P-FEM Based on Meshless Trial and Test Functions: Part I-MLS Approximation
In this paper the essential features of the P-FEM methods for solving linear elliptic equations using variational principles was addressed from the point of view of approximation space enrichment using meshless approximation. As meshless trial and test functions, MLS approximation was used as generalized p-version convergence. By using this generalized p-version convergence, along with the FEM paradigm, a new numerical approach is proposed to deal with differential equations. Through numerical examples, convergence tests are performed and numerical results are compared with MLPG and analytical solutions. The analysis has shown that the numerical solution obtained by using this method will converge as the order of MLS approximation increases. P-FEM can be directly used for higher order equations because there are no difficulties in construction shape function of any regularity. Adaptive procedures can be realized through the adaptive construction of meshless trial and test functions. The present method possesses a tremendous potential for convergent improved compared with traditional h- or p-version FEM.
KeywordsP-FEM MLS Galerkin approximation Meshless trial function Meshless test function
This research is supported by National Natural Science Foundation of China (grant No. 51405066) and Natural Science Foundation of China (grant No. 51405063).
- 2.Dong, L., Atluri, S.N.: Development of T-Trefftz four-node quadrilateral and Voronoi cell finite elements for macro-& micromechanical modeling of solids. Comput. Model. Eng. Sci. (CMES). 81(1), 69–118 (2011)Google Scholar
- 3.Šolín, P., Segeth, K. et al.: Higher-Order Finite Element Methods, Taylor & Francis (2003)Google Scholar
- 4.Melenk, J.M., Wihler, T.P.: A Posteriori error analysis of $ hp $-FEM for singularly perturbed problems. arXiv:1408.6037 (2014)
- 10.Atluri, S.N., Shen, S.: The meshless local Petrov-Galerkin (MLPG) method, Crest (2002)Google Scholar
- 11.Shen, S.N.A., Shengping.: The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. Comput. Model. Eng. Sci.3, 11–51 (2002)Google Scholar