DeepBND: a Machine Learning approach to enhance Multiscale Solid Mechanics.
2021
Effective properties of materials with random heterogeneous structures are
typically determined by homogenising the mechanical quantity of interest in a
window of observation. The entire problem setting encompasses the solution of a
local PDE and some averaging formula for the quantity of interest in such
domain. There are relatively standard methods in the literature to completely
determine the formulation except for two choices: i) the local domain itself
and the ii) boundary conditions. Hence, the modelling errors are governed by
the quality of these two choices. The choice i) relates to the degree of
representativeness of a microscale sample, i.e., it is essentially a
statistical characteristic. Naturally, its reliability is higher as the size of
the observation window becomes larger and/or the number of samples increases.
On the other hand, excepting few special cases there is no automatic guideline
to handle ii). Although it is known that the overall effect of boundary
condition becomes less important with the size of the microscale domain, the
computational cost to simulate such large problem several times might be
prohibitive even for relatively small accuracy requirements. Here we introduce
a machine learning procedure to select most suitable boundary conditions for
multiscale problems, particularly those arising in solid mechanics. We propose
the combination Reduced-Order Models and Deep Neural Networks in an offline
phase, whilst the online phase consists in the very same homogenisation
procedure plus one (cheap) evaluation of the trained model for boundary
conditions. Hence, the method allows an implementation with minimal changes in
existing codes and the use of relatively small domains without losing accuracy,
which reduces the computational cost by several orders of magnitude.
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