The finite element method

목차 (Table of Contents)

• [Volume. 2]----------
• CONTENTS
• Preface to Volume 2 = xiii
• 1. General problems in solid mechanics and non-linearity = 1
• 1.1 Introduction = 1
• 1.2 Small deformation non-linear solid mechanics problems = 3
• 1.3 Non-linear quasi-harmonic field problems = 12
• 1.4 Some typical examples of transient non-linear calculations = 14
• 1.5 Concluding remarks = 20
• References = 20
• 2. Solution of non-linear algebraic equations = 22
• 2.1 Introduction = 22
• 2.2 Iterative techniques = 23
• References = 36
• 3. Inelastic and non-linear materials = 38
• 3.1 Introduction = 38
• 3.2 Viscoelasticity - history dependence of deformation = 39
• 3.3 Classical time-independent plasticity theory = 48
• 3.4 Computation of stress increments = 56
• 3.5 isotropic plasticity models = 61
• 3.6 Generalized plasticity - non-associative case = 68
• 3.7 Some examples of plastic computation = 71
• 3.8 Basic formulation of creep problems = 75
• 3.9 Viscoplasticity - a generalization = 78
• 3.10 Sonic special problems of brittle materials = 84
• 3.11 Non-uniqueness and localization in elasto-plastic deformations = 88
• 3.12 Adaptive refinement and localization(slip-line) capture = 93
• 3.13 Non-linear quasi-harmonic field problems = 101
• References = 104
• 4. Plate bending approximation : thin(Kirchhoff) plates and C₁ continuity requirements = 111
• 4.1 Introduction = 111
• 4.2 The plate problem : thick and thin formulations = 113
• 4.3 Rectangular element with corner nodes(12 degrees of freedom) = 124
• 4.4 Quadrilateral and parallelogram elements = 128
• 4.5 Triangular element with corner nodes(9 degrees of freedom) = 128
• 4.6 Triangular element of the simplest form(6 degrees of freedom) = 133
• 4.7 The patch test - an analytical requirement = 134
• 4.8 Numerical examples = 138
• 4.9 General remarks = 145
• 4.10 Singular shape functions for the simple triangular element = 145
• 4.11 An 18degree-of-freedom triangular element with conforming shape functions = 148
• 4.12 Compatible quadrilateral elements = 149
• 4.13 Quasi-conforming elements = 150
• 4.14 Hermitian rectangle shape function = 151
• 4.15 The 21 and 18 degree-of-freedom triangle = 153
• 4.16 Mixed formulations - general remarks = 155
• 4.17 Hybrid plate elements = 157
• 4.18 Discrete Kirchhoff constraints = 158
• 4.19 Rotation-free elements = 162
• 4.20 Inelastic material behaviour = 164
• 4.21 Concluding remarks - which elements? = 166
• References = 167
• 5. 'Thick' Reissner-Mindlin plates - irreducible and mixed formulations = 173
• 5.1 Introduction = 173
• 5.2 The irreducible formulation - reduced integration = 176
• 5.3 Mixed formulation for thick plates = 180
• 5.4 The patch test for plate bending elements = 183
• 5.5 Elements with discrete collocation constraints = 187
• 5.6 Elements with rotational bubble or enhanced modes = 196
• 5.7 Linked interpolation - an improvement of accuracy = 199
• 5.8 Discrete 'exact' thin plate limit = 202
• 5.9 Performance of various 'thick' plate elements - limitations of thin plate theory = 203
• 5.10 Forms without rotation parameters = 208
• 5.11 Inelastic material behaviour = 210
• 5.12 Concluding remarks - adaptive refinement = 211
• References = 212
• 6. Shells as an assembly of flat elements = 216
• 6.1 Introduction = 216
• 6.2 Stiffness of a plane element in local coordinates = 218
• 6.3 Transformation to global coordinates and assembly of elements = 219
• 6.4 Local direction cosines = 221
• 6.5 'Drilling' rotational stiffness - 6 degree-of-freedom assembly = 225
• 6.6 Elements with mid-side slope connections only = 230
• 6.7 Choice of element = 230
• 6.8 Practical examples = 231
• References = 240
• 7. Axisymmetric shells = 244
• 7.1 Introduction = 244
• 7.2 Straight element = 245
• 7.3 Curved elements = 251
