김경종
368 lines
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Markdown
368 lines
10 KiB
Markdown
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Klaus-Jürgen Bathe
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# Finite Element Procedures
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Second Edition
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# Finite Element Procedures
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Klaus-Jürgen Bathe
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Professor of Mechanical Engineering
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Massachusetts Institute of Technology
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Second Edition
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The author/publisher of this book has used his best efforts in preparing this book. These efforts include the development, research, and testing of the theories and educational computer programs given in the book. The author/publisher makes no warranty of any kind, expressed or implied, with regard to any text and the programs contained in this book. The author/publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of this text and these programs.
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© 2006 by Klaus-Jürgen Bathe, 1st edition
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© 2014 by Klaus-Jürgen Bathe, 2nd edition
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All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the author/publisher.
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The first edition of this book was previously published by:
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Prentice Hall, Pearson Education, Inc.
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Printed and distributed by:
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K.J. Bathe, Watertown, MA
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Printed in the United States of America
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2nd edition: fourth printing 2016
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10 9 8 7 6 5 4
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ISBN 978-0-9790049-5-7
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— Finite element analysis is an art to predict the future —
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# To my students
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... Progress in design of new structures seems to be unlimited.
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Last sentence of article: "The Use of the Electronic Computer in Structural Analysis," by K. J. Bathe (undergraduate student), published in Impact, Journal of the University of Cape Town Engineering Society, pp. 57–61, 1967.
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# Contents
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# Preface
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xiii
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# CHAPTER ONE
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# An Introduction to the Use of Finite Element Procedures 1
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1.1 Introduction 1
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1.2 Physical Problems, Mathematical Models, and the Finite Element Solution 2
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1.3 Finite Element Analysis as an Integral Part of Computer-Aided Engineering 11
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1.4 Some Recent Research Accomplishments 14
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# CHAPTER TWO
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# Vectors, Matrices, and Tensors 17
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2.1 Introduction 17
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2.2 Introduction to Matrices 18
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2.3 Vector Spaces 34
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2.4 Definition of Tensors 40
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2.5 The Symmetric Eigenproblem $\mathbf{A}\mathbf{v} = \lambda \mathbf{v}$ 51
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2.6 The Rayleigh Quotient and the Minimax Characterization of Eigenvalues 60
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2.7 Vector and Matrix Norms 66
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2.8 Exercises 72
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# CHAPTER THREE
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# Some Basic Concepts of Engineering Analysis and an Introduction to the Finite Element Method
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77
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# 3.1 Introduction 77
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# 3.2 Solution of Discrete-System Mathematical Models 78
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3.2.1 Steady-State Problems, 78
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3.2.2 Propagation Problems, 87
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3.2.3 Eigenvalue Problems, 90
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3.2.4 On the Nature of Solutions, 96
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3.2.5 Exercises, 101
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# 3.3 Solution of Continuous-System Mathematical Models 105
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3.3.1 Differential Formulation, 105
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3.3.2 Variational Formulations, 110
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3.3.3 Weighted Residual Methods; Ritz Method, 116
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3.3.4 An Overview: The Differential and Galerkin Formulations, the Principle of Virtual Displacements, and an Introduction to the Finite Element Solution, 124
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3.3.5 Finite Difference Differential and Energy Methods, 129
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3.3.6 Exercises, 138
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# 3.4 Imposition of Constraints 143
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3.4.1 An Introduction to Lagrange Multiplier and Penalty Methods, 143
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3.4.2 Exercises, 146
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# CHAPTER FOUR
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# Formulation of the Finite Element Method—Linear Analysis in Solid and Structural Mechanics
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148
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# 4.1 Introduction 148
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# 4.2 Formulation of the Displacement-Based Finite Element Method 149
