This new approach allows us to solve the equations that we encounter (when constructing models) as accurately as possible, thereby enabling us to approximate the solutions of the problems that we are studying. These approximate solutions are typically calculated by computers using suitable algorithms.
Practical experience has shown that, compared to standard numerical approaches, a carefully planned and optimized methodology can improve the speed of computation by a factor of 100 or even higher. This can transform a completely unreasonable calculation into a perfectly routine computation, hence our great interest in numerical methods! Clearly, it is important for researchers and engineers to understand the methods that they are using and, in particular, the limitations and advantages associated with each approach. The computations needed by most scientific fields require techniques to represent functions as well as algorithms to calculate derivatives and integrals, solve differential equations, locate zeros, find the eigenvectors and eigenvalues of a matrix, and much more.
The objective of this book is to present and study the fundamental numerical methods that allow scientific computations to be executed. This involves implementing a suitable methodology for the scientific problem at hand, whether derived from physics (e.g. meteorology, pollution) or engineering (e.g. structural mechanics, fluid mechanics, signal processing).
This book is divided into two parts, with two appendices. The first part contains two chapters dedicated to solving nonlinear equations and differential equations. The second part consists of four chapters on the various numerical methods that are used to solve partial differential equations: finite differences, finite elements, finite volumes and meshless methods.
Each chapter starts with a brief overview of relevant theoretical concepts and definitions, with a range of illustrative numerical examples and graphics. At the end of each chapter, we introduce the reader to the various Matlab commands for implementing the methods that have been discussed. As is often the case, practical applications play an essential role in understanding and mastering these methods. There is little hope of being able to assimilate them without the opportunity to apply them to a range of concrete examples. Accordingly, we will present various examples and explore them with Matlab. These examples can be used as a starting point for practical exploration.
Matlab is currently widely used in teaching, industry and research. It has become a standard tool in various fields thanks to its integrated toolboxes (e.g. optimization, statistics, control, image processing). Graphical interfaces have been improved considerably in recent versions. One of our appendices is dedicated to introducing readers to Matlab.
Bouchaib RADI
Abdelkhalak EL HAMI
March 2018