• 7.4 Independent slope-displacement interpolation with penalty functions(thick or thin shell formulations) = 261
• References = 264
• 8. Shells as a special case of three-dimensional analysis - Reissner-Mindlin assumptions = 266
• 8.1 Introduction = 266
• 8.2 Shell element with displacement and rotation parameters = 266
• 8.3 Special case of axisymmetric, curved, thick shells = 275
• 8.4 Special case of thick plates = 277
• 8.5 Convergence = 278
• 8.6 Inelastic behaviour = 279
• 8.7 Some shell examples = 280
• 8.8 Concluding remarks = 285
• References = 286
• 9. Semi-analytical finite element processes - use of orthogonal functions and 'finite strip' methods = 289
• 9.1 Introduction = 289
• 9.2 Prismatic bar = 292
• 9.3 Thin membrane box structures = 295
• 9.4 Plates and boxes with flexure = 296
• 9.5 Axisymmetric solids with non-symmetrical load = 297
• 9.6 Axisymmetric shells with non-symmetrical load = 303
• 9.7 Finite strip method - incomplete decoupling = 305
• 9.8 Concluding remarks = 308
• References = 309
• 10. Geometrically non-linear problems - finite deformation = 312
• 10.1 Introduction = 312
• 10.2 Governing equations = 314
• 10.3 Variational description for finite deformation = 319
• 10.4 A three-field mixed finite deformation formulation = 328
• 10.5 A mixed-enhanced finite deformation formulation = 332
• 10.6 Forces dependent on deformation - pressure loads = 336
• 10.7 Material constitution for finite deformation = 338
• 10.8 Contact problems = 347
• 10.9 Numerical examples = 355
• 10.10 Concluding remarks = 359
• References = 360
• 11. Non-linear structural problems - large displacement and instability = 365
• 11.1 Introduction = 365
• 11.2 Large displacement theory of beams = 365
• 11.3 Elastic stability energy interpretation = 373
• 11.4 Large displacement theory of thick plates = 375
• 11.5 Large displacement theory of thin plates = 381
• 11.6 Solution of large deflection problems = 383
• 11.7 Shells = 386
• 11.8 Concluding remarks = 391
• References = 392
• 12. Pseudo-rigid and rigid-flexible bodies = 396
• 12.1 Introduction = 396
• 12.2 Pseudo-rigid motions = 396
• 12.3 Rigid motions = 398
• 12.4 Connecting a rigid body to a flexible body = 402
• 12.5 Multibody coupling by joints = 404
• 12.6 Numerical examples = 409
• References = 410
• 13. Computer procedures for finite element analysis = 413
• 13.1 introduction = 413
• 13.2 Description of additional program features = 414
• 13.3 Solution of non-linear problems = 415
• 13.4 Restart option = 428
• 13.5 Solution of example problems = 429
• 13.6 Concluding remarks = 430
• References = 430
• Appendix A : Invariants of second-order tensors = 432
• A.1 Principal invariants = 432
• A.2 Moment invariants = 433
• A.3 Derivatives of invariants = 434
• Author index = 437
• Subject index = 445
• [Volume. 3]----------
• CONTENTS
• Preface to Volume 3 = xiii
• 1 Introduction and the equations of fluid dynamics = 1
• 1.1 General remarks and classification of fluid mechanics problems discussed in this book = 1
• 1.2 The governing equations of fluid dynamics = 4
• 1.3 Incompressible(or nearly incompressible) flows = 10
• 1.4 Concluding remarks = 12
• References = 12
• 2 Convection dominated problems - finite element approximations to the convection-diffusion equation = 13
• 2.1 Introduction = 13
• 2.2 The steady-state problem in one dimension = 15
• 2.3 The steady-state problem in two(or three) dimensions = 26
• 2.4 Steady state - concluding remarks = 30
• 2.5 Transients - introductory remarks = 32
• 2.6 Characteristic-based methods = 35
• 2.7 Taylor-Galerkin procedures for scalar variables = 47
• 2.8 Steady-state condition = 48
• 2.9 Non-linear waves and shocks = 48
• 2.10 Vector-valued variables = 52
• 2.11 Summary and concluding remarks = 59
• References = 59
