This paper presents the fundamental concepts behind the moving particle finite element method (MPFEM), which combines salient features of finite element and meshfree methods. The proposed method alleviates certain problems that plague meshfree techniques, such as essential boundary condition enforcement and the use of a separate background mesh to integrate the weak form. The method is illustrated via two-dimensional linear elastic problems. Numerical examples are provided to show the capability of the method in benchmark problems.

. For a collection of sets in R d we consider the task of nding all sets in the collection that cover or contain a given point. The algorithms introduced in this paper are based on quadtrees and their generalizations to R d . The advantages of our new splitting algorithm to nd the covering sets of a point over the basic algorithm are detailed by means of hit rates and the expected depth traversed in the quadtree search, numerically and theoretically. This solves a dicult problem faced by meshfree discretizations of partial dierential equations. Key words. Quadtree and octree search methods, computational geometry. AMS subject classications. Primary: 68U05, 65D18; Secondary: 65N30 1. Introduction. The task of this paper is, given a collection of support sets S i # R d , i = 1, 2, . . . , N , to eciently answer the query Given a point x # R d , nd all S i such that x # S i . This task arises when assembling the matrix for a meshfree discretization of a partial d...

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