Finite Element Method: Linear Static and Dynamic Finite Element Analysis

Finite Element Method: Linear Static and Dynamic Finite Element Analysis

by Thomas J. R. Hughes, Ted Hughes

ISBN-10: 0486411818

ISBN-13: 9780486411811

Pub. Date: 08/16/2000

Publisher: Dover Publications

This text is geared toward assisting engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Based on courses taught at Stanford University and the California Institute of Technology, it ranges from fundamental concepts to practical computer implementations. Additional sections touch…  See more details below


This text is geared toward assisting engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Based on courses taught at Stanford University and the California Institute of Technology, it ranges from fundamental concepts to practical computer implementations. Additional sections touch upon the frontiers of research, making the book of potential interest to more experienced analysts and researchers working in the finite element field.
In addition to its examination of numerous standard aspects of the finite element method, the volume includes many unique components, including a comprehensive presentation and analysis of algorithms of time-dependent phenomena, plus beam, plate, and shell theories derived directly from three-dimensional elasticity theory. It also contains a systematic treatment of "weak," or variational, formulations for diverse classes of initial/boundary-value problems.
Directed toward students without in-depth mathematical training, the text incorporates introductory material on the mathematical theory of finite elements and many important mathematical results, making it an ideal primer for more advanced works on this subject.

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Product Details

Dover Publications
Publication date:
Dover Civil and Mechanical Engineering Series
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Product dimensions:
6.51(w) x 9.22(h) x 1.34(d)

