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# Meshless Regular Hybrid Boundary Node Method

Jianming Zhang, Zhenhan Yao1
Department of Engineering Mechanics, Tsinghua University, Beijing, 100084, China. Telephone: +8610-62772913. E-mail: demyzh@tsinghua.edu.cn

Computer Modeling in Engineering & Sciences 2001, 2(3), 307-318. https://doi.org/10.3970/cmes.2001.002.307

### Abstract

The Meshless Local Boundary Integral Equation (MLBIE) method is a truly meshless one as it does not need a 'finite element or boundary element mesh', either for variable interpolation or for 'energy' integration. The Boundary Node Method (BNM) further reduces the dimensionality of the problem by one, i.e. it only requires nodes constructed on the surface. However, the BNM is not truly meshless, as a background mesh is needed for boundary integration; and the MLBIE does not have the advantage of reduced dimensionality as the BNM. A new Regular Hybrid Boundary Node method based on a modified functional and the Moving Least Squares (MLS) approximation, and combining the advantages of both the MLBIE and the BNM, is presented in this paper.
The Regular Hybrid Boundary Node Method is formulated in terms of the domain and boundary variables. The domain variables are interpolated by classical fundamental solutions with the source points located outside the domain; and the boundary variables are interpolated by Moving Least-Squares (MLS) approximation. The main idea is to retain the dimensionality advantages of the BNM, and localize the integrationdomain to a regular sub-domain, as in the MLBIE, such that no mesh is needed for integration. All integrals can be easily evaluated over regular shaped domains (in general, semi-circle in the 2-D problem) and their boundaries.
Numerical examples for the solution of 2-D Laplace equation show that the high convergence rates with mesh reﬁnement and the high accuracy with a small node number are achievable. The treatment of singularities and further integrations required for the computation of the unknown domain variables, as in the conventional BEM and BNM, can be avoided.

Zhang, J., Yao, Z. (2001). Meshless Regular Hybrid Boundary Node Method. CMES-Computer Modeling in Engineering & Sciences, 2(3), 307–318.

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