Point placement for meshless methods using Sphere packing and Advancing Front methods

For simulation of numerical problems, meshless methods have emerged as an alternative to mesh based methods and become popular due to several reasons. Primarily, mesh generation is a difficult problem, Also these methods have generated promising results in the accuracy of the simulations. Similar to mesh generation problems, meshless methods also induce challenging geometric problems. Partition of Unity Method (PUM) is one of the meshless methods that is based on the definition of overlapping patches covering the domain. Some overlap among the patches is required. These patches should overlap to cover the domain, however, it should not be to the extend that deteriorates the accuracy. Also any single point in the domain should not be covered by more than a certain number of patches. We formalize this criteria to define a good point set for meshless methods in the second section the paper. Earlier implementations employed arbitrary point set generation which could not support required structure of PUM. The alternative ”mesh free” approach uses a mesh to obtain the point set. Biting Method is an advancing front technique that generates a good sphere packing [6]. The centers of the spheres, a well-spaced point set, can in turn be used to generate a provably good simplicial mesh. In this study, we use this approach to generate a good point set for meshless methods. However, instead of relating the radii of the spheres with the mesh spacing function, size of the spheres (patches) is defined by the support function and the overlap criteria described above. We prove that biting method can be used to generate a good point set for meshless methods.