A Manifold Learning Approach for Integrated Computational Materials Engineering

Abstract

Image-based simulation is becoming an appealing technique to homogenize properties of real microstructures of heterogeneous materials. However fast computation techniques are needed to take decisions in a limited timescale. Techniques based on standard computational homogenization are seriously compromised by the real-time constraint. The combination of model reduction techniques and high performance computing contribute to alleviate such a constraint but the amount of computation remains excessive in many cases. In this paper we consider an alternative route that makes use of techniques traditionally considered for machine learning purposes in order to extract the manifold in which data and fields can be interpolated accurately and in real-time and with minimum amount of online computation. Locallly Linear Embedding-LLE-is considered in this work for the real-time thermal homogenization of heterogeneous microstructures.

Consistent data-driven computational mechanics

“…These methods eliminate the need for phenomenological constitutive equation fitting, a process that is frequently cumbersome. Instead, they work operating solely on experimental data [1,2,3,4]. However, the need to guarantee thermodynamic consistency continues to be an issue for these methods.…”

New Trends in Computational Mechanics: Model Order Reduction, Manifold Learning and Data‐Driven

“…This strategy was successfully considered in [LOP16] for addressing models involving parametrized microstructures and shapes. However, there is a strong assumption in the rationale just described.…”

“…where here x denotes the space coordinates involved in usual models and their associated partial differential equations, t the time involved in transient models and y j are the latent variables grouped in vector Y (defining the slow manifold). This procedure was successfully applied in [GON16] for addressing the same problems that were addressed in [LOP16].…”

Learning Corrections for Hyperelastic Models From Data

“…In some of our previous works, this approach is further generalized by defining the concept of constitutive manifold, a low-dimensional embedding for the stress-strain pairs (see Lopez et al, 2018). Thus, by alternating between stress-strain pairs that satisfy either equilibrium or the constitutive equation, the solution that satisfies the three families of equations is found, regardless of the non-linearity of the behavior.…”