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Basic Theory of Finite Element Method , livre ebook

EDP Sciences - Yongliang WANG

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224 pages

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This book is an introduction to the basic theory of the finite element method. In addition, it formulates a scheme for finite element computation of elasticity problems and provides typical computer programs for implementing the finite element method. The basic concepts and key techniques involved in finite element computation are presented to enable readers to quickly grasp the knowledge system of the finite element method. Main contents include: (1) introduction of finite element method; (2) fundamentals of elasticity mechanics; (3) weak form of equivalent integration; (4) elements and shape functions; (5) isoparametric element and numerical integration; (6) finite element computation scheme of elasticity problems; (7) solutions of linear algebraic equations; (8) error estimation and adaptive analysis; (9) programs of finite element method. This book is intended for undergraduate and postgraduate students majoring in engineering mechanics, civil engineering, and mining engineering. It provides also reference for researchers in related fields. To facilitate teaching and learning, this book provides selected homework exercises and programming projects with well-designed solutions. Online course and the source code of the related programs are also available.

Preface..................................................... V

About the Author............................................ VII

CHAPTER 1

Introduction of Finite Element Method............................. 1

1.1 Development Process of Finite Element Method ................. 1

1.2 Computation Procedure of Finite Element Method ............... 3

1.3 Main Contents of the Book................................. 5

1.4 Exercises...............................................6

CHAPTER 2

Fundamentals of Elasticity Mechanics.............................. 7

2.1 Displacements........................................... 8

2.2 Strains................................................ 9

2.3 Stresses................................................9

2.4 GeometricEquations...................................... 11

2.5 Constitutive Equations.................................... 12

2.6 Equilibrium Equations.................................... 14

2.6.1 Three-Dimensional Problems.......................... 14

2.6.2 Two-Dimensional Plane Stress and Strain Problems......... 14

2.6.3 Two-Dimensional Axisymmetric Problems................ 15

2.7 Boundary Conditions..................................... 15

2.8Exercises............................................... 16

CHAPTER 3

Weak Form of Equivalent Integration.............................. 17

3.1 Weak Form of Equivalent Integration for Differential Equations ..... 17

3.2 Weak Form of One-Dimensional Elasticity Problems .............. 17

3.3 Finite Element Computation Based on Weak Form ............... 20

3.3.1 Galerkin Method................................... 20

3.3.2 Finite Element Computation.......................... 24

3.4 Global Assembly from One-Dimensional Elements................ 26

3.5 Treatments on Boundary Conditions.......................... 28

3.6 Exercises............................................... 32

CHAPTER 4

Elements and Shape Functions................................... 33

4.1 One-Dimensional Lagrange Element.......................... 33

4.1.1 Linear Element with Two Nodes....................... 33

4.1.2 Higher-Order Lagrange Element....................... 34

4.1.3 Quadratic Lagrange Element.......................... 36

4.2 Two-Dimensional Triangle Element........................... 37

4.2.1 Triangle with Three Nodes............................ 37

4.2.2 Higher-Order Triangle Element........................ 40

4.2.3 Quadratic Triangle Element........................... 43

4.2.4 Cubic Triangle Element.............................. 43

4.3 Two-Dimensional Rectangle Element......................... 43

4.3.1 Linear Rectangle Element with Four Nodes............... 43

4.3.2 Higher-Order Rectangle Element....................... 46

4.3.3 Quadratic Rectangle Element......................... 47

4.4 Three-Dimensional Tetrahedron Element....................... 48

4.4.1 Linear Tetrahedron Element with Four Nodes ............. 48

4.4.2 Higher-Order Tetrahedron Element..................... 51

4.4.3 Quadratic Tetrahedron Element........................ 51

4.4.4 Cubic Tetrahedron Element........................... 52

4.5 Three-Dimensional Hexahedron Element....................... 52

4.5.1 Hexahedron with Eight Nodes......................... 52

4.5.2 Higher-Order Hexahedron Element..................... 54

4.5.3 Quadratic Hexahedron Element........................ 55

