Mixing Data-Driven and Physics-Based Constitutive Models using Uncertainty-Driven Phase Fields

TL;DR

This work tackles the high cost of multiscale simulations by blending a fast Gaussian Process surrogate with a high-fidelity elasto-plastic model through a phase-field that is driven by surrogate uncertainty. A staggered, uncertainty-guided scheme facilitates a smooth, spatially varying mixture that switches to the HF model only where needed, thereby maintaining accuracy with fewer HF evaluations. The method is shown to be robust under various phase-field parameters and outperforms purely local switching in stability, with results highlighting the importance of surrogate data quality and interface width. Practically, this framework enables data-efficient acceleration of FE2-like simulations by adaptively leveraging data-driven surrogates without sacrificing fidelity, and it can incorporate a range of surrogate models beyond Gaussian Processes.

Abstract

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. Here, we introduce an alternative adaptive mixture approach that uses a fast probabilistic surrogate model as constitutive model when possible, but resorts back 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 the requirements of this approach to achieve accurate and numerically stable results and compare the phase-field model to a purely local approach that does not enforce spatial smoothness for the 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.

Figures (40)

• Figure 1: Overview of how the mixture constitutive model interacts with the other models. The tangent stiffness $\boldsymbol{D}$ is omitted for clarity.
• Figure 2: As $\phi$ increases, the constitutive model $\mathcal{C}_{mix}$ gradually switches from $\mathcal{C}_{\mathrm{GP}}$ to $\mathcal{C}_{\mathrm{HF}}$.
• Figure 3: The staggered updating approach used in this work. A prescribed displacement $\Delta u^*$ is selected using an adaptive time-step function, before updating the phase field based on $\mathcal{U}$. Then the mechanical problem is solved. The phase-field and mechanical problem are updated iteratively until convergence, or $k_{max}$ is reached.
• Figure 4: $\sigma_{xx}$
• Figure 5: $\sigma_{yy}$