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.

Abstract

The article has a dual historical and educational theme. It is a tutorial on finite elements templates for two-dimensional structural problems. The exposition focused on the four-node plane stress element of flat rectangular geometry, called here the “rectangular panel” for brevity. This is one of two oldest two-dimensional structural elements, soon to reach its gold anniversary. On the other hand, the concept of finite element templates is a recent development. Interweaving the old and the new throws historical perspective into the “golden age” of discovery of finite elements. Templates provide a framework in which diverse element development methods can be fitted, compared and traced back to the sources. On the technical side templates have the virtue of facilitating the unified implementation of back to the sources. On the technical side templates have the virtue of facilitating the unified implementation of element families as well as the construction of custom elements. As an illustration of customization power, the Appendix presents the construction of a four noded bending optimal trapezoid that has eluded FEM investigators for several decades.