Keywords: meshfree method, method of finite spheres, interpolation, partition of unity, numerical integration, boundary conditions.

Over 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". 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, meshfree methods have been developed. Meshfree methods not only overcome the longstanding problems associated with mesh generation and remeshing but also allow flexibility in the generation of approximation spaces.

However, the current literature is rife with "pseudo meshfree" methods since in most of these methods the interpolation is performed in a meshfree manner, but numerical integration is performed using a background mesh. A truly meshfree method is one that allows interpolation as well as numerical integration to be performed without a mesh. We introduced the method of finite spheres (De and Bathe [1]) as a truly meshfree method for the solution of problems in computational solid and fluid mechanics.

While the principle of generation of a truly meshfree method is straightforward, there are many problems in successfully implementing these methods. The first and foremost issue, from a practical point of view, is computational efficiency. This depends on the techniques of generation of nodal points, the proper choice of the computational subdomains, effective schemes of implementing boundary conditions and performing numerical integration without using a background mesh.

Most of the currently available meshfree methods are inefficient. However, for a meshfree method to be generally useful, it must be computationally competitive with the traditional finite element methods. The method of finite spheres is being developed with this very important requirement in mind.

We have developed an efficient technique for generating the nodal points using hierarchical partitioning of space using an octree data structure (Macri, et.al. [3]). An one-level-adjusted octree together with the use of face neighbor points allows neighbor search in constant time. Such a scheme also allows adaptive computations to be performed in a straightforward manner.

As computational primitives we have chosen spherical subdomains as well as intersections of spheres with the boundaries. The major advantage of such a choice is that one needs to store only the center and radius of each sphere together with a list of faces and edges of the model geometry with which it intersects.

We have developed very efficient numerical integration schemes (Macri and De [4]) which allow the method of finite spheres to be comparable in efficiency with the finite element method. This is a significant achievement and will be discussed in detail in the paper.

Finally, we have developed efficient techniques of applying Dirichlet boundary conditions as well as coupling the method of finite spheres with the finite element method.

1

S. De and K.J. Bathe, "The method of finite spheres with improved numerical integration", Computers & Structures, 79 (22-25), 2183-2196, 2001. doi:10.1016/S0045-7949(01)00124-9

2

K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1996.

3

M. Macri, S. De and M.S. Shephard, "Hierarchical tree-based discretization in the method of finite spheres", Computers & Structures, 81, 789-803, 2003. doi:10.1016/S0045-7949(02)00475-3

4

M. Macri, and S. De, "Towards an Automatic Discretization Scheme for the Method of Finite Spheres and its Coupling with the Finite Element Method", Computers & Structures (in press). doi:10.1016/j.compstruc.2004.09.019

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