A model order reduction approach to construct efficient and reliable virtual charts in computational homogenisation

Computational homogenisation is a widely spread technique to calculate the overall properties
of a composite material from the knowledge of the constitutive laws of its microscopic constituents
[1, 2]. Indeed, it relies on fewer assumptions than analytical or semi-analytical homogenisation approaches
and can be used to coarse-grain a large range of micro-mechanical models. However, this accuracy comes
at large computational costs, which prevents computational homogenisation from being used routinely in
optimisation, even in the context of linear elastic materials. Indeed, a unit cell problem has to be solved
for each microscopic distribution of interest in order to obtain the corresponding homogenised material
constants. In the context of nonlinear, time-dependant problem, the computational effort becomes even
greater as computational homogenisation requires solving for the time-evolution of the microstructure at
every point of the macroscopic domain. In this paper, we propose to address these two issues within the unified framework of projection-based
model order reduction (see for instance [3, 4, 5, 6]). The smoothness of the solution of the unit cell problem with respect to parameter or time variations is used to create a reduced order model with very
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