4.6 Exercises............................................... 56

CHAPTER 5

Isoparametric Element and Numerical Integration..................... 59

5.1 Isoparametric Element.................................... 59

5.1.1 One-Dimensional Isoparametric Lagrange Element .......... 59

5.1.2 Two-Dimensional Isoparametric Triangle Element .......... 60

5.1.3 Two-Dimensional Isoparametric Rectangle Element ......... 63

5.1.4 Three-Dimensional Isoparametric Tetrahedron Element ...... 64

5.1.5 Three-Dimensional Isoparametric Hexahedron Element ...... 67

5.1.6 Requirements of Isoparametric Element.................. 69

5.2 Numerical Integration..................................... 70

5.2.1 One-Dimensional Integration for Lagrange Element ......... 70

5.2.2 Two-Dimensional Integration for Triangle Element ......... 73

5.2.3 Two-Dimensional Integration for Rectangle Element ........ 74

5.2.4 Three-Dimensional Integration for Tetrahedron Element ..... 75

5.2.5 Three-Dimensional Integration for Hexahedron Element...... 76

5.2.6 Required Order of Numerical Integration................. 78

5.3 Exercises............................................... 78

CHAPTER 6

Finite Element Computation Scheme of Elasticity Problems ............. 81

6.1 Weak Form for General Elasticity Problems.................... 81

6.2 Finite Element Method for Solving Elasticity Problems ............ 84

6.3 Global Assembly from High-Dimensional Elements ............... 86

6.4 Treatments on Boundary Conditions.......................... 93

6.5 Exercises............................................... 98

CHAPTER 7

Solutions of Linear Algebraic Equations............................ 101

7.1 LU Decomposition Method................................. 101

7.2 Exercises............................................... 106

CHAPTER 8

Error Estimation and Adaptive Analysis............................ 107

8.1 Error Estimation of Finite Element Solutions ................... 107

8.1.1 Error of Finite Element Solutions...................... 107

8.1.2 Superconvergent Patch Recovery Method................. 108

8.2 Adaptive Finite Element Method............................ 110

8.2.1 Categories of Adaptive Finite Element Method ............ 110

8.2.2 h-Version Adaptive Finite Element Method............... 111

8.2.3 hp-Version Adaptive Finite Element Method.............. 112

8.3 Exercises............................................... 114

CHAPTER 9

Programs of Finite Element Method............................... 115

9.1 One-Dimensional Program of Beam Deformation................ 115

9.1.1 Main Program..................................... 116

9.1.2 Numerical Example................................. 118

9.1.3 Interactive Interface................................. 118

9.2 Two-Dimensional Program of Plane Strain Problem .............. 128

9.2.1 Main Program..................................... 128

9.2.2 Numerical Example................................. 132

9.2.3 Interactive Interface................................. 134

9.3 Three-Dimensional Program of Solid Compression................ 145

9.3.1 Main Program..................................... 145

9.3.2 Numerical Example................................. 148

9.3.3 Interactive Interface................................. 151

9.4 Exercises............................................... 162

Contents XI

Appendix A. Keyword Index ....................................165

Appendix B. Matrix Calculation.................................. 169

B.1 Definition.............................................. 169

B.2 Matrix Addition or Subtraction............................. 170

B.3 Transpose ..............................................171

B.4 Transpose of a Product................................... 172

B.5 Inverse................................................ 172

B.6 Symmetric Matrices...................................... 172

B.7 Partitioning ............................................172

Appendix C. Summary of Elements and Shape Functions ............... 175

C.1 One-Dimensional Lagrange Element.......................... 175

C.2 Two-Dimensional Triangle Element.......................... 175

C.3 Two-Dimensional Rectangle Element......................... 176

C.4 Three-Dimensional Tetrahedron Element...................... 177

C.5 Three-Dimensional Hexahedron Element...................... 178

Appendix D. Gaussian Integration Points and Weights ................. 179

Appendix E. Exercise Solutions.................................. 189

References.................................................. 207

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Publié par	EDP Sciences
Date de parution	20 juillet 2023
Nombre de lectures	0
EAN13	9782759829392
Langue	English
Poids de l'ouvrage	14 Mo

Informations légales : prix de location à la page 0,9550€. Cette information est donnée uniquement à titre indicatif conformément à la législation en vigueur.

