Learning mesh motion techniques with application to fluid-structure interaction

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shapeoptimization via the method of mappings. In both cases, an appropriate mesh motion techniqueis required. The choice is typically based on heuristics, e.g., the solution operators of partialdifferential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter,which shows good numerical performance for large displacements, is expensive. Moreover,from a continuous perspective, choosing the mesh motion technique is to a certain extentarbitrary and has no influence on the physically relevant quantities. Therefore, we considerapproaches inspired by machine learning. We present a hybrid PDE-NN approach, where theneural network (NN) serves as parameterization of a coefficient in a second order nonlinearPDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neuralnetwork architecture. Moreover, we present an approach where a neural network corrects theharmonic extension such that the boundary displacement is not changed. In order to avoidtechnical difficulties in coupling finite element and machine learning software, we work witha splitting of the monolithic FSI system into three smaller subsystems. This allows to solve themesh motion equation in a separate step. We assess the quality of the learned mesh motiontechnique by applying it to a FSI benchmark problem. In addition, we discuss generalizabilityand computational cost of the learned mesh motion operators