3.3.3 Symmetry-Aware Mesh Processing

Symmetry-aware mesh processing is a popular area of research in the field of computer graphics, for example symmetrization [97][137]. Symmetries and similarities are used in re-construction and mesh healing processes after 3D scanning [30][68][123]. As mentioned in the introduction (cf. Chapter 1), one of the main approaches for this work is a technique in the area of computer graphics called instancing: With instancing, a high number of instances of a single object template is rendered into a scene at different locations [37][47][71][92].

However, most algorithms and concepts in the field of computer graphics only work for surface geometries or surface meshes. Some of the properties vital for simulation in science and engineering, like conformity, are, however, not important in the field of computer graphics. For example, in a rendering process, meshes are not required to be watertight or intersection-free [132].

A more general approach for symmetry-aware mesh processing, which also supports volumetric meshes, uses group theory and the generalized Fourier transform [72]. This work, however, only covers symmetries and lacks support of general similarities. Benchmarks for rotational symmetry-aware 3D volumetric mesh generation yield possible performance gains of factors of up to $50$ [48]. Improvements of these concepts and approaches for supporting other types of symmetries and similarities are presented and discussed in Section 5.1.2.

A topic related to similarity in meshes concerns periodic meshes, where one part of the mesh is repeated in several directions. Generation of periodic meshes has been investigated and implemented in the CGAL software library [29][86].

As motivated in the introduction (cf. Chapter 1), mesh generation and mesh adaptation algorithms would benefit from a theoretical approach which considers symmetries and/or similarities. However, there is no literature for such an approach which at the same time takes symmetries and similarities in account and additionally supports multi-region geometries and meshes.