The finite element method for engineers

목차 (Table of Contents)

• CONTENTS
• PART Ⅰ
• 1 Meet the Finite Element Method = 3
• 1.1 What Is the Finite Element Method? = 3
• 1.2 How the Method Works = 5
• 1.3 A Brief History of the Method = 8
• 1.4 Range of Applications = 11
• 1.5 The Future of the Finite Element Method = 12
• References = 14
• 2 The Direct Approach : A Physical Interpretation = 15
• 2.1 Introduction = 15
• 2.2 Defining Elements and Their Properties = 16
• 2.2.1 Linear Spring Systems = 16
• 2.2.2 Flow Systems = 19
• 2.2.3 Simple Elements From Structural Mechanics = 23
• 2.2.4 Coordinate Transformations = 35
• 2.3 Assembling the Parts = 38
• 2.3.1 Assembly Rules Derived From an Example = 38
• 2.3.2 General Assembly Procedure = 44
• 2.3.3 Features of the Assembled Matrix = 45
• 2.3.4 Introducing Boundary Conditions = 46
• 2.4 Closure = 53
• References = 54
• Problems = 55
• 3 The Mathematical Approach : A Variational Interpretation = 65
• 3.1 Introduction = 65
• 3.2 Continuum Problems = 66
• 3.2.1 Introduction = 66
• 3.2.2 Problem Statement = 67
• 3.2.3 Classification of Differential Equations = 69
• 3.3 Some Methods for Solving Continuum Problems = 70
• 3.3.1 An Overview = 70
• 3.3.2 The Variational Approach = 71
• 3.3.3 The Ritz Method = 72
• 3.4 The Finite Element Method = 77
• 3.4.1 Relation to the Ritz Method = 77
• 3.4.2 Generalizing the Definition of an Element = 77
• 3.4.3 Example of a Piecewise Approximation = 78
• 3.4.4 Element Equations From a Variational Principle = 82
• 3.4.5 Requirements for Interpolation Functions = 84
• 3.4.6 Domain Discretization = 90
• 3.4.7 Example of a Complete Finite Element Solution = 92
• 3.5 Closure = 98
• References = 98
• Problems = 99
• 4 The Mathematical Approach : A Generalized Interpretation = 103
• 4.1 Introduction = 103
• 4.2 Deriving Finite Element Equations from the Method of Weighted Residuals = 104
• 4.2.1 Example : One-Dimensional Poisson Equation = 108
• 4.2.2 Example : Two-Dimensional Heat Conduction = 111
• 4.2.3 Example : Time-Dependent Heat Conduction = 115
• 4.3 Closure = 116
• References = 117
• Problems = 117
• 5 Elements and Interpolation Functions = 121
• 5.1 Introduction = 122
• 5.2 Basic Element Shapes = 123
• 5.3 Terminology and Preliminary Considerations = 128
• 5.3.1 Types of Nodes = 128
• 5.3.2 Degrees of Freedom = 128
• 5.3.3 Interpolation Functions - Polynomials = 128
• 5.4 Generalized Coordinates and the Order of the Polynomial = 131
• 5.4.1 Generalized Coordinates = 131
• 5.4.2 Geometric Isotropy = 131
• 5.4.3 Deriving Interpolation Functions = 133
• 5.5 Natural Coordinates = 135
• 5.5.1 Natural Coordinates in One Dimension = 136
• 5.5.2 Natural Coordinates in Two Dimensions = 137
• 5.5.3 Natural Coordinates in Three Dimensions = 141
• 5.6 Interpolation Concepts in One Dimension = 145
• 5.6.1 Lagrange Polynomials = 145
• 5.6.2 Hermite Polynomials = 147
• 5.7 Internal Nodes - Condensation／ Substructuring = 149
• 5.8 Two-Dimensional Elements = 153
• 5.8.1 Elements for $$C^0$$ Problems = 153
• 5.8.2 Elements for $$C^1$$ Problems = 162
• 5.9 Three-Dimensional Elements = 168
• 5.9.1 Elements for $$C^0$$ Problems = 168
• 5.9.2 Elements for $$C^1$$ Problems = 173
• 5.10 Curved Isoparametric Elements for $$C^0$$ Problems = 173
• 5.10.1 Coordinate Transformation = 173
• 5.10.2 Evaluation of Element Matrices = 176
• 5.10.3 Example of Isoparametric Element Matrix Evaluation = 179
• 5.11 Closure = 180
• References = 180
• Problems = 182
• PART Ⅱ
• 6 Elasticity Problems = 189
• 6.1 Introduction = 190
• 6.2 General Formulation for Three-Dimensional Problems = 190
• 6.2.1 Problem Statement = 190
• 6.2.2 The Variational Method = 190
• 6.2.3 The Galerkin Method = 198
• 6.2.4 The System Equations = 201
• 6.3 Application to Plane Stress and Plane Strain = 203
• 6.3.1 Displacement Model for a Triangular Element = 203
• 6.3.2 Element Stiffness Matrix for a Triangle = 205
• 6.3.3 Element Force Vectors for a Triangle = 208
