Abstract

Volumetric mesh generation plays an important role in computer-aided engineering processes. Often, objects used in computer-aided engineering, for example a gear, show symmetries or similarities. So far, available volumetric mesh generation and adaptation algorithms do not consider symmetries or similarities and thus ignore potential memory and algorithm optimizations. For example, instead of generating and storing a mesh of a regular $16$-polygon, the mesh of one single slice can be generated and stored together with the information that this so-called mesh template is copied and rotated $16$ times to yield the desired $16$-polygon mesh. In this particular case, improvements in memory and algorithm runtime of a factor of $16$ are expected.

This work investigates how the generation, usage, and storage of volumetric meshes can benefit from symmetries and similarities. In particular, impacts and optimizations in memory usage, algorithm runtime, and mesh element quality are investigated. For this reason, a theory based on so-called templated structures is developed. These templated structures contain mesh templates which are instanced (potentially) multiple times using geometrical transformations to obtain the resulting mesh. The proposed theory uses an abstract approach to support both, symmetries as well as similarities. Furthermore, theoretical mechanisms are developed to tackle potential conformity issues at mesh instance interfaces.

Based on these theoretical approaches, data structures and algorithms for adapting and generating templated meshes are developed in this thesis. In particular, two different algorithms for templated mesh generation are proposed and investigated. These algorithms are also specialized for symmetries, being reflective and rotational symmetries and their combinations. Additionally, a selection of popular mesh adaptation algorithms is investigated for their use with templated structures.

The benefits of the proposed data structures and algorithms are investigated and discussed in a benchmark-based survey. Expected memory savings and runtime speedups of the templated mesh generation process are indeed achieved for all considered two-dimensional and most three-dimensional objects. For three-dimensional objects with high rotational symmetry orders, the improvements are lower than expected but at least a factor of $15$. Memory savings drop significantly when including memory requirements for the system matrix of a finite element method. Therefore, a templated matrix data structure is developed, which compensates these losses. Mesh element qualities of templated meshes are as good as conventionally generated meshes in most scenarios and minimally worse otherwise.

Additionally, effects of symmetric and non-symmetric meshes on finite element based simulations are investigated. If the simulation domain is symmetric, the mathematical solution to the boundary value problem with boundary conditions transformed by the symmetry transformation is equal to the transformed solution of the initial problem. However, an analysis shows, that a symmetric mesh is required for this statement to hold for the numerical solutions with a finite element method.