Introduction to Boundary Elements: Theory and Applications

1 Fundamentals.- 1.1 Notation.- 1.2 The basic idea.- 1.2.1 The basic idea from a mathematical point of view.- 1.2.2 The basic idea from an engineering point of view.- 1.3 Influence functions.- 1.3.1 Betti's principle.- 1.3.2 Betti data.- 1.3.3 Fundamental solutions.- 1.4 Coupling on the boundary.- 1.5 Boundary elements.- 1.6 Conforming and non-conforming solutions.- 1.7 The interpretation of the solution.- 1.8 Symmetric formulations.- 1.9 The integral operators and their shifts.- 1.10 Galerkin, collocation and least square.- 1.11 Potentials.- 1.12 The indirect method.- 1.13 Weighted residuals.- 1.14 Influence functions and finite elements.- 1.15 The scale.- 1.16 Trefftz's method.- 1.17 Construction of fundamental solutions.- 1.18 Mixed methods.- 1.19 Shells.- Exercises.- 2 One-dimensional problems.- 2.1 Rods.- 2.2 Beams.- 2.3.Transfer matrices.- 2.4 Matrix-displacement method.- 2.5 The general principle.- Exercises.- 3 Membranes.- 3.1 The influence function for the deflection u(x).- 3.2 Discretization.- 3.3 Element matrices.- 3.4 The master element.- 3.5 Singular integrals.- 3.6 The treatment of the system of equations.- 3.7 The domain integral.- 3.8 Internal actions.- 3.9 Examples.- 3.9.1 Shear and torsion.- 3.9.2 Temperature distribution.- 3.9.3 Brownian motion.- 3.10 The maximum principle.- 3.11 The influence function for the normal derivative.- 3.12 Substructures.- 3.13 Alternatives to substructures.- 3.14 Singularities.- 3.15 Three-dimensional problems.- Exercises.- 4 Elastic plates and bodies.- 4.1 Introduction.- 4.2 The influence functions.- 4.3 Discretization.- 4.4 Element matrices for plates.- 4.5 Boundary conditions.- 4.6 Stresses.- 4.7 The domain integrals.- 4.8 Double nodes.- 4.9 Infinite domains.- 4.10 Examples.- 4.10.1 Beam with openings.- 4.10.2 Spring element.- 4.10.3 The search for cavities an inverse problem.- 4.11 Singularities.- 4.12 Concentrated forces.- 4.13 Three-dimensional problems.- 4.14 Axisymmetric problems.- 4.15 Examples.- Exercises.- 5 Nonlinear problems.- 5.1 The principle of virtual forces.- 5.2 The calculation of the singular integrals.- 5.3 The system of differential equations.- 5.4 Numerical treatment.- 5.4.1 Initial strain algorithm.- 5.4.2 Initial stress algorithm.- 5.4.3 Examples.- 6 Plates.- 6.1 Introduction.- 6.2 Fundamentals.- 6.3 Influence functions for ? and ??/?n.- 6.4 Coupling on the boundary.- 6.5 Discretization.- 6.6 Singular integrals.- 6.7 Element matrices.- 6.8 Degrees-of-freedom.- 6.9 The domain integrals.- 6.10 Actions on the boundary.- 6.11 Internal actions.- 6.12 Internal supports and subdomain loads.- 6.13 Examples.- 6.14 Singularities.- 6.15 Influence surfaces.- 6.16 Special problems.- 6.16.1 Plates on elastic foundations.- 6.16.2 Plate buckling.- Exercises.- 7 Boundary elements and finite elements.- 7.1 Theory.- 7.2 Practice.- 7.3 Experience.- 8 Harmonic oscillations.- 8.1 Rods.- 8.2 Beams.- 8.3 Elastic plates and bodies.- 8.4 Kirchhoff plates.- 8.5 Natural frequencies.- 8.6 Helmholtz equation (membrane).- 8.6.1 The radiation resistance of a rotating engine.- 8.7 Algebraization of the eigenvalue problem.- 9 Transient problems.- 9.1 Finite elements and boundary elements.- 9.1.1 Finite elements.- 9.1.2 Boundary elements.- 9.1.3 Summary.- 9.2 The wave equation.- 9.3 The heat equation.- 9.4 Dynamic displacement fields.- 9.5 Numerical treatment.- 9.6 Fourier-and Laplace transforms.- 9.7 Dynamic stiffness matrices.- 10 Computer programs.- 10.1 BE-LAPLACE.- 10.2 BE-PLATES.- 10.3 BE-PLATE-BENDING.- 10.4 Service.- Appendix A.- Appendix B.- Literature.