23. Září 2014
Eduard Rohan, University of West Bohemia, Pilsen, Czech Republic Joint work with Vladimír Lukeš The two-scale homogenization is well suited for modeling periodic media described by linear PDEs. Nonlinear problems can also be treated, however, the separation of the local autonomous problems from the global ones relevant to the upscaled medium is not possible, in general. For problems related to deforming media, a linearization is always used to solve the problem using an incremental procedure, it is therefore natural to consider such an incremental problem for homogenization. We propose a homogenized model which is based upon the Eulerian rate formulation for large deforming fluid saturated porous medium. For problems characterized by moderate deformations, a weakly nonlinear homogenized model is proposed which involves linear expansions of the homogenized coefficients using their sensitivity w.r.t. macroscopic fields; the local problems and their sensitivities are solved for the initial configuration. In this case, computational costs are only slightly affected by the two-scale character of the problem, in contrast with solving a fully nonlinear problem requiring subsequent updates of local microstructures and, consequently, solving the local problems at almost any point of the macroscopic domain. Numerical illustrations are given.
MOdelling REvisited + MOdel REduction Modeling, analysis and computing in nonlinear PDEs. September 21-26, 2014, Chateau Liblice, Czech Republic. Recently developed implicit constitutive theory allows one to describe nonlinear response of complex materials in complicated processes and to model phenomena in both fluid and solid mechanics that have hitherto remained unexplained. The theory also provides a thermodynamically consistent framework for technologically important but so far only ad hoc engineering models without sound footing. The overall goal of the project is to develop accurate, efficient and robust numerical methods that allow one to perform large-scale simulations for the models arising from the new theoretical framework. A natural part of the goal is rigorous mathematical analysis of the models. The nonstandard structure of constitutive relations arising from the new framework requires the reconsideration of many existing approaches in the mathematical theory of partial differential equations and the development of new ones. In particular, basic notions such as the concept of the solution and its well-posedness need to be reconsidered. Further, the complexity of the constitutive relations calls for rigorous investigation on model reduction - the identification of simplified models that capture the chosen (practically relevant) information about the behaviour of the system and disregard irrelevant information. Reliable numerical simulations require the derivation of sharp a posteriori error estimates to control all possible sources of errors, including rarely studied but important algebraic errors. We believe that in solving difficult problems in mathematical modelling the individual aspects discussed above - physics, mathematical analysis and numerical analysis - are so closely interrelated that no breakthrough can be achieved without emphasising the holistic approach as the main principle. Our vision is to follow this principle: the entire process of modelling of complex materials will be revisited in an innovative manner.
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