## Adaptive Methods of Heterogeneous Materials

- Jan. 24, 2001
- 3:30 p.m.
- LeConte 412

## 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.