Multiscale Problems in Materials Science: A Mathematical Approach to the Role of Uncertainty
Abstract
The bottom line of this work is to develop affordable numerical methods in the context of stochastic homogenization. Many partial differential equations of materials science indeed involve highly oscillatory coefficients and small length-scales. Homogenization theory is concerned with the derivation of averaged equations from the original oscillatory equations, and their treatment by adequate numerical approaches. Stationary ergodic random problems (and the associated stochastic homogenization theory) are one instance for modelling uncertainty in continuous media. The theoretical aspects of these problems are now well-understood, at least for a large variety of situations. On the other hand, the numerical aspects have received less attention from the mathematics community. Standard methods available in the literature often lead to very, and sometimes prohibitively, costly computations. In this report, we first review an approach popular in particular in the computational mechanics community, which is to try and obtain bounds on the homogenized matrix, rather than computing it. Only computations of moderate difficulty are then required. However, we will show that, not unexpectedly, this method has strong limitations. We will next introduce a class of materials of significant practical relevance, that of random materials where the amount of randomness is small. They can be considered as stochastic perturbations of deterministic materials, in a sense made precise below. We will adapt to such a case the well-known Multiscale Finite Element Method (MsFEM), and design a method which is much more affordable than, and as accurate as, the original method.
Document Details
- Document Type
- Technical Report
- Publication Date
- Oct 01, 2010
- Accession Number
- ADA534993
Entities
People
- Claude Le Bris
- Florian Thomines
- Frederic Legoll
Organizations
- École des Ponts ParisTech