Numerical methods and software tools in industrial mathematics

목차 (Table of Contents)

• CONTENTS
• Ⅰ NUMERICAL SOFTWARE TOOLS = 5
• 1 Object-Oriented Numerics / E. Arge ; A. M. Bruaset ; H. P. Langtangen = 7
• 1.1 Introduction = 7
• 1.2 A Motivating Example = 9
• 1.3 A Simple Matrix Class = 11
• 1.4 Flexible User Interfaces = 18
• 1.5 Abstraction and Top-Down Design = 20
• 1.6 Concluding Remarks = 24
• 2 Basic Tools for Linear Algebra / A. M. Bruaset ; H. P. Langtangen = 27
• 2.1 Introduction = 27
• 2.2 The Basic Building Blocks = 28
• 2.3 Representation of Linear Systems = 36
• 2.4 Solving Linear Systems = 39
• 2.5 Concluding Remarks = 43
• 3 Software Tools for Modelling Scattered Data / E. Arge ; Ø. Hjelle = 45
• 3.1 Introduction = 45
• 3.2 Scattered Data Characteristics = 46
• 3.3 Software Requirements = 49
• 3.4 Elements of Software Design = 51
• 3.5 Concluding Remarks = 58
• 4 A Comprehensive Set of Tools for Solving Partial Differential Equations ; Diffpack / A. M. Bruaset ; H. P. Langtangen = 61
• 4.1 Introduction = 61
• 4.2 Current Applications = 63
• 4.3 The Need for Abstractions = 67
• 4.4 Overview = 68
• 4.5 General Representation of Fields = 73
• 4.6 Some Details on Finite Element Methods = 76
• 4.7 An Example = 79
• 4.8 Code Extension Based on OOP = 82
• 4.9 Controllers = 84
• 4.10 Combining Simulators = 85
• 4.11 Conclusions = 87
• 5 On the Numerical Efficiency of C＋＋ in Scientific Computing / E. Arge et al. = 91
• 5.1 Introduction = 92
• 5.2 Optimizing C＋＋ for Numerical Applications = 93
• 5.3 Low-level Linear Algebra Operations = 100
• 5.4 Finite Element Applications = 106
• 5.5 Concluding Remarks = 114
• Ⅱ PARTIAL DIFFERENTIAL EQUATIONS = 119
• 6 Basic Equations in Eulerian Continuum Mechanics / E. Haug ; H. P. Langtangen = 121
• 6.1 Introduction = 121
• 6.2 Mass, Momentum and Energy Balance = 123
• 6.3 Constitutive Laws = 128
• 6.4 Boundary Conditions = 135
• 6.5 Initial and Boundary Value Problems = 136
• 6.6 Numerical Solution Procedures = 145
• 6.7 Mixtures = 150
• 6.8 Concluding Remarks = 155
• References = 155
• 7 A Mathematical Model of Macrosegregation Formation in Binary Alloy Solidification / E. Haug ; T. Rusten ; H. J. Thevik = 157
• 7.1 Introduction = 157
• 7.2 Microscopic Conservation Equations = 158
• 7.3 Fundamentals of Volume Averaging = 160
• 7.4 Macroscopic Conservation Equations = 163
• 7.5 Macroscopic Constitutive Relations = 164
• 7.6 A Mixture Formulation of the Governing Equations = 167
• 7.7 Transformation to Dimensionless Form = 169
• 7.8 Dimensionless Mixture Equations = 172
• 7.9 Concluding Remarks = 174
• 8 Computation of Macrosegregation due to Solidification Shrinkage / A. Mo ; T. Rusten ; H. J. Thevik = 177
• 8.1 Introduction = 177
• 8.2 The Mathematical Model = 178
• 8.3 The Finite Element Method = 181
• 8.4 Implementation of the Finite Element Method in Diffpack = 185
• 8.5 Results and Discussion = 190
• 8.6 Conclusion = 193
• 9 A Mathematical Model for the Melt Spinning of Polymer Fibers / E. Andreassen et al. = 195
• 9.1 Introduction = 195
• 9.2 Model Description = 196
• 9.3 Additional Constitutive Relations = 204
• 9.4 Numerical Methods = 206
