Automatic generation of octree-based three-dimensional discretizations for Partition of Unity methods

Abstract

The Partition of Unity Method (PUM) can be used to numerically solve a set of differential equations on a domain Ω. The method is based on the definition of overlapping patches Ω i comprising a cover {Ω i } of the domain Ω. For an efficient implementation it is important that the interaction between the patches themselves, and between the patches and the boundary, is well understood and easily accessible during runtime of the program. We will show that an octree representation of the domain with a tetrahedral mesh at the boundary is an efficient means to provide the needed information. It subdivides an arbitrary domain into simply shaped topological objects (cubes, tetrahedrons) giving a non-overlapping discrete representation of the domain on which efficient numerical integration schemes can be employed. The octants serve as the basic unit to construct the overlapping partitions. The structure of the octree allows the efficient determination of patch interactions.