A Microstructure-based Graph Neural Network for Accelerating Multiscale Simulations

TL;DR

The paper tackles the high cost of concurrent multiscale FE analyses by introducing a GNN-based surrogate that predicts full-field microscopic strains and uses an embedded constitutive model to compute stresses, thereby preserving multiscale physics while enabling interchange with FE solvers. It proposes a dual-graph GNN architecture with physics-based material updates, trained end-to-end on time-stepped data, and demonstrates strong performance on monotonic and non-monotonic elasto-plastic paths with good generalization to unseen microstructures. Key contributions include embedding the material model inside the surrogate, autoregressive time stepping, and showing scalability across increasing microstructure size, with explicit comparisons to FE$^2$ in computation time. The results indicate that hybrid data-physics surrogates can accelerate multiscale simulations while maintaining essential microscopic information, offering practical speedups for engineering applications.

Abstract

Simulating the mechanical response of advanced materials can be done more accurately using concurrent multiscale models than with single-scale simulations. However, the computational costs stand in the way of the practical application of this approach. The costs originate from microscale Finite Element (FE) models that must be solved at every macroscopic integration point. A plethora of surrogate modeling strategies attempt to alleviate this cost by learning to predict macroscopic stresses from macroscopic strains, completely replacing the microscale models. In this work, we introduce an alternative surrogate modeling strategy that allows for keeping the multiscale nature of the problem, allowing it to be used interchangeably with an FE solver for any time step. Our surrogate provides all microscopic quantities, which are then homogenized to obtain macroscopic quantities of interest. We achieve this for an elasto-plastic material by predicting full-field microscopic strains using a graph neural network (GNN) while retaining the microscopic constitutive material model to obtain the stresses. This hybrid data-physics graph-based approach avoids the high dimensionality originating from predicting full-field responses while allowing non-locality to arise. By training the GNN on a variety of meshes, it learns to generalize to unseen meshes, allowing a single model to be used for a range of microstructures. The embedded microscopic constitutive model in the GNN implicitly tracks history-dependent variables and leads to improved accuracy. We demonstrate for several challenging scenarios that the surrogate can predict complex macroscopic stress-strain paths. As the computation time of our method scales favorably with the number of elements in the microstructure compared to the FE method, our method can significantly accelerate FE2 simulations.

Figures (23)

• Figure 1: An overview of an FE$^\mathrm{2}$ simulation. A microscopic model $\omega$ is solved for every integration point of the macroscopic model $\Omega$.
• Figure 2: A dual graph is created over the mesh, coinciding with the integration points. Each node of this graph has unique features providing information about a number of the closest surrounding voids through their distances $\Delta x$ and $\Delta y$. We consider the nine closest voids, but we only show distances to three here for clarity. Periodicity is considered in both the void features and the edge connections - nodes at opposite sides of the microstructure are connected.
• Figure 3: Abstract overview of the multi-step GNN predictions.
• Figure 4: Visualization of the network architecture. The rectangles represent weight layers, where the height roughly indicates their number of weights. The center layer in $F^U$ represents dropout.
• Figure 5: Examples of stress-strain curves obtained from simulating one to nine void microstructures with a monotonic strain increase.