Numerical Methods and Software Tools in Industrial Mathematics

Front Cover
Springer Science & Business Media, 1997 - Computers - 400 pages
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 301 N. P. Fremming, @. Hjelle, C. Tarrou 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 317 G. Misund 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 329 J. G. Bjaalie, M. Dtllhlen, T. V. Stensby 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 347 E. Arge, M. Dtllhlen 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 365 T. Dokken 18. 1 Introduction . . . . . . . . . . . . . . . 365 18. 2 Basic Concepts Used . . . . . . . . . .
 

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Contents

ObjectOriented Numerics
7
12 A Motivating Example
9
13 A Simple Matrix Class
11
14 Flexible User Interfaces
18
15 Abstraction and TopDown Design
20
16 Concluding Remarks
24
Basic Tools for Linear Algebra
27
22 The Basic Building Blocks
28
Finite Element Methods for TwoPhase Flow in Heterogeneous Porous Media
213
102 Mathematical Model
215
103 Numerical Methods
217
104 Onedimensional Flow
221
105 Twodimensional Flow
223
106 Summary and Conclusion
231
Splines and Ocean Wave Modelling
235
112 Spline Functions Summary
236

23 Representation of Linear Systems
36
24 Solving Linear Systems
39
25 Concluding Remarks
43
Software Tools for Modelling Scattered Data
45
32 Scattered Data Characteristics
46
33 Software Requirements
49
34 Elements of Software Design
51
35 Concluding Remarks
58
A Comprehensive Set of Tools for Solving Partial Differential Equations Diffpack
61
42 Current Applications
63
43 The Need for Abstractions
67
44 Overview
68
45 General Representation of Fields
73
46 Some Details on Finite Element Methods
76
47 An Example
79
48 Code Extension Based on OOP
82
49 Controllers
84
410 Combining Simulators
85
411 Conclusions
87
On the Numerical Efficiency of C++ in Scientific Computing
91
51 Introduction
92
52 Optimizing C++ for Numerical Applications
93
53 Lowlevel Linear Algebra Operations
100
54 Finite Element Applications
106
55 Concluding Remarks
114
PARTIAL DIFFERENTIAL EQUATIONS
119
Basic Equations in Eulerian Continuum Mechanics
121
62 Mass Momentum and Energy Balance
123
63 Constitutive Laws
128
64 Boundary Conditions
135
65 Initial and Boundary Value Problems
136
66 Numerical Solution Procedures
145
67 Mixtures
150
68 Concluding Remarks
155
A Mathematical Model of Macrosegregation Formation in Binary Alloy Solidification
157
72 Microscopic Conservation Equations
158
73 Fundamentals of Volume Averaging
160
74 Macroscopic Conservation Equations
163
75 Macroscopic Constitutive Relations
164
76 A Mixture Formulation of the Governing Equations
167
77 Transformation to Dimensionless Form
169
79 Concluding Remarks
174
Computation of Macrosegregation due to Solidification Shrinkage
177
82 The Mathematical Model
178
83 The Finite Element Method
181
84 Implementation of the Finite Element Method in Diffpack
185
85 Results and Discussion
190
86 Conclusion
193
A Mathematical Model for the Melt Spinning of Polymer Fibers
195
92 Model Description
196
93 Additional Constitutive Relations
204
94 Numerical Methods
206
95 A Case Study
208
96 Summary
210
a Worked Example
237
a Worked Example
240
115 Splines in Water Wave Equations
245
A Worked Example
248
Krylov Subspace Iterations for Sparse Linear Systems
255
122 Krylov Subspace Methods
257
123 Preconditioning Techniques
270
124 Concluding Remarks
276
Preconditioning Linear Saddle Point Problems
281
132 Abstract Saddle Point Problems
282
133 Preconditioned Iterative Methods
283
134 Examples of Saddle Point Problems
286
135 Discretizations of Saddle Point Problems
290
136 Numerical Results
295
GEOMETRIC MODELLING
299
Surface Modelling from Scattered Geological Data
301
142 Description of Geological Data
302
143 Triangulations
304
144 Regular Grid Models
306
145 A Composite Surface Model
307
146 Examples
312
147 Concluding Remarks
314
Varioscale Surfaces in Geographic Information Systems
317
152 Surfaces of Variable Resolution
318
153 Surface Varioscaling by Normalization
320
154 Examples
323
155 Final Remarks
327
Surface Modelling from Biomedical
329
161 Boundary Polygons
332
162 Curve Approximation
333
163 Reducing Twist in the Closed Surface
336
164 Surface Approximation
337
165 Open Surfaces
339
166 Examples
340
167 Concluding Remarks
344
Data Reduction of Piecewise Linear Curves
347
172 Preliminaries
349
173 The Intersecting Cones Method
351
174 The Improved Douglas Method
353
175 Numerical Examples
360
176 Resolution Sorting
361
Aspects of Algorithms for Manifold Intersection
365
182 Basic Concepts Used
367
183 Geometric Tolerance and Intersection
369
185 Finding all Intersection Occurrences
372
186 Loop Elimination
377
Surface Editing
381
192 Requirements for Surface Editing
382
194 Surface Modification
384
195 Boundary Conditions of the Surface
389
197 Examples
391
198 Conclusion
394
Index
397
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