Optimization of Meshfree Method with Distance Fields using Localized Solution Structure and Radial Basis Function Collocation Method

Abstract

To avoid the meshing problems with traditional engineering analysis methods many meshfree methods have been developed, and one of the most powerful methods proposed is meshfree method with distance fields. This research work aims at reducing the computational cost of meshfree method with distance fields. The main idea of the proposed approach is to apply a solution structure operator only to those basis functions whose supports are in the vicinity of the boundary and leave other basis functions unaffected. Unfortunately, straightforward implementation of this approach leads to elevated errors in partial derivatives of the solution. To overcome this drawback I propose to modify distance fields in such a way that they behave as a Euclidean distance in the region near the boundary and have smooth transition to a constant value within some distance away from the boundary. The uniqueness of the proposed method over other approaches is its adaptability to all kinds of boundary conditions. This narrow band technique improves the computational cost of meshfree method with distance fields with reasonable impact on accuracy. Another technique proposed in this work is to glue the global solution structures of meshfree method with distance fields with radial basis functions (RBF) and collocation technique. RBF with collocation method itself is proved to give good accurate results with less computational cost [12]. So using RBF-collocation technique with meshfree method with distance fields demands for more accuracy even with less computational cost. Later, these techniques are applied to solve heat transfer problems and the results are compared with global solution techniques to show that the proposed methods are close to global approach and computationally very effective.