• 6.4 Application to Axisymmetric Stress Analysis = 210
• 6.4.1 Displacement Model for Triangular Toroid = 211
• 6.4.2 Element Stiffness Matrix for Triangular Toroid = 212
• 6.4.3 Element Force Vectors for Triangular Toroid = 215
• 6.5 Application to Plate-Bending Problems = 220
• 6.5.1 Requirements for the Displacement Interpolation Functions = 222
• 6.5.2 Rectangular Plate-Bending Elements = 223
• 6.6 Three-Dimensional Problems = 226
• 6.6.1 Introduction = 226
• 6.6.2 Formulation for the Linear Tetrahedral Element = 227
• 6.6.3 Higher-Order Elements = 228
• 6.7 Introduction to Structural Dynamics = 229
• 6.7.1 Formulation of Equations = 229
• 6.7.2 Free Undamped Vibrations = 232
• 6.7.3 Finding Transient Motion via Mode Superposition = 234
• 6.7.4 Finding Transient Motion via Recurrence Relations = 237
• 6.8 Closure = 243
• References = 244
• Problems = 246
• 7 General Field Problems = 254
• 7.1 Introduction = 254
• 7.2 Equilibrium Problems = 255
• 7.2.1 Quasiharmonic Equations = 255
• 7.2.2 Boundary Conditions = 256
• 7.2.3 Variational Principle = 258
• 7.2.4 Element Equations = 259
• 7.2.5 Element Equations in Two Dimensions = 261
• 7.3 Eigenvalue Problems = 268
• 7.3.1 Helmholtz Equations = 268
• 7.3.2 Variational Principle = 270
• 7.3.3 Element Equations = 270
• 7.3.4 Examples = 271
• 7.3.5 Sample Problem = 273
• 7.4 Propagation Problems = 276
• 7.4.1 General Time-Dependent Field Problems = 276
• 7.4.2 Finite Element Equations = 281
• 7.4.3 Element Equations in One Space Dimension = 282
• 7.5 Solving the Discretized Time-Dependent Equations = 284
• 7.5.1 Solution Methods for First-Order Equations = 285
• 7.5.2 Finding Transient Response via Mode Superposition = 286
• 7.5.3 Finding Transient Response via Recurrence Relations = 288
• 7.5.4 Oscillation and Stability of Transient Response = 291
• 7.5.5 Algorithm Order = 295
• 7.5.6 Sample Problem = 296
• 7.6 Closure = 299
• References = 300
• Problems = 301
• 8 Heat Transfer Problems = 313
• 8.1 Introduction = 313
• 8.2 Conduction = 314
• 8.2.1 Problem Statement = 315
• 8.2.2 Finite Element Formulation = 317
• 8.2.3 Element Equations = 322
• 8.2.4 Linear Steady-State and Transient Solutions = 330
• 8.2.5 Nonlinear Steady-State Solutions = 336
• 8.2.6 Nonlinear Transient Solutions = 340
• 8.3 Conduction With Surface Radiation = 344
• 8.3.1 Problem Statement = 345
• 8.3.2 Element Equations With Radiation = 347
• 8.3.3 Steady-State Solutions = 349
• 8.3.4 Transient Solutions = 351
• 8.4 Convective-Diffusion Equation = 355
• 8.4.1 Problem Statement = 356
• 8.4.2 Finite Element Formulation = 357
• 8.4.3 One-Dimensional Problem = 358
• 8.4.4 Two-Dimensional Solutions = 361
• 8.5 Free and Forced Convection = 363
• 8.5.1 Problem Statement = 363
• 8.5.2 Finite Element Formulation = 364
• 8.5.3 Solution Techniques = 366
• 8.5.4 Free Convection Example = 367
• 8.5.5 Forced Convection = 368
• 8.6 Closure = 368
• References = 371
• Problems = 375
• 9 Fluid Mechanics Problems = 387
• 9.1 Introduction = 387
• 9.2 Inviscid Incompressible Flow = 388
• 9.2.1 Problem Statement = 389
• 9.2.2 Finite Element Formulation = 390
• 9.2.3 Velocity Component Smoothing = 393
• 9.2.4 Example With Unstructured Mesh = 394
• 9.2.5 The Kutta Condition = 398
• 9.3 Viscous Incompressible Flow Without Inertia = 399
• 9.3.1 Problem Statement = 399
• 9.3.2 Stream Function Formulation = 401
• 9.3.3 Velocity and Pressure Formulation = 402
• 9.4 Viscous Incompressible Flow With Inertia = 406
• 9.4.1 Mixed Velocity and Pressure Formulation = 407
• 9.4.2 Penalty Function Formulation = 413
• 9.4.3 Equal-Order Velocity and Pressure Formulation = 415
• 9.5 Compressible Flow = 423
• 9.5.1 Problem Statement = 424
• 9.5.2 Low-Speed Flow With Variable Density = 426
• 9.5.3 High-Speed Flow = 433
• 9.6 Closure = 444
• References = 444
• Problems = 450
• 10 A Sample Computer Code and Other Practical Considerations = 459
• 10.1 Introduction = 460
