Mathematical Research & Services

Projekt im Fokus

Scientific engineering and physical questions are often formulated by partial differential equations and solved using the Finite Element Method (FEM). In this method, the given calculation domain is first meshed, i.e. broken down into simple geometric elements, such as triangles or quadrilaterals in case of surface models, or tetrahedra or hexahedra in case of volume models. These elements are used as the basis for defining the solution function, the coefficients of which are to be determined by the FE procedure. Depending on the quality of the resulting mesh, this is followed by a mesh optimisation step, which may be integrated, if applicable, into the mesh generation procedure. Then, taking into account boundary conditions such as loads, fixations, etc., a linear system of equations is set up during the FE simulation process to determine the solution coefficients. Subsequently, the resulting FE system is solved using special numerical methods and the solution of the simulation problem is evaluated.

Starting point: Obtaining high-quality meshes at geometric complexity

In this process, the quality of the meshing has a decisive influence on the efficiency and accuracy of FE simulation. As a rule, meshes with elements that are as regular as possible are desirable in order to avoid element angles that are too small or too large, since these lead to an increase in the condition number of the stiffness matrix and thus to poorer solvability of the resulting FE system or to inaccuracies in the resulting solution. The generation of high-quality meshes becomes more and more problematic with increasing geometric complexity.

TWT approach to solving the problem: GETMe

TWT's Mathematical Research & Services department developed the Geometric Element Transformation Method (GETMe) for smoothing finite element meshes. In this method, the quality improvement is achieved exclusively by repositioning mesh nodes without changing the mesh topology, i.e. the connectivity structure of the mesh elements is maintained. This is exemplified in the figure below for an outer mesh consisting of hexahedrons of the Aletis open passenger car developed by TWT. In the figure, the elements are coloured according to their regularity. Regularity was measured using a regularity measure, taking the value 0 (red) for degenerated elements and 1 (blue) for regular elements. In particular, elements with small quality numbers should be avoided, as these can lead to instabilities and inaccuracies in the finite element calculation. 

Mesh smoothing by GETMe is based on the use of geometric transformations for polygons and polyhedra, which, when applied iteratively, successively transform problematic elements into regular and thus higher-order elements. In principle, the procedure is suitable for the improvement of the most common FE mesh types.

Outcome

In the publications authored by TWT, it was possible to demonstrate through numerous numerical tests and mathematical proofs that GETMe achieves mesh qualities previously only attainable with global optimisation methods. However, GETMe yields a significant speed advantage, since global optimisation methods require considerably more computing power due to their mathematical optimisation approach.