Numerical Methods and Software Tools in Industrial Mathematics, 1997

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 . . . . . . . . . .

I Numerical Software Tools.- 1 Object-Oriented Numerics.- 1.1 Introduction.- 1.2 A Motivating Example.- 1.3 A Simple Matrix Class.- 1.4 Flexible User Interfaces.- 1.5 Abstraction and Top-Down Design.- 1.6 Concluding Remarks.- 2 Basic Tools for Linear Algebra.- 2.1 Introduction.- 2.2 The Basic Building Blocks.- 2.3 Representation of Linear Systems.- 2.4 Solving Linear Systems.- 2.5 Concluding Remarks.- 3 Software Tools for Modelling Scattered Data.- 3.1 Introduction.- 3.2 Scattered Data Characteristics.- 3.3 Software Requirements.- 3.4 Elements of Software Design.- 3.5 Concluding Remarks.- 4 A Comprehensive Set of Tools for Solving Partial Differential Equations; Diffpack.- 4.1 Introduction.- 4.2 Current Applications.- 4.3 The Need for Abstractions.- 4.4 Overview.- 4.5 General Representation of Fields.- 4.6 Some Details on Finite Element Methods.- 4.7 An Example.- 4.8 Code Extension Based on OOP.- 4.9 Controllers.- 4.10 Combining Simulators.- 4.11 Conclusions.- 5 On the Numerical Efficiency of C++ in Scientific Computing.- 5.1 Introduction.- 5.2 Optimizing C++ for Numerical Applications.- 5.3 Low-level Linear Algebra Operations.- 5.4 Finite Element Applications.- 5.5 Concluding Remarks.- II Partial Differential Equations.- 6 Basic Equations in Eulerian Continuum Mechanics.- 6.1 Introduction.- 6.2 Mass, Momentum and Energy Balance.- 6.3 Constitutive Laws.- 6.4 Boundary Conditions.- 6.5 Initial and Boundary Value Problems.- 6.6 Numerical Solution Procedures.- 6.7 Mixtures.- 6.8 Concluding Remarks.- References.- 7 A Mathematical Model of Macrosegregation Formation in Binary Alloy Solidification.- 7.1 Introduction.- 7.2 Microscopic Conservation Equations.- 7.3 Fundamentals of Volume Averaging.- 7.4 Macroscopic Conservation Equations.- 7.5 Macroscopic Constitutive Relations.- 7.6 A Mixture Formulation of the Governing Equations.- 7.7 Transformation to Dimensionless Form.- 7.8 Dimensionless Mixture Equations.- 7.9 Concluding Remarks.- 8 Computation of Macrosegregation due to Solidification Shrink age.- 8.1 Introduction.- 8.2 The Mathematical Model.- 8.3 The Finite Element Method.- 8.4 Implementation of the Finite Element Method in Diffpack..- 8.5 Results and Discussion.- 8.6 Conclusion.- 9 A Mathematical Model for the Melt Spinning of Polymer Fibers.- 9.1 Introduction.- 9.2 Model Description.- 9.3 Additional Constitutive Relations.- 9.4 Numerical Methods.- 9.5 A Case Study.- 9.6 Summary.- 10 Finite Element Methods for Two-Phase Flow in Heterogeneous Porous Media.- 10.1 Introduction.- 10.2 Mathematical Model.- 10.3 Numerical Methods.- 10.4 One-dimensional Flow.- 10.5 Two-dimensional Flow.- 10.6 Summary and Conclusion.- 11 Splines and Ocean Wave Modelling.- 11.1 Introduction and Background.- 11.2 Spline Functions Summary.- 11.3 Splines and Ordinary Differential Equations: a Worked Example.- 11.4 Splines and Partial Differential Equations: a Worked Example.- 11.5 Splines in Water Wave Equations.- 11.6 Nonlinear Waves in a Box: A Worked Example.- 12 Krylov Subspace Iterations for Sparse Linear Systems.- 12.1 Introduction.- 12.2 Krylov Subspace Methods.- 12.3 Preconditioning Techniques.- 12.4 Concluding Remarks.- 13 Preconditioning Linear Saddle Point Problems.- 13.1 Introduction.- 13.2 Abstract Saddle Point Problems.- 13.3 Preconditioned Iterative Methods.- 13.4 Examples of Saddle Point Problems.- 13.5 Discretizations of Saddle Point Problems.- 13.6 Numerical Results.- III Geometric Modelling.- 14 Surface Modelling from Scattered Geological Data.- 14.1 Introduction.- 14.2 Description of Geological Data.- 14.3 Triangulations.- 14.4 Regular Grid Models.- 14.5 A Composite Surface Model.- 14.6 Examples.- 14.7 Concluding Remarks.- 15 Varioscale Surfaces in Geographic Information Systems.- 15.1 Introduction.- 15.2 Surfaces of Variable Resolution.- 15.3 Surface Varioscaling by Normalization.- 15.4 Examples.- 15.5 Final Remarks.- 16 Surface Modelling from Biomedical Data.- 16.1 Boundary Polygons.- 16.2 Curve Approximation.- 16.3 Reducing Twist in the Closed Surface.- 16.4 Surface Approximation.- 16.5 Open Surfaces.- 16.6 Examples.- 16.7 Concluding Remarks.- 17 Data Reduction of Piecewise Linear Curves.- 17.1 Introduction.- 17.2 Preliminaries.- 17.3 The Intersecting Cones Method.- 17.4 The Improved Douglas Method.- 17.5 Numerical Examples.- 17.6 Resolution Sorting.- 18 Aspects of Algorithms for Manifold Intersection.- 18.1 Introduction.- 18.2 Basic Concepts Used.- 18.3 Geometric Tolerance and Intersection.- 18.4 Representation of the ?-Intersection.- 18.5 Finding all Intersection Occurrences.- 18.6 Loop Elimination.- 19 Surface Editing.- 19.1 Introduction.- 19.2 Requirements for Surface Editing.- 19.3 Surface Interrogation.- 19.4 Surface Modification.- 19.5 Boundary Conditions of the Surface.- 19.6 Introducing New Degrees of Freedom.- 19.7 Examples.- 19.8 Conclusion.