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The main goal of this research is to develop, analyze, and implement robust numerical upscaling algorithms for emerging multiphysics problems of flows in deformable porous media. The physical problem under consideration at the fine scale is the strongly coupled, nonlinear Fluid-Structure Interaction (FSI) problem subject to large pore-level deformations and/or a nonlinear hysteretic solid. A key scientific contribution is the design of a Multiscale Finite Element Method (MsFEM), which bypasses the explicit homogenisation step by building fine-scale information directly into a coarse-scale computational grid. The approach allows accurate numerical simulations at several tightly coupled scales, with the fine scale physics being properly incorporated at the coarser scales. We have also investigated the applicability of various fine-scale solvers in a Multigrid (MG) context, where a direct numerical simulation is needed.

The work is motivated by clear scientific, technological, and socio-economic reasons. Developing robust upscaling algorithms for multiscale problems is a hot topic in mathematics, physics, and computational engineering. A large number of articles are published every year in highly ranked journals. The work extends the numerical upscaling approach to a very general poroelastic model based on nonlinear FSI at the microscale. It also expands current homogenisation methods for poroelasticity to the case of inseparable scales. Iterative upscaling of nonlinear FSI problems across inseparable scales addresses important bio-medical and technological needs. An example is hydro-mechanical modeling of human bone tissue, which has highly heterogeneous porous microstructure. Standard homogenisation techniques cannot be used due to lack of scale separation and nonlinear fluid-structure interactions are rarely considered. Similarly, loading filtering media with captured solid particles during solid/liquid separation in industrial filtration can easily destroy scale separation. These two examples alone have significant technological and socio-economical impacts. Others include geomechanics and environmental problems such as carbon sequestration, unconventional oil recovery and waste water management.

We have made considerable progress towards the goals of this grant. We have developed two different MsFEM algorithms for FSI problems with multiple length scales. The methods make innovative use of the concept of iterative homogenisation. They incorporate nonlinear macroscopic equations for conservation of mass and momentum into an iterative homogenisation framework. The equations are not stated explicitly. Instead, given an iterative approximation to the coarse and downscaled quantities, effective constitutive relations are computed and used as means to define the next approximation of the coarse and downscaled quantities.

In a separate development we have also successfully developed analytical tools which allow very fast approximation of cell solutions. While these analytical approximations to cell solutions do not yield very accurate downscaled solutions, they turned to be extremely useful in an MG context. When one is interested in the actual fine-scale solution, rather then homogenisation based approximations, a very large system of equations needs to be solved. The natural approach is to use a MG method. We tested several highly heterogeneous diffusive flow problems, where standard MG approaches do not work well and we obtained excellent acceleration when computing the actual fine-scale solution. The work was also extended to using MG cycles are preconditioners

We have successfully applied our algorithms to several model problems involving flow past linear elastic obstacles. In all cases our iterative multiscale algorithm did converge and provided accurate approximation of the fine-scale reference solution. This proves the iterative MsFEM convert when applied to FSI problems. Moreover, we have performed mathematical analysis of the two algorithms and showed that they indeed do provide accurate approximation under certain assumptions. We have also developed tools for processing voxel-based data and reconstructing the fluid-solid interface of complex CAT images of human femur. With this we have enabled the use of realistic CAT data into our multiscale algorithms. We have made computations of cell problems on realistic bone tissue samples.

We also paid considerable attention to software development and parallelisation of our MsFEM algorithms. Iterative upscaling requires significant computational resources, hence the development of proper parallel algorithms is crucial to the success of this proposal. We have implemented our methods for heterogeneous and distributed computing environments. We have run the codes on both traditional high-performance clusters and on distributed computing environments. In particular, the macroscopic discretisation is run on a single workstation and the downscaling tasks on a heterogeneous computing grid. The communication between the macro solver and the downscaling tasks is done in ways similar to modern peer-to-peer networks. This resulted in a flexible and fault-tolerant implementation, suitable to a variety of heterogeneous computing architectures.

In addition to the directly planned activities we have also executed a number of other tasks, related to our grant. One is an extension of the framework to problems of flow in media with porous component and free flow regions. This research direction allows to add yet another, finer length-scale and will be pursued both as part of the current project and beyond its planned duration. Also, we have developed a fine-scale solver for numerical simulations of electrochemical diffusion processes in Li-Ion batteries. This activity can be combined with fluid-structure problems, for example in modeling processes in fuel cells, as well as other multifunctional devices.

As part of this project we also hired two MS students who participated actively in the project and successfully defended in June 2011. They have now continued their professional development as PhD students at Fraunhofer ITWM, Kaiserslautern. We have also been active in disseminating the results, by publishing a number of papers and taking part in many international conferences and workshops.

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