The best of two worlds: The expedite boundary element method

• Published in: Engineering structures Vol. 43; pp. 235 - 244
• Main Authors: Dumont, Ney Augusto, Aguilar, Carlos Andrés
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.10.2012; Elsevier
• Subjects: Applied sciences; Boundary element method; Boundary elements; Buildings. Public works; Collocation; Computation; Computation methods. Tables. Charts; Exact sciences and technology; Integrals; Large-scale problems; Mathematical analysis; Mathematical models; Matrices; Matrix methods; Meshless methods; Structural analysis. Stresses; Variational methods; Boundary elements; Variational methods; Meshless methods; Large-scale problems; Structural design; Formulation; Meshless method; Example; Variational calculus; Numerical simulation; Large scale; Boundary element method
• ISSN: 0141-0296; 1873-7323
• DOI: 10.1016/j.engstruct.2012.04.042

► The method provides approximations of the matrices of the collocation BEM. ► Generally curved, 2D and 3D elements for potential and elasticity problems are used. ► No singular or quasi-singular numerical integration is necessary. ► The evaluation of results at any internal or boundary points requires no integrations. ► The method can be advantageously used in combination with the fast multi-pole method. The present developments result from the combination of the variationally-based, hybrid boundary element method and a consistent formulation of the conventional, collocation boundary element method. The procedure is simple to implement and turns out to be computationally faster than the mentioned, preceding numerical methods – and almost as accurate – for the analysis of large-scale, two-dimensional and three-dimensional problems of potential and elasticity of general shape and topology, also applicable to time-dependent problems. Both the double-layer and the single-layer potential matrices of the collocation boundary element method, H and G, respectively, whose standard evaluation requires dealing with singular and improper integrals, are obtained in an expedite way that circumvents almost any numerical integration – except for a few regular integrals. Since the resultant matrices do not differ in nature from the ones of the conventional, collocation boundary element method, the developments are suited for a matrix solution in terms of a GMRES algorithm, for example, and in the framework of the fast multi-pole method, so that very large problems can be ultimately dealt with efficiently. A few numerical examples are shown to assess the applicability of the method, its computational effort and some convergence issues.