The consistent boundary element method for potential and elasticity: Part III — Topologically challenging numerical assessments for 2D problems. (June 2023)

• Record Type: Journal Article
• Authors: Dumont, Ney Augusto
• Is Part Of: Engineering analysis with boundary elements. Volume 151(2023)
• Page Start: 548
• Page End: 564
• Publication Date: 2023-06
• Subjects: Collocation boundary element method -- Conceptual aspects -- Quasi-singularities -- Curved elements -- High precision computation -- Convergence assessments; Boundary element methods -- Periodicals; Engineering mathematics -- Periodicals; Équations intégrales de frontière, Méthodes des -- Périodiques; Mathématiques de l'ingénieur -- Périodiques; Boundary element methods; Engineering mathematics; Periodicals; 620.00151
• DOI: 10.1016/j.enganabound.2023.03.026
• Languages: English
• ISSNs: 0955-7997

Abstract

Abstract: The collocation boundary element method is consistently outlined in a companion (Part I) paper for the general three-dimensional, static case of elasticity on the basis of a weighted-residuals statement that leads to the Somigliana's identity. Arbitrary rigid-body displacements, as for elasticity, are naturally taken into account, and traction force parameters are always in balance independently of problem scale and mesh discretization. For generally curved boundaries, the correct definition of traction force interpolation functions enables the enunciation of a general convergence theorem, the introduction of patch and cut-out tests and, not least, a considerable simplification of the numerical implementations. Simple code schemes are proposed in a second companion (Part II) paper for 2D problems of potential and elasticity, which rely exclusively on Gauss–Legendre quadrature and lead to arbitrarily high – actually only machine-precision dependent – computational accuracy of all results of interest independently of a problem's geometry and topology. We present here numerical results and convergence assessments – including numerical illustration of convergence Theorem 1 of the companion paper I – for 2D potential and elasticity problems with very challenging topology issues and even for subnanometer source–field distances — maybe with approximations due to a coarse mesh discretization but never introducing unduly singularities. We dare say the present results cannot be replicated by any code implementation other than the ones of our own, which resort to no ad hoc means beyond the problem's correct mathematics. In fact, we present the objective and controllable tools of separately assessing a mesh discretization for a given practical problem in terms of precision, accuracy and liability to round-off-errors. We also evaluate the distance threshold for a source point to be considered sufficiently close to demand the proposed exact mathematical treatment, as, for large distances, only Gauss–Legendre quadrature is required, be it in terms of quadrature adaptivity of even a fast multipole scheme.