Computational Physics Group

Karel Matous










A nonlinear data-driven reduced order model for computational homogenization with physics/pattern-guided sampling

S. Bhattacharjee and K. Matous

Center for Shock Wave-processing of Advanced Reactive Materials

Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame,
Indiana 46556, USA.


   Developing an accurate nonlinear reduced order model from simulation data has been an outstanding research topic for many years. For many physical systems, data collection is very expensive and the optimal data distribution is not known in advance. Thus, maximizing the information gain remains a grand challenge. In a recent paper, Bhattacharjee and Matous (2016) proposed a manifold-based nonlinear reduced order model for multiscale problems in mechanics of materials. Expanding this work here, we develop a novel sampling strategy based on the physics/pattern-guided data distribution. Our adaptive sampling strategy relies on enrichment of sub-manifolds based on the principal stretches and rotational sensitivity analysis. This novel sampling strategy substantially decreases the number of snapshots needed for accurate reduced order model construction (i.e., ~ 5x reduction of snapshots over Bhattacharjee and Matous (2016)). Moreover, we build the nonlinear manifold using the displacement rather than deformation gradient data. We provide rigorous verification and error assessment. Finally, we demonstrate both localization and homogenization of the multiscale solution on a large particulate composite unit cell.


   We have designed and implemented a novel physics-guided adaptive sequential sampling technique in the context of the manifold-based reduced order model, MNROM, for multiscale modeling of nonlinear hyperelastic materials. The MNROM relies on different nonlinear maps, established through Isomap for dimension reduction, kernel inverse/reconstruction map and a NN. This manifold-based ROM not only drastically speeds up the CH process (i.e., by avoiding additional large parallel finite element simulations in the FE2 setting), but it also provides both homogenization and localization of the multiscale analysis for complex three-dimensional hyperelastic materials.
    Although MNROM is a promising data-driven approach, it lacks an efficient means of data sampling. Thus, we have developed a novel physics-driven sampling strategy that mitigates this issue. Our approach is a stepwise data enrichment method, intended to minimize the sparse regions of the high-dimensional manifolds which effectively maximizes information gain in each step. The method couples input and output space and explores the pattern in the output space, which is directly guided by the inherent physics in the input space (i.e., the macroscale loading condition). We learn the pattern in each step based on the current dataset which determines how extra simulations are decided and added to the database. In this work, we have used the same framework of MNROM, where the multiscale loading conditions are simulated in terms of the macroscopic principal stretches as well as the orthogonal
principal directions, while the high-dimensional manifold has been constructed from the microscopic displacement fields. Moreover, we have shown the effectiveness of the proposed sampling technique through meticulous numerical simulations.
    In the numerical example, we observed that the data distribution becomes a smooth stepwise data enrichment and accordingly, all the maps improved significantly. This physics-driven deterministic sampling strategy also appreciably reduced the number of required simulations by eliminating redundant data points. Thus, the computational complexity of MNROM is reduced enormously. Also, we have shown that with the proper enrichment the additional simulations can be added to the dataset seamlessly using a greedy algorithm. Moreover, we can observe that even the localization process produced extremely small error with a very small number of simulations. This sampling strategy is not only helpful for CH and FE2 acceleration, but can potentially be applied to many other physical problems by identifying the influential parameters. The development of MNROM that considers also path-dependent material nonlinearity (e.g., visco-plasticity) is an important future direction.


This work was supported by the Department of Energy, National Nuclear Security Administration, under the reward No. DE-NA0002377 as part of the Predictive Science Academic Alliance Program II. We would like to thank two anonymous reviewers for providing useful suggestions and comments that improved the quality of the manuscript.

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