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
previous works, such models were constructed in a data-driven manner using
methods such as Neural Networks (NN) or Gaussian Process Regression (GPR).
However, these approaches currently suffer from issues, such as need for large
amounts of training data, lack of physics, and considerable extrapolation
errors. 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