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 propose a splitting of the monolithic FSI system into three smaller subsystems, in order to segregate the mesh motion. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem.
翻译:电磁变形是流体结构互动模拟和通过绘图方法优化形状的瓶颈。在这两种情况下,都需要适当的网状运动技术。选择通常基于湿度,例如部分差异方程式(PDE)的溶液操作者,如Laplace 或双调方程式。特别是后者,对于大型置换而言,其数字性能良好,是昂贵的。此外,从持续的角度来看,选择网状运动技术在某种程度上是任意的,对物理相关数量没有影响。因此,我们考虑了机器学习所启发的方法。我们提出了一个混合的PDE-NNN方法,即神经网络(NN)在二级非线性PDE中作为系数的参数化。我们通过选择神经网络结构,确保非线性PDE存在解决办法。此外,我们提议将单线型FSI系统分成三个较小的子系统,以便分离网状运动。我们通过将它应用于FSI基准问题来评估所学过的网状运动技术的质量。