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AuthorDe, S., Hong, J. W., Bathe, K.J.
TitleThe method of finite spheres: A generalization of the finite element technique
Year2002
EditorSecond International Conference on Advances in Structural Engineering and Mechanics
AbstractOver the past several decades the finite element technique has become a powerful numerical method for the solution of a variety of engineering problems. In this technique, the continuum is discretized using "elements" which are connected together at special points called "nodes" (see figure 1). The technique of discretizing a continuum by elements is known as "mesh generation". In spite of its popularity, one of the major problems of the traditional finite element scheme is associated with mesh generation. The finite elements need to satisfy certain stringent conditions on the aspect ratios of sides and included angles which make the automatic generation of a good quality mesh a nontrivial task, especially in three-dimensions. To alleviate this problem, we introduced the method of finite spheres (De and Bathe, 2001) as a generalization of the traditional finite element technique. In the finite element technique, the mesh is generated to define the shape functions which are piece-wise Lagrange polynomials. The support of the shape function defined at a node corresponds to the union of the elements that contain the node. This results in a banded stiffness matrix with desirable properties. However, since the support is an n-dimensional polytope, certain element shapes should not be used, namely those for which Jacobians are singular or almost singular, or the accuracy of analysis is inadequate (Bathe, 1996). In the method of finite spheres the discretization is performed using functions that are compactly supported on n-dimensional spheres. The compact support of the functions results in banded stiffness matrices just as in the finite element method. However, since the supports of the shape functions are regular, the element Jacobians are well behaved. The only important criterion is that the spheres cover the entire domain. Therefore, generating an acceptable nodal arrangement in the method of finite spheres is not as difficult as generating a good quality mesh for a traditional finite element analysis. This is a definite advantage for the analysis of many problems, in the linear and nonlinear analysis of solids and structures and the analysis of fluid-structure systems. The method of finite spheres may be viewed as a generalized finite element technique in which the spheres behave conceptually as finite elements. However, unlike the traditional finite elements, the spheres are not constrained to abut each other. In the traditional finite element technique we are used to employ Galerkin formulations and polynomial shape functions. In the method of finite spheres the shape functions may be easily chosen to be non-polynomials to improve convergence. Moreover, it is straightforward to use weighted residual techniques other than the Galerkin method, e.g. the point collocation procedure. In this paper we discuss important features of the method of finite spheres and give insight into the formulation and implementation of the method. The topics include the selection of effective shape functions, effective numerical integration schemes, the imposition of the boundary conditions, the selection of the size and spacing of the spheres, the efficient implementation of the method for general analysis, and the current computational expense of the method when compared to using the traditional finite element method. Finally, we discuss applications of the method of finite spheres to interesting engineering problems as well as to problems in virtual surgery.