• 9.5 A Case Study = 208
• 9.6 Summary = 210
• 10 Finite Element Methods for Two-Phase Flow in Heterogeneous Porous Media / E. Gundersen ; H. P. Langtangen = 213
• 10.1 Introduction = 213
• 10.2 Mathematical Model = 215
• 10.3 Numerical Methods = 217
• 10.4 One-dimensional Flow = 221
• 10.5 Two-dimensional Flow = 223
• 10.6 Summary and Conclusion = 231
• 11 Splines and Ocean Wave Modelling / E. Mehlum = 235
• 11.1 Introduction and Background = 235
• 11.2 Spline Functions Summary = 236
• 11.3 Splines and Ordinary Differential Equations : a Worked Example = 237
• 11.4 Splines and Partial Differential Equations : a Worked Example = 240
• 11.5 Splines in Water Wave Equation = 245
• 11.6 Nonlinear Waves in a Box : A Worked Example = 248
• 12 Krylov Subspace Iterations for Sparse Linear Systems / A. M. Bruaset = 255
• 12.1 Introduction = 255
• 12.2 Krylov Subspace Methods = 257
• 12.3 Preconditioning Techniques = 270
• 12.4 Concluding Remarks = 276
• 13 Preconditioning Linear Saddle Point Problems / T. Rusten ; R. Winther = 281
• 13.1 Introduction = 281
• 13.2 Abstract Saddle Point Problems = 282
• 13.3 Preconditioned Iterative Methods = 283
• 13.4 Examples of Saddle Point Problems = 286
• 13.5 Discretizations of Saddle Point Problems = 290
• 13.6 Numerical Results = 295
• Ⅲ GEOMETRIC MODELLING = 299
• 14 Surface Modelling from Scattered Geological Data / N. P. Fremming ; Ø. Hjelle ; C. Tarrou = 301
• 14.1 Introduction = 301
• 14.2 Description of Geological Data = 302
• 14.3 Triangulations = 304
• 14.4 Regular Grid Models = 306
• 14.5 A Composite Surface Model = 307
• 14.6 Examples = 312
• 14.7 Concluding Remarks = 314
• 15 Varioscale Surfaces in Geographic Information Systems / G. Misund = 317
• 15.1 Introduction = 317
• 15.2 Surfaces of Variable Resolution = 318
• 15.3 Surface Varioscaling by Normalization = 320
• 15.4 Examples = 323
• 15.5 Final Remarks = 327
• 16 Surface Modelling from Biomedical Data / J. G. Bjaalie ; M. Dœhlen ; T. V. Stensby = 329
• 16.1 Boundary Polygons = 332
• 16.2 Curve Approximation = 333
• 16.3 Reducing Twist in the Closed Surface = 336
• 16.4 Surface Approximation = 337
• 16.5 Open Surfaces = 339
• 16.6 Examples = 340
• 16.7 Concluding Remarks = 344
• 17 Data Reduction of Piecewise Linear Curves / E. Arge ; M. Dœhlen = 347
• 17.1 Introduction = 347
• 17.2 Preliminaries = 349
• 17.3 The Intersecting Cones Method = 351
• 17.4 The Improved Douglas Method = 353
• 17.5 Numerical Examples = 360
• 17.6 Resolution Sorting = 361
• 18 Aspects of Algorithms for Manifold Intersection / T. Dokken = 365
• 18.1 Introduction = 365
• 18.2 Basic Concepts Used = 367
• 18.3 Geometric Tolerance and Intersection = 369
• 18.4 Representation of the - Intersection = 369
• 18.5 Finding all Intersection Occurrences = 372
• 18.6 Loop Elimination = 377
• 19 Surface Editing / E. Mehlum ; V. Skytt = 381
• 19.1 Introduction = 381
• 19.2 Requirements for Surface Editing = 382
• 19.3 Surface Interrogation = 382
• 19.4 Surface Modification = 384
• 19.5 Boundary Conditions of the Surface = 389
• 19.6 Introducing New Degrees of Freedom = 389
• 19.7 Examples = 391
• 19.8 Conclusion = 394
• Index = 397