# Mesh Generation

**Overview:**

The generation of numerical meshes for discretizing arbitrary physical domains is one of the most challenging and time consuming tasks within the simulation process of the Finite Element Method (FEM). There exists a large variety of methods for the generation of Finite Element meshes of different types and dimensions, but not all methods are generally valid and therefore applicable for all types of geometric problems. Especially the generation of good quality all-hexahedral Finite Element meshes for arbitrary structures is considered to be the Holy Grail in the fields of mesh generation and remains an unsolved and ongoing research task.

The chair for Computation in Engineering (CiE) has been involved in research in the field of mesh generation since the early 1990ies with developing a surface mesh generation algorithm for discretizing arbitrary freeform surfaces with triangular and quadrilateral elements (DoMesh). In the later stages of research activities in this fields the surface mesh generator has been enhanced to a mesh generation framework discretizing arbitrary BRep-volumes with curved high-order tetrahedral elements and thin walled shell-like structures with curved high-order hexahedral elements (TUM.GeoFrame).

**Curved high-order tetrahedral mesh generation for arbitrary BRep-volumes:**

The generation of all-tetrahedral Finite Element meshes for arbitrary BRep-volumes is achieved by coupling the Delauney-based open source mesh generation library *Netgen* to the framework. *Netgen* requires a surface mesh consisting of pure triangular elements with normal vectors pointing outside the volumetric domain as input data and produces trilinear tetrahedral elements. The initial triangle mesh is generated using the surface mesh generator DoMesh and the normal vectors of the elements are normalized by computing the Nullspace of the topology matrix on the mesh. After the tetrahedral element generation the surfaces of the single tets can be discretized by high-order polynomials to approximate the curved shapes of the original structure.

**Figure 1: **Steps of the high-order tetrahedral mesh generation on a screw model

**Curved high-order hexahedral mesh generation for shell-like structures:**

The generation of all-hexahedral Finite Element meshes for shell-like structures is based on traditional sweeping techniques combining different single-shell sweeping models intersecting in parallel, perpendicular and skew angled directions. Single-shells can be described by a set of surfaces and a thickness parameter in case the final shell-like structure has a constant thickness. Structures with varying thickness are described by three surface sets, one describing the top-surfaces, one the bottom-surfaces and one a set of reference surfaces to be meshed with quadrilateral elements.

**Figure 2: **Mulitple-shell surface model of a violin body

In a first step, the boundary of the original geometry is manipulated and the reference surfaces are disconnected. In a second step the single disconnected surfaces are meshed with quadrilateral elements introducing references to ensure a conformal subdivision of the original boundary edges. In a final step the disconnected surfaces meshes are extruded individually and connected to an interface mesh to result in a conforming mesh of all hexahedral Finite Elements.

**Figure 3: **Steps of the mesh generation process on a violin model

This procedure allows the generation of all hexahedral finite element meshes on complex shell-like structures with curved surfaces and arbitrary thickness of the structure. The violin body shown below is composed of a top and a bottom plate with a thickness varying between 3.5 and 5.0 mm and side walls with a constant thickness of 2 mm.

**Figure 4: **All hexahedral mesh of a violin model

**Publications:**

**C. Sorger**, S. Kollmannsberger, E.Rank:

Visual DoMesh: Hexahedral meshing for thin curved solid structures. In: Proceedings of the 11th US National Congress of Computational Mechanics, Minneapolis, USA, 2011.**C. Sorger**, A. Duester und E. Rank:

Generation of curved high-order hexahedral finite element meshes for thin-walled structures. In: Proceedings of the 11th ISGG Conference on Grid Generation, Montreal, Canada. 2009**C. Sorger**, S. Kollmannsberger:*Refaktorisierung des Netz-Generators DoMesh.*In: Proceedings of the Forum Bauinformatik, Dresden, Germany 2008