Boundary Element Techniques in Computer-Aided Engineering

1. Weighted Residual Formulation of Approximate Methods.- 1.1. Introduction.- 1.2. Basic Definition.- 1.3. Approximate Solutions.- 1.4. Method of Weighted Residuals.- 1.5. Weak Formulations.- 1.6. The Inverse Problem.- 1.7. Conclusions.- References.- 2. Boundary Element Methods.- 2.1. Fundamentals of Functional Analysis.- 2.2. Generalized Green’s Formula.- 2.3. Variational Formulation.- 2.4. Weighted Residual Scheme.- 2.5. Boundary Element Formulation of Poisson’s Equation.- 2.6. Fundamental Solutions.- 2.7. Boundary Discretisation and Systems Equations.- 2.8. Computation of Integrals — 2D case.- 3. Boundary Integral Equations.- 3.1. Simple-layer Formulations.- 3.2. Double-layer Formulations.- 3.3. Direct Formulations.- 3.4. Indirect Vector Formulations.- 3.5. Direct Formulations.- References.- 4. Scalar and Vector Potential Theory.- 4.1. The Simple-layer Potential.- 4.2. The Double-layer Potential.- 4.3. Green’s Formula.- 4.4. Identification of Scalar and Vector Symbolism.- 4.5. Somigliana’s Identity.- 4.6. Rigid-body Displacement Field.- References.- 5. Potential Problems in Two Dimensions.- 5.1. Introduction.- 5.2. Flow past an Obstacle.- 5.3. Discretisation.- 5.4. Green’s Boundary Formula.- 5.5. Applications.- 5.6. Boundary Singularities.- 5.7. Composite Domains.- 5.8. Conclusion.- References.- 6. Three-dimensional Axisymmetrical Potential Problems.- 6.1. Introduction.- 6.2. The Newtonian Potential.- 6.3. Discretisation.- 6.4. General Domain.- 6.5. Axisymmetric Problems.- 6.6. Conclusion.- References.- 7. Heat Transfer Applications.- 7.1. Introduction.- 7.2. Integral Equations associated with Steady Heat Conduction Problems.- 7.3. Numerical Solution of the Integral Equations.- 7.4. Poisson’s Equation.- 7.5. Non-homogeneous Bodies; Method ofSubregions.- 7.6. Anisotropic Bodies.- References.- 8. Numerical Integration and other Computational Techniques.- 8.1. Introduction.- 8.2. Isoparametric Elements.- 8.3. Numerical Integration.- References.- 9. Starting to work with Boundary Elements.- 9.1. Introduction.- 9.2. The Boundary Element Method.- 9.3. Advantages and Disadvantages of the BEM compared to FEM.- 9.4. Introduction to BEASY.- 9.5. Examples.- 9.6. Conclusions.- References.- 10. Experiences in Boundary Element Applications.- 10.1. Introduction.- 10.2. Pre- and Post Processing.- 10.3. C.A.D. Coupling.- 10.4. Installation on Different Computers.- 10.5. Recommendations for BEM use.- 11. Electrostatics Problems.- 11.1. Introduction.- 11.2. Theoretical Basis.- 11.3. Boundary Elements.- 11.4. Applications.- 11.5. Conclusions.- References.- 12. A Boundary Element Solution of the Wave Equation.- 12.1. Introduction.- 12.2. Theoretical Development.- 12.3. Boundary Conditions.- 12.4. Numerical Implementation.- 12.5. Velocities and Pressures.- 12.6. Identification of Areas in Shadow.- 12.7. Test Example.- 12.8. Conclusions.- References.- 13. Elasticity Problems.- 13.1. Introduction.- 13.2. Governing Equations.- 13.3. Boundary Integral Formulation.- 13.4. Two Dimensional Elasticity Problems.- 13.5. Three Dimensional Elasticity Problems.- 13.6. Axisymmetric Elasticity Problems.- References.- 14. Elasticity Problems with Body Forces.- 14.1. Introduction.- 14.2. Transformation to Boundary Integrals.- 14.3. 2D Body Forces.- 14.4. 3D Body Forces.- 14.5. Axisymmetric Body Forces.- References.- 15. Time Dependent Problems.- 15.1. Introduction.- 15.2. Time Dependent Diffusion.- 15.3. The Scalar Wave Equation.- 15.4. Transient Elastodynamics.- 15.5. Mass Matrix Representation.- 15.6. Conclusions.- References.- 16. Time Dependent Potential Problems.- 16.1. Introduction.- 16.2. Integral Formulation of Heat Conduction Problems.- 16.3. Numerical Solution of the Integral Equations.- 16.4. Conclusions.- References.- 17. Plate Bending Problems.- 17.1. Preliminaries.- 17.2. Reciprocal Work Relation.- 17.3. Boundary Integral Representations.- 17.4. Concluding Remarks.- References.- 18. A Choice of Fundamental Solutions.- 18.1. Introductory Remarks.- 18.2. A simple example: 2D Heat Conduction.- 18.3. A more significant example: Plane Elastostatics.- 18.4. Concluding Remarks.- References.- 19. Formulation for Cracks in Plate Bending.- 19.1. Fundamental Solutions for Cracks.- 19.2. Augmented Boundary Integral Equations.- 19.3. Concluding Remarks.- References.- 20. Fracture Mechanics Stress Analysis, I..- 20.1. Introduction.- 20.2. Stress Intensity Factors.- 20.3. Integral Equation Methods for Crack Tip Stress Analysis.- References.- 21. Fracture Mechanics Stress Analysis, II.- 21.1. Introduction.- 21.2. Invariant Integral based on the Energy Momentum Tensor.- 21.3 Invariant Integrals deduced from Betti’s Reciprocal Theorem.- 21.4. Some Numerical Results for a Nocht Problem.- 21.5. A Problem of Debond Stress Analysis.- References.- 22. BEM in Geomechanics.- 22.1. Introduction.- 22.2. Notation and some Basic Ideas.- 22.3. BEM applied to the Interaction between Structures and the Supporting Ground.- 22.4. Inhomogeneity, Zoning and Layering.- 22.5. Elastoplasticity.- 22.6. Concluding Remarks.- References.- 23. An Asymptotic Error Analysis and Underlying Mathematical Principles for Boundary Element Methods.- 23.1. Projection Methods and Garding’s Inequality.- 23.2. Examples of Strongly Elliptic Boundary Integral Equations.- 23.3. Asymptotic Convergence of Galerkin type Boundary Element Methods.- 23.4.Asymptotic Convergence of Collocation Methods.- References.