An efficient and accurate implementation of the Localized Regular Dual Reciprocity Method

A note on the use of the Companion Solution (Dirichlet Green's function) on meshless boundary element methods

“…Results for different sizes of integration subregions for the LRDRM where reported in [3], when using the integral representation formula based on the superposition of single and double layer potentials, i.e. in terms of the fundamental solution and its normal derivative.…”

“…The main objective of this work is not to compare different implementation of meshless DRMs (see Caruso et al [3] for such a comparison) but instead, we highlight the misreference to the so called Companion Solution (Green's function) in meshless BEMs and the lost in versatility of the approach when the Green's function is only evaluated at the center of the integration subregions. As we will show in the next section, by keeping the number of interpolation stencils fixed and increasing the number of collocation points (different to the center of the circle) at the local integration formula in regions of high variations of the field variable u a kind of P-adaptive scheme is implemented.…”

“…To compare the performance of the so called CS using a single evaluation point at the centre of the circle and the GF using more than one evaluation point per integration subregion, when required, we present in this section numerical results obtained by the LRDRM [3] meshless approach based on the integral representation formula (10) with Q as the normal derivative of the CS or GF, respectively. In our implementation of the LRDRM approach, we used as interpolant function the multiquadric RBF and the shape parameter is defined at each stencil to be proportional to the average distance between stencil points, as suggested in the original work, changes on the value of the shape parameter are obtained by changing the value of the proportionality constant.…”

“…In order to be able to implement the above integral representation in terms of only double layer potentials, with the unknown potential u and the auxiliary particular solution ϕ as densities, we use here the version of the meshless DRM approach proposed by Caruso et al [3], i.e. the LRDRM, where the boundary conditions of the problem are imposed locally at interpolation schemes in contact with the problem boundary, in terms of a Hermite interpolation, i.e.…”

Stabilizing radial basis functions techniques for a local boundary integral method

“…Smooth ϕ(r, ε) e −(εr) 2 Gaussian GA [2,28,16,31] 1 + (εr) 2 Multiquadric MQ [1,4,6,25] 1 + (εr) 2 β , β ∈ R\N 0 Generalized Multiquadric GMQ [19] 1/ 1 + (εr) 2 Inverse Multiquadric IMQ [6,12,19,34] Piecewise smooth ϕ(r) r β , β / ∈ 2N Radial Potential RP [30] r 2β log(r), β ∈ N Thin Plate Spline TPS [4,32,33] r 2β−1 or r 2β log(r), β ∈ N Polyharmonic Spline PHS [3,11,28,29] Table 1. Some well-known RBFs; ε is the shape parameter.…”