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In recent years, there has been a growing interest in understanding complexmicrostructures and their effect on macroscopic properties. In general, it isdifficult to derive an effective constitutive law for such microstructures withreasonable accuracy and meaningful parameters. One numerical approach to bridgethe scales is computational homogenization, in which a microscopic problem issolved at every macroscopic point, essentially replacing the effectiveconstitutive model. Such approaches are, however, computationally expensive andtypically infeasible in multi-query contexts such as optimization and materialdesign. To render these analyses tractable, surrogate models that canaccurately approximate and accelerate the microscopic problem over a largedesign space of shapes, material and loading parameters are required. In thiswork, we develop a reduced order model based on Proper Orthogonal Decomposition(POD), Empirical Cubature Method (ECM) and a geometrical transformation methodwith the following key features: (i) large shape variations of themicrostructure are captured, (ii) only relatively small amounts of trainingdata are necessary, and (iii) highly non-linear history-dependent behaviors aretreated. The proposed framework is tested and examined in two numericalexamples, involving two scales and large geometrical variations. In both cases,high speed-ups and accuracies are achieved while observing good extrapolationbehavior.

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