A hybrid numerical methodology coupling Reduced Order Modeling and Graph Neural Networks for non-parametric geometries: applications to structural dynamics problems

TL;DR

The paper addresses the computational burden of time-domain elastodynamics simulations on non-parametric geometries by coupling reduced-order modeling (ROM) with Graph Neural Networks (GNNs).A novel GNN-PGD generator uses a GNN to predict a reduced-order basis (ROB) for a new geometry from a heterogeneous database, followed by Galerkin projection to obtain the space-time field efficiently, with a PGD-based offline compression forming each geometry's ROB.The approach is demonstrated on a crash-load seat design problem, achieving sub-5% space-time L2 errors for new seat geometries while significantly reducing training time compared to autoregressive GNNs, and showing generalization to topologies beyond the training set.The work offers a practical framework for rapid pre-dimensioning in engineering design where geometries vary widely, with potential extensions to incorporate FE operators into graph inputs and to address nonlinear material behavior.

Abstract

This work introduces a new approach for accelerating the numerical analysis of time-domain partial differential equations (PDEs) governing complex physical systems. The methodology is based on a combination of a classical reduced-order modeling (ROM) framework and recently-introduced Graph Neural Networks (GNNs), where the latter is trained on highly heterogeneous databases of varying numerical discretization sizes. The proposed techniques are shown to be particularly suitable for non-parametric geometries, ultimately enabling the treatment of a diverse range of geometries and topologies. Performance studies are presented in an application context related to the design of aircraft seats and their corresponding mechanical responses to shocks, where the main motivation is to reduce the computational burden and enable the rapid design iteration for such problems that entail non-parametric geometries. The methods proposed here are straightforwardly applicable to other scientific or engineering problems requiring a large number of finite element-based numerical simulations, with the potential to significantly enhance efficiency while maintaining reasonable accuracy.

Figures (20)

• Figure 1: Example illustrations of $N$ non-parameterized $\bm{\Omega}_p$ geometries with possibly different discretization sizes.
• Figure 2: A flowchart diagram introducing the steps of the proposed GNN-PGD methodology (detailed in Section \ref{['sec:methodology']}).
• Figure 3: Illustration of a Message-Passing layer on a three-stage graph. Neural networks symbolize functions whose parameters can be adjusted by learning (here applied to node three of the example graph of $n=7$ nodes).
• Figure 4: Effect of the presence of noise in the ROB on the resolution of the projected equation. Three modes have been used to generate the ROB.
• Figure 5: Loading of interest