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Model reduction of porous‐media problems using proper orthogonal decomposition
Author(s) -
Otto Davina,
Ehlers Wolfgang
Publication year - 2014
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201410213
Subject(s) - porous medium , finite element method , subspace topology , nonlinear system , proper orthogonal decomposition , computation , context (archaeology) , degrees of freedom (physics and chemistry) , reduction (mathematics) , computer science , consolidation (business) , porosity , mathematics , algorithm , mathematical analysis , mechanics , structural engineering , materials science , engineering , physics , geometry , composite material , paleontology , turbulence , business , biology , accounting , quantum mechanics
In the context of finite‐element simulations of porous media, computing time and numerical effort is an important issue because the number of degrees of freedom of such coupled problems can become very large. Following this, model reduction plays an important role. A broad variety of materials exhibit a porous microstructure. In order to evaluate the overall response of these materials, a macroscopic continuum‐mechanical modelling approach is used. Therefore, the complex inner structure of porous media is regarded in a multi‐phasic and multi‐component manner by means of the well‐founded Theory of Porous Media (TPM). The mechanical behaviour of porous media is solved using the Finite‐Element Method (FEM). The basic idea of model reduction is to transform a high dimensional system, in terms of the system's degrees of freedom, to a low dimensional subspace to minimise the computational effort while maintaining the accuracy of the solution. The method of proper orthogonal decomposition (POD) can be seen as a method to approximate a given data set with a low dimensional subspace. Furthermore, the POD method is independent of the type of the model and can be used for nonlinear systems as well as for systems of second order. In several applications, such as consolidation problems of partially saturated soils, commonly occurring motion sequences can be found, which can be used as typical “snapshots” of the system. Therefore, the application of the POD method to the simulation of porous media is discussed in the present contribution. Investigated computations of a biphasic standard problem show that the POD method reduces the numerical effort to solve the linearised system of equations in each iteration step. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)