Application and modification of the POD method and the POD‐DEIM for model reduction in porous‐media simulations

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 continuummechanical 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 properorthogonal 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.

Computational Modeling in Liver Surgery

“…The overall goal is to reduce complexity without compromising the aspects relevant for the question at hand. Different approaches of model reduction have been applied in the field of liver simulations, e.g., representative sinusoids (Schwen et al, 2015 ), method of proper orthogonal decomposition (Fink and Ehlers, 2015 ), or the use of an energy function (Holzapfel et al, 2000 ; Humphrey, 2003 ; Balzani et al, 2006 ).…”