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4.2.1 General Derivation of Finite Element Equilibrium Equations, 153
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4.2.2 Imposition of Displacement Boundary Conditions, 187
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4.2.3 Generalized Coordinate Models for Specific Problems, 193
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4.2.4 Lumping of Structure Properties and Loads, 212
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4.2.5 Exercises, 214
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# 4.3 Convergence of Analysis Results 225
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4.3.1 The Model Problem and a Definition of Convergence, 225
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4.3.2 Criteria for Monotonic Convergence, 229
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4.3.3 The Monotonically Convergent Finite Element Solution: A Ritz Solution, 234
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4.3.4 Properties of the Finite Element Solution, 236
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4.3.5 Rate of Convergence, 244
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4.3.6 Calculation of Stresses and the Assessment of Error, 254
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4.3.7 Exercises, 259
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# 4.4 Incompatible and Mixed Finite Element Models 261
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4.4.1 Incompatible Displacement-Based Models, 262
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4.4.2 Mixed Formulations, 268
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4.4.3 Mixed Interpolation—Displacement/Pressure Formulations for Incompressible Analysis, 276
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4.4.4 Exercises, 296
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4.5 The Inf-Sup Condition for Analysis of Incompressible Media and Structural Problems 300
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4.5.1 The Inf-Sup Condition Derived from Convergence Considerations, 301
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4.5.2 The Inf-Sup Condition Derived from the Matrix Equations, 312
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4.5.3 The Constant (Physical) Pressure Mode, 315
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4.5.4 Spurious Pressure Modes—The Case of Total Incompressibility, 316
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4.5.5 Spurious Pressure Modes—The Case of Near Incompressibility, 318
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4.5.6 The Inf-Sup Test, 322
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4.5.7 An Application to Structural Elements: The Isoparametric Beam Elements, 330
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4.5.8 Exercises, 335
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# CHAPTER FIVE
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# Formulation and Calculation of Isoparametric Finite Element Matrices 338
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5.1 Introduction 338
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5.2 Isoparametric Derivation of Bar Element Stiffness Matrix 339
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5.3 Formulation of Continuum Elements 341
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5.3.1 Quadrilateral Elements, 342
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5.3.2 Triangular Elements, 363
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5.3.3 Convergence Considerations, 376
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5.3.4 Element Matrices in Global Coordinate System, 386
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5.3.5 Displacement/Pressure Based Elements for Incompressible Media, 388
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5.3.6 Exercises, 389
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5.4 Formulation of Structural Elements 397
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5.4.1 Beam and Axisymmetric Shell Elements, 399
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5.4.2 Plate and General Shell Elements, 420
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5.4.3 Exercises, 450
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5.5 Numerical Integration 455
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5.5.1 Interpolation Using a Polynomial, 456
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5.5.2 The Newton-Cotes Formulas (One-Dimensional Integration), 457
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5.5.3 The Gauss Formulas (One-Dimensional Integration), 461
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5.5.4 Integrations in Two and Three Dimensions, 464
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5.5.5 Appropriate Order of Numerical Integration, 465
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5.5.6 Reduced and Selective Integration, 476
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5.5.7 Exercises, 478
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5.6 Computer Program Implementation of Isoparametric Finite Elements 480
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# CHAPTER SIX
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# Finite Element Nonlinear Analysis in Solid and Structural Mechanics 485
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6.1 Introduction to Nonlinear Analysis 485
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6.2 Formulation of the Continuum Mechanics Incremental Equations of Motion 497
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6.2.1 The Basic Problem, 498
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6.2.2 The Deformation Gradient, Strain, and Stress Tensors, 502
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6.2.3 Continuum Mechanics Incremental Total and Updated Lagrangian Formulations, Materially-Nonlinear-Only Analysis, 522
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6.2.4 Exercises, 529
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6.3 Displacement-Based Isoparametric Continuum Finite Elements 538
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6.3.1 Linearization of the Principle of Virtual Work with Respect to Finite Element Variables, 538
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6.3.2 General Matrix Equations of Displacement-Based Continuum Elements, 540
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6.3.3 Truss and Cable Elements, 543
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6.3.4 Two-Dimensional Axisymmetric, Plane Strain, and Plane Stress Elements, 549
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6.3.5 Three-Dimensional Solid Elements, 555
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6.3.6 Exercises, 557
