Heterogeneous systems exist in a variety of scientific and engineering fields. Modeling of these complex systems often relies on the structural information acquired from imaging techniques. In recent decades, the development of imaging and material processing techniques has led to great improvements in material characterization and modeling. However, image-based models often contain a prohibitively large amount of data from high-resolution images. To this end, part of this dissertation focuses on developing an innovative image-based modeling technique, based on Google Earth like algorithms, to effectively resolve intricate morphologies and address the computational complexity associated with heterogeneous systems. The statistical and mechanical robustness of data compression is demonstrated through the corresponding numerical analysis.
The heterogeneity between phases increases the numerical difficulty by producing ill-conditioned algebraic problems that often cause severe stagnation of iterative solvers. To resolve this issue, part of this dissertation focuses on developing a robust computational framework for systems of algebraic equations associated with highly heterogeneous systems. A novel image-based multiscale multigrid solver is developed based on the aforementioned image-based multiresolution model, which enables reliable data flow between corresponding computational grids and provides significant data compression. A set of inter-grid operators is constructed based on the microstructural data which remedies the issue of missing coarse grid information. Moreover, an image-based multiscale preconditioner is developed from the multiscale coarse images which does not traverse through any intermediate grid levels and thus leads to a faster solution process. Finally, an image-based reduced order model is designed by prolongating the coarse-scale solution to approximate the fine-scale one with improved accuracy. The numerical robustness and efficiency of this image-based computational framework is demonstrated on a two-dimensional example with high degrees of data heterogeneity and geometrical complexity.
The data heterogeneity often results in highly localized solution characteristics that require resolution and accuracy well beyond what is required for the rest of the domain. To resolve this issue, the last part of this dissertation focuses on the development of an error-driven adaptive multiscale solver. This solver solves a macroscopic quasi-static boundary value problem in a domain that is composed of multiple multi-resolution microstructures. A dynamic refinement strategy is designed based on the local a posteriori error estimates, which enables adaptive model reduction. A staggered algorithm is developed to realize semi-synchronized convergence between the subdomain and the common-refinement-based interface. Preliminary numerical results are included to demonstrate this multiscale adaptive solution process.