| Author | De, S., Hong, J.W., Bathe, K.J. |
|---|---|
| Title | Further developments and some applications in the method of finite spheres |
| Year | 2002 |
| Editor | Fifth World Congress on Computational Mechanics |
| Abstract | The conceptual development of a meshless computational technique is rather straightforward. However, for the success of such a technique, it is essential that it be efficient. The efficiency of a truly meshless technique depends on the choice of the computational subdomains, the interpolation functions, the techniques used to impose the boundary conditions and perform numerical integration.The method of finite spheres was introduced as a truly meshless technique with the goal of achieving computational efficiency. In the method of finite spheres, discretization is performed using low cost partition of unity functions that are compactly supported on n-dimensional spheres. Dirichlet boundary conditions are imposed efficiently and the numerical integration of the terms in the Galerkin weak form is performed using specialized numerical integration rules. Unlike in the traditional finite element methods, the integrands in the method of finite spheres are rational functions, the integration domains are general spheres or spheres truncated by the domain boundary or general "lens" shaped regions of overlap of two spheres. Hence the development of efficient numerical integration schemes for the terms in the weak form without using a mesh is the key to achieving computational efficiency in the method of finite spheres. We have developed a set of numerical integration rules on disks, sectors and the "lens" shaped regions of overlap of two disks which result in a significant improvement in computational efficiency. The automatic placement of nodal points, the choice of the radii of the spheres for optimal accuracy and such other implementational issues are also important in deciding the overall efficiency of our computational scheme. In this paper we present some of our recent developments in the method of finite spheres with applications to simple problems in two-dimensional linear elastostatics. We also present an interesting application of a special version of the method of finite spheres, using point collocation, to the simulation of soft tissues in surgery. In this application the goal is to develop a virtual environment where the user can interact with virtual organs in real time using virtual laparoscopic tools. For real time graphical display, an update rate of 30Hz is sufficient, while a much higher update rate of 1kHz is required for real time display of interaction forces through a haptic interface device (Phantom). |