A new boundary meshfree method for potential problems

• A new singularity removal technique.• A new regularized boundary integral equation, removing the singularity computation.• Average source technique, resulting in no need of boundary integral.• A new truly meshless method, termed average source method.

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 off-diagonal 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.