• 10.2 Setting up a Simple Heat Conduction Problem = 461
• 10.2.1 Problem Formulation = 461
• 10.2.2 The Finite Element Equations = 463
• 10.2.3 The Overall Program Logic = 464
• 10.3 The Computer Program and Its Explanation = 465
• 10.3.1 Structure of the Program = 465
• 10.3.2 Preparation of the Finite Element Model = 468
• 10.3.3 Preparation of Input Data = 470
• 10.3.4 Description of Output = 472
• 10.3.5 FORTRAN Listing = 472
• 10.4 Example Problems With Input and Output = 487
• 10.4.1 Example 1 : Axisymmetric Heat Flow in an Insulated Hollow Cylinder = 487
• 10.4.2 Example 2 : Heat Transfer with Steep Local Gradients = 488
• 10.5 Mesh Generation = 504
• 10.6 Adaptive Mesh Refinement = 507
• 10.6.1 Mesh Refinement Methods = 507
• 10.6.2 Error Indicators = 509
• 10.6.3 Adaptive Remeshing = 514
• 10.7 Numerical Integration Formulas = 517
• 10.7.1 Newton-Cotes = 518
• 10.7.2 Legendre-Gauss = 521
• 10.8 Solving Algebraic Equations = 521
• 10.8.1 Linear Matrix Equations = 524
• 10.8.2 Nonlinear Matrix Equations = 528
• 10.9 Closure = 530
• References = 530
• Problems = 534
• Appendix A Matrices = 540
• A.1 Definitions = 541
• A.2 Special Types of Square Matrices = 542
• A.3 Matrix Operations = 542
• A.4 Special Matrix Products = 544
• A.4.1 Product of a Square Matrix and a Column Matrix = 544
• A.4.2 Product of a Row Matrix and a Square Matrix = 544
• A.4.3 Product of a Row Matrix and a Column Matrix = 545
• A.4.4 Product of the Identity Matrix and Any Other Matrix = 545
• A.5 Matrix Transpose = 545
• A.6 Quadratic Forms = 545
• A.7 Matrix Inverse = 547
• A.8 Matrix Partitioning = 547
• A.9 The Calculus of Matrices = 549
• A.9.1 Differentiation of a Matrix = 549
• A.9.2 Integration of a Matrix = 549
• A.9.3 Differentiation of a Quadratic Functional = 549
• A.10 Norms = 550
• Appendix B Variational Calculus = 552
• B.1 Introduction = 552
• B.2 Calculus - The Minima of a Function = 552
• B.2.1 Definitions = 552
• B.2.2 Functions of One Variable = 553
• B.2.3 Functions of Two or More Variables = 554
• B.3 Variational Calculus - The Minima of Functionals = 554
• B.3.1 Definitions = 554
• B.3.2 Functionals of One Variable = 555
• B.3.3 More General Functionals = 559
• References = 560
• Appendix C Basic Equations from Linear Elasticity Theory = 561
• C.1 Introduction = 561
• C.2 Stress Components = 562
• C.3 Strain Components = 563
• C.4 Generalized Hooke's Law (Constitutive Equations) = 564
• C.5 Static Equilibrium Equations = 566
• C.6 Compatibility Conditions = 567
• C.7 Differential Equations for Displacements = 569
• C.8 Minimum Potential Energy Principle = 569
• C.9 Plane Strain and Plane Stress = 572
• C.10 Thermal Effects = 574
• C.11 Thin-Plate Bending = 575
• References = 577
• Appendix D Basic Equations from Fluid Mechanics = 578
• D.1 Introduction = 578
• D.2 Definitions and Concepts = 579
• D.3 Laws of Motion = 581
• D.3.1 Differential Continuity Equation = 581
• D.3.2 Differential Momentum Equation (Navier-Stokes Equations) = 582
• D.3.3 Thermal Energy Equation = 584
• D.3.4 Conservation Form of Equations = 584
• D.3.5 Supplementary Equations = 585
• D.3.6 Problem Statement = 586
• D.4 Stream Functions and Vorticity = 587
• D.5 Potential Flow = 589
• D.6 Viscous Incompressible Flow = 590
• D.6.1 Primitive Variable Formulation = 590
• D.6.2 Vorticity and Stream Function Formulation = 591
• D.7 Boundary Layer Flow = 593
• References = 594
• Appendix E Basic Equations from Heat Transfer = 596
• E.1 Introduction = 596
• E.2 Conduction = 597
• E.2.1 Heat Conduction Equation = 598
• E.2.2 Boundary Conditions = 599
• E.2.3 Nondimensional Parameters = 601
• E.3 Convection = 601
• E.3.1 Convection Equations = 602
• E.3.2 Boundary Conditions = 603
• E.3.3 Nondimensional Parameters = 604
• E.4 Radiation = 605
• E.4.1 Surface Radiation = 606
• E.4.2 Radiation Exchange Between Surfaces = 608
• E.5 Heat Transfer Units = 612
• References = 612
• Author Index = 615
• Subject Index = 620