Bob Svendsen

Iman Peivaste, Nima H. Siboni, Ghasem Alahyarizadeh et al.

Phase-field-based models have become common in material science, mechanics, physics, biology, chemistry, and engineering for the simulation of microstructure evolution. Yet, they suffer from the drawback of being computationally very costly when applied to large, complex systems. To reduce such computational costs, a Unet-based artificial neural network is developed as a surrogate model in the current work. Training input for this network is obtained from the results of the numerical solution of initial-boundary-value problems (IBVPs) based on the Fan-Chen model for grain microstructure evolution. In particular, about 250 different simulations with varying initial order parameters are carried out and 200 frames of the time evolution of the phase fields are stored for each simulation. The network is trained with 90% of this data, taking the $i$-th frame of a simulation, i.e. order parameter field, as input, and producing the $(i+1)$-th frame as the output. Evaluation of the network is carried out with a test dataset consisting of 2200 microstructures based on different configurations than originally used for training. The trained network is applied recursively on initial order parameters to calculate the time evolution of the phase fields. The results are compared to the ones obtained from the conventional numerical solution in terms of the errors in order parameters and the system's free energy. The resulting order parameter error averaged over all points and all simulation cases is 0.005 and the relative error in the total free energy in all simulation boxes does not exceed 1%.

Sarthak Kapoor, Jaber Rezaei Mianroodi, Mohammad Khorrami et al.

The purpose of this work is the systematic comparison of the application of two artificial neural networks (ANNs) to the surrogate modeling of the stress field in materially heterogeneous periodic polycrystalline microstructures. The first ANN is a UNet-based convolutional neural network (CNN) for periodic data, and the second is based on Fourier neural operators (FNO). Both of these were trained, validated, and tested with results from the numerical solution of the boundary-value problem (BVP) for quasi-static mechanical equilibrium in periodic grain microstructures with square domains. More specifically, these ANNs were trained to correlate the spatial distribution of material properties with the equilibrium stress field under uniaxial tensile loading. The resulting trained ANNs (tANNs) calculate the stress field for a given microstructure on the order of 1000 (UNet) to 2500 (FNO) times faster than the numerical solution of the corresponding BVP. For microstructures in the test dataset, the FNO-based tANN, or simply FNO, is more accurate than its UNet-based counterpart; the normalized mean absolute error of different stress components for the former is 0.25-0.40% as compared to 1.41-2.15% for the latter. Errors in FNO are restricted to grain boundary regions, whereas the error in U-Net also comes from within the grain. In comparison to U-Net, errors in FNO are more robust to large variations in spatial resolution as well as small variations in grain density. On other hand, errors in U-Net are robust to variations in boundary box aspect ratio, whereas errors in FNO increase as the domain becomes rectangular. Both tANNs are however unable to reproduce strong stress gradients, especially around regions of stress concentration.

Mohammad S. Khorrami, Pawan Goyal, Jaber R. Mianroodi et al.

The purpose of the current work is the development of a so-called physics-encoded Fourier neural operator (PeFNO) for surrogate modeling of the quasi-static equilibrium stress field in solids. Rather than accounting for constraints from physics in the loss function as done in the (now standard) physics-informed approach, the physics-encoded approach incorporates or "encodes" such constraints directly into the network or operator architecture. As a result, in contrast to the physics-informed approach in which only training is physically constrained, both training and output are physically constrained in the physics-encoded approach. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the proposed PeFNO, a heterogeneous polycrystalline material consisting of isotropic elastic grains subject to uniaxial extension is considered. Stress field data for training are obtained from the numerical solution of a corresponding boundary-value problem for quasi-static mechanical equilibrium. This data is also employed to train an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO) for comparison. As confirmed by this comparison and as expected on the basis of their differences, the output of the trained PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the trained PgFNO or the trained PiFNO.

Mohammad S. Khorrami, Pawan Goyal, Soroush Motahari et al.

The purpose of the current work is the development of an approach to account for quasi-static mechanical equilibrium in empirical (i.e., data-based) models for the stress field employing neural approximations (NAs), which include neural networks (NNs) and neural operators (NOs), in particular Fourier NOs (FNOs). Rather than including such constraints from physics in the loss function as done in the (now standard) physics-informed approach, the current approach incorporates or "encodes" such constraints directly into the architecture of the NA. As a result, both NA training and output are physically constrained in the physics-encoded approach, in contrast to the physics-informed approach, in which only training is physically constrained. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the current approach, a physics-encoded FNO (PeFNO) is developed for a heterogeneous polycrystalline material consisting of isotropic elastic grains and subject to uniaxial extension. Stress field data for this purpose are obtained from the numerical solution of corresponding boundary-value problems for quasi-static mechanical equilibrium. For comparison with the PeFNO, this data is also employed to develop an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO). As expected theoretically, and confirmed by this computational comparison, for comparable accuracy of the stress field itself as compared to the data, the stress field output by the trained and tested PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the PgFNO or the PiFNO.