Extrait

Y. WANG
Basic Theory of Finite Element Method
Textbooks for Tomorrow’s Scientists Textbooks for Tomorrow’s Scientists
NUMERICAL NUMERICAL Basic Theory of
ANALYSIS ANALYSIS
Finite Element Method
Yongliang WANG
This book is an introduction to the basic theory of the finite
element method. In addition, it formulates a scheme for finite
element computation of elasticity problems and provides typical
computer programs for implementing the finite element method.
Yongliang WANGThe basic concepts and key techniques involved in finite element
computation are presented to enable readers to quickly grasp the
knowledge system of the finite element method. Main contents
include: (1) introduction of finite element method; (2) fundamentals
of elasticity mechanics; (3) weak form of equivalent integration;
(4) elements and shape functions; (5) isoparametric element and
numerical integration; (6) finite element computation scheme of
elasticity problems; (7) solutions of linear algebraic equations; Basic Theory of
(8) error estimation and adaptive analysis; (9) programs of finite
element method. Finite Element Method
This book is intended for undergraduate and postgraduate
students majoring in engineering mechanics, civil engineering,
and mining engineering. It provides also reference for researchers
in related fields. To facilitate teaching and learning, this book
provides selected homework exercises and programming projects
with well-designed solutions. Online course and the source code
of the related programs are also available.
ISBN : 978-2-7598-2938-5
www.edpsciences.org
9 782759 829385Y. WANG
Basic Theory of Finite Element Method
Textbooks for Tomorrow’s Scientists Textbooks for Tomorrow’s Scientists
NUMERICAL NUMERICAL Basic Theory of
ANALYSIS ANALYSIS
Finite Element Method
Yongliang WANG
This book is an introduction to the basic theory of the finite
element method. In addition, it formulates a scheme for finite
element computation of elasticity problems and provides typical
computer programs for implementing the finite element method.
Yongliang WANGThe basic concepts and key techniques involved in finite element
computation are presented to enable readers to quickly grasp the
knowledge system of the finite element method. Main contents
include: (1) introduction of finite element method; (2) fundamentals
of elasticity mechanics; (3) weak form of equivalent integration;
(4) elements and shape functions; (5) isoparametric element and
numerical integration; (6) finite element computation scheme of
elasticity problems; (7) solutions of linear algebraic equations; Basic Theory of
(8) error estimation and adaptive analysis; (9) programs of finite
element method. Finite Element Method
This book is intended for undergraduate and postgraduate
students majoring in engineering mechanics, civil engineering,
and mining engineering. It provides also reference for researchers
in related fields. To facilitate teaching and learning, this book
provides selected homework exercises and programming projects
with well-designed solutions. Online course and the source code
of the related programs are also available.
ISBN : 978-2-7598-2938-5
www.edpsciences.org
9 782759 829385Textbooks for Tomorrow’s Scientists
Yongliang WANG
Basic Theory of
Finite Element MethodPrinted in France
EDP Sciences – ISBN(print): 978-2-7598-2938-5 – ISBN(ebook): 978-2-7598-2939-2
DOI: 10.1051/978-2-7598-2938-5
All rights relative to translation, adaptation and reproductionbyany means whatsoever
arereserved,worldwide.Inaccordancewiththetermsofparagraphs2and3ofArticle41
of the French Act dated March 11, 1957, “copies or reproductions reserved strictly for
private use and not intended for collective use” and, on the other hand, analyses and
short quotations for example or illustrative purposes, are allowed. Otherwise, “any
representation or reproduction – whether in full or in part – without the consent of the
author or of his successors or assigns, is unlawful” (Article 40, paragraph 1). Any
representation or reproduction, by any means whatsoever, will therefore be deemed an
infringement of copyright punishable under Articles 425 and following of the French
Penal Code.
The printed edition is not for sale in Chinese mainland. Customers in Chinese mainland
please order the print book from Science Press. ISBN of the China edition: Science Press
978-7-03-075291-8
Science Press, EDP Sciences, 2023To My StudentsPreface
This book covers the finite element method and is intended to be used in courses
taught in English or in bilingual courses. It is suitable for the knowledge system
of undergraduate and postgraduate students majoring in engineering mechanics
and similar majors. Most of the material in the book is based on the lecture
notes of a bilingual course in computational mechanics given by the author. The
lecture notes were used in undergraduate engineering mechanics majors from
2017 to 2022 and the related graduate civil engineering, energy power, and
