Improvement of Boundary Meshfree Method with Boundary Distributed Sources

Most practical engineering problems are currently solved by the well-developed standard finite element method (FEM). However, the FEM has its inherent shortcomings as it is a numerical method that relies on meshes. While conventional mesh-based techniques such as FEM can accurately approximate governing equations, labor and computational costs associated with creating the conforming mesh undermine the efficiency of such methods. The Boundary Element Method (BEM) is another technique that can alleviate the difficulties associated with the implementation of the standard FEM, but it still relies on meshes.

Thus, an idea of getting rid of meshes in the process of numerical treatments has naturally evolved, and the concepts of meshfree or boundary meshfree methods have been shaped up.

Boundary meshfree methods can be roughly sorted into two: MFS (Method of Fundamental Solutions)-based type and the BIE (Boundary Integral Equation)-based type. The former is based on the concept of the MFS. The traditional MFS uses only a fundamental solution, which is a response due to a concentrated point source, in the construction of the solution of a problem without using any integrals. It is a natural boundary meshfree method.

Studies show that the MFS has been developed by improving the locations or types of sources. In the traditional MFS, a fictitious boundary is required to have the source points on it to avoid the singularity of fundamental solutions. The determination of the distance between the real boundary and the fictitious boundary is based on experience and therefore it is troublesome. In recent years, a number of efforts have been made aiming to remove this barrier in the MFS so that the source points can directly be placed on the real boundary.

And the MFS has been improved by replacing concentrated point sources of the traditional MFS with area-distributed sources covering the source points for 2D problems (volume-distributed sources for 3D problems).

Kim Un Ok, a lecturer at the Faculty of Applied Mathematics, has successfully improved the boundary meshfree method with area-distributed sources placed on the real boundary called boundary distributed source (BDS) method, by moving distributed sources outside the boundary.

To show the effectiveness of the new improved method, she has plotted the relative errors in the computed results using the BDS method and her new improved method respectively.

She has demonstrated the accuracy of the new method by solving 2D potential problems in a square domain covering 0≤x and y≤1 with Dirichlet BC,Ф(x, y)=x2-y2. She has moved sources 1.5 times d, average distance between the original boundary distributed sources, outside the boundary in the direction of outward normal vectors.

When BDS method is used, the solution is inaccurate near the boundary regions. However, the proposed method works well in improving the accuracy of the numerical solution. The results say that moving distributed sources outside the boundary can improve the BDS method.