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Reproducing kernel element method. I: Theoretical formulation. (English) Zbl 1060.74670
Summary: In this paper and its sequels [for part II see Li, Shaofan et al., Comput. Methods Appl. Mech. Eng. 193, No. 12–14, 953–987 (2004; Zbl 1093.74062); for part III see the following entry] we introduce and analyze a new class of methods, collectively called the reproducing kernel element method (RKEM). The central idea in the development of the new method is to combine the strengths of both finite element methods (FEM) and meshfree methods. Two distinguished features of RKEM are: the arbitrarily high order smoothness and the interpolation property of shape functions. These properties are desirable, especially in solving Galerkin weak forms of higher-order partial differential equations and in treating Dirichlet boundary conditions. So, unlike the FEM, there is no need for special treatment, with the RKEM, of solution of high-order equations. Compared to meshfree methods, Dirichlet boundary conditions do not present any difficulty in using the RKEM. A rigorous error analysis and convergence study of the method are presented. The performance of the method is illustrated and assessed through some numerical examples.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
74S05 Finite element methods applied to problems in solid mechanics
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