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laboutique.edpsciences.fr-R001917 03 01 01 R001917 00 ED E107 00 01 01 Basic Theory of Finite Element Method 01 eng 08 224 03 22 15313054 17 01 P1 EDP Sciences 01 01 P1 EDP Sciences 17 avenue du Hoggar - PA de Courtaboeuf - 91944 Les Ulis cedex A http://publications.edpsciences.org/ 11 00 20230720 02 01 002300 03 9782759829392 15 9782759829392 03 01 D1 EDP Sciences 45 03 laboutique.edpsciences.fr-002300 03 01 01 002300 03 9782759829392 15 9782759829392 10 EA E107 10 00 01 R001917 ED E107 1 10 01 02 Textbooks for Tomorrow's Scientists Textbooks for Tomorrow’s Scientists introduces advanced textbooks on a wide range of topics in STM subject areas. These books offer students, researchers, faculty and professionals the resources they need to learn and succeed in graduate courses. 01 01 Basic Theory of Finite Element Method 1 A01 01 A2163 Yongliang WANG WANG, Yongliang Yongliang WANG Dr. Yongliang Wang is currently an associate professor 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 the group of computational mechanics. He obtained his Ph.D. degree from the Department of Civil Engineering at Tsinghua University in 2014. Dr. Yongliang Wang mainly engaged in the research of high-performance adaptive finite element method, computation and analysis of mine-rock mass damage and fracture, and structural vibration stability. He published four monographs in English, and more than 150 academic papers; furthermore, he obtained more than 50 software copyrights. 01 eng 08 222 03 01 24 Izibook:Subject Ingénierie 24 Izibook:SubjectAndCategoryAndTags |Ingénierie| 20 finite element;elasticity;isoparametric;Finite Element Method;finite element computation of elasticity problems;civil engineering;engineering mechanics;isoparametric element;elasticity mechanics;computer program 10 2011 TEC009020 01 620 06 06 05 03 00 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. 02 00 This book is an introduction to the basic theory of the finite element method. 04 00 Preface..................................................... V<o:p></o:p>About the Author............................................ VII<o:p></o:p>CHAPTER 1<o:p></o:p>Introduction of Finite Element Method............................. 1<o:p></o:p>1.1 Development Process of Finite Element Method ................. 1<o:p></o:p>1.2 Computation Procedure of Finite Element Method ............... 3<o:p></o:p>1.3 Main Contents of the Book................................. 5<o:p></o:p>1.4 Exercises...............................................6<o:p></o:p>CHAPTER 2<o:p></o:p>Fundamentals of Elasticity Mechanics.............................. 7<o:p></o:p>2.1 Displacements........................................... 8<o:p></o:p>2.2 Strains................................................ 9<o:p></o:p>2.3 Stresses................................................9<o:p></o:p>2.4 GeometricEquations...................................... 11<o:p></o:p>2.5 Constitutive Equations.................................... 12<o:p></o:p>2.6 Equilibrium Equations.................................... 14<o:p></o:p>2.6.1 Three-Dimensional Problems.......................... 14<o:p></o:p>2.6.2 Two-Dimensional Plane Stress and Strain Problems......... 14<o:p></o:p>2.6.3 Two-Dimensional Axisymmetric Problems................ 15<o:p></o:p>2.7 Boundary Conditions..................................... 15<o:p></o:p>2.8Exercises............................................... 16<o:p></o:p>CHAPTER 3<o:p></o:p>Weak Form of Equivalent Integration.............................. 17<o:p></o:p>3.1 Weak Form of Equivalent Integration for Differential Equations ..... 17<o:p></o:p>3.2 Weak Form of One-Dimensional Elasticity Problems .............. 17<o:p></o:p>3.3 Finite Element Computation Based on Weak Form ............... 20<o:p></o:p>3.3.1 Galerkin Method................................... 20<o:p></o:p>3.3.2 Finite Element Computation.......................... 24<o:p></o:p>3.4 Global Assembly from One-Dimensional Elements................ 26<o:p></o:p>3.5 Treatments on Boundary Conditions.......................... 28<o:p></o:p>3.6 Exercises............................................... 