There is a high interest in accelerating multiscale models using data-driven surrogate modeling techniques. Creating a large training dataset encompassing all relevant load scenarios is essential for a good surrogate, yet the computational cost of producing this data quickly becomes a limiting factor. Commonly, a pre-trained surrogate is used throughout the computational domain. We introduce an alternative adaptive mixture approach that uses a fast probabilistic surrogate model as a constitutive model when possible, but resorts to the true high-fidelity model when necessary. The surrogate is thus not required to be accurate for every possible load condition, enabling a significant reduction in the data collection time. We achieve this by creating phases in the computational domain corresponding to the different models. These phases evolve using a phase-field model driven by the surrogate uncertainty. When the surrogate uncertainty becomes large, the phase-field model causes a local transition from the surrogate to the high-fidelity model, maintaining a highly accurate simulation. We discuss requirements for accuracy and numerical stability and compare the phase-field model to a local approach that does not enforce spatial smoothness in phase mixing. Using a Gaussian Process surrogate for an elasto-plastic material, we demonstrate the potential of this mixture of models to accelerate multiscale simulations.

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. In addition, this approach introduces beneficial inductive bias to the model by encoding microscopic geometrical features. 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. While the microscopic stresses are fully dependent on the microscopic strains, we found it crucial to include both microscopic strains and stresses in the loss function. 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.

MUDE stands for Modelling, Uncertainty and Data for Engineers, a required module in the MSc programs from the faculty of Civil Engineering and Geosciences at Delft University of Technology in the Netherlands.

The current version of the MUDE Textbook can be found at mude.citg.tudelft.nl/book and the most recent "complete" version is mude.citg.tudelft.nl/book/2024. Additional information about the book and its contents can be found on the Credits Page from 2024; technical information about the book and its source code can be found in the README of the GitHub repository TUDelft-MUDE/book. General information about MUDE can be found at mude.citg.tudelft.nl.

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