دانلود رایگان مقاله روش بدون شبکه مرزی جدید برای مشکلات احتمالی

Abstract

This work presents a new boundary meshfree method, named the average source method (ASM), for solving two-dimensional potential problems. The method is based on combining a ‘completely’ regularized boundary integral equation (CRBIE) with indirect unknowns developed in this paper, removing the singularity computation, and an average source technique (AST). In this approach there are two critical developments. One is the presentation of a new removal singularity technique that results in the CRBIE, and therefore all diagonal coefficients of influence matrices can be evaluated analytically by the offdiagonal ones, unlike some existing meshless boundary approaches that determine diagonal coefficients from the fundamental solution by using a known solution, thereby doubling the solution procedure. The other is to introduce an AST, by which the distributed source on a segment/cell can be reduced to the concentrated point source and therefore the boundary integrals in the CRBIE are not necessary. Hence, in the ASM only boundary nodes are required for computation without involving any integration and element notion. Several benchmark test examples are presented to demonstrate the accuracy, convergence, efficiency and robustness of this new meshfree boundary-node methodology.

4. Conclusions

In this paper, a new boundary-type meshfree method, termed average source method (ASM), is proposed to solve 2D Laplace problems. This method is based on coupling the RBIEs and the AST, which has the following key features: 1. The proposed ASM develops a new strategy to analytically compute diagonal coefficients of influence matrices, which are weakly singular or strongly singular, so that the problem can be solved only once, unlike the foregoing MFSbased type methods that apply a known solution to determine such diagonal coefficients, thereby doubling the solution procedure. This is an essentially crucial but very dif- ficult issue for the boundary-type collocation schemes, and therefore great advance has been achieved by present work in this area. 2. Due to the using of the AST, the proposed ASM is a truly boundary-node method without involving any element or integration concept, unlike the aforementioned MLS-based methods that introduce the MLS-based meshless shape functions constructed elaborately into BIE to exploit the meshless attribute of MLS approximation, and but still need the calculation of boundary integrals. It is worth noting that this is a general methodology and also readily applicable to other or even singular BIEs. Some applications of the AST to other BIEs are already underway and will be reported in some subsequent papers.