The Finite Element Method Linear Static and Dynamic Finite Element Analysis

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  • Format: Paperback
  • Copyright: 2000-08-16
  • Publisher: Dover Publications

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Directed towards students without in-depth mathematical training, this text is intended to assist engineering and physical science students in cultivating comprehensive skills in linear static and dynamic finite element methodology. Included are a comprehensive presentation and analysis of algorithms of time-dependent phenomena plus beam, plate, and shell theories derived directly from 3-dimensional elasticity theory. An ideal primer for more advanced works on this subject. Brief Glossary of Notations. Solution guide available upon request.

Table of Contents

Prefacep. XV
A Brief Glossary of Notationsp. XXII
Linear Static Analysis
Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problemp. 1
Introductory Remarks and Preliminariesp. 1
Strong, or Classical, Form of the Problemp. 2
Weak, or Variational, Form of the Problemp. 3
Eqivalence of Strong and Weak Forms; Natural Boundary Conditionsp. 4
Galerkin's Approximation Methodp. 7
Matrix Equations; Stiffness Matrix Kp. 9
Examples: 1 and 2 Degrees of Freedomp. 13
Piecewise Linear Finite Element Spacep. 20
Properties of Kp. 22
Mathematical Analysisp. 24
Interlude: Gauss Elimination; Hand-calculation Versionp. 31
The Element Point of Viewp. 37
Element Stiffness Matrix and Force Vectorp. 40
Assembly of Global Stiffness Matrix and Force Vector; LM Arrayp. 42
Explicit Computation of Element Stiffness Matrix and Force Vectorp. 44
Exercise: Bernoulli-Euler Beam Theory and Hermite Cubicsp. 48
An Elementary Discussion of Continuity, Differentiability, and Smoothnessp. 52
Referencesp. 55
Formulation of Two- and Three-Dimensional Boundary-Value Problemsp. 57
Introductory Remarksp. 57
Preliminariesp. 57
Classical Linear Heat Conduction: Strong and Weak Forms; Equivalencep. 60
Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of Kp. 64
Heat Conduction: Element Stiffness Matrix and Force Vectorp. 69
Heat Conduction: Data Processing Arrays ID, IEN, and LMp. 71
Classical Linear Elastostatics: Strong and Weak Forms; Equivalencep. 75
Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of Kp. 84
Elastostatics: Element Stiffness Matrix and Force Vectorp. 90
Elastostatics: Data Processing Arrays ID, IEN, and LMp. 92
Summary of Important Equations for Problems Considered in Chapters 1 and 2p. 98
Axisymmetric Formulations and Additional Exercisesp. 101
Referencesp. 107
Isoparametric Elements and Elementary Programming Conceptsp. 109
Preliminary Conceptsp. 109
Bilinear Quadrilateral Elementp. 112
Isoparametric Elementsp. 118
Linear Triangular Element; An Example of "Degeneration"p. 120
Trilinear Hexahedral Elementp. 123
Higher-order Elements; Lagrange Polynomialsp. 126
Elements with Variable Numbers of Nodesp. 132
Numerical Integration; Gaussian Quadraturep. 137
Derivatives of Shape Functions and Shape Function Subroutinesp. 146
Element Stiffness Formulationp. 151
Additional Exercisesp. 156
Triangular and Tetrahedral Elementsp. 164
Methodology for Developing Special Shape Functions with Application to Singularitiesp. 175
Referencesp. 182
Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimesp. 185
"Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does notp. 185
Incompressible Elasticity and Stokes Flowp. 192
Prelude to Mixed and Penalty Methodsp. 194
A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limitp. 197
Strong Formp. 198
Weak Formp. 198
Galerkin Formulationp. 200
Matrix Problemp. 200
Definition of Element Arraysp. 204
Illustration of a Fundamental Difficultyp. 207
Constraint Countsp. 209
Discontinuous Pressure Elementsp. 210
Continuous Pressure Elementsp. 215
Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methodsp. 217
Pressure Smoothingp. 226
An Extension of Reduced and Selective Integration Techniquesp. 232
Axisymmetry and Anisotropy: Prelude to Nonlinear Analysisp. 232
Strain Projection: The B-approachp. 232
The Patch Test; Rank Deficiencyp. 237
Nonconforming Elementsp. 242
Hourglass Stiffnessp. 251
Additional Exercises and Projectsp. 254
Mathematical Preliminariesp. 263
Basic Properties of Linear Spacesp. 263
Sobolev Normsp. 266
Approximation Properties of Finite Element Spaces in Sobolev Normsp. 268
Hypotheses on a(.,.)p. 273
Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimatesp. 276
Pressure Modes, Spurious and Otherwisep. 276
Existence and Uniqueness of Solutions in the Presence of Modesp. 278
Two Sides of Pressure Modesp. 281
Pressure Modes in the Penalty Formulationp. 289
The Big Picturep. 292
Error Estimates and Pressure Smoothingp. 297
Referencesp. 303
The C[superscript 0]-Approach to Plates and Beamsp. 310
Introductionp. 310
Reissner-Mindlin Plate Theoryp. 310
Main Assumptionsp. 310
Constitutive Equationp. 313
Strain-displacement Equationsp. 313
Summary of Plate Theory Notationsp. 314
Variational Equationp. 314
