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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
The finite element method, which emerged in the 1950s to deal with structural mechanics problems, has since undergone continuous development. Using partial differential equation models, it is now present in such fields of application as mechanics, physics, chemistry, economics, finance and biology. It is also used in most scientific computing software, and many engineers become adept at using it in their modeling and numerical simulation activities. This book presents all the essential elements of the finite element method in a progressive and didactic way: the theoretical foundations, practical considerations of implementation, algorithms, as well as numerical illustrations created in MATLAB. Original exercises with detailed answers are provided at the end of each chapter.
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Seitenzahl: 462
Veröffentlichungsjahr: 2023
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Table of Contents
Cover
Title Page
Copyright Page
Preface
Chapter 1. Theoretical Aspects of Elliptic Equations
1.1. Some vocabulary
1.2. Classical solutions to Laplace’s equation
1.3. Weak solutions
1.4. Reminders on Banach and Hilbert spaces
1.5. Sobolev spaces
1.6. Existence of the trace and integration by parts formulae
1.7. Exercises – statements
Chapter 2. Variational Formulations and Their Solutions
2.1. Variational formulations of the Dirichlet, Neumann and Fourier problems
2.2. Other variational formulations examples
2.3. Existence and uniqueness of weak solutions
2.4. Existence and uniqueness – general framework
2.5. Some properties of weak solutions
2.6. Exercises – statements
Chapter 3. Introduction to the Finite Element Method
3.1. Galerkin’s approximation
3.2. Principles of the affine finite element method in two dimensions
3.3. General procedure of constructing finite elements
3.4. Extension of finite elements
3.5. Exercises – statements
Chapter 4. Numerical Analysis of the Finite Element Method
4.1. Convergence of finite element methods
4.2. Error estimators and mesh refinement
4.3. Non-polyhedral domains and approximations of the data
4.4. Exercises – statements
Chapter 5. Concrete Aspects of the Finite Element Method
5.1. Implementation
5.2. Algorithmic considerations
5.3. Some numerical illustrations
Appendices
Appendix 1. Solving Linear Systems
Appendix 2. Solutions
Appendix 3. Formulas
References
Index
Other titles from iSTE in Numerical Methods in Engineering
End User License Agreement
List of Tables
Chapter 2
Table 2.1. Spaces and their respective forms
Chapter 4
Table 4.1. Quadrature of seven points in two dimensions: points and weights
Table 4.2. Quadrature of three points in one dimension: points and weights
Guide
Cover
Table of Contents
Title Page
Copyright Page
Preface
Begin Reading
Appendix 1
Appendix 2
Appendix 3
References
Index
Other titles from iSTE in Numerical Methods in Engineering
End User License Agreement
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The Finite Element Method
From Theory to Practice
Patrick Ciarlet
Eric Lunéville
First published 2023 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 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2023The rights of Patrick Ciarlet and Eric Lunéville to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.
Library of Congress Control Number: 2023933918
British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-768-2
Preface
Numerical simulation is becoming a powerful means of investigation that plays a very important role in science and technology, second only to the classical experimental approaches. In physics, chemistry and mechanics, models based upon systems of partial differential equations are used frequently (elastodynamics, acoustics, electromagnetism, aerodynamics, hydrodynamics, ab initio chemistry, etc.). More recently, this type of modeling has started to appear in other fields such as biology and economics (finance, in particular). Apart from a few special cases, it is not possible to solve these systems of equations analytically, and therefore it is necessary to use approximation techniques instead. Semi-analytical techniques, probabilistic methods, finite difference methods, spectral methods and finite element methods are just a few principle ones. Since being introduced in the 1950s, the finite element method has undergone several developments and today is used in numerical simulation software. Any engineer who works in an environment where numerical simulation plays an important role will have encountered, and moreover should understand, its fundamental principles, up to the most recent refinements.
Similar to other approximation techniques, the finite element method must be employed while obeying certain rules, which include usage constraints in addition to various kinds of quality factors in particular. The aim of this book is to provide the necessary components, from the most theoretical to the most concrete, so that one can master these constraints and factors. With this in mind, we want to present the different aspects of the method: the mathematical framework, numerical analysis and efficient implementation, highlighting each’s importance, all while providing the key aspects in addition to applying them to situations typically encountered in practice. This work is devoted to presenting the fundamental concepts of the finite element method applied to stationary elliptical problems. The body of the text presents the basic notions that one should understand completely, and some additional results are also provided. Exercises along with their answers can be found at the end of each chapter and in the appendix, to allow the reader to further their understanding and also introduce them to more advanced subjects.
Chapters 1 and 2 are devoted to studying elliptical problems and the variational theory of elliptical equations (in the context of the finite element method), which are covered in a rigorous functional framework and only require a basic knowledge of analysis. Most of the operational tools of functional analysis (Hilbert analysis and distribution theory) will be recalled to allow for an easier reading for those unfamiliar with these areas. The finite element method is then the subject of Chapters 3 and 4, for which we start by providing a concrete presentation of the method by applying it to a simple yet important example, before presenting the formal framework that allows for a wide range of finite elements to be constructed, with the main results of error estimation being discussed in detail. In Chapter 5, we will study the practical and algorithmic aspects of the method that are used in its computer implementation, for which digital illustrations made with MATLAB1 are provided as examples. In Appendix 1, we will present some elementary – albeit essential – applications to the resolution of linear systems and, in particular, sparse linear systems that follow from the finite element approximation. Detailed answers to the exercises are provided in Appendix 2, and the definitions for the usual differential operators are in Appendix 3.
March 2023
Note
1
MATLAB is a trademark of MathWorks, Inc.
