FASTMM · FAST SEMI-ANALYTIC MULTISCALE METHODS FOR MULTISCALE ELLIPTIC PROBLEMS

FAST SEMI-ANALYTIC MULTISCALE METHODS FOR MULTISCALE ELLIPTIC PROBLEMS

The main goal of this research is to substantially reduce the computational requirements of multilevel methods for heterogeneous media by incorporating state of the art analytical techniques from homogenization theory into multilevel methods. A key scientific contribution is the coupling of analytical concepts from homogenization theory, previously published by the Researcher, to state of the art numerical methods for large scale systems in order to decrease substantially the computational complexity. The work is motivated by clear scientific and technological basis: developing fast and reliable multilevel methods for general diffusion (scalar) and elasticity (coupled) problems is an important topic in applied mathematics, computational physics and engineering. The result of the performed work is a significant reduction of computational resources needed for solving problems with complex multiscale morphology. The rationale of the research plan is that multilevel iterative schemes, such as multigrid, provide a very natural and efficient framework for incorporating fine-scale multiscale phenomena into coarse scales. This study brings an approach to make these schemes more feasible to large scale simulations, by achieving substantial speed up. The obtained results cover all planned tasks. Since our goal was to show that our method is accurate and less computationally demanding than standard calculations, considerable time and attention was dedicated to solve the same problems using some of the most popular standard algorithms of this class. Iterative upscaling requires significant computational resources, hence the development of proper parallel algorithms is crucial to the success of such kind of project. We have implemented our methods on heterogeneous and distributed computing environments. The codes are run on high-performance clusters. Throughout the applications, highly heterogeneous media were considered. The finite element method (FEM) was applied for discretization of the related elliptic boundary value problems. The scalar elliptic case is straightforwardly related to computer simulation of flows in porous media that appear in many industrial, scientific, engineering, and environmental applications. The selected test cases include typical multiscale geometries with islands and channels. In the studied elasticity problems, the bulk modulus was variable through the material. Displacement decomposition is used as a basic preconditioning scheme. The new feature of this work is the block diagonal preconditioner incorporating an analytical effective tensor into the simulation, avoiding costly numerical solutions of local problems that are usually inherent in methods for multiscale problems. The reliability/robustness of the proposed algorithm is measured by comparing with other known techniques. An extension of the work discussed in the previous paragraph is performed, by having not only spatially variable bulk modulus but also the Poisson ratio approaching the incompressibility limit. Moreover, the technique is applied to more general system beyond pure displacement. This scenario is more realistic than the one presented above, also because the media resemble trabecular bone tissues. The method is reliable with respect to the heterogeneity ratio as well in the limiting case, where it is know that the linear system becomes additionally very ill-conditioned. One promising continuation of this work is to apply the algorithm on 3-D simulator for micro-structure FEM analysis based on fully realistic 3-D tomography images.

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