Computational Physics Group

Karel Matous










A Nonlinear Manifold-based Reduced Order Model for Multiscale Analysis of Heterogeneous Hyperelastic Materials

S. Bhattacharjee and K. Matous

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


    A new manifold-based reduced order model for nonlinear problems in multiscale modeling of heterogeneous hyperelastic materials is presented. The model relies on a global geometric framework for nonlinear dimensionality reduction (Isomap), and the macroscopic loading parameters are linked to the reduced space using a Neural Network. The proposed model provides both homogenization and localization of the multiscale solution in the context of computational homogenization. To construct the manifold, we perform a number of large three-dimensional simulations of a statistically representative unit cell using a parallel finite strain finite element solver. The manifold-based reduced order model is verified using common principles from the machine-learning community. Both homogenization and localization of the multiscale solution are demonstrated on a large three-dimensional example and the local microscopic fields as well as the homogenized macroscopic potential are obtained with acceptable engineering accuracy.


   We have proposed here a novel manifold-based reduced order model for multiscale modeling of nonlinear hyperelastic materials in finite strain setting. The reduced order model is built on top of Isomap, which provides dimensionality reduction from the high-dimensional data stemming from large parallel computational homogenization simulations. The map between the set of macroscopic loading parameters and the reduced space is accomplished by the Neural Network. Once constructed, the manifold-based reduced order model provides both homogenization and localization of the multiscale solution for complex three-dimensional material domains. Thus, solution of the large microscale boundary value problem to arbitrary multiscale loading conditions is approximated quickly without the need for additional large parallel finite element simulations.
    In order to build the manifold, we employ the large parallel three-dimensional finite strains solver, PGFem3D, and perform several parallel simulations using a statistically representative unit cell. This statistically representative unit cell is generated by a packing algorithm. The multiscale loading conditions are simulated in terms of the macroscopic principal stretches as well as the orthogonal principal directions. Therefore, arbitrary multiscale loading conditions can be applied. In order to obtain a uniform discretization for rotational parameters, we employ the HEALPix grid. The manifold-based reduced order model is verified step by step using traditional machine-learning procedures. Next, we perform homogenization and localization of the multiscale solution for data points not in the digital database. Rigorous assessment of both homogenization and localization errors is performed. We show that the manifold-based reduced order model discovers the homogenized material response as well as provides microscopic fields of interest. In this work, we select the deformation gradient as the building block of the manifold, since the other physical fields (i.e. Almansi strain, Piola-Kirchhoff stress) can be easily derived.
    The proposed manifold-based reduced order model can be used for material classification and lends itself to predictive scientific studies as well as Virtual Materials Testing. It can also be used to accelerate fully coupled multiscale computational homogenization simulations (i.e. FE2). Future studies on pattern/physics based discretization are needed to improve this reduced order model. Moreover, the manifold-based reduced order model for problems with both geometric and material nonlinearities (e.g. plasticity, damage) is a natural next step.


This work was supported by the Multiscale Design Systems, LLC under the contract no. FA8651-14-C-0045 (Eglin Air Force Base, STTR Phase II project) by the Department of Defense, and by the Department of Energy, National Nuclear Security Administration, under the award no. DE-NA0002377.

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2015 Notre Dame and Dr. Karel Matous