32<o:p></o:p>CHAPTER 4<o:p></o:p>Elements and Shape Functions................................... 33<o:p></o:p>4.1 One-Dimensional Lagrange Element.......................... 33<o:p></o:p>4.1.1 Linear Element with Two Nodes....................... 33<o:p></o:p>4.1.2 Higher-Order Lagrange Element....................... 34<o:p></o:p>4.1.3 Quadratic Lagrange Element.......................... 36<o:p></o:p>4.2 Two-Dimensional Triangle Element........................... 37<o:p></o:p>4.2.1 Triangle with Three Nodes............................ 37<o:p></o:p>4.2.2 Higher-Order Triangle Element........................ 40<o:p></o:p>4.2.3 Quadratic Triangle Element........................... 43<o:p></o:p>4.2.4 Cubic Triangle Element.............................. 43<o:p></o:p>4.3 Two-Dimensional Rectangle Element......................... 43<o:p></o:p>4.3.1 Linear Rectangle Element with Four Nodes............... 43<o:p></o:p>4.3.2 Higher-Order Rectangle Element....................... 46<o:p></o:p>4.3.3 Quadratic Rectangle Element......................... 47<o:p></o:p>4.4 Three-Dimensional Tetrahedron Element....................... 48<o:p></o:p>4.4.1 Linear Tetrahedron Element with Four Nodes ............. 48<o:p></o:p>4.4.2 Higher-Order Tetrahedron Element..................... 51<o:p></o:p>4.4.3 Quadratic Tetrahedron Element........................ 51<o:p></o:p>4.4.4 Cubic Tetrahedron Element........................... 52<o:p></o:p>4.5 Three-Dimensional Hexahedron Element....................... 52<o:p></o:p>4.5.1 Hexahedron with Eight Nodes......................... 52<o:p></o:p>4.5.2 Higher-Order Hexahedron Element..................... 54<o:p></o:p>4.5.3 Quadratic Hexahedron Element........................ 55<o:p></o:p>4.6 Exercises............................................... 56<o:p></o:p>CHAPTER 5<o:p></o:p>Isoparametric Element and Numerical Integration..................... 59<o:p></o:p>5.1 Isoparametric Element.................................... 59<o:p></o:p>5.1.1 One-Dimensional Isoparametric Lagrange Element .......... 59<o:p></o:p>5.1.2 Two-Dimensional Isoparametric Triangle Element .......... 60<o:p></o:p>5.1.3 Two-Dimensional Isoparametric Rectangle Element ......... 63<o:p></o:p>5.1.4 Three-Dimensional Isoparametric Tetrahedron Element ...... 64<o:p></o:p>5.1.5 Three-Dimensional Isoparametric Hexahedron Element ...... 67<o:p></o:p>5.1.6 Requirements of Isoparametric Element.................. 69<o:p></o:p>5.2 Numerical Integration..................................... 70<o:p></o:p>5.2.1 One-Dimensional Integration for Lagrange Element ......... 70<o:p></o:p>5.2.2 Two-Dimensional Integration for Triangle Element ......... 73<o:p></o:p>5.2.3 Two-Dimensional Integration for Rectangle Element ........ 74<o:p></o:p>5.2.4 Three-Dimensional Integration for Tetrahedron Element ..... 75<o:p></o:p>5.2.5 Three-Dimensional Integration for Hexahedron Element...... 76<o:p></o:p>5.2.6 Required Order of Numerical Integration................. 78<o:p></o:p>5.3 Exercises............................................... 78CHAPTER 6<o:p></o:p>Finite Element Computation Scheme of Elasticity Problems ............. 81<o:p></o:p>6.1 Weak Form for General Elasticity Problems.................... 81<o:p></o:p>6.2 Finite Element Method for Solving Elasticity Problems ............ 84<o:p></o:p>6.3 Global Assembly from High-Dimensional Elements ............... 86<o:p></o:p>6.4 Treatments on Boundary Conditions.......................... 93<o:p></o:p>6.5 Exercises............................................... 98<o:p></o:p>CHAPTER 7<o:p></o:p>Solutions of Linear Algebraic Equations............................ 101<o:p></o:p>7.1 LU Decomposition Method................................. 101<o:p></o:p>7.2 Exercises............................................... 106<o:p></o:p>CHAPTER 8<o:p></o:p>Error Estimation and Adaptive Analysis............................ 107<o:p></o:p>8.1 Error Estimation of Finite Element Solutions ................... 107<o:p></o:p>8.1.1 Error of Finite Element Solutions...................... 107<o:p></o:p>8.1.2 Superconvergent Patch Recovery Method................. 108<o:p></o:p>8.2 Adaptive Finite Element Method............................ 110<o:p></o:p>8.2.1 Categories of Adaptive Finite Element Method ............ 