A first course in finite elements

Developed from the authors, combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems governed by partial differential equations. Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, the book is both introductory and self-contained, as well as being a hands-on experience for any student. This authoritative text on Finite Elements: Adopts a generic approach to the subject, and is not application specific In conjunction with a web-based chapter, it integrates code development, theory, and application in one book Provides an accompanying Web site that includes ABAQUS Student Edition, Matlab data and programs, and instructor resources Contains a comprehensive set of homework problems at the end of each chapter Produces a practical, meaningful course for both lecturers, planning a finite element module, and for students using the text in private study. Accompanied by a book companion website housing supplementary material that can be found at http://www.wileyeurope.com/college/Fish A First Course in Finite Elements is the ideal practical introductory course for junior and senior undergraduate students from a variety of science and engineering disciplines. The accompanying advanced topics at the end of each chapter also make it suitable for courses at graduate level, as well as for practitioners who need to attain or refresh their knowledge of finite elements through private study. A First Course in Finite Elements......Page 4 Contents......Page 8 Preface......Page 14 1.1 Background......Page 18 1.2 Applications of Finite elements......Page 24 References......Page 26 2.1 Describing the Behavior of a Single Bar Element......Page 28 2.2 Equations for a System......Page 32 2.2.1 Equations for Assembly......Page 35 2.2.2 Boundary Conditions and System Solution......Page 37 2.3 Applications to Other Linear Systems......Page 41 2.4 Two-Dimensional Truss Systems......Page 44 2.5 Transformation Law......Page 47 2.6 Three-Dimensional Truss Systems......Page 52 References......Page 53 Problems......Page 54 3 Strong and Weak Forms for One-Dimensional Problems......Page 58 3.1.1 The Strong Form for an Axially Loaded Elastic Bar......Page 59 3.1.2 The Strong Form for Heat Conduction in One Dimension......Page 61 3.1.3 Diffusion in One Dimension......Page 63 3.2 The Weak Form in One Dimension......Page 64 3.3 Continuity......Page 67 3.4 The Equivalence Between the Weak and Strong Forms......Page 68 3.5.1 Strong Form for One-Dimensional Stress Analysis......Page 75 3.5.2 Weak Form for One-Dimensional Stress Analysis......Page 76 3.6.1 Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions......Page 77 3.6.2 Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions......Page 78 3.7.1 Strong Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions......Page 79 3.7.2 Weak Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions......Page 80 3.8 Advection–Diffusion......Page 81 3.8.1 Strong Form of Advection–Diffusion Equation......Page 82 3.8.2 Weak Form of Advection–Diffusion Equation......Page 83 3.9 Minimum Potential Energy......Page 84 3.10 Integrability......Page 88 Problems......Page 89 4 Approximation of Trial Solutions, Weight Functions and Gauss Quadrature for One-Dimensional Problems......Page 94 4.1 Two-Node Linear Element......Page 96 4.2 Quadratic One-Dimensional Element......Page 98 4.3 Direct Construction of Shape Functions in One Dimension......Page 99 4.5 Global Approximation and Continuity......Page 101 4.6 Gauss Quadrature......Page 102 Problems......Page 107 5.1 Development of Discrete Equation: Simple Case......Page 110 5.2 Element Matrices for Two-Node Element......Page 114 5.3 Application to Heat Conduction and Diffusion Problems......Page 116 5.4 Development of Discrete Equations for Arbitrary Boundary Conditions......Page 122 5.5 Two-Point Boundary Value Problem with Generalized Boundary Conditions......Page 128 5.6 Convergence of the FEM......Page 130 5.6.1 Convergence by Numerical Experiments......Page 132 5.6.2 Convergence by Analysis......Page 135 5.7 FEM for Advection–Diffusion Equation......Page 137 References......Page 139 Problems......Page 140 6 Strong and Weak Forms for Multidimensional Scalar Field Problems......Page 148 6.1 Divergence Theorem and Green’s Formula......Page 150 6.2 Strong Form......Page 156 6.3 Weak Form......Page 159 6.4 The Equivalence Between Weak and Strong Forms......Page 161 6.5 Generalization to Three-Dimensional Problems......Page 162 6.6 Strong and Weak Forms of Scalar Steady-State Advection–Diffusion in Two Dimensions......Page 163 Problems......Page 165 7 Approximations of Trial Solutions, Weight