Learning Markovian Homogenized Models in Viscoelasticity

Margaret Trautner, California Institute of Technology

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The macroscopic behavior of materials is governed in part by small-scale rapidly-varying material properties. Fully resolving these features within the balance laws thus involves expensive fine-scale computations which need to be conducted on macroscopic scales. The theory of homogenization provides an approach to derive effective macroscopic equations which eliminates the small scales by exploiting scale separation. An accurate homogenized model avoids the computationally-expensive task of numerically solving the underlying balance laws at a fine scale, thereby rendering a numerical solution of the balance laws more computationally tractable. In simple settings the homogenization produces an explicit formula for a macroscopic constitutive model, but in more complex settings it may only define the constitutive model implicitly. In these complex settings machine learning can be used to learn the constitutive model from localized fine-scale simulations. In the case of one-dimensional viscoelasticity, the linearity of the model allows for a complete analysis. For this case, we derive a homogenized constitutive model and develop a theory to prove that the model may be approximated by a recurrent neural network (RNN) model that captures the memory; this may be thought of as discovering appropriate internal variables. Simulations are presented which validate the theory, and additional numerical experiments demonstrate extension of the methodology to higher dimensions and to nonlinear viscoplasticity.

Abstract Author(s): Kaushik Bhattacharya, Burigide Liu, Andrew Stuart, Margaret Trautner