Self‐regular boundary integral equation formulations for Laplace's equation in 2‐D

Abstract

SUMMARYThe purpose of this work is to demonstrate the application of the self-regular formulation strategy using Green's identity (potential-BIE) and its gradient form ( ux-BIE) for Laplace's equation. Selfregular formulations lead to highly e ective BEM algorithms that utilize standard conforming boundary elements and low-order Gaussian integrations. Both formulations are discussed and implemented for two-dimensional potential problems, and numerical results are presented. Potential results show that the use of quartic interpolations is required for the ux-BIE to show comparable accuracy to the potential-BIE using quadratic interpolations. On the other hand, ux error results in the potential-BIE implementation can be dominated by the numerical integration of the logarithmic kernel of the remaining weakly singular integral. Accuracy of these ux results does not improve beyond a certain level when using standard quadrature together with a special transformation, but when an alternative logarithmic quadrature scheme is used these errors are shown to reduce abruptly, and the ux results converge monotonically to the exact answer. In the ux-BIE implementation, where all integrals are regularized, ux results accuracy improves systematically, even with some oscillations, when reÿning the mesh or increasing the order of the interpolating function. The ux-BIE approach presents a great numerical sensitivity to the mesh generation scheme and reÿnement. Accurate results for the potential and the ux were obtained for coarse-graded meshes in which the rate of change of the tangential derivative of the potential was better approximated. This numerical sensitivity and the need for graded meshes were not found in the elasticity problem for which self-regular formulations have also been developed using a similar approach. Logarithmic quadrature to evaluate the weakly singular integral is implemented in the self-regular potential-BIE, showing that the magnitude of the error is dependent only on the standard Gauss integration of the regularized integral, but not on this logarithmic quadrature of the weakly singular integral. The self-regular potential-BIE is compared with the standard ( and computational algorithms are established as robust alternatives to singular BIE formulations for potential problems.

Evaluation of non-singular BEM algorithms for potential problems

“…In this section these formulations are shown and further details about the analytical development are presented in Jorge et al (2001) and in Cruse and Richardson (2000). The first formulation is the self-regular potential formulation wherein the unique potential at an arbitrary boundary point is used to regularize the singular integral identity.…”

“…The Green's identity for problems in which the potential field is C 0,α continuous in the Hölder sense can be regularized. The procedure of regularization is shown in Jorge et al (2001) and consists in subtracting and adding back the integral…”

“…According to the detailed procedure shown in Cruse and Richardson (2000) and Jorge et al (2001), the following expression is obtained:…”

“…For 3-D potential theory, Cruse and Richardson (2000) have presented two self-regular formulations of the BIE, while Jorge et al (2001) in a similar fashion have developed the correspondent self-regular forms for 2-D problems.…”

Numerical calculations of the potential on the rectangular and ellpitic domains with various aspect ratios

“…Direct numerical identification of boundary values in the Laplace equation is investigated by Hayashi et al [29]. Jorge et al [30] focused on self-regular boundary integral equation formulations for Laplace's equation in a 2D domain. In this work, application of the self-regular formulation strategy using Greens identity and its gradient form for Laplace's equation is demonstrated.…”