The Finite Element Method: Linear Static and Dynamic Finite Element Analysis

by Thomas J. R. Hughes

Overview

Originally developed to address specific areas of structural mechanics and elasticity, the finite element method is applicable to problems throughout applied mathematics, continuum mechanics, engineering, and physics. This text elucidates the method's broader scope, bridging the gap between mathematical foundations and practical applications. Intended for students as well as professionals, it is an excellent companion for independent study, with numerous illustrative examples and problems.
The authors trace the method's development and explain the technique in clearly understandable stages. Topics include solving problems involving partial differential equations, with a thorough finite element analysis of Poisson's equation; a step-by-step assembly of the master matrix; various numerical techniques for solving large systems of equations; and applications to problems in elasticity and the bending of beams and plates. Additional subjects include general interpolation functions, numerical integrations, and higher-order elements; applications to second- and fourth-order partial differential equations; and a variety of issues involving elastic vibrations, heat transfer, and fluid flow. The displacement model is fully developed, in addition to the hybrid model, of which Dr. Tong was an originator. The text concludes with numerous helpful appendixes.

Product Details

ISBN-13:	9780486135021
Publisher:	Dover Publications
Publication date:	04/25/2012
Series:	Dover Civil and Mechanical Engineering
Sold by:	Barnes & Noble
Format:	eBook
Pages:	704
File size:	41 MB
Note:	This product may take a few minutes to download.

