TY - JOUR
T1 - Reproducing kernel element method. Part I
T2 - Theoretical formulation
AU - Liu, Wing Kam
AU - Han, Weimin
AU - Lu, Hongsheng
AU - Li, Shaofan
AU - Cao, Jian
N1 - Funding Information:
This work is made possible by support from NSF under the grant DMI-0115079 to Northwestern University, DMS-0106781 to University of Iowa, and CMS-0239130 to University of California (Berkeley), which are greatly appreciated.
PY - 2004/3/26
Y1 - 2004/3/26
N2 - In this paper and its sequels, 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 the 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 in solving 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.
AB - In this paper and its sequels, 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 the 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 in solving 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.
KW - Approximation theory
KW - Finite element method
KW - Meshfree method
KW - Reproducing kernel element method
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U2 - 10.1016/j.cma.2003.12.001
DO - 10.1016/j.cma.2003.12.001
M3 - Article
AN - SCOPUS:1842523085
SN - 0374-2830
VL - 193
SP - 933
EP - 951
JO - Computer Methods in Applied Mechanics and Engineering
JF - Computer Methods in Applied Mechanics and Engineering
IS - 12-14
ER -