Solution of non‐linear boundary integral equations in complex geometries with auxiliary integral subtraction

Abstract

SUMMARYThe boundary integral equation that results from the application of the reciprocity theorem to nonlinear or non-homogeneous di erential equations generally contains a domain integral. While methods exist for the meshless evaluation of these integrals, mesh-based domain integration is generally more accurate and can be performed more quickly with the application of fast multipole methods. However, polygonalization of complex multiply-connected geometries can become a costly task, especially in three-dimensional analyses. In this paper, a method that allows a mesh-based integration in complex domains, while retaining a simple mesh structure, is described. Although the technique is intended for the numerical solution of more complex di erential equations, such as the Navier-Stokes equations, for simplicity the method is applied to the solution of a Poisson equation, in domains of varying complexity. It is shown that the error introduced by the auxiliary domain subtraction method is comparable to the discretization error, while the complexity of the mesh is signiÿcantly reduced. The behaviour of the error in the boundary solution observed with the application of the new method is analogous to the behaviour observed with conventional cell-based domain integration.

The evaluation of domain integrals in complex multiply-connected three-dimensional geometries for boundary element methods

“…For example, RBFs of the 1 þ r type, as well as some compactly-supported RBFs, would not be suitable, since they are not smooth at all points in the domain. On the other hand, Gaussian radial basis functions satisfy the first requirement identically, and their parameters can be adjusted so that the second condition is also met (Mammoli, 2002). In fact, the aforementioned study of similar problems in two-dimensions shows that the condition number of the matrix that must be inverted to obtain the a coefficients is a strong function of the location of the RBF centers, the radius of influence r 0 and the type of collocation that is used (function or derivative).…”

“…An auxiliary domain embedding method was developed by Khadra et al (2000) to solve twodimensional viscous fluid flow problems with obstacles using a finite volume method. Mammoli (2002) developed an auxiliary domain method for two-dimensional Poisson and Helmholtz problems in multiply-connected domains. The current approach is an extension of Mammoli's development for three-dimensional domains.…”

Maximizing Storage Rate and Capacity and Insuring the Environmental Integrity of Carbon Dioxide Sequestration in Geological Reservoirs

“…A novel method to overcome this problem was devised by Mammoli (2002). It consists of evaluating the integral over a complex domain by Boolean operations involving integrals over simpler domains.…”

A Comparison of Splittings and Integral Equation Solvers for a Nonseparable Elliptic Equation

“…For problems with source terms that admit smooth extrapolation, one could solve an extended Poisson equation with any standard technique on a larger, more regular, domain. See Mammoli [24] and Biros, Ying, and Zorin [3] for attempts in this direction.…”