The Kriging integration method applied to the boundary element analysis of Poisson problems

A new radial basis integration method applied to the boundary element analysis of 2D scalar wave equations

“…The present method is based on our most recent work, i.e. the Kriging Integration Method [27]. The main difference is that the new approach contemplates the use of other radial basis functions different to the Gaussian but without free parameters, and they are also complemented with polynomial terms to ensure greater convergence.…”

“…In RBI-BEM, in the same way as in the Cartesian Integration Method (CTM) [26] and the combined approach called DIBEM-RIM [37], regularization techniques can be used to treat weakly singular integrals, however, experience has shown that the results obtained using regularization can be even less precise than the achieved by the untreated ones. Additionally, it was inferred from the mathematical development carried out in [27] that the presence of logarithmic singularities does not significantly affect the direct evaluation of domain integrals, therefore, in this work these integrals are calculated using a quadrature obtained from the Radial Basis Integration Method and discarding the possible integration point where the radial distance between this and the source point becomes zero.…”

“…On the other hand, within the category of methods based on meshless techniques, the following stand out: The Monte Carlo Method (MCM) [23,24], The Quasi-Monte Carlo Method [25],The Cartesian Transformation Method (CTM) [26] and The Kriging Integration Method (KIM) [27]. All of them are based on a quadrature defined for integration points distributed within the domain, however, except in the CMT such distributions can be random and allow solving problems with singularities without any special treatment, although in KIM this only seems to be true for weak singularities.…”

“…In this paper, a meshless-based integration method as in [26,27,38], named the Radial Basis Integration Method (RBIM) has been developed for the computation of inertial domain integrals that are found in the BEM formulation of the 2D scalar wave equation. Using this quadrature, it is possible to compute them directly, therefore, it is a potential alternative in relation to the existent methods.…”

Application of boundary element method with convolution core in solving carrier evolution equation

“…Poisson equation is one of the most typical elliptic partial differential equation, which has wide application in various scientific and mechanic fields, such as elastic mechanics [1] and electromagnetics [2]. Nowadays, plenty of methods, such as finite element methods [3,4], boundary element method [5,6], deep-learning methods [7], are able to solve a Poisson equation quickly and accurately. By optimizing and integrating these algorithms, many commercial multi-physics software, including COMSOL, Ansys, and Abaqus can simulate complex physical fields using different methods.…”