Strong Formp. 317
Weak Formp. 317
Matrix Formulationp. 319
Finite Element Stiffness Matrix and Load Vectorp. 320
Plate-bending Elementsp. 322
Some Convergence Criteriap. 322
Shear Constraints and Lockingp. 323
Boundary Conditionsp. 324
Reduced and Selective Integration Lagrange Plate Elementsp. 327
Equivalence with Mixed Methodsp. 330
Rank Deficiencyp. 332
The Heterosis Elementp. 335
T1: A Correct-rank, Four-node Bilinear Elementp. 342
The Linear Trianglep. 355
The Discrete Kirchhoff Approachp. 359
Discussion of Some Quadrilateral Bending Elementsp. 362
Beams and Framesp. 363
Main Assumptionsp. 363
Constitutive Equationp. 365
Strain-displacement Equationsp. 366
Definitions of Quantities Appearing in the Theoryp. 366
Variational Equationp. 368
Strong Formp. 371
Weak Formp. 372
Matrix Formulation of the Variational Equationp. 373
Finite Element Stiffness Matrix and Load Vectorp. 374
Representation of Stiffness and Load in Global Coordinatesp. 376
Reduced Integration Beam Elementsp. 376
Referencesp. 379
The C[superscript 0]-Approach to Curved Structural Elementsp. 383
Introductionp. 383
Doubly Curved Shells in Three Dimensionsp. 384
Geometryp. 384
Lamina Coordinate Systemsp. 385
Fiber Coordinate Systemsp. 387
Kinematicsp. 388
Reduced Constitutive Equationp. 389
Strain-displacement Matrixp. 392
Stiffness Matrixp. 396
External Force Vectorp. 396
Fiber Numerical Integrationp. 398
Stress Resultantsp. 399
Shell Elementsp. 399
Some References to the Recent Literaturep. 403
Simplifications: Shells as an Assembly of Flat Elementsp. 404
Shells of Revolution; Rings and Tubes in Two Dimensionsp. 405
Geometric and Kinematic Descriptionsp. 405
Reduced Constitutive Equationsp. 407
Strain-displacement Matrixp. 409
Stiffness Matrixp. 412
External Force Vectorp. 412
Stress Resultantsp. 413
Boundary Conditionsp. 414
Shell Elementsp. 414
Referencesp. 415
Linear Dynamic Analysis
Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problemsp. 418
Parabolic Case: Heat Equationp. 418
Hyperbolic Case: Elastodynamics and Structural Dynamicsp. 423
Eigenvalue Problems: Frequency Analysis and Bucklingp. 429
Standard Error Estimatesp. 433
Alternative Definitions of the Mass Matrix; Lumped and Higher-order Massp. 436
Estimation of Eigenvaluesp. 452
Error Estimates for Semidiscrete Galerkin Approximationsp. 456
Referencesp. 457
Algorithms for Parabolic Problemsp. 459
One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Methodp. 459
Analysis of the Generalized Trapezoidal Methodp. 462
Modal Reduction to SDOF Formp. 462
Stabilityp. 465
Convergencep. 468
An Alternative Approach to Stability: The Energy Methodp. 471
Additional Exercisesp. 473
Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysisp. 479
Element-by-element (EBE) Implicit Methodsp. 483
Modal Analysisp. 487
Referencesp. 488
Algorithms for Hyperbolic and Parabolic-Hyperbolic Problemsp. 490
One-step Algorithms for the Semidiscrete Equation of Motionp. 490
The Newmark Methodp. 490
Analysisp. 492
Measures of Accuracy: Numerical Dissipation and Dispersionp. 504
Matched Methodsp. 505
Additional Exercisesp. 512
Summary of Time-step Estimates for Some Simple Finite Elementsp. 513
Linear Multistep (LMS) Methodsp. 523
LMS Methods for First-order Equationsp. 523
LMS Methods for Second-order Equationsp. 526
Survey of Some Commonly Used Algorithms in Structural Dynamicsp. 529
Some Recently Developed Algorithms for Structural Dynamicsp. 550
Algorithms Based upon Operator Splitting and Mesh Partitionsp. 552
Stability via the Energy Methodp. 556
Predictor/Multicorrector Algorithmsp. 562
Mass Matrices for Shell Elementsp. 564
Referencesp. 567
Solution Techniques for Eigenvalue Problemsp. 570
The Generalized Eigenproblemp. 570
Static Condensationp. 573
Discrete Rayleigh-Ritz Reductionp. 574
Irons-Guyan Reductionp. 576
Subspace Iterationp. 576
Spectrum Slicingp. 578
Inverse Iterationp. 579
The Lanczos Algorithm for Solution of Large Generalized Eigenproblemsp. 582
Introductionp. 582
Spectral Transformationp. 583
Conditions for Real Eigenvaluesp. 584
The Rayleigh-Ritz Approximationp. 585
Derivation of the Lanczos Algorithmp. 586
Reduction to Tridiagonal Formp. 589
Convergence Criterion for Eigenvaluesp. 592
Loss of Orthogonalityp. 595
Restoring Orthogonalityp. 598
Referencesp. 601
Dlearn--A Linear Static and Dynamic Finite Element Analysis Programp. 603
Introductionp. 603
Description of Coding Techniques Used in DLEARNp. 604
Compacted Column Storage Schemep. 605
Crout Eliminationp. 608
Dynamic Storage Allocationp. 616
Program Structurep. 622
Global Controlp. 623
Initialization Phasep. 623
Solution Phasep. 625
Adding an Element to DLEARNp. 631
DLEARN User's Manualp. 634
Remarks for the New Userp. 634
Input Instructionsp. 635
Examplesp. 663
Planar Trussp. 663
Static Analysis of a Plane Strain Cantilever Beamp. 666
Dynamic Analysis of a Plane Strain Cantilever Beamp. 666
Implicit-explicit Dynamic Analysis of a Rodp. 668
Subroutine Index for Program Listingp. 670
Referencesp. 675
Indexp. 676
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