Abstract

The present work is concerned with the application of projection-based,
model reduction techniques to the ecient solution of the cell equilibrium
equation appearing in (otherwise prohibitively costly) two-scale, computational
homogenization problems. The main original elements of the proposed
Reduced-Order Model (ROM) are fundamentally three. Firstly, the reduced
set of empirical, globally-supported shape functions are constructed from
pre-computed Finite Element (FE) snapshots by applying, rather than the
standard Proper Orthogonal Decomposition (POD), a partitioned version
of the POD that accounts for the elastic/inelastic character of the solution.
Secondly, we show that, for purposes of fast evaluation of the nonane
term (in this case, the stresses), the widely adopted approach of replacing
such a term by a low-dimensional interpolant constructed from POD modes,
obtained, in turn, from FE snapshots, leads invariably to ill-posed formulations.
To safely avoid this ill-posedness, we propose a method that consists
in expanding the approximation space for the interpolant so that it embraces
also the gradient of the global shape functions. A direct consequence
of such an expansion is that the spectral properties of the Jacobian matrix
of the governing equation becomes a ected by the number and particular
placement of sampling points used in the interpolation. The third innovative
ingredient of the present work is a points selection algorithm that does
acknowledge this peculiarity and chooses the sampling points guided, not
only by accuracy requirements, but also by stability considerations. The
eciency of the proposed approach is critically assessed in the solution of
the cell problem corresponding to a highly complex porous metal material
under plane strain conditions. Results obtained convincingly show that the
computational complexity of the proposed ROM is virtually independent of
the size and geometrical complexity of the considered representative volume,
and this a ords gains in performance with respect to nite element analyses
of above three orders of magnitude without signi cantly sacri cing accuracy
|hence the appellation High-Performance ROM.