| Author | De, S., Bathe, K.J. |
|---|---|
| Title | Analysis of incompressible media using the method of finite spheres and some improvements in efficiency |
| Year | 2002 |
| Editor | Fourteenth U.S. National Congress on Theoretical and Computational Mechanics |
| Abstract | Over the past few decades, the finite element method has emerged as a highly effective and successful numerical technique for the solution of a wide variety of boundary value problems in Engineering. However, in these techniques, a great deal of effort is associated with the generation of a good quality mesh. For this reason there is much interest in the development of so-called meshless techniques. The method of finite spheres [1] was introduced as a truly meshless technique with the goal of achieving computational efficiency. In the method of finite spheres interpolation is performed using functions that are compactly supported on n-dimensional spheres (n=1,2 or 3), which form a covering for the analysis domain. It was observed that for incompressible or nearly incompressible media, the pure displacement-based formulation exhibits a degradation of accuracy and convergence rate. This phenomenon is known as "volumetric locking". In order to remedy the problem of locking, we present a mixed formulation based on displacement and pressure interpolations. However, unlike a pure displacement-based formulation a displacement/pressure mixed formulation may behave reasonably for certain problems and completely fail for certain others unless the displacement and pressure approximation spaces are properly chosen. To obtain a stable and optimal procedure for the selected interpolation, the mixed formulation should satisfy the ellipticity condition and the inf-sup condition. We have identified several displacement/pressure mixed interpolation schemes that pass the numerical inf-sup test. We also discuss several new integration schemes that have allowed us to reduce the computational cost substantially. |