• 3 A general algorithm for compressible and incompressible flows - the characteristic-based split(CBS) algorithm = 64
• 3.1 Introduction = 64
• 3.2 Characteristic-based split(CBS) algorithm = 67
• 3.3 Explicit, semi-implicit and nearly implicit forms = 76
• 3.4 'Circumventing' the Babu$${\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}}\over s}}$$ka-Brezzi(BB) restrictions = 7
• 3.5 A single-step version = 80
• 3.6 Boundary conditions = 81
• 3.7 The performance of two- and single-step algorithms on an inviscid problem = 85
• 3.8 Concluding remarks = 87
• References = 87
• 4 Incompressible laminar flow - newtonian and non-newtonian fluids = 91
• 4.1 Introduction and the basic equations = 91
• 4.2 Inviscid, incompressible flow(potential flow) = 93
• 4.3 Use of the CBS algorithm for incompressible or nearly incompressible flows = 97
• 4.4 Boundary-exit conditions = 100
• 4.5 Adaptive mesh refinement = 102
• 4.6 Adaptive mesh generation for transient problems = 113
• 4.7 Importance of stabilizing convective terms = 113
• 4.8 Slow flows - mixed and penalty formulations = 113
• 4.9 Non-newtonian flows - metal and polymer forming = 118
• 4.10 Direct displacement approach to transient metal forming = 132
• 4.11 Concluding remarks = 133
• References = 134
• 5 Free surfaces, buoyancy and turbulent incompressible flows = 143
• 5.1 Introduction = 143
• 5.2 Free surface flows = 144
• 5.3 Buoyancy driven flows = 153
• 5.4 Turbulent flows = 161
• References = 165
• 6 Compressible high-speed gas flow = 169
• 6.1 Introduction = 169
• 6.2 The governing equations = 170
• 6.3 Boundary conditions - subsonic and supersonic flow = 171
• 6.4 Numerical approximations and the CBS algorithm = 173
• 6.5 Shock capture = 174
• 6.6 Some preliminary examples for the Euler equation = 176
• 6.7 Adaptive refinement and shock capture in Euler problems = 180
• 6.8 Three-dimensional inviscid examples in steady state = 188
• 6.9 Transient two and three-dimensional problems = 195
• 6.10 Viscous problems in two dimensions = 197
• 6.11 Three-dimensional viscous problems = 207
• 6.12 Boundary layer-inviscid Euler solution coupling = 209
• 6.13 Concluding remarks = 212
• References = 212
• 7 Shallow-water problems = 218
• 7.1 Introduction = 218
• 7.2 The basis of the shallow-water equations = 219
• 7.3 Numerical approximation = 223
• 7.4 Examples of application = 224
• 7.5 Drying areas = 236
• 7.6 Shallow-water transport = 237
• References = 239
• 8 Waves = 242
• 8.1 Introduction and equations = 242
• 8.2 Waves in closed domains - finite element models = 243
• 8.3 Difficulties in modelling surface waves = 245
• 8.4 Bed friction and other effects = 245
• 8.5 The short-wave problem = 245
• 8.6 Waves in unbounded domains(exterior surface wave problems) = 250
• 8.7 Unbounded problems = 253
• 8.8 Boundary dampers = 253
• 8.9 Linking to exterior solutions = 255
• 8.10 Infinite elements = 259
• 8.11 Mapped periodic infinite elements = 260
• 8.12 Ellipsoidal type infinite elements of Burnett and Holford = 261
• 8.13 Wave envelope infinite elements = 262
• 8.14 Accuracy of infinite elements = 264
• 8.15 Transient problems = 265
• 8.16 Three-dimensional effects in surface waves = 266
• References = 270
• 9 Computer implementation of the CBS algorithm = 274
• 9.1 Introduction = 274
• 9.2 The data input module = 275
• 9.3 Solution module = 278
• 9.4 Output module = 289
• 9.5 Possible extensions to CBSflow = 289
• References = 289
• Appendix A : Non-conservative form of Navier-Stokes equations = 291
• Appendix B : Discontinuous Galerkin methods in the solution of the convection-diffusion equation = 293
• Appendix C : Edge-based finite element formulation = 298
• Appendix D : Multigrid methods = 300
• Appendix E : Boundary layer-inviscid flow coupling = 302
• Author index = 307
• Subject index = 315