First Course in Finite Elements

rong>Jacob Fish The Rosalind and John J. Redfern, Jr. '33 Chaired Professor in Engineering Rensselaer Polytechnic Institute, Troy, NYDr. Fish has 20 years of experience (both industry and academia) in the field of multi-scale computational engineering, which bridges the gap between modeling, simulation and design of products based on multi-scale principles. Dr. Fish has published over one hundred journal articles and book chapters. Two of his papers, one on development of multilevel solution techniques for large scale systems presented at the 1995 ASME International Computers in Engineering Conference and the second one, on fatigue crack growth in aging aircraft presented at the 1993 Structures, Structural Dynamics, and Materials Conference have won the Best Paper Awards. Dr. Fish is a recipient of 2005 USACM Computational Structural Mechanics Award given "in recognition of outstanding and sustained contributions to the broad field of Computational Structural Mechanics". He is editor of the International Journal for Multiscale Computational Engineering.Ted Belytschko, Department of Mechanical Engineering, Northwestern University, Evanston, ILTed Belytschko's main interests lie in the development of computational methods for engineering problems. He has developed explicit finite element methods that are widely used in crashworthiness analysis and virtual prototyping. He is also interested in engineering education, and he chaired the committee that developed the "Engineering First Program" at Northwestern.  He obtained his B.S. and Ph.D. at Illinois Institute of Technology in 1965 and 1968, respectively.  He has been at Northwestern since 1977 where he is currently Walter P. Murphy Professor and McCormick Distinguished Professor of Computational Mechanics. He is co-author of the book NONLINEAR FINITE ELEMENTS FOR CONTINUA AND STRUCTURES with W.K.Liu and B. Moran (published by Wiley and in the third printing) and he has edited more than 10 other books. n January 2004, he was listed as the 4th most cited researcher in engineering. He is past Chairman of the Engineering Mechanics Division of the ASCE, the Applied Mechanics Division of ASME, past President of USACM, and a member of the National Academy of Engineering (elected in 1992) and the American Academy of Arts and Sciences (elected in 2002). He is the editor of Numerical Methods in Engineering.

