Abstract:
The selection of a mathematical or computational model to describe a
physical event remains the most arbitrary step in computational
engineering, judgement, and experience of the modeler and is often based
on heuristics or empirical data. Yet, the choice of an appropriate model
is the most critical and important step in mathematical and computational
science.
In this lecture, the notion of hierarchical modeling is presented in which
an appropriate model is systematically selected from a broad class of
plausible models. The lecture focuses on a particular class of
multi-scale problems in the mechanics of solids: heterogeneous elastic
materials. We address a classical and largely unsolved problem: given a
structural component constructed of heterogeneous elastic material that is
in equilibrium under the action of applied loads, determine local
micro-mechanical features of its response (e.g. local stresses and
displacements in or around phase boundaries or in inclusions) to an
arbitrary present level of accuracy, it being understood that the
microstructure is a "priori" unknown, may be randomly distributed, may
exist at multiple spatial scales, and may contain millions, even billions,
of microscale components. An approach is presented wihc leads to results
of arbitrary accuracy independently of the number of microscale components
of constituents.
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