Finite Element Method

Linear Static and Dynamic Finite Element Analysis

By: THOMAS J. R. HUGHES

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Paperback | 28 March 2003

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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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Non-Fiction Mathematics Calculus & Mathematical Analysis Numerical Analysis Differential Calculus & Equations Engineering & Technology Civil Engineering Technology in General Engineering in General

Preface	p. XV
A Brief Glossary of Notations	p. XXII
Linear Static Analysis
Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem	p. 1
Introductory Remarks and Preliminaries	p. 1
Strong, or Classical, Form of the Problem	p. 2
Weak, or Variational, Form of the Problem	p. 3
Eqivalence of Strong and Weak Forms; Natural Boundary Conditions	p. 4
Galerkin's Approximation Method	p. 7
Matrix Equations; Stiffness Matrix K	p. 9
Examples: 1 and 2 Degrees of Freedom	p. 13
Piecewise Linear Finite Element Space	p. 20
Properties of K	p. 22
Mathematical Analysis	p. 24
Interlude: Gauss Elimination; Hand-calculation Version	p. 31
The Element Point of View	p. 37
Element Stiffness Matrix and Force Vector	p. 40
Assembly of Global Stiffness Matrix and Force Vector; LM Array	p. 42
Explicit Computation of Element Stiffness Matrix and Force Vector	p. 44
Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics	p. 48
An Elementary Discussion of Continuity, Differentiability, and Smoothness	p. 52
References	p. 55
Formulation of Two- and Three-Dimensional Boundary-Value Problems	p. 57
Introductory Remarks	p. 57
Preliminaries	p. 57
Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence	p. 60
Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K	p. 64
Heat Conduction: Element Stiffness Matrix and Force Vector	p. 69
Heat Conduction: Data Processing Arrays ID, IEN, and LM	p. 71
Classical Linear Elastostatics: Strong and Weak Forms; Equivalence	p. 75
Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K	p. 84
Elastostatics: Element Stiffness Matrix and Force Vector	p. 90
Elastostatics: Data Processing Arrays ID, IEN, and LM	p. 92
Summary of Important Equations for Problems Considered in Chapters 1 and 2	p. 98
Axisymmetric Formulations and Additional Exercises	p. 101
References	p. 107
Isoparametric Elements and Elementary Programming Concepts	p. 109
Preliminary Concepts	p. 109
Bilinear Quadrilateral Element	p. 112
Isoparametric Elements	p. 118
Linear Triangular Element; An Example of "Degeneration"	p. 120
Trilinear Hexahedral Element	p. 123
Higher-order Elements; Lagrange Polynomials	p. 126
Elements with Variable Numbers of Nodes	p. 132
Numerical Integration; Gaussian Quadrature	p. 137
Derivatives of Shape Functions and Shape Function Subroutines	p. 146
Element Stiffness Formulation	p. 151
Additional Exercises	p. 156
Triangular and Tetrahedral Elements	p. 164
Methodology for Developing Special Shape Functions with Application to Singularities	p. 175
References	p. 182
Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes	p. 185
"Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not	p. 185
Incompressible Elasticity and Stokes Flow	p. 192
Prelude to Mixed and Penalty Methods	p. 194
A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit	p. 197
Strong Form	p. 198
Weak Form	p. 198
Galerkin Formulation	p. 200
Matrix Problem	p. 200
Definition of Element Arrays	p. 204
Illustration of a Fundamental Difficulty	p. 207
Constraint Counts	p. 209
Discontinuous Pressure Elements	p. 210
Continuous Pressure Elements	p. 215
Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods	p. 217
Pressure Smoothing	p. 226
An Extension of Reduced and Selective Integration Techniques	p. 232
Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis	p. 232
Strain Projection: The B-approach	p. 232
The Patch Test; Rank Deficiency	p. 237
Nonconforming Elements	p. 242
Hourglass Stiffness	p. 251
Additional Exercises and Projects	p. 254
Mathematical Preliminaries	p. 263
Basic Properties of Linear Spaces	p. 263
Sobolev Norms	p. 266
Approximation Properties of Finite Element Spaces in Sobolev Norms	p. 268
Hypotheses on a(.,.)	p. 273
Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates	p. 276
Pressure Modes, Spurious and Otherwise	p. 276
Existence and Uniqueness of Solutions in the Presence of Modes	p. 278
Two Sides of Pressure Modes	p. 281
Pressure Modes in the Penalty Formulation	p. 289
The Big Picture	p. 292
Error Estimates and Pressure Smoothing	p. 297
References	p. 303
The C[superscript 0]-Approach to Plates and Beams	p. 310
Introduction	p. 310
Reissner-Mindlin Plate Theory	p. 310
Main Assumptions	p. 310
Constitutive Equation	p. 313
Strain-displacement Equations	p. 313
Summary of Plate Theory Notations	p. 314
Variational Equation	p. 314
Strong Form	p. 317
Weak Form	p. 317
Matrix Formulation	p. 319
Finite Element Stiffness Matrix and Load Vector	p. 320
Plate-bending Elements	p. 322
Some Convergence Criteria	p. 322
Shear Constraints and Locking	p. 323
Boundary Conditions	p. 324
