Multi-scale modelling of heterogeneous materials with fixed and evolving microstructures

Many engineering and scientific problems require a full understanding of physical phenomena that span a wide spectrum of spatial length scales. These multi-scale problems cannot be analysed readily under the classical continuum mechanics framework. While classical methods have been proposed to study the physical phenomena on smaller scales and the resulting information has been transferred by homogenization techniques onto larger scales, effective methods to explicitly couple information at multiple length scales are still lacking. Passing information between physical models at different length scales requires mathematically consistent and physically meaningful formulation and numerical techniques. This paper presents a class of multi-scale mathematical and computational formulations as well as homogenization and localization numerical procedures for multi-scale modelling of (1) materials with fixed microstructures, and (2) problems with evolving microstructures such as stressed grain growth processes in polycrystalline materials. Waveletbased computational methods are introduced for multi-scale modelling and homogenization of materials with fixed microstructures. Two wavelet-based methods, the wavelet Galerkin method and wavelet projection method, are presented. For problems with evolving microstructures, a multi-scale variational formulation based on an asymptotic expansion method and a double-grid numerical method is proposed.