110<o:p></o:p>8.2.2 h-Version Adaptive Finite Element Method............... 111<o:p></o:p>8.2.3 hp-Version Adaptive Finite Element Method.............. 112<o:p></o:p>8.3 Exercises............................................... 114<o:p></o:p>CHAPTER 9<o:p></o:p>Programs of Finite Element Method............................... 115<o:p></o:p>9.1 One-Dimensional Program of Beam Deformation................ 115<o:p></o:p>9.1.1 Main Program..................................... 116<o:p></o:p>9.1.2 Numerical Example................................. 118<o:p></o:p>9.1.3 Interactive Interface................................. 118<o:p></o:p>9.2 Two-Dimensional Program of Plane Strain Problem .............. 128<o:p></o:p>9.2.1 Main Program..................................... 128<o:p></o:p>9.2.2 Numerical Example................................. 132<o:p></o:p>9.2.3 Interactive Interface................................. 134<o:p></o:p>9.3 Three-Dimensional Program of Solid Compression................ 145<o:p></o:p>9.3.1 Main Program..................................... 145<o:p></o:p>9.3.2 Numerical Example................................. 148<o:p></o:p>9.3.3 Interactive Interface................................. 151<o:p></o:p>9.4 Exercises............................................... 162<o:p></o:p>Contents XI<o:p></o:p>Appendix A. Keyword Index ....................................165<o:p></o:p>Appendix B. Matrix Calculation.................................. 169<o:p></o:p>B.1 Definition.............................................. 169<o:p></o:p>B.2 Matrix Addition or Subtraction............................. 170<o:p></o:p>B.3 Transpose ..............................................171<o:p></o:p>B.4 Transpose of a Product................................... 172<o:p></o:p>B.5 Inverse................................................ 172<o:p></o:p>B.6 Symmetric Matrices...................................... 172<o:p></o:p>B.7 Partitioning ............................................172<o:p></o:p>Appendix C. Summary of Elements and Shape Functions ............... 175<o:p></o:p>C.1 One-Dimensional Lagrange Element.......................... 175<o:p></o:p>C.2 Two-Dimensional Triangle Element.......................... 175<o:p></o:p>C.3 Two-Dimensional Rectangle Element......................... 176<o:p></o:p>C.4 Three-Dimensional Tetrahedron Element...................... 177<o:p></o:p>C.5 Three-Dimensional Hexahedron Element...................... 178<o:p></o:p>Appendix D. Gaussian Integration Points and Weights ................. 179<o:p></o:p>Appendix E. Exercise Solutions.................................. 189<o:p></o:p>References.................................................. 207<o:p></o:p> Preface..................................................... V<o:p></o:p>About the Author............................................ VII<o:p></o:p>CHAPTER 1<o:p></o:p>Introduction of Finite ElementMethod............................. 1<o:p></o:p>1.1 Development Process of Finite ElementMethod ................. 1<o:p></o:p>1.2 Computation Procedure of Finite ElementMethod ............... 3<o:p></o:p>1.3 Main Contents of the Book................................. 5<o:p></o:p>1.4 Exercises...............................................6<o:p></o:p>CHAPTER 2<o:p></o:p>Fundamentals of Elasticity Mechanics.............................. 7<o:p></o:p>2.1 Displacements........................................... 8<o:p></o:p>2.2 Strains................................................ 9<o:p></o:p>2.3 Stresses................................................9<o:p></o:p>2.4 GeometricEquations...................................... 11<o:p></o:p>2.5 Constitutive Equations.................................... 12<o:p></o:p>2.6 Equilibrium Equations.................................... 14<o:p></o:p>2.6.1 Three-Dimensional Problems.......................... 14<o:p></o:p>2.6.2 Two-Dimensional Plane Stress and StrainProblems......... 14<o:p></o:p>2.6.3 Two-Dimensional Axisymmetric Problems................ 15<o:p></o:p>2.7 Boundary Conditions..................................... 15<o:p></o:p>2.8Exercises............................................... 