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

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.

Figures (13)

• Figure 1: Two-scale formulation in second-order computational homogenization. At every macroscopic point, the deformation gradient $\mkern 1.5mu\overline{\mkern-1.5mu\bm{F}\mkern-1.5mu}\mkern 1.5mu$ and its gradient $\mkern 1.5mu\overline{\mkern-1.5mu\bm{\mathcal{G}}\mkern-1.5mu}\mkern 1.5mu$ are used to prescribe boundary conditions for the microscopic problem which, after solving, returns an effective stress $\mkern 1.5mu\overline{\mkern-1.5mu\bm{P}\mkern-1.5mu}\mkern 1.5mu$ and higher-order stress $\mkern 1.5mu\overline{\mkern-1.5mu\bm{Q}\mkern-1.5mu}\mkern 1.5mu$. The parameter $\bm{\mu}$ describes the geometry of the RVE. More information on $\bm{\mu}$ are provided in \ref{['subsec:second_order_micro']}.
• Figure 2: Family of RVE domains $\Omega^{\bm{\mu}}$ parameterized through parameters $\bm{\mu}$. A parent domain $\Omega^{\text{p}}$ can be defined which can be transformed through transformations $\bm{\Phi}_{\bm{\mu}}$ into any of the RVE domains $\Omega^{\bm{\mu}}$.
• Figure 3: Example geometries for (a) $\zeta=-0.075mm$, (b) $\zeta=-0.035mm$, (c) $\zeta=0.025mm$, and (d) $\zeta=0.055mm$. The control points defining the hole shapes are shown in orange and the matrix material in blue. Depending on $\zeta$, the shape of the holes is more circular or square-like, and the RVE more prone to local or global buckling. The parent domain $\Omega^{\text{p}}$ is chosen for $\zeta=0.025mm$ with a simulation mesh consisting of 1066 six-noded triangular elements with 4882 DOFs.
• Figure 4: DNS solutions for $\zeta=-0.035mm$ in (a), $\zeta=0.03mm$ in (b). In each panel, the undeformed (left) and deformed states at 4% (middle) and 7.5% (right) compression are shown. For $\zeta=-0.035mm$ the structure first buckles locally and then globally, while for $\zeta=0.03mm$ the structure first buckles globally with subsequent local patterning.
• Figure 5: Decay of standardized norm of residuals $r_1$, $r_2$, $r_3$ and $r_4$, defined in \ref{['eq:ecm_tolerances']}, over the number of selected integration points $Q$ with $N=48$, $M=28$, $L=28$ and $\varepsilon_1=\varepsilon_2=\varepsilon_3=\varepsilon_4={1}\times 10^{-4}$. The hyperparameters $c_1$, $c_2$ and $c_3$ control the rate of decay for each $r_i$, which results in a different number of required quadrature points to reach the same level of accuracy: (a) $Q=543$, (b) $Q=358$, whereas (c) only $Q=297$ quadrature points are selected. The same $x$-axis range is used for all plots to highlight that much fewer quadrature points are selected for different combinations of $c_i$.

Theorems & Definitions (1)

• Remark