| Abstract |
Numerical modeling of structural elements and engineering materials is mostly based on the
so-called phenomenological approach, where the connection between stress and strain is
established via empirical, constitutive laws that are believed to capture certain experimentally
observed (at the macroscopic level) phenomena. The task of formulating appropriate
constitutive equations and designing and conducting experiments to calibrate the
corresponding constitutive parameters, while trivial for isotropic, linear materials, can become
extremely complicated, if not outright impossible, for anisotropic, history-dependent materials
well into their nonlinear regime.
An alternative in such cases is to adopt the so-called multi-scale approach, whereby
microscopic information can be incorporated into the constitutive description of the
macroscopic behaviour by means of homogenization techniques, where the macroscopic
strains and stresses at each point of the macroscopic continuum are volume averages of their
counterparts over a certain “microscopic cell" whose structure and composition
(heterogeneities, porosity, micro-agents …) is supposed to be statistically representative of the
material microstructure. In most cases, the microscopic cell, termed also representative
volume element (RVE), can be regarded as a continuum body itself.
In practical terms, however, the homogenization technique as such possesses a serious
inconvenient that detract from its applicability in daily engineering design: the enormous
computational cost. Indeed, this theory tells us that, in a typical finite element analysis, one
should solve, for each gauss or quadrature point a finite element problem for the RVE
geometry (using as driving variable the macroscopic strain). To alleviate the computational
burden associated with the solution of the finite element analysis at each gauss point, this
project proposes to exploit the advantages offered by the so-called model reduction
techniques, which, as its name indicates, permit to simplify models characterized by highdimensional
state or input parameter spaces to their essential dimensions or fundamental
modes, with a significantly reduced number of degrees of freedom and no substantial
reduction of accuracy. Model reduction allows precomputing certain information in a
preprocessing phase — off-line computations. In the case at hand, such precomputations will
consist in, first, several finite element analysis of the microcell structure under varying
macroscopic strains (the basic parameter of the problem); and, then, in the application of the
corresponding model reduction algorithm to condense the information generated by the finite
element simulations. Somehow, these off-line computations replace the experiments in the
laboratory.
The present project intends also to develop a new double-reduction (or hyperreduction)
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technique that it is hoped to diminish the computational cost of a multiscale analysis to the
same order to that of a phenomenological analysis. In order to check the performance of the
method it will be applied to a number of mechanical, civil and construction engineering
problems, i.e.:
• Modelling of the structural and failure behavior of concrete reinforced with fibers.
• Simulation of granular flows, like those occurring in silos or in soil instabilities
involving soil slides and avalanches of granular or quasi-granular materials (snow or
debris).
• Simulation of the compaction of powder materials like in powder metallurgy and
pharmaceutics fabrication. |
