The generalized finite point method

Abstract

In this paper we propose a new mesh-less method based on a sub-domain collocation approach. By reducing the size of the sub-domains the method becomes similar to the well-known finite point method (FPM) and thus it can be regarded as the generalized form of finite point method (GFPM). However, unlike the FPM, the equilibrium equations are weakly satisfied on the sub-domains. It is shown that the accuracy of the results is dependent on the sizes of the sub-domains. To find an optimal size for a sub-domain we propose a patch test procedure in which a set of polynomials of higher order than those chosen for the approximations/interpolations are used as the exact solution and a suitable error norm is minimized through a size tuning procedure. In this paper we have employed the GFPM in elasto-static problems. We give the results of the size optimization in a series of tables for further use. Also the results of the integrations on a generic sub-domain are given as a series of library functions for those who want to use GFPM as a cheap and fast integral-based mesh-less method. The performance of GFPM has been demonstrated by solving several sample problems.

Exponential basis functions in solution of static and time harmonic elastic problems in a meshless style

“…The readers may note that collocation methods are growing fast in two forms; those using global shape functions such as radial basis functions in [6][7][8] and those using local shape functions like the ones in [23] for instance (for the former set 982 B. BOROOMAND, S. SOGHRATI AND B. MOVAHEDIAN mathematical basis can be found for stability and convergence of the method [24][25][26]; however, for the latter set just a few studies are traceable in the literature, for example [27], in this regard). It has also been experienced that other integral-based meshless methods may show better stable behaviour (see [28,29]). In the method presented in this paper there is no problem pertaining to interpolation; however, there may be some discussions on the treatment of the boundary condition regarding the stability of the solution.…”

The finite point method for the p-Laplace equation

“…The finite point method (FPM) proposed in [7,[26][27][28][29] is a truly meshfree procedure. The approximation around each point is obtained using the standard moving least squares techniques similarly as in DE and EFG methods.…”

A Review on the Modified Finite Point Method

“…Contrarily, weak-form methods with MLS or WLS approaches [28][29][30][31][32][33][34][35], which need background cells for local integration, yield much higher stability and are applied more widely in many numerical implementations, such as viscous flow problems [31], nonlinear water wave problems [32,33], and wave breaking problems [34,35]. In some studies efforts are made to generalize the FPM to weak-form formulation by applying alternative weights [36] or alternative derivative approximation [37,38] in the FPM. A new meshless method, based on subdomain collocation approach, is also present and named "the generalized finite point method" (GFPM) [38].…”