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6.4 Displacement/Pressure Formulations for Large Deformations 561
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6.4.1 Total Lagrangian Formulation, 561
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6.4.2 Updated Lagrangian Formulation, 565
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6.4.3 Exercises, 566
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6.5 Structural Elements 568
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6.5.1 Beam and Axisymmetric Shell Elements, 568
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6.5.2 Plate and General Shell Elements, 575
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6.5.3 Exercises, 578
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6.6 Use of Constitutive Relations 581
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6.6.1 Elastic Material Behavior—Generalization of Hooke's Law, 583
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6.6.2 Rubberlike Material Behavior, 592
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6.6.3 Inelastic Material Behavior; Elastoplasticity, Creep, and Viscoplasticity, 595
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6.6.4 Large Strain Elastoplasticity, 612
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6.6.5 Exercises, 617
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6.7 Contact Conditions 622
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6.7.1 Continuum Mechanics Equations, 622
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6.7.2 A Solution Approach for Contact Problems: The Constraint Function Method, 626
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6.7.3 Exercises, 628
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6.8 Some Practical Considerations 628
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6.8.1 The General Approach to Nonlinear Analysis, 629
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6.8.2 Collapse and Buckling Analyses, 630
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6.8.3 The Effects of Element Distortions, 636
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6.8.4 The Effects of Order of Numerical Integration, 637
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6.8.5 Exercises, 640
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# CHAPTER SEVEN
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# Finite Element Analysis of Heat Transfer, Field Problems, and Incompressible Fluid Flows
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642
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7.1 Introduction 642
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7.2 Heat Transfer Analysis 642
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7.2.1 Governing Heat Transfer Equations, 642
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7.2.2 Incremental Equations, 646
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7.2.3 Finite Element Discretization of Heat Transfer Equations, 651
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7.2.4 Exercises, 659
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# 7.3 Analysis of Field Problems 661
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7.3.1 Seepage, 662
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7.3.2 Incompressible Inviscid Flow, 663
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7.3.3 Torsion, 664
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7.3.4 Acoustic Fluid, 666
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7.3.5 Exercises, 670
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# 7.4 Analysis of Viscous Incompressible Fluid Flows 671
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7.4.1 Continuum Mechanics Equations, 675
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7.4.2 Finite Element Governing Equations, 677
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7.4.3 High Reynolds and High Péclet Number Flows, 682
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7.4.4 Fluid-Structure Interactions, 690
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7.4.5 Exercises, 691
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# CHAPTER EIGHT
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# Solution of Equilibrium Equations in Static Analysis 695
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# 8.1 Introduction 695
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# 8.2 Direct Solutions Using Algorithms Based on Gauss Elimination 696
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8.2.1 Introduction to Gauss Elimination, 697
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8.2.2 The $\mathbf{LDL}^T$ Solution, 705
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8.2.3 Computer Implementation of Gauss Elimination—The Active Column Solution, 708
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8.2.4 Cholesky Factorization, Static Condensation, Substructures, and Frontal Solution, 717
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8.2.5 Positive Definiteness, Positive Semidefiniteness, and the Sturm Sequence Property, 726
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8.2.6 Solution Errors, 734
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8.2.7 Exercises, 741
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# 8.3 Iterative Solution Methods 745
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8.3.1 The Gauss-Seidel Method, 747
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8.3.2 Conjugate Gradient Method with Preconditioning, 749
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8.3.3 Exercises, 752
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# 8.4 Solution of Nonlinear Equations 754
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8.4.1 Newton-Raphson Schemes, 755
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8.4.2 The BFGS Method, 759
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8.4.3 Load-Displacement-Constraint Methods, 761
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8.4.4 Convergence Criteria, 764
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8.4.5 Exercises, 765
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# CHAPTER NINE
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# Solution of Equilibrium Equations in Dynamic Analysis 768
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# 9.1 Introduction 768
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# 9.2 Direct Integration Methods 769
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9.2.1 The Central Difference Method, 770
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9.2.2 The Houbolt Method, 774
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9.2.3 The Newmark Method, 777
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