16<o:p></o:p>CHAPTER 3<o:p></o:p>Weak Form of Equivalent Integration.............................. 17<o:p></o:p>3.1 Weak Form of Equivalent Integration forDifferential Equations ..... 17<o:p></o:p>3.2 Weak Form of One-Dimensional ElasticityProblems .............. 17<o:p></o:p>3.3 Finite Element Computation Based on WeakForm ............... 20<o:p></o:p>3.3.1 Galerkin Method................................... 20<o:p></o:p>3.3.2 Finite Element Computation.......................... 24<o:p></o:p>3.4 Global Assembly from One-DimensionalElements................ 26<o:p></o:p>3.5 Treatments on Boundary Conditions.......................... 28<o:p></o:p>3.6Exercises............................................... 32<o:p></o:p>CHAPTER 4<o:p></o:p>Elements and Shape Functions................................... 33<o:p></o:p>4.1 One-Dimensional Lagrange Element.......................... 33<o:p></o:p>4.1.1 Linear Element with Two Nodes....................... 33<o:p></o:p>4.1.2 Higher-Order Lagrange Element....................... 34<o:p></o:p>4.1.3 Quadratic Lagrange Element.......................... 36<o:p></o:p>4.2 Two-Dimensional Triangle Element........................... 37<o:p></o:p>4.2.1 Triangle with ThreeNodes............................ 37<o:p></o:p>4.2.2 Higher-Order Triangle Element........................ 40<o:p></o:p>4.2.3 Quadratic Triangle Element........................... 43<o:p></o:p>4.2.4 Cubic Triangle Element.............................. 43<o:p></o:p>4.3 Two-Dimensional Rectangle Element......................... 43<o:p></o:p>4.3.1 Linear Rectangle Element with Four Nodes............... 43<o:p></o:p>4.3.2 Higher-Order Rectangle Element....................... 46<o:p></o:p>4.3.3 Quadratic Rectangle Element......................... 47<o:p></o:p>4.4 Three-Dimensional TetrahedronElement....................... 48<o:p></o:p>4.4.1 Linear Tetrahedron Element with FourNodes ............. 48<o:p></o:p>4.4.2 Higher-Order Tetrahedron Element..................... 51<o:p></o:p>4.4.3 Quadratic TetrahedronElement........................ 51<o:p></o:p>4.4.4 Cubic Tetrahedron Element........................... 52<o:p></o:p>4.5 Three-Dimensional Hexahedron Element....................... 52<o:p></o:p>4.5.1 Hexahedron with Eight Nodes......................... 52<o:p></o:p>4.5.2 Higher-Order Hexahedron Element..................... 54<o:p></o:p>4.5.3 Quadratic Hexahedron Element........................ 55<o:p></o:p>4.6Exercises............................................... 56<o:p></o:p>CHAPTER 5<o:p></o:p>Isoparametric Element and NumericalIntegration..................... 59<o:p></o:p>5.1 Isoparametric Element.................................... 59<o:p></o:p>5.1.1 One-Dimensional Isoparametric LagrangeElement .......... 59<o:p></o:p>5.1.2 Two-Dimensional Isoparametric TriangleElement .......... 60<o:p></o:p>5.1.3 Two-Dimensional Isoparametric RectangleElement ......... 63<o:p></o:p>5.1.4 Three-Dimensional IsoparametricTetrahedron Element ...... 64<o:p></o:p>5.1.5 Three-Dimensional IsoparametricHexahedron Element ...... 67<o:p></o:p>5.1.6 Requirements of Isoparametric Element.................. 69<o:p></o:p>5.2 Numerical Integration..................................... 70<o:p></o:p>5.2.1 One-Dimensional Integration for LagrangeElement ......... 70<o:p></o:p>5.2.2 Two-Dimensional Integration for TriangleElement ......... 73<o:p></o:p>5.2.3 Two-Dimensional Integration for RectangleElement ........ 74<o:p></o:p>5.2.4 Three-Dimensional Integration forTetrahedron Element ..... 75<o:p></o:p>5.2.5 Three-Dimensional Integration forHexahedron Element...... 76<o:p></o:p>5.2.6 Required Order of Numerical Integration................. 78<o:p></o:p>5.3Exercises............................................... 78<o:p></o:p>X Contents<o:p></o:p>CHAPTER 6<o:p></o:p>Finite Element Computation Scheme of ElasticityProblems ............. 81<o:p></o:p>6.1 Weak Form for General Elasticity Problems.................... 81<o:p></o:p>6.2 Finite Element Method for SolvingElasticity Problems ............ 84<o:p></o:p>6.3 Global Assembly from High-DimensionalElements ............... 86<o:p></o:p>6.4 Treatments on Boundary Conditions.......................... 93<o:p></o:p>6.5Exercises............................................... 