Analysis and optimization of nested meshes for adaptive mesh refinement

Abstract

Adaptive mesh refinement is a numerical method which locally increases the mesh resolution within existing meshes in order to reduce the computational effort of numerical simulations. The mesh resolution is only increased where required, for example, in areas with a high estimated error. This enables accurate simulation of problems with highly varying spatial scales, which would be unfeasible with fixed mesh resolutions. Focusing on adaptive refinement of Cartesian meshes, the nested meshes have to be properly embedded in a mesh hierarchy according to specific nesting criteria. The creation of such a nested mesh hierarchy requires a priori defined points of interest to be suitably clustered. In line with the goal of reducing computational effort of subsequent numerical simulations, the number of meshes as well as the number of individual mesh cells have to be optimally minimized. The challenge in creating an efficient mesh hierarchy is finding a trade-off between these quantities and the computational cost for the adaptation of the hierarchical data structure, including interpolation of the data associated with the mesh cells. In this thesis, two approaches to mesh clustering and nested mesh generation are evaluated and a novel algorithm for transforming a given Cartesian nested mesh configuration according to given points of interest is discussed. The algorithm produces efficient nested mesh hierarchies that are guaranteed to conform to extended nesting criteria, e.g., each mesh also has a unique parent mesh. An implementation of the adaptive mesh refinement procedure, including the mesh generation algorithms, is presented together with numerical examples. For these examples, the novel algorithm reduces the total number of cells for a single refinement level by up to 10%, depending on the problem geometry.