Mesh optimization based on a Neo-Hookean hyperelasticity model

Abstract

For industrial applications CFD-simulations have become an important addition to experiments. To perform them the underlying geometry has to be created on a computer and embedded into a computational mesh. This mesh needs to meet certain quality criteria. This mesh needs to meet certain quality criteria, which are not commonly met by many conventional mesh-generation tool. However, automated tools are necessary to simulate experiments with a time-dependent geometry, for example a rising bubble or fluid-structure interaction. In this thesis, we study the automatic optimization of a mesh by minimizing a neohookean hyperelasticity model. The aim of our mesh optimization is to either smoothen a mesh, or to adapt a mesh to a given geometry. In the first part of the thesis we propose a specific energy model from this class and investigate if a solution to the minimization of this specific energy exists or not. To solve this minimization problem we need to develop an algorithm. After this we have to investigate if this specific energy function fulfills all requirements of this algorithm. The chosen algorithm is an adaptation of Newton’s method in a function space. To globalize the convergence of Newton’s method we use operator-adaption techniques in a Hilbert space. This makes the algorithm a Quasi-Newton-Method. We proceed to add other elements that are known from optimization so that the algorithm becomes even more robust. Finally we perform several numerical tests to investigate the performance of this method. In our studies we find that for a certain set of parameters the solution of the minimization problem exists. This set of parameters is limited, but the limits are reasonable for most practical use-cases. During our numerical tests we find the method to be stable and robust enough to automatically smoothen a mesh, but to adapt a given mesh to a given geometry our results are unclear: For simulations in two dimensions, the developed method seems to perform well and we get promising results with even just a type of Picard-iteration. For simulations in three dimensions, some adaptations might be necessary and more tests are required.