Table of Contents

	Preface	XV
	A Brief Glossary of Notations	XXII
Part 1	Linear Static Analysis
1	Fundamental Concepts; A Simple One-Dimensional Boundary-Value Problem	1
1.1	Introductory Remarks and Preliminaries	1
1.2	Strong, or Classical, Form of the Problem	2
1.3	Weak, or Variational, Form of the Problem	3
1.4	Eqivalence of Strong and Weak Forms; Natural Boundary Conditions	4
1.5	Galerkin's Approximation Method	7
1.6	Matrix Equations; Stiffness Matrix K	9
1.7	Examples: 1 and 2 Degrees of Freedom	13
1.8	Piecewise Linear Finite Element Space	20
1.9	Properties of K	22
1.10	Mathematical Analysis	24
1.11	Interlude: Gauss Elimination; Hand-calculation Version	31
1.12	The Element Point of View	37
1.13	Element Stiffness Matrix and Force Vector	40
1.14	Assembly of Global Stiffness Matrix and Force Vector; LM Array	42
1.15	Explicit Computation of Element Stiffness Matrix and Force Vector	44
1.16	Exercise: Bernoulli-Euler Beam Theory and Hermite Cubics	48
Appendix 1.I	An Elementary Discussion of Continuity, Differentiability, and Smoothness	52
	References	55
2	Formulation of Two- and Three-Dimensional Boundary-Value Problems	57
2.1	Introductory Remarks	57
2.2	Preliminaries	57
2.3	Classical Linear Heat Conduction: Strong and Weak Forms; Equivalence	60
2.4	Heat Conduction: Galerkin Formulation; Symmetry and Positive-definiteness of K	64
2.5	Heat Conduction: Element Stiffness Matrix and Force Vector	69
2.6	Heat Conduction: Data Processing Arrays ID, IEN, and LM	71
2.7	Classical Linear Elastostatics: Strong and Weak Forms; Equivalence	75
2.8	Elastostatics: Galerkin Formulation, Symmetry, and Positive-definiteness of K	84
2.9	Elastostatics: Element Stiffness Matrix and Force Vector	90
2.10	Elastostatics: Data Processing Arrays ID, IEN, and LM	92
2.11	Summary of Important Equations for Problems Considered in Chapters 1 and 2	98
2.12	Axisymmetric Formulations and Additional Exercises	101
	References	107
3	Isoparametric Elements and Elementary Programming Concepts	109
3.1	Preliminary Concepts	109
3.2	Bilinear Quadrilateral Element	112
3.3	Isoparametric Elements	118
3.4	Linear Triangular Element; An Example of "Degeneration"	120
3.5	Trilinear Hexahedral Element	123
3.6	Higher-order Elements; Lagrange Polynomials	126
3.7	Elements with Variable Numbers of Nodes	132
3.8	Numerical Integration; Gaussian Quadrature	137
3.9	Derivatives of Shape Functions and Shape Function Subroutines	146
3.10	Element Stiffness Formulation	151
3.11	Additional Exercises	156
Appendix 3.I	Triangular and Tetrahedral Elements	164
Appendix 3.II	Methodology for Developing Special Shape Functions with Application to Singularities	175
	References	182
4	Mixed and Penalty Methods, Reduced and Selective Integration, and Sundry Variational Crimes	185
4.1	"Best Approximation" and Error Estimates: Why the standard FEM usually works and why sometimes it does not	185
4.2	Incompressible Elasticity and Stokes Flow	192
4.2.1	Prelude to Mixed and Penalty Methods	194
4.3	A Mixed Formulation of Compressible Elasticity Capable of Representing the Incompressible Limit	197
4.3.1	Strong Form	198
4.3.2	Weak Form	198
4.3.3	Galerkin Formulation	200
4.3.4	Matrix Problem	200
4.3.5	Definition of Element Arrays	204
4.3.6	Illustration of a Fundamental Difficulty	207
4.3.7	Constraint Counts	209
4.3.8	Discontinuous Pressure Elements	210
4.3.9	Continuous Pressure Elements	215
4.4	Penalty Formulation: Reduced and Selective Integration Techniques; Equivalence with Mixed Methods	217
4.4.1	Pressure Smoothing	226
4.5	An Extension of Reduced and Selective Integration Techniques	232
4.5.1	Axisymmetry and Anisotropy: Prelude to Nonlinear Analysis	232
4.5.2	Strain Projection: The B-approach	232
4.6	The Patch Test; Rank Deficiency	237
4.7	Nonconforming Elements	242
4.8	Hourglass Stiffness	251
4.9	Additional Exercises and Projects	254
Appendix 4.I	Mathematical Preliminaries	263
4.I.1	Basic Properties of Linear Spaces	263
4.I.2	Sobolev Norms	266
4.I.3	Approximation Properties of Finite Element Spaces in Sobolev Norms	268
4.I.4	Hypotheses on a(.,.)	273
Appendix 4.II	Advanced Topics in the Theory of Mixed and Penalty Methods: Pressure Modes and Error Estimates	276
4.II.1	Pressure Modes, Spurious and Otherwise	276
4.II.2	Existence and Uniqueness of Solutions in the Presence of Modes	278
4.II.3	Two Sides of Pressure Modes	281
4.II.4	Pressure Modes in the Penalty Formulation	289
4.II.5	The Big Picture	292
4.II.6	Error Estimates and Pressure Smoothing	297
	References	303
5	The C[superscript 0]-Approach to Plates and Beams	310
5.1	Introduction	310
5.2	Reissner-Mindlin Plate Theory	310
5.2.1	Main Assumptions	310
5.2.2	Constitutive Equation	313
5.2.3	Strain-displacement Equations	313
5.2.4	Summary of Plate Theory Notations	314
5.2.5	Variational Equation	314
5.2.6	Strong Form	317
5.2.7	Weak Form	317
5.2.8	Matrix Formulation	319
5.2.9	Finite Element Stiffness Matrix and Load Vector	320
5.3	Plate-bending Elements	322
5.3.1	Some Convergence Criteria	322
5.3.2	Shear Constraints and Locking	323
5.3.3	Boundary Conditions	324
5.3.4	Reduced and Selective Integration Lagrange Plate Elements	327
5.3.5	Equivalence with Mixed Methods	330
5.3.6	Rank Deficiency	332
