Reduced-order modeling for second-order computational homogenization with applications to geometrically parameterized elastomeric metamaterials

TL;DR

This paper introduces a reduced-order modeling approach for second-order computational homogenization of elastomeric metamaterials, significantly decreasing computational costs while maintaining accuracy in microstructure simulations.

Contribution

It develops a novel hyperreduction method based on proper orthogonal decomposition and empirical cubature, tailored for second-order homogenization problems.

Findings

Achieves speed-ups of about 100 times compared to full simulations.

Maintains high accuracy when the training data is representative.

Effectively approximates full second-order homogenization results.

Abstract

The structural properties of mechanical metamaterials are typically studied with two-scale methods based on computational homogenization. Because such materials have a complex microstructure, enriched schemes such as second-order computational homogenization are required to fully capture their non-linear behavior, which arises from non-local interactions due to the buckling or patterning of the microstructure. In the two-scale formulation, the effective behavior of the microstructure is captured with a representative volume element (RVE), and a homogenized effective continuum is considered on the macroscale. Although an effective continuum formulation is introduced, solving such two-scale models concurrently is still computationally demanding due to the many repeated solutions for each RVE at the microscale level. In this work, we propose a reduced-order model for the microscopic problem arising in second-order computational homogenization, using proper orthogonal decomposition and a novel hyperreduction method that is specifically tailored for this problem and inspired by the empirical cubature method. Two numerical examples are considered, in which the performance of the reduced-order model is carefully assessed by comparing its solutions with direct numerical simulations (entirely resolving the underlying microstructure) and the full second-order computational homogenization model. The reduced-order model is able to approximate the result of the full computational homogenization well, provided that the training data is representative for the problem at hand. Any remaining errors, when compared with the direct numerical simulation, can be attributed to the inherent approximation errors in the computational homogenization scheme. Regarding run times for one thread, speed-ups on the order of 100 are achieved with the reduced-order model as compared to direct numerical simulations.