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Application and modification of the POD method and the POD‐DEIM for model reduction in porous‐media simulations
Author(s) -
Fink Davina,
Ehlers Wolfgang
Publication year - 2015
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201510183
Subject(s) - point of delivery , finite element method , proper orthogonal decomposition , nonlinear system , reduction (mathematics) , interpolation (computer graphics) , mathematics , computation , subspace topology , mathematical optimization , displacement (psychology) , porous medium , algorithm , computer science , porosity , mathematical analysis , structural engineering , materials science , engineering , geometry , computer graphics (images) , biology , quantum mechanics , agronomy , animation , physics , psychotherapist , composite material , psychology
Abstract Researchers with a continuum‐mechanical background typically use a multi‐phasic and multi‐component modelling approach for materials with a saturated porous microstructure. Therefore, the mechanical behaviour is considered in a continuum‐mechanical manner and solved using the finite‐element method (FEM). The developed models need to be complex enough to capture the relevant properties of the considered materials, what often results in expensive simulations with a very large number of degrees of freedom (DOF). The aim of the present contribution is to reduce the computing time of these simulations through model‐reduction methods, while the accuracy of the solution needs to be maintained. Therefore, the method of proper‐orthogonal decomposition (POD) for linear problems and the discrete‐empirical‐interpolation method (DEIM) in combination with the POD method (POD‐DEIM) for nonlinear problems are investigated. Using the POD method, a given data set is approximated with a low‐dimensional subspace. To generate this data set, the vector of unknowns of the FE simulation is stored in a pre‐computation in the full (unreduced) system in each time‐step (so‐called “snapshots” of the system). Dealing with porous‐media problems, the primary variables are the solid displacement, the pore pressure and, depending on the particular problem, other primary variables. Following this, the primary variables have entries with very huge differences in their absolute values. As a result, non‐negligible rounding errors may occur when applying the POD method. To overcome this problems, modifications of the classical POD method need to be performed for such problems. The present contribution discusses this issue and presents results for the reduced simulations of porous media. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)