Consistency, precision, and accuracy assessment of the collocation boundary element method for two-dimensional problems of potential and elasticity

• Published in: Archive of applied mechanics (1991) Vol. 94; no. 9; pp. 2489 - 2518
• Main Author: Dumont, Ney Augusto
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 2024; Springer Nature B.V
• Subjects: Accuracy; Boundary element method; Classical Mechanics; Collocation; Collocation methods; Domains; Elasticity; Engineering; Fracture mechanics; Original; Quadratures; Roundoff error; Singularities; Subdivisions; Theoretical and Applied Mechanics; Topology; Elasticity; Complex-variable formulation; Consistent formulation; Boundary elements; Machine-precision evaluation
• ISSN: 0939-1533; 1432-0681
• DOI: 10.1007/s00419-024-02592-8

The collocation boundary element method, as developed and taught in the traditional books, suffers from severe inconsistencies, partly responsible for the method’s lack of clarity and broader applicability (not to mention the uncontrollable proliferation of misleading alternatives that only add to more confusion). This has been recently corrected, as summarized in this review paper, in which we report the proposition of a convergence theorem for the general, just consistent, three-dimensional isoparametric formulation of potential and elasticity problems ( https://doi.org/10.1016/j.enganabound.2023.01.017 ), and also introduce the not interchangeable concepts of nodes and loci , for boundary displacements and tractions in elasticity, respectively, as well as of points , for domain sources. We have implemented, for both two-dimensional potential and elasticity problems, real-variable ( https://doi.org/10.1016/j.enganabound.2023.01.015 , https://doi.org/10.1016/j.enganabound.2023.03.026 ) and—still better—complex-variable ( https://doi.org/10.1016/j.enganabound.2023.04.024 ) codes for generally high-order, curved elements, which effortlessly enable evaluations with numerical precision that is only machine-limited and only resorts to the problem’s mathematics—plus Gauss–Legendre quadrature of all integrals’ regular parts. (Ad hoc mechanical interpretations, “regularizations,” interval subdivisions, special quadrature formulas, or variable transformations are neither strictly speaking correct nor generally applicable nor required.) Despite the singular kernels that enter diverse forms of Somigliana’s identity (as for evaluating stress results at a domain point), problem-unrelated singularities do not come up in a mathematically consistent formulation and implementation: singularities are just man-made. A few examples of highly challenging topological issues are presented. We show that fracture mechanics problems may be dealt with seamlessly, and we manage to objectively assess a problem’s topological consistency, precision, accuracy, and round-off errors related to a given mesh discretization and software-conditioned computational digits.