A reduced order model for geometrically parameterized two-scale simulations of elasto-plastic microstructures under large deformations

TL;DR

This paper introduces a reduced order modeling framework using POD, ECM, and geometric transformations to efficiently simulate large deformation elasto-plastic microstructures with significant shape variations, enabling faster and accurate multiscale analysis.

Contribution

The work presents a novel reduced order model that captures large microstructural shape variations with minimal training data and handles non-linear behaviors, improving computational efficiency in multiscale simulations.

Findings

Achieved high speed-ups in simulations.

Maintained accuracy with large shape variations.

Demonstrated good extrapolation behavior.

Abstract

In recent years, there has been a growing interest in understanding complex microstructures and their effect on macroscopic properties. In general, it is difficult to derive an effective constitutive law for such microstructures with reasonable accuracy and meaningful parameters. One numerical approach to bridge the scales is computational homogenization, in which a microscopic problem is solved at every macroscopic point, essentially replacing the effective constitutive model. Such approaches are, however, computationally expensive and typically infeasible in multi-query contexts such as optimization and material design. To render these analyses tractable, surrogate models that can accurately approximate and accelerate the microscopic problem over a large design space of shapes, material and loading parameters are required. In this work, we develop a reduced order model based on Proper Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a geometrical transformation method with the following key features: (i) large shape variations of the microstructure are captured, (ii) only relatively small amounts of training data are necessary, and (iii) highly non-linear history-dependent behaviors are treated. The proposed framework is tested and examined in two numerical examples, involving two scales and large geometrical variations. In both cases, high speed-ups and accuracies are achieved while observing good extrapolation behavior.