hp-Meshless cloud method

A methodology to build discrete models of boundary-value problems (BVP) is presented. The method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. A version of moving least-square interpolation and the collocation method are used to discretize BVP equations, which results in a truly meshless method (i.e. without a background mesh of integration points). h- and p-adaptive strategies are tested and very good convergence of the method was observed. The improvements in the methodology, in particular the introduction of spectral degrees of freedom, result in a fast and accurate method, significantly more efficient than the Finite Element Method or Element Free Galerkin Method. Several practical applications of the method to solve various engineering problems are presented.