AuthorDe, S., Bathe, K.J.
TitleAnalysis of incompressible media using the method of finite spheres and some improvements in efficiency
EditorFourteenth 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.