Table of Contents

A Brief Glossary of NotationsXXII
Part 1Linear Static Analysis
1Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem1
1.1Introductory Remarks and Preliminaries1
1.2Strong, or Classical, Form of the Problem2
1.3Weak, or Variational, Form of the Problem3
1.4Eqivalence of Strong and Weak Forms; Natural Boundary Conditions4
1.5Galerkin's Approximation Method7
1.6Matrix Equations; Stiffness Matrix K9
1.7Examples: 1 and 2 Degrees of Freedom13
1.8Piecewise Linear Finite Element Space20
1.9Properties of K22
1.10Mathematical Analysis24
1.11Interlude: Gauss Elimination; Hand-calculation Version31
1.12The Element Point of View37
1.13Element Stiffness Matrix and Force Vector40
1.14Assembly of Global Stiffness Matrix and Force Vector; LM Array42
1.15Explicit Computation of Element Stiffness Matrix and Force Vector44
1.16Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics48
Appendix 1.IAn Elementary Discussion of Continuity, Differentiability, and Smoothness52
2Formulation of Two- and Three-Dimensional Boundary-Value Problems57
2.1Introductory Remarks57
2.3Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence60
2.4Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K64
2.5Heat Conduction: Element Stiffness Matrix and Force Vector69
2.6Heat Conduction: Data Processing Arrays ID, IEN, and LM71
2.7Classical Linear Elastostatics: Strong and Weak Forms; Equivalence75
2.8Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K84
2.9Elastostatics: Element Stiffness Matrix and Force Vector90
2.10Elastostatics: Data Processing Arrays ID, IEN, and LM92
2.11Summary of Important Equations for Problems Considered in Chapters 1 and 298
2.12Axisymmetric Formulations and Additional Exercises101
3Isoparametric Elements and Elementary Programming Concepts109
3.1Preliminary Concepts109
3.2Bilinear Quadrilateral Element112
3.3Isoparametric Elements118
3.4Linear Triangular Element; An Example of "Degeneration"120
3.5Trilinear Hexahedral Element123
3.6Higher-order Elements; Lagrange Polynomials126
3.7Elements with Variable Numbers of Nodes132
3.8Numerical Integration; Gaussian Quadrature137
3.9Derivatives of Shape Functions and Shape Function Subroutines146
3.10Element Stiffness Formulation151
3.11Additional Exercises156
Appendix 3.ITriangular and Tetrahedral Elements164
Appendix 3.IIMethodology for Developing Special Shape Functions with Application to Singularities175
4Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes185
4.1"Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not185
4.2Incompressible Elasticity and Stokes Flow192
4.2.1Prelude to Mixed and Penalty Methods194
4.3A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit197
4.3.1Strong Form198
4.3.2Weak Form198
4.3.3Galerkin Formulation200
4.3.4Matrix Problem200
4.3.5Definition of Element Arrays204
4.3.6Illustration of a Fundamental Difficulty207
4.3.7Constraint Counts209
4.3.8Discontinuous Pressure Elements210
4.3.9Continuous Pressure Elements215
4.4Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods217
4.4.1Pressure Smoothing226
4.5An Extension of Reduced and Selective Integration Techniques232
4.5.1Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis232
4.5.2Strain Projection: The B-approach232
4.6The Patch Test; Rank Deficiency237
4.7Nonconforming Elements242
4.8Hourglass Stiffness251
4.9Additional Exercises and Projects254
Appendix 4.IMathematical Preliminaries263
4.I.1Basic Properties of Linear Spaces263
4.I.2Sobolev Norms266
4.I.3Approximation Properties of Finite Element Spaces in Sobolev Norms268
4.I.4Hypotheses on a(.,.)273
Appendix 4.IIAdvanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates276
4.II.1Pressure Modes, Spurious and Otherwise276
4.II.2Existence and Uniqueness of Solutions in the Presence of Modes278
4.II.3Two Sides of Pressure Modes281
4.II.4Pressure Modes in the Penalty Formulation289
4.II.5The Big Picture292
4.II.6Error Estimates and Pressure Smoothing297
5The C[superscript 0]-Approach to Plates and Beams310
5.2Reissner-Mindlin Plate Theory310
5.2.1Main Assumptions310
5.2.2Constitutive Equation313
5.2.3Strain-displacement Equations313
5.2.4Summary of Plate Theory Notations314
5.2.5Variational Equation314
5.2.6Strong Form317
5.2.7Weak Form317
5.2.8Matrix Formulation319
5.2.9Finite Element Stiffness Matrix and Load Vector320
5.3Plate-bending Elements322
5.3.1Some Convergence Criteria322
5.3.2Shear Constraints and Locking323
5.3.3Boundary Conditions324
5.3.4Reduced and Selective Integration Lagrange Plate Elements327
5.3.5Equivalence with Mixed Methods330
5.3.6Rank Deficiency332
5.3.7The Heterosis Element335
5.3.8T1: A Correct-rank, Four-node Bilinear Element342
5.3.9The Linear Triangle355
5.3.10The Discrete Kirchhoff Approach359
5.3.11Discussion of Some Quadrilateral Bending Elements362
5.4Beams and Frames363
5.4.1Main Assumptions363
5.4.2Constitutive Equation365
5.4.3Strain-displacement Equations366
5.4.4Definitions of Quantities Appearing in the Theory366
5.4.5Variational Equation368
5.4.6Strong Form371
5.4.7Weak Form372
5.4.8Matrix Formulation of the Variational Equation373
5.4.9Finite Element Stiffness Matrix and Load Vector374
5.4.10Representation of Stiffness and Load in Global Coordinates376
5.5Reduced Integration Beam Elements376
The C[superscript 0]-Approach to Curved Structural Elements383
6.2Doubly Curved Shells in Three Dimensions384
6.2.2Lamina Coordinate Systems385
6.2.3Fiber Coordinate Systems387
6.2.5Reduced Constitutive Equation389
6.2.6Strain-displacement Matrix392
6.2.7Stiffness Matrix396
6.2.8External Force Vector396
6.2.9Fiber Numerical Integration398
6.2.10Stress Resultants399
6.2.11Shell Elements399
6.2.12Some References to the Recent Literature403
6.2.13Simplifications: Shells as an Assembly of Flat Elements404
6.3Shells of Revolution; Rings and Tubes in Two Dimensions405
6.3.1Geometric and Kinematic Descriptions405
6.3.2Reduced Constitutive Equations407
6.3.3Strain-displacement Matrix409
6.3.4Stiffness Matrix412
6.3.5External Force Vector412
6.3.6Stress Resultants413
6.3.7Boundary Conditions414
6.3.8Shell Elements414
Part 2Linear Dynamic Analysis
7Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems418
7.1Parabolic Case: Heat Equation418
7.2Hyperbolic Case: Elastodynamics and Structural Dynamics423
7.3Eigenvalue Problems: Frequency Analysis and Buckling429
7.3.1Standard Error Estimates433
7.3.2Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass436
7.3.3Estimation of Eigenvalues452
Appendix 7.IError Estimates for Semidiscrete Galerkin Approximations456
8Algorithms for Parabolic Problems459
8.1One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method459
8.2Analysis of the Generalized Trapezoidal Method462
8.2.1Modal Reduction to SDOF Form462
8.2.4An Alternative Approach to Stability: The Energy Method471
8.2.5Additional Exercises473
8.3Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis479
8.4Element-by-element (EBE) Implicit Methods483
8.5Modal Analysis487
9Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems490
9.1One-step Algorithms for the Semidiscrete Equation of Motion490
9.1.1The Newmark Method490
9.1.3Measures of Accuracy: Numerical Dissipation and Dispersion504
9.1.4Matched Methods505
9.1.5Additional Exercises512
9.2Summary of Time-step Estimates for Some Simple Finite Elements513
9.3Linear Multistep (LMS) Methods523
9.3.1LMS Methods for First-order Equations523
9.3.2LMS Methods for Second-order Equations526
9.3.3Survey of Some Commonly Used Algorithms in Structural Dynamics529
9.3.4Some Recently Developed Algorithms for Structural Dynamics550
9.4Algorithms Based upon Operator Splitting and Mesh Partitions552
9.4.1Stability via the Energy Method556
9.4.2Predictor/Multicorrector Algorithms562
9.5Mass Matrices for Shell Elements564
10Solution Techniques for Eigenvalue Problems570
10.1The Generalized Eigenproblem570
10.2Static Condensation573
10.3Discrete Rayleigh-Ritz Reduction574
10.4Irons-Guyan Reduction576
10.5Subspace Iteration576
10.5.1Spectrum Slicing578
10.5.2Inverse Iteration579
10.6The Lanczos Algorithm for Solution of Large Generalized Eigenproblems582
10.6.2Spectral Transformation583
10.6.3Conditions for Real Eigenvalues584
10.6.4The Rayleigh-Ritz Approximation585
10.6.5Derivation of the Lanczos Algorithm586
10.6.6Reduction to Tridiagonal Form589
10.6.7Convergence Criterion for Eigenvalues592
10.6.8Loss of Orthogonality595
10.6.9Restoring Orthogonality598
11Dlearn--A Linear Static and Dynamic Finite Element Analysis Program603
11.2Description of Coding Techniques Used in DLEARN604
11.2.1Compacted Column Storage Scheme605
11.2.2Crout Elimination608
11.2.3Dynamic Storage Allocation616
11.3Program Structure622
11.3.1Global Control623
11.3.2Initialization Phase623
11.3.3Solution Phase625
11.4Adding an Element to DLEARN631
11.5DLEARN User's Manual634
11.5.1Remarks for the New User634
11.5.2Input Instructions635
1.Planar Truss663
2.Static Analysis of a Plane Strain Cantilever Beam666
3.Dynamic Analysis of a Plane Strain Cantilever Beam666
4.Implicit-explicit Dynamic Analysis of a Rod668
11.5.4Subroutine Index for Program Listing670

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