mining engineering majors from 2019 to 2022 at China University of Mining and
Technology (Beijing). The lecture notes have been revised and optimised several
times to form this book.
The book introduces the basic theoryof the finite element method, formulates a
finite element computation scheme for elasticity problems, and provides typical
computer programs. To enable readers to quickly master the knowledge system and
solution process of the finite element method, the book focuses on the basic
concepts, algorithmtheory, and key techniques of this method. Fromthe perspective of
reforming teaching methods and enriching the curriculum system, this book meets
the following four requirements:
(1) English instruction: The book is written in English. Simple and common
English words are used to facilitate the interpretation of keywords, thus
reducing the “double challenges” in the English barrier of non-native readers
and the knowledge comprehension of new curriculum.
(2) Systematic knowledge: The book only requires basic knowledge of elasticity.
Starting with the simple weak form of the one-dimensional elasticity problem
andthefiniteelementsolutionprocess,itintroducesthebasicconceptsandkey
techniques, namely, element, isoparametric transformation, numerical
integration, and computation scheme of two-dimensional and three-dimensional
DOI: 10.1051/978-2-7598-2938-5.c901
Science Press, EDP Sciences, 2023VI Preface
elasticity problems, to enable readers to quickly grasp the knowledge system of
the finite element method.
(3) Practice exercises: Carefully selected homework exercises at the end of each
chapter and programming projects are well designed to form a set of exercise
libraries matched with the knowledge topics of the book for testing the
student’s understanding of the material and the corresponding exercise solutions
are provided. Mastering the basic theory of the finite element method and
having certain programming skills are the basic requirements of a
computational mechanics course. In addition to the theoretical knowledge, this book
provides typical application programs so that readers can further practice by
implementing the finite element method in specific problems. The electronic
courseware related to this book can be available by contacting the author
through the email: wangyl@cumtb.edu.cn.
(4) Advanced applications: The book briefly introduces error estimation and the
adaptivefiniteelementmethod.Thiscanserveasareferenceformoreadvanced
readers interested in high-performance computation and analysis techniques.
This work was supported by the National Natural Science Foundation of China
(Grant Nos. 41877275 and 51608301), Beijing Foundation (Grant
No. L212016), the Teaching Reform and Research Projects of Undergraduate
Education of China University of Mining and Technology, Beijing (Grant Nos.
J210613,J200709,andJ190701),andtheYueQiYoungScholarProjectFoundation
of China University of Mining and Technology, Beijing (Grant No. 2019QN14).
Some of the material in this book refers to classical textbooks on the finite element
method, and the pioneer authors and researchers are sincerely respected and
appreciated.
Because this book is restricted by the limited knowledge of its author, a few
errorsareunavoidable.Theauthorhopesthatexperts,scholars,andotherreadersof
this book will provide helpful suggestions for the book’s improvement.
Dr. Yongliang Wang
DepartmentofEngineeringMechanics
SchoolofMechanicsandCivilEngineering
State Key Laboratory of Coal Resources and Safe Mining
ChinaUniversityofMiningandTechnology (Beijing)
July2022About the Author
Dr. Yongliang Wang is currently a researcher in the Department of Engineering
Mechanics, School of Mechanics and Civil Engineering, State Key Laboratory of
Coal Resources and Safe Mining, at China University of Mining and Technology
(Beijing), and the head of computational mechanics group. He obtained his Ph.D.
degreefromtheDepartmentofCivilEngineeringatTsinghuaUniversityin2014.In
2015, 2016, 2017, and 2019, he successively visited the Zienkiewicz Centre for
Computational Engineering at Swansea University, UK, the Applied and
Computational Mechanics Center at Cardiff University, UK, and the Rockfield Software
Ltd, UK, to carryout cooperative research. In 2022 and 2023, he visited University
of California, Berkeley and San Diego, USA, as visiting scholar.
His research interests include high-performance adaptive finite element method,
computationandanalysisofrockdamageandfracture,andstructuralvibrationand
stability. He has also taught computational mechanics at the undergraduate level
and the basic theory of the finite element method, computational solid mechanics,
androckfracturemechanics,andfrontierandprogressinmechanicsatthegraduate
level.HeisaboardmemberoftheSoftRockBranchoftheChineseSocietyforRock
MechanicsandEngineeringandamemberoftheChineseSocietyofTheoreticaland
Applied Mechanics, China Civil Engineering Society, and China Coal