Numerical methods and software tools in industrial mathematics

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Bibliographic Information

Numerical methods and software tools in industrial mathematics

Morten Dæhlen, Aslak Tveito, editors

Birkhäuser, 1997

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Note

Includes bibliographical references and index

Description and Table of Contents

Volume

ISBN 9780817639730

Description

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 . . . . . . . . . .

Table of Contents

  • 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.
Volume

ISBN 9783764339739

Description

The main feature of this text is the discussion on how modern concepts in computer science can be applied in order to develop scientific software that is easier to extend, maintain and use than the more traditional counterparts. In addition, models and methods for challenging problems in industrial mathematics ranging from multiphase flow problems and modelling of solidification processes to surface modelling based on geological or medical data, are derived and discussed in the light of the software concepts.

Table of Contents

  • Part 1 Numerical software tools: object-oriented numerics
  • basic tools for linear algebra
  • software tools for modelling scattered data
  • a comprehensive set of tools for solving partial differential equations - Diffpack
  • on the numerical efficiency of C++ in scientific computing. Part 2 Partial differential equations: basic equations in Eulerian continuum mechanics
  • a mathematical model of macrosegregation formation in binary alloy solidification
  • computation of macrosegregation due to solidification shrinkage
  • a mathematical model for the melt spinning of polymer fibres
  • finite element methods for two-phase flow in heterogeneous porous media
  • splines and ocean wave modelling
  • Krylov sybspace iterations for sparse linear systems
  • preconditioning linear saddle point problems. Part 3 Geometric modelling: surface modelling from scattered geological data
  • varioscale surfaces in geographic information systems
  • surface modelling from biomedical data
  • data reduction of Piecewise linear curves
  • aspects of algorithms for manifold intersection
  • surface editing.

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