5.3.7	The Heterosis Element	335
5.3.8	T1: A Correct-rank, Four-node Bilinear Element	342
5.3.9	The Linear Triangle	355
5.3.10	The Discrete Kirchhoff Approach	359
5.3.11	Discussion of Some Quadrilateral Bending Elements	362
5.4	Beams and Frames	363
5.4.1	Main Assumptions	363
5.4.2	Constitutive Equation	365
5.4.3	Strain-displacement Equations	366
5.4.4	Definitions of Quantities Appearing in the Theory	366
5.4.5	Variational Equation	368
5.4.6	Strong Form	371
5.4.7	Weak Form	372
5.4.8	Matrix Formulation of the Variational Equation	373
5.4.9	Finite Element Stiffness Matrix and Load Vector	374
5.4.10	Representation of Stiffness and Load in Global Coordinates	376
5.5	Reduced Integration Beam Elements	376
	References	379
	The C[superscript 0]-Approach to Curved Structural Elements	383
6.1	Introduction	383
6.2	Doubly Curved Shells in Three Dimensions	384
6.2.1	Geometry	384
6.2.2	Lamina Coordinate Systems	385
6.2.3	Fiber Coordinate Systems	387
6.2.4	Kinematics	388
6.2.5	Reduced Constitutive Equation	389
6.2.6	Strain-displacement Matrix	392
6.2.7	Stiffness Matrix	396
6.2.8	External Force Vector	396
6.2.9	Fiber Numerical Integration	398
6.2.10	Stress Resultants	399
6.2.11	Shell Elements	399
6.2.12	Some References to the Recent Literature	403
6.2.13	Simplifications: Shells as an Assembly of Flat Elements	404
6.3	Shells of Revolution; Rings and Tubes in Two Dimensions	405
6.3.1	Geometric and Kinematic Descriptions	405
6.3.2	Reduced Constitutive Equations	407
6.3.3	Strain-displacement Matrix	409
6.3.4	Stiffness Matrix	412
6.3.5	External Force Vector	412
6.3.6	Stress Resultants	413
6.3.7	Boundary Conditions	414
6.3.8	Shell Elements	414
	References	415
Part 2	Linear Dynamic Analysis
7	Formulation of Parabolic, Hyperbolic, and Elliptic-Elgenvalue Problems	418
7.1	Parabolic Case: Heat Equation	418
7.2	Hyperbolic Case: Elastodynamics and Structural Dynamics	423
7.3	Eigenvalue Problems: Frequency Analysis and Buckling	429
7.3.1	Standard Error Estimates	433
7.3.2	Alternative Definitions of the Mass Matrix; Lumped and Higher-order Mass	436
7.3.3	Estimation of Eigenvalues	452
Appendix 7.I	Error Estimates for Semidiscrete Galerkin Approximations	456
	References	457
8	Algorithms for Parabolic Problems	459
8.1	One-step Algorithms for the Semidiscrete Heat Equation: Generalized Trapezoidal Method	459
8.2	Analysis of the Generalized Trapezoidal Method	462
8.2.1	Modal Reduction to SDOF Form	462
8.2.2	Stability	465
8.2.3	Convergence	468
8.2.4	An Alternative Approach to Stability: The Energy Method	471
8.2.5	Additional Exercises	473
8.3	Elementary Finite Difference Equations for the One-dimensional Heat Equation; the von Neumann Method of Stability Analysis	479
8.4	Element-by-element (EBE) Implicit Methods	483
8.5	Modal Analysis	487
	References	488
9	Algorithms for Hyperbolic and Parabolic-Hyperbolic Problems	490
9.1	One-step Algorithms for the Semidiscrete Equation of Motion	490
9.1.1	The Newmark Method	490
9.1.2	Analysis	492
9.1.3	Measures of Accuracy: Numerical Dissipation and Dispersion	504
9.1.4	Matched Methods	505
9.1.5	Additional Exercises	512
9.2	Summary of Time-step Estimates for Some Simple Finite Elements	513
9.3	Linear Multistep (LMS) Methods	523
9.3.1	LMS Methods for First-order Equations	523
9.3.2	LMS Methods for Second-order Equations	526
9.3.3	Survey of Some Commonly Used Algorithms in Structural Dynamics	529
9.3.4	Some Recently Developed Algorithms for Structural Dynamics	550
9.4	Algorithms Based upon Operator Splitting and Mesh Partitions	552
9.4.1	Stability via the Energy Method	556
9.4.2	Predictor/Multicorrector Algorithms	562
9.5	Mass Matrices for Shell Elements	564
	References	567
10	Solution Techniques for Eigenvalue Problems	570
10.1	The Generalized Eigenproblem	570
10.2	Static Condensation	573
10.3	Discrete Rayleigh-Ritz Reduction	574
10.4	Irons-Guyan Reduction	576
10.5	Subspace Iteration	576
10.5.1	Spectrum Slicing	578
10.5.2	Inverse Iteration	579
10.6	The Lanczos Algorithm for Solution of Large Generalized Eigenproblems	582
10.6.1	Introduction	582
10.6.2	Spectral Transformation	583
10.6.3	Conditions for Real Eigenvalues	584
10.6.4	The Rayleigh-Ritz Approximation	585
10.6.5	Derivation of the Lanczos Algorithm	586
10.6.6	Reduction to Tridiagonal Form	589
10.6.7	Convergence Criterion for Eigenvalues	592
10.6.8	Loss of Orthogonality	595
10.6.9	Restoring Orthogonality	598
	References	601
11	Dlearn--A Linear Static and Dynamic Finite Element Analysis Program	603
11.1	Introduction	603
11.2	Description of Coding Techniques Used in DLEARN	604
11.2.1	Compacted Column Storage Scheme	605
11.2.2	Crout Elimination	608
11.2.3	Dynamic Storage Allocation	616
11.3	Program Structure	622
11.3.1	Global Control	623
11.3.2	Initialization Phase	623
11.3.3	Solution Phase	625
11.4	Adding an Element to DLEARN	631
11.5	DLEARN User's Manual	634
11.5.1	Remarks for the New User	634
11.5.2	Input Instructions	635
11.5.3	Examples	663
1.	Planar Truss	663
2.	Static Analysis of a Plane Strain Cantilever Beam	666
3.	Dynamic Analysis of a Plane Strain Cantilever Beam	666
4.	Implicit-explicit Dynamic Analysis of a Rod	668
11.5.4	Subroutine Index for Program Listing	670
	References	675
	Index	676