Automatic domain decomposition for radial basis meshless methods in solving problems with concavity and singularity

Radial basis functions(RBFs) has been used for interpolating scattered points before they are used in solving PDEs. RBFs have the advantage that they do not require a mesh which can be difficult and time consuming to construct especially in three dimensional cases. However, being global smooth functions, RBFs have problems in modelling entities with discontinuity and concavity. Domain decomposition is a solution to both problems. In this paper, we introduce a method to automatically decompose any object into a number of domains so that each domain would have no concavity. The method is to form a constrained Delaunay triangulation of nodal points on the boundary of the object. The concavity can then be discovered by identifying groups of external triangles that are enclosed by the object. Then each domain is modelled by a separate RBF. Smoothness between different domains is maintained by using the overlapping domains method.