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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 |
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