# Mathematical Research & Services

Objective: High quality industrial meshes

Problems in engineering and phyics are often formulated using partial differential equations and solved via the finite element method (FEM). In this method, the domain is first meshed, that is, decomposed in simple geometric elements like triangles and quadrangles for surfaces and tetrahedra and hexahedra for three-dimensional models. These elements form the basis for the definition of a solution function, whose coefficients are to be determined by the FE method. Depending on the quality of the resulting mesh, these meshes might need to be optimized, which could also be integrated in the mesh generation process. This is followed by setting up a linear system of equations in the FE simulation process by incorporating boundary conditions like loads, fixations etc. After the resulting FE system is solved using certain numerical methods, the solution to the simulation problem is evaluated.

In this process, the mesh quality is a critical factor for the efficiency and accurateness of the FE simulation. Usually, regular elements are preferred in order to avoid small or big angles. These extreme angles would increase the condition number of the stiffness matrix and result in inaccurate solutions. The problems in the generation of high quality meshes increase the geometric complexity.

TWT approach: "GETMe"

The Department of Mathematical Research & Services of TWT developed the geometric element transformation method (GETMe) for smoothing of finite element meshes. In the procedure the quality is improved by repositioning the nodes of the mesh and preserving the mesh connectivity. This is exemplified by the figure below for a hexahedral decomposition of the complement of the car Aletis developed by TWT. In the figure, the elements are colored in terms of the regularity. These were measured with the help of a regularity measure, which vanishes for degenerate elements (red) and assumes the value 1 for regular elements (blue). Elements with small numbers need to be avoided, because they can lead to instability and inaccurateness of the finite element computation.

GETMe mesh smoothing is based upon the use of geometric transformations for polygons and polyhedra, which iteratively converge towards regular and therefore higher quality elements. In principle, this method is suitable for the optimization of the most common FE mesh types. Result

TWT has proved through numerous numerical tests and mathematical proofs, that GETMe yields meshes of superior quality, which could thus far only be acchieved by global optimization-based methods. However, GETMe shows a significant run-time advantage, because global optimization require more computations by its mathematical optimization approach.