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Recent developments in stabilized Galerkin and collocation meshfree methods

Chen J-S., Chi S-W., Hu H-Y.

Computer Assisted Mechanics and Engineering Sciences

2011

Vol. 18, no. 1-2

3-21

Meshfree methods have been developed based on Galerkin type weak formulation and strong formulationwith collocation. Galerkin type formulation in conjunction with the compactly supported approximation functions and polynomial reproducibility yields algebraic convergence, while strong form collocationmethod with nonlocal approximation such as radial basis functions offers exponential convergence. In thiswork, we discuss rank instability resulting from the nodal integration of Galerkin type meshfree methodas well as the ill-conditioning type instability in the radial basis collocation method. We present the recentadvances in resolving these diffculties in meshfree methods, and demonstrate how meshfree methods can be applied to problems di?cult to be modeled by the conventional ?nite element methods due to their intrinsic regularity constraints.