98<o:p></o:p>CHAPTER 7<o:p></o:p>Solutions of Linear Algebraic Equations............................ 101<o:p></o:p>7.1 LU Decomposition Method................................. 101<o:p></o:p>7.2Exercises............................................... 106<o:p></o:p>CHAPTER 8<o:p></o:p>Error Estimation and AdaptiveAnalysis............................ 107<o:p></o:p>8.1 Error Estimation of Finite ElementSolutions ................... 107<o:p></o:p>8.1.1 Error of Finite Element Solutions...................... 107<o:p></o:p>8.1.2 Superconvergent Patch Recovery Method................. 108<o:p></o:p>8.2 Adaptive Finite Element Method............................ 110<o:p></o:p>8.2.1 Categories of Adaptive Finite ElementMethod ............ 110<o:p></o:p>8.2.2 h-Version Adaptive Finite Element Method............... 111<o:p></o:p>8.2.3 hp-Version Adaptive Finite Element Method.............. 112<o:p></o:p>8.3Exercises............................................... 114<o:p></o:p>CHAPTER 9<o:p></o:p>Programs of Finite Element Method............................... 115<o:p></o:p>9.1 One-Dimensional Program of Beam Deformation................ 115<o:p></o:p>9.1.1 Main Program..................................... 116<o:p></o:p>9.1.2 Numerical Example................................. 118<o:p></o:p>9.1.3 Interactive Interface................................. 118<o:p></o:p>9.2 Two-Dimensional Program of Plane StrainProblem .............. 128<o:p></o:p>9.2.1 Main Program..................................... 128<o:p></o:p>9.2.2 Numerical Example................................. 132<o:p></o:p>9.2.3 Interactive Interface................................. 134<o:p></o:p>9.3 Three-Dimensional Program of SolidCompression................ 145<o:p></o:p>9.3.1 Main Program..................................... 145<o:p></o:p>9.3.2 Numerical Example................................. 148<o:p></o:p>9.3.3 Interactive Interface................................. 151<o:p></o:p>9.4Exercises............................................... 162<o:p></o:p>Contents XI<o:p></o:p>Appendix A. Keyword Index ....................................165<o:p></o:p>Appendix B. Matrix Calculation.................................. 169<o:p></o:p>B.1 Definition.............................................. 169<o:p></o:p>B.2 Matrix Addition or Subtraction............................. 170<o:p></o:p>B.3 Transpose ..............................................171<o:p></o:p>B.4 Transpose of a Product................................... 172<o:p></o:p>B.5 Inverse................................................ 172<o:p></o:p>B.6 Symmetric Matrices...................................... 172<o:p></o:p>B.7 Partitioning ............................................172<o:p></o:p>Appendix C. Summary of Elements and ShapeFunctions ............... 175<o:p></o:p>C.1 One-Dimensional Lagrange Element.......................... 175<o:p></o:p>C.2 Two-Dimensional Triangle Element.......................... 175<o:p></o:p>C.3 Two-Dimensional Rectangle Element......................... 176<o:p></o:p>C.4 Three-Dimensional Tetrahedron Element...................... 177<o:p></o:p>C.5 Three-Dimensional Hexahedron Element...................... 178<o:p></o:p>Appendix D. Gaussian Integration Points andWeights ................. 179<o:p></o:p>Appendix E. Exercise Solutions.................................. 189<o:p></o:p>References.................................................. 207<o:p></o:p> 21 00 06 01 https://laboutique.edpsciences.fr/produit/1352/9782759829392/basic-theory-of-finite-element-method 01 00 03 02 https://laboutique.edpsciences.fr/system/product_pictures/data/009/982/970/original/9782759829385-Basic_Theory_of_Finite_Element_Method_couv-sofedis.jpg 17 14 20240913T181910+0200 01 P1 EDP Sciences 01 01 P1 EDP Sciences 17 avenue du Hoggar - PA de Courtaboeuf - 91944 Les Ulis cedex A http://publications.edpsciences.org/ 04 11 00 20230720 01 00 20230720 19 00 20230720 2026 06 3052868830012 01 WORLD 13 03 9782759829385 15 9782759829385 WORLD 08 EDP Sciences 04 01 00 20230720 03 01 D1 EDP Sciences 20 04 05 01 02 1 00 53.99 01 5.50 2.81 EUR 04 06 01 02 1 00 190.99 01 5.50 2.81 EUR 01 04 06 01 02 1 00 209.99 01 5.50 10.95 USD 01