Functions and Gauss Quadrature for Multidimensional Problems......Page 168 7.1 Completeness and Continuity......Page 169 7.2 Three-Node Triangular Element......Page 171 7.2.1 Global Approximation and Continuity......Page 174 7.2.2 Higher Order Triangular Elements......Page 176 7.2.3 Derivatives of Shape Functions for the Three-Node Triangular Element......Page 177 7.3 Four-Node Rectangular Elements......Page 178 7.4 Four-Node Quadrilateral Element......Page 181 7.4.2 Derivatives of Isoparametric Shape Functions......Page 183 7.5 Higher Order Quadrilateral Elements......Page 185 7.6.1 Linear Triangular Element......Page 189 7.6.2 Isoparametric Triangular Elements......Page 191 7.6.3 Cubic Element......Page 192 7.6.4 Triangular Elements by Collapsing Quadrilateral Elements......Page 193 7.7 Completeness of Isoparametric Elements......Page 194 7.8 Gauss Quadrature in Two Dimensions......Page 195 7.8.1 Integration Over Quadrilateral Elements......Page 196 7.8.2 Integration Over Triangular Elements......Page 197 7.9.1 Hexahedral Elements......Page 198 7.9.2 Tetrahedral Elements......Page 200 References......Page 202 Problems......Page 203 8.1 Finite Element Formulation for Two-Dimensional Heat Conduction Problems......Page 206 8.2 Verification and Validation......Page 218 8.3 Advection–Diffusion Equation......Page 224 Problems......Page 226 9.1 Linear Elasticity......Page 232 9.1.1 Kinematics......Page 234 9.1.2 Stress and Traction......Page 236 9.1.3 Equilibrium......Page 237 9.1.4 Constitutive Equation......Page 239 9.2 Strong and Weak Forms......Page 240 9.3 Finite Element Discretization......Page 242 9.4 Three-Node Triangular Element......Page 245 9.4.1 Element Body Force Matrix......Page 246 9.4.2 Boundary Force Matrix......Page 247 9.5 Generalization of Boundary Conditions......Page 248 9.6 Discussion......Page 256 9.7 Linear Elasticity Equations in Three Dimensions......Page 257 Problems......Page 258 10.1.1 Kinematics of Beam......Page 266 10.1.2 Stress–Strain Law......Page 269 10.1.3 Equilibrium......Page 270 10.1.4 Boundary Conditions......Page 271 10.2 Strong Form to Weak Form......Page 272 10.2.1 Weak Form to Strong Form......Page 274 10.3.1 Trial Solution and Weight Function Approximations......Page 275 10.3.2 Discrete Equations......Page 277 10.4 Theorem of Minimum Potential Energy......Page 278 10.5 Remarks on Shell Elements......Page 282 Problems......Page 286 11.2 Preliminaries......Page 292 11.3 Creating a Part......Page 293 11.4 Creating a Material Definition......Page 295 11.5 Defining and Assigning Section Properties......Page 296 11.8 Applying a Boundary Condition and a Load to the Model......Page 297 11.9 Meshing the Model......Page 299 11.12 Solving the Problem Using Quadrilaterals......Page 301 11.13 Refining the Mesh......Page 302 11.16 Configuring the Analysis......Page 304 11.17 Applying a Boundary Condition and a Load to the Model......Page 305 11.18 Meshing the Model......Page 306 11.20.1 Plate with a Hole in Tension......Page 307 11.22 Creating a Part......Page 309 11.23 Creating a Material Definition......Page 310 11.24 Defining and Assigning Section Properties......Page 311 11.27 Applying a Boundary Condition and a Load to the Model......Page 312 11.28 Meshing the Model......Page 314 11.29 Creating and Submitting an Analysis Job......Page 315 11.31 Refining the Mesh......Page 316 A.1 Rotation of Coordinate System in Three Dimensions......Page 320 A.3 Taylor’s Formula with Remainder and the Mean Value Theorem......Page 321 A.4 Green’s Theorem......Page 322 A.5 Point Force (Source)......Page 324 A.6 Static Condensation......Page 325 A.7 Solution Methods......Page 326 Iterative Solvers......Page 327 Conditioning......Page 328 Problem......Page 329 Index......Page 330 Colour Section......Page 337 Developed From The Author's Combined Total Of 50 Years Undergraduate And Graduate Teaching Experience, This Book Presents The Finite Element Method Formulated As A General-purpose Numerical Procedure For Solving Engineering Problems, Governed By Partial Differential Equations. Focusing On The Formulation And Application Of The Finite Element Method Through The Integration Of Finite Element Theory, Code Development, And Software Application, This Book Is Both Introductory And Self-contained, As Well As Being A Hands-on Experience For Any Student. A First Course In Finite Elements Is The Ideal Practical Introductory Course For Junior And Senior Undergraduate Students From A Variety Of Science And Engineering Disciplines. The Accompanying Advanced Topics At The End Of Each Chapter Also Make It Suitable For Courses At Graduate Level, As Well As For Practitioners