Boundary Collocation Techniques and Their Application in Engineering

Methods of mathematical modelling applied in contemporary computational mechanics can be either purely numerical or analytical-numerical procedures. Purely analytical solutions lose their popularity because of strong limitations connected with simple regions and the mostly linear equations to which they can be applied. Obviously, the fundamental monographs (for example, insert those on elastic solids, fluid mechanics or heat exchange) are always popular and often quoted, but rather as sources of comparative benchmarks confirming correctness and accuracy of computer solutions. This volume can be divided into two parts. In the first part is a general presentation of the boundary collocation approach and its numerous variants. In the second part the method is applied to many different engineering problems, showing its properties, accuracy and convergence. Both evident advantages and also limitations of the approach are clearly presented. The observations are based mainly on investigations carried out in the last two decades by the authors and their co-operators. The monograph includes figures and tables that present results of numerical examples. A considerable number (above 1000) of papers and monographs concerning the discussed approach are quoted. They are listed separately in each chapter, which makes the literature survey easier to use.

Chapter 1: Introduction Boundary collocation versus other modelling methods; Different names of presented formulation; Short review of present monograph Chapter 2: General information Description of the method - basic versions of boundary collocation; Relation between boundary collocation, boundary integral Trefftz approach and regular boundary element method; Conditioning of the Trefftz-type formulations Chapter 3: Analytical trial T-functions applied in the boundary collocation method Complete systems of general solutions; Influence functions with singularities outside an investigated region; Influence T-functions represented by rigonometric series; Heuristic T-functions; Special purpose T-functions Chapter 4: Substructuring and T-elements Chapter 5: Basic numerical investigations of the boundary collocation method Simply connected regions; Doubly connected regions Chapter 6: 2D harmonic problems - the Laplace-type equations Torsion of prismatic bars. Formulation of the boundary value problem; Application of the boundary collocation method to twisted bars - review; Torsion of a bar made of different materials; The Poiseuille flow; Singular membrane problem; The Laplace-type solutions with special T-complete functions; Recent observations Chapter 7: Problems described by two-dimensional, biharmonic equations Typical problems; Plane creeping flow. Boundary collocation as a modification of the Trefftz method; Special purpose trial functions for 2D creeping flow; Thin plate bending Chapter 8: Two-dimensional problem of elastostatics Formulation of the 2D elastic problem; Example: tension of square plate with a central hole; Application of the T-complete Herrera and Kupradze functions; Optimization of helical spring cross-section Chapter 9: Inverse problems in 2D elasticity Introduction; The Trefftz formulation of the boundary value problem; Three methods of solving inverse problems; Numerical illustration; Conclusions Chapter 10: Three-dimensional problem described by elliptic equations General information; 3D potential problems; 3D creeping flow; Spatial (3D) elastostatic problems; Other applications of boundary collocation method to 3D problems Chapter 11: Non-homogenous equations and nonlinear boundary conditions Non-homogenous harmonic problems; Non-homogenous biharmonic problems; Boundary collocation method applied to non-linear conditions Chapter 12: Physically non-linear, elastic-plastic problems Scheme of solution; Numerical illustration of the algorithm proposed; Final remarks Chapter 13: Problems described by parabolic equations The boundary collocation method in boundary-initial problems; The boundary collocation for non-stationary temperature field; comparison with the FEM Chapter 14: Eigenvalue problems Problem formulation and general description of the method; Example. Determination of eigenvalues for a silencer Chapter 15: Final remarks, conclusions and perspectives