# Mesh generation for mixed structures

Project team leader |
Dr.-Ing. Stefan Kollmannsberger |

PhD students |
Benjamin Wassermann, MSc, Dr.-Ing. Christian Sorger MSc |

Principal investigators |
Prof. Dr. rer. nat. Ernst Rank Prof. Dr.-Ing. Casimir Katz (SOFiSTiK AG) |

Project partners |
SOFiSTiK AG |

**Project description**

The Finite Element Method (FEM) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as integral equations as they arise in many problem formulations in fluid and in solid mechanics. The basic idea of the FEM is to discretize complex structural domains by a surface or volume mesh consisting of a finite number of triangular, quadrilateral, tetrahedral, pentahedral and hexahedral elements and to render the governing PDEs into an approximating system of ordinary differential equations, which are then solved using numerical integration and solution techniques.

In the *h*-version of the FEM complex domains are discretized by introducing a large number of linear finite elements sharing the same boundary nodes to ensure conformity. All physical quantities are defined in the nodes of an element; the distribution of the physical quantity inside the elements is determined by linear interpolation. In the *h*-version of the FEM the results of the simulation and the geometry of the structure are exactly described in the nodes, an improvement of the accuracy is achieved by locally or globally refining the mesh.

The idea of the *p*-version of the FEM is to increase the accuracy of the solution by increasing the polynomial degree of the solution space locally defined on each single finite element. By this it is possible to compute accurate solutions for large and complex domain with only a few numbers of finite elements. In case of complex and curved structural domains therefore it is necessary that the single elements describe the geometry of the original structure exactly. This can be achieved either by coupling the geometric description of the original domain with the geometric description of each element, or by approximating the original structure with curved finite elements using polynomial formulations.

In terms of surface meshing the research fields have been widely exploited and also the generation of tetrahedral elements is mostly solved by many different efficient and powerful algorithms. Nevertheless, the generally valid computation of qualitatively good hexahedral elements is still afflicted with difficulties and lots of research projects have been investigated in finding a general solution.

Mesh example of a multistory building |

**Publications**

- C. Sorger, F. Frischmann, S. Kollmannsberger and E. Rank:

"TUM.GeoFrame: Automated high-order hexahedral mesh generation for shell-like structures",*Engineering with Computers*, online 13.09.2012, 2012. [More] [Digital version] -
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

Contact: Benjamin Wassermann, M.Sc.