Who Need To Attain Or Refresh Their Knowledge Of Finite Elements Through Private Study--p. 4 Of Cover. Introduction -- Direct Approach For Discrete Systems -- Strong And Weak Forms For One-dimensional Problems -- Approximation Of Trial Solutions, Weight Functions And Gauss Quadrature For One-dimensional Problems -- Finite Element Formulation For One-dimensional Problems -- Strong And Weak Forms For Multidimensional Scalar Field Problems -- Approximations Of Trial Solutions, Weight Functions And Gauss Quadrature For Multidimensional Problems -- Finite Element Formulation For Multidimensional Scalar Field Problems -- Finite Element Formulation For Vector Field Problems--linear Elasticity -- Finite Element Formulation For Beams -- Commercial Finite Element Program Abaqus Tutorials -- Appendix. Jacob Fish, Ted Belytschko. Title On Cd-rom: Abaqus Student Edition. Abaqus68se, Version 6.8--cd-rom. Includes Bibliographical References (p. 312) And Index. System Requirements: Windows Xp Professional Edition, Windows Xp Home Edition, Or Windows 2000 Professional. See Abaqus Answer 3198 For Windows Vista (login To My Abaque On Www.abaqus.com) ; Web Browser: Internet Explorer 6.0, Netscape 7.0, Mozilla 1.2, Or Firefox 1.0.1 ; Monitor: 17 In. Or Larger Is Recommended ; Minimum Memory: 128 Mb ; Minimum Disk Space For Installation: 1 Gb. Developed from the authors, combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems governed by partial differential equations.

Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, the book is both introductory and self-contained, as well as being a hands-on experience for any student.

This authoritative text on Finite Elements:
* Adopts a generic approach to the subject, and is not application specific
* In conjunction with a web-based chapter, it integrates code development, theory, and application in one book
* Provides an accompanying Web site that includes ABAQUS Student Edition, Matlab data and programs, and instructor resources
* Contains a comprehensive set of homework problems at the end of each chapter
* Produces a practical, meaningful course for both lecturers, planning a finite element module, and for students using the text in private study.
* Accompanied by a book companion website housing supplementary material that can be found at http://www.wileyeurope.com/college/Fish


A First Course in Finite Elements is the ideal practical introductory course for junior and senior undergraduate students from a variety of science and engineering disciplines. The accompanying advanced topics at the end of each chapter also make it suitable for courses at graduate level, as well as for practitioners who need to attain or refresh their knowledge of finite elements through private study. "Developed from the author's combined total of 50 years undergraduate and graduate teaching experience, this book presents the finite element method formulated as a general-purpose numerical procedure for solving engineering problems, governed by partial differential equations. Focusing on the formulation and application of the finite element method through the integration of finite element theory, code development, and software application, this book is both introductory and self-contained, as well as being a hands-on experience for any student." "A First Course in Finite Elements is the ideal practical introductory course for junior and senior undergraduate students from a variety of science and engineering disciplines. The accompanying advanced topics at the end of each chapter also make it suitable for courses at graduate level, as well as for practitioners who need to attain or refresh their knowledge of finite elements through private study"--Page 4 of cover The text material evolved from over 50 years of combined teaching experience it deals with a formulation and application of the finite element method. A meaningful course can be constructed from a subset of the chapters in this book for a quarter course; instructions for such use are given in the preface. The course material is organized in three chronological units of one month each: 1) the finite element formulation for one-dimensional problems, 2) the finite element formulation for scalar field problems in two dimensions and 3) finite element programming and application to scalar field problems; and finite element formulation for vector field problems in two dimensions and beams. In conjunction with the book there will be the access and use of ABAQUS software and MATLAB exercises. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The Publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought.