Please use this identifier to cite or link to this item: http://dx.doi.org/10.18419/opus-10689
Authors: Fink, Davina
Title: Model reduction applied to finite-element techniques for the solution of porous-media problems
Issue Date: 2019
Publisher: Stuttