A mesh-based Graph Neural Network approach for surrogate modeling of Lagrangian free surface fluid flows

The study of free surface fluid flows is of significant interest across various research fields, including civil, aerospace, and biomedical engineering. Among the numerical methods used to address free surface problems, the Particle Finite Element Method (PFEM) stands out as a robust and efficient approach. PFEM solves the governing equations using the standard finite element method while addressing mesh distortion through a fast and efficient remeshing procedure. In recent years, deep learning (DL) algorithms have demonstrated remarkable successes in learning from examples, and their application to datasets generated from numerical simulations could result in surrogate models able to reduce the computational cost of classical numerical methods. In the context of free surface fluid simulations, particularly noteworthy are attempts to employ Graph Neural Networks (GNNs) given their ability to process unstructured data that cannot be represented as structured grids, which are typical of these applications. In this work, we introduce NeuralPFEM (NPFEM), a GNN-based approach for surrogate modeling of free surface fluid simulations. NPFEM learns the system's temporal evolution in an autoregressive manner, preserving the same structure of a standard numerical solver. It inherits its hybrid nature from PFEM, combining features of particle-based and mesh-based methods. This hybrid approach distinguishes NPFEM from existing methods, such as the Graph Neural Simulator (GNS), which are purely particle-based. As a result, to construct the graph during training, NPFEM exploits the mesh connectivity already available in the dataset, while GNS must reconstruct graph connectivity at every training step based on particle distributions. During prediction, NPFEM employs PFEM mesh generation algorithm and particle redistribution tools to build the graph connectivity, ensuring a more uniform particle distribution within the domain and producing a mesh-based output solution. This approach preserves mesh quality and mitigates undesirable effects like particle clustering. We evaluate the results both qualitatively and quantitatively, comparing them with those obtained from PFEM. Moreover, we compute physical quantities out of the learned solution. In particular, the output mesh structure, combined with the joint prediction of the velocity and the pressure fields, facilitates the calculation of forces and stresses, a first step in the direction of applying this kind of tool to Fluid–Structure Interaction (FSI) problems.