Learning Mesh Motion Techniques with Application to Fluid-Structure Interaction

TL;DR

This paper introduces a machine learning-inspired hybrid PDE-neural network approach for mesh motion in fluid-structure interaction simulations, aiming to improve efficiency and flexibility over traditional PDE-based methods.

Contribution

It proposes a novel hybrid PDE-NN method for mesh motion, ensuring solution existence and boundary condition preservation, with a modular solution approach for better integration.

Findings

The learned mesh motion technique performs well on FSI benchmark problems.

The approach offers potential improvements in computational cost and generalizability.

The method maintains boundary conditions while adapting to large displacements.

Abstract

Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential 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 extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem. In addition, we discuss generalizability and computational cost of the learned mesh motion operators.