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# A Local Meshless Shepard and Least Square Interpolation Method Based on Local Weak Form

Y.C. Cai1 and H.H. Zhu1
Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Department of Geotechnical Engineering, School of Civil Engineering, Tongji University, 200092, P.R.China.

Computer Modeling in Engineering & Sciences 2008, 34(2), 179-204. https://doi.org/10.3970/cmes.2008.034.179

### Abstract

The popular Shepard PU approximations are easy to construct and have many advantages, but they have several limitations, such as the difficulties in handling essential boundary conditions and the known problem of linear dependence regarding PU-based methods, and they are not the good choice for MLPG method. With the objective of alleviating the drawbacks of Shepared PU approximations, a new meshless PU-based Shepard and Least Square (SLS) interpolation is employed here to develop a new type of MLPG method, which is named as Local Meshless Shepard and Least Square (LMSLS) method. The SLS interpolation possesses the much desired Kronecker-delta property, hence the prescribed nodal displacement boundary conditions can be implemented as easily as in FEM. Based on the local Petrov-Galerkin weak form, the present LMSLS method utilizes a local polygonal domain to simplify the integration and the discrete equations and is a \textit {truly} meshless method which constructs interpolation without using mesh and integrates the local weak form without a background mesh. Additionally, the orthogonal basis functions are used to totally eliminate the matrix inversion and matrix multiplication in the computation of the SLS interpolation. Numerical examples show that the present method has a high accuracy and convergence rate.

### Keywords

Meshless, MLPG, point interpolation, kronecker property, partition of unity.

Cai, Y. (2008). A Local Meshless Shepard and Least Square Interpolation Method Based on Local Weak Form. CMES-Computer Modeling in Engineering & Sciences, 34(2), 179–204.

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