The method of finite spheres

The objective of this paper is to present some of our recent developments in meshless methods. In particular, a technique is given ± the method of ®nite spheres ± that is truly meshless in nature in the sense that the nodes are placed and the numerical integration is performed without a mesh. The method can be viewed as a special case of the general formulation known as the meshless local Petrov±Galerkin (MLPG) procedure. Some of the novel features of the method of ®nite spheres are the numerical integration scheme and the way in which the Dirichlet boundary conditions are incorporated. A new way of modeling doubly-connected domains is also presented. Various example problems are solved to demonstrate the method. List of symbols A Interior of a set in R; n ˆ 1; 2; 3: oA Boundary of set A A = A [ oA Closure of set A supp…v† = fx 2 A; v…x† 6ˆ 0g Support of a function v X Open bounded domain in R; n ˆ 1; 2; 3: C = oX Boundary of X (assumed to be Lipschitz continuous) n Outward unit normal de®ned on C B…xI ; rI† = fx 2 A; kxÿ xIk0 < rIg Open sphere of radius rI centered at xI in n-dimensional Euclidean space (n ˆ 1; 2 or 3) S…xI ; rI† = fx 2 A; kxÿ xIk0 ˆ rIg Surface of the n-dimensional sphere of radius rI centered at xI . h a global measure of the support radii (determined by the values of rI) hIm…x† = The mth shape function at node I Cm…A† Space of functions with continuous derivatives up to order m on A C 0 …A† = fvjv 2 Cm…A†; supp(v) is a compact subset of Ag Qm…A† Space of polynomials of degree m de®ned on A V The two-parameter global approximation space (depending on parameters h and p). v 2 V a… ; † A bilinear form from V V # R V h;p I The two-parameter local approximation space. v I 2 V I