Reduced and Selective Integration Lagrange Plate Elements	p. 327
Equivalence with Mixed Methods	p. 330
Rank Deficiency	p. 332
The Heterosis Element	p. 335
T1: A Correct-rank, Four-node Bilinear Element	p. 342
The Linear Triangle	p. 355
The Discrete Kirchhoff Approach	p. 359
Discussion of Some Quadrilateral Bending Elements	p. 362
Beams and Frames	p. 363
Main Assumptions	p. 363
Constitutive Equation	p. 365
Strain-displacement Equations	p. 366
Definitions of Quantities Appearing in the Theory	p. 366
Variational Equation	p. 368
Strong Form	p. 371
Weak Form	p. 372
Matrix Formulation of the Variational Equation	p. 373
Finite Element Stiffness Matrix and Load Vector	p. 374
Representation of Stiffness and Load in Global Coordinates	p. 376
Reduced Integration Beam Elements	p. 376
References	p. 379
The C[superscript 0]-Approach to Curved Structural Elements	p. 383
Introduction	p. 383
Doubly Curved Shells in Three Dimensions	p. 384
Geometry	p. 384
Lamina Coordinate Systems	p. 385
Fiber Coordinate Systems	p. 387
Kinematics	p. 388
Reduced Constitutive Equation	p. 389
Strain-displacement Matrix	p. 392
Stiffness Matrix	p. 396
External Force Vector	p. 396
Fiber Numerical Integration	p. 398
Stress Resultants	p. 399
Shell Elements	p. 399
Some References to the Recent Literature	p. 403
Simplifications: Shells as an Assembly of Flat Elements	p. 404
Shells of Revolution; Rings and Tubes in Two Dimensions	p. 405
Geometric and Kinematic Descriptions	p. 405
Reduced Constitutive Equations	p. 407
Strain-displacement Matrix	p. 409
Stiffness Matrix	p. 412
External Force Vector	p. 412
Stress Resultants	p. 413
Boundary Conditions	p. 414
Shell Elements	p. 414
References	p. 415
Linear Dynamic Analysis
Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems	p. 418
Parabolic Case: Heat Equation	p. 418
Hyperbolic Case: Elastodynamics and Structural Dynamics	p. 423
Eigenvalue Problems: Frequency Analysis and Buckling	p. 429
Standard Error Estimates	p. 433
Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass	p. 436
Estimation of Eigenvalues	p. 452
Error Estimates for Semidiscrete Galerkin Approximations	p. 456
References	p. 457
Algorithms for Parabolic Problems	p. 459
One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method	p. 459
Analysis of the Generalized Trapezoidal Method	p. 462
Modal Reduction to SDOF Form	p. 462
Stability	p. 465
Convergence	p. 468
An Alternative Approach to Stability: The Energy Method	p. 471
Additional Exercises	p. 473
Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis	p. 479
Element-by-element (EBE) Implicit Methods	p. 483
Modal Analysis	p. 487
References	p. 488
Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems	p. 490
One-step Algorithms for the Semidiscrete Equation of Motion	p. 490
The Newmark Method	p. 490
Analysis	p. 492
Measures of Accuracy: Numerical Dissipation and Dispersion	p. 504
Matched Methods	p. 505
Additional Exercises	p. 512
Summary of Time-step Estimates for Some Simple Finite Elements	p. 513
Linear Multistep (LMS) Methods	p. 523
LMS Methods for First-order Equations	p. 523
LMS Methods for Second-order Equations	p. 526
Survey of Some Commonly Used Algorithms in Structural Dynamics	p. 529
Some Recently Developed Algorithms for Structural Dynamics	p. 550
Algorithms Based upon Operator Splitting and Mesh Partitions	p. 552
Stability via the Energy Method	p. 556
Predictor/Multicorrector Algorithms	p. 562
Mass Matrices for Shell Elements	p. 564
References	p. 567
Solution Techniques for Eigenvalue Problems	p. 570
The Generalized Eigenproblem	p. 570
Static Condensation	p. 573
Discrete Rayleigh-Ritz Reduction	p. 574
Irons-Guyan Reduction	p. 576
Subspace Iteration	p. 576
Spectrum Slicing	p. 578
Inverse Iteration	p. 579
The Lanczos Algorithm for Solution of Large Generalized Eigenproblems	p. 582
Introduction	p. 582
Spectral Transformation	p. 583
Conditions for Real Eigenvalues	p. 584
The Rayleigh-Ritz Approximation	p. 585
Derivation of the Lanczos Algorithm	p. 586
Reduction to Tridiagonal Form	p. 589
Convergence Criterion for Eigenvalues	p. 592
Loss of Orthogonality	p. 595
Restoring Orthogonality	p. 598
References	p. 601
Dlearn--A Linear Static and Dynamic Finite Element Analysis Program	p. 603
Introduction	p. 603
Description of Coding Techniques Used in DLEARN	p. 604
Compacted Column Storage Scheme	p. 605
Crout Elimination	p. 608
Dynamic Storage Allocation	p. 616
Program Structure	p. 622
Global Control	p. 623
Initialization Phase	p. 623
Solution Phase	p. 625
Adding an Element to DLEARN	p. 631
DLEARN User's Manual	p. 634
Remarks for the New User	p. 634
Input Instructions	p. 635
Examples	p. 663
Planar Truss	p. 663
Static Analysis of a Plane Strain Cantilever Beam	p. 666
Dynamic Analysis of a Plane Strain Cantilever Beam	p. 666
Implicit-explicit Dynamic Analysis of a Rod	p. 668
Subroutine Index for Program Listing	p. 670
References	p. 675
Index	p. 676
Table of Contents provided by Syndetics. All Rights Reserved.

Finite Element Method

Linear Static and Dynamic Finite Element Analysis

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