• Preface xi1 Introduction 11.1 Background 11.2 Applications of Finite elements 7References 92 Direct Approach for Discrete Systems 112.1 Describing the Behavior of a Single Bar Element 112.2 Equations for a System 152.2.1 Equations for Assembly 182.2.2 Boundary Conditions and System Solution 202.3 Applications to Other Linear Systems 242.4 Two-Dimensional Truss Systems 272.5 Transformation Law 302.6 Three-Dimensional Truss Systems 35References 36Problems 373 Strong and Weak Forms for One-Dimensional Problems 413.1 The Strong Form in One-Dimensional Problems 423.1.1 The Strong Form for an Axially Loaded Elastic Bar 423.1.2 The Strong Form for Heat Conduction in One Dimension 443.1.3 Diffusion in One Dimension 463.2 The Weak Form in One Dimension 473.3 Continuity 503.4 The Equivalence Between the Weak and Strong Forms 513.5 One-Dimensional Stress Analysis with Arbitrary Boundary Conditions 583.5.1 Strong Form for One-Dimensional Stress Analysis 583.5.2 Weak Form for One-Dimensional Stress Analysis 593.6 One-Dimensional Heat Conduction with Arbitrary Boundary Conditions 603.6.1 Strong Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 603.6.2 Weak Form for Heat Conduction in One Dimension with Arbitrary Boundary Conditions 613.7 Two-Point Boundary Value Problem with Generalized Boundary Conditions 623.7.1 Strong Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions 623.7.2 Weak Form for Two-Point Boundary Value Problems with Generalized Boundary Conditions 633.8 Advection–Diffusion 643.8.1 Strong Form of Advection–Diffusion Equation 653.8.2 Weak Form of Advection–Diffusion Equation 663.9 Minimum Potential Energy 673.10 Integrability 71References 72Problems 724 Approximation of Trial Solutions, Weight Functions and Gauss Quadrature for One-Dimensional Problems 774.1 Two-Node Linear Element 794.2 Quadratic One-Dimensional Element 814.3 Direct Construction of Shape Functions in One Dimension 824.4 Approximation of the Weight Functions 844.5 Global Approximation and Continuity 844.6 Gauss Quadrature 85Reference 90Problems 905 Finite Element Formulation for One-Dimensional Problems 935.1 Development of Discrete Equation: Simple Case 935.2 Element Matrices for Two-Node Element 975.3 Application to Heat Conduction and Diffusion Problems 995.4 Development of Discrete Equations for Arbitrary Boundary Conditions 1055.5 Two-Point Boundary Value Problem with Generalized Boundary Conditions 1115.6 Convergence of the FEM 1135.6.1 Convergence by Numerical Experiments 1155.6.2 Convergence by Analysis 1185.7 FEM for Advection–Diffusion Equation 120References 122Problems 1236 Strong and Weak Forms for Multidimensional Scalar Field Problems 1316.1 Divergence Theorem and Green’s Formula 1336.2 Strong Form 1396.3 Weak Form 1426.4 The Equivalence Between Weak and Strong Forms 1446.5 Generalization to Three-Dimensional Problems 1456.6 Strong and Weak Forms of Scalar Steady-State Advection–Diffusion in Two Dimensions 146References 148Problems 1487 Approximations of Trial Solutions, Weight Functions and Gauss Quadrature for Multidimensional Problems 1517.1 Completeness and Continuity 1527.2 Three-Node Triangular Element 1547.2.1 Global Approximation and Continuity 1577.2.2 Higher Order Triangular Elements 1597.2.3 Derivatives of Shape Functions for the Three-Node Triangular Element 1607.3 Four-Node Rectangular Elements 1617.4 Four-Node Quadrilateral Element 1647.4.1 Continuity of Isoparametric Elements 1667.4.2 Derivatives of Isoparametric Shape Functions 1667.5 Higher Order Quadrilateral Elements 1687.6 Triangular Coordinates 1727.6.1 Linear Triangular Element 1727.6.2 Isoparametric Triangular Elements 1747.6.3 Cubic Element 1757.6.4 Triangular Elements by Collapsing Quadrilateral Elements 1767.7 Completeness of Isoparametric Elements 1777.8 Gauss Quadrature in Two Dimensions 1787.8.1 Integration Over Quadrilateral Elements 1797.8.2 Integration Over Triangular Elements 1807.9 Three-Dimensional Elements 1817.9.1 Hexahedral Elements 1817.9.2 Tetrahedral Elements 183References 185Problems 1868 Finite Element Formulation for Multidimensional Scalar Field Problems 1898.1 Finite Element Formulation for Two-Dimensional Heat Conduction Problems 1898.2 Verification and Validation 2018.3 Advection–Diffusion Equation 207References 209Problems 2099 Finite Element Formulation for Vector Field Problems – Linear Elasticity 2159.1 Linear Elasticity 2159.1.1 Kinematics 2179.1.2 Stress and Traction 2199.1.3 Equilibrium 2209.1.4 Constitutive Equation 2229.2 Strong and Weak Forms 2239.3 Finite Element Discretization 2259.4 Three-Node Triangular Element 2289.4.1 Element Body Force Matrix 2299.4.2 Boundary Force Matrix 2309.5 Generalization of Boundary Conditions 2319.6 Discussion 2399.7 Linear Elasticity Equations in Three Dimensions 240Problems 24110 Finite Element Formulation for Beams 24910.1 Governing Equations of the Beam 24910.1.1 Kinematics of Beam 24910.1.2 Stress–Strain Law 25210.1.3 Equilibrium 25310.1.4 Boundary Conditions 25410.2 Strong Form to Weak Form 25510.2.1 Weak Form to Strong Form 25710.3 Finite Element Discretization 25810.3.1 Trial Solution and Weight Function Approximations 25810.3.2 Discrete Equations 26010.4 Theorem of Minimum Potential Energy 26110.5 Remarks on Shell Elements 265Reference 269Problems 26911 Commercial Finite Element Program ABAQUS Tutorials 27511.1 Introduction 27511.1.1 Steady-State Heat Flow Example 27511.2 Preliminaries 27511.3 Creating a Part 27611.4 Creating a Material Definition 27811.5 Defining and Assigning Section Properties 27911.6 Assembling the Model 28011.7 Configuring the Analysis 28011.8 Applying a Boundary Condition and a Load to the Model 28011.9 Meshing the Model 28211.10 Creating and Submitting an Analysis Job 28411.11 Viewing the Analysis Results 28411.12 Solving the Problem Using Quadrilaterals 28411.13 Refining the Mesh 28511.13.1 Bending of a Short Cantilever Beam 28711.14 Copying the Model 28711.15 Modifying the Material Definition 28711.16 Configuring the Analysis 28711.17 Applying a Boundary Condition and a Load to the Model 28811.18 Meshing the Model 28911.19 Creating and Submitting an Analysis Job 29011.20 Viewing the Analysis Results 29011.20.1 Plate with a Hole in Tension 29011.21 Creating a New Model 29211.22 Creating a Part 29211.23 Creating a Material Definition 29311.24 Defining and Assigning Section Properties 29411.25 Assembling the Model 29511.26 Configuring the Analysis 29511.27 Applying a Boundary Condition and a Load to the Model 29511.28 Meshing the Model 29711.29 Creating and Submitting an Analysis Job 29811.30 Viewing the Analysis Results 29911.31 Refining the Mesh 299Appendix 303A. 1 Rotation of Coordinate System in Three Dimensions 303A. 2 Scalar Product Theorem 304A. 3 Taylor’s Formula with Remainder and the Mean Value Theorem 304A. 4 Green’s Theorem 305A. 5 Point Force (Source) 307A. 6 Static Condensation 308A. 7 Solution Methods 309Direct Solvers 310Iterative Solvers 310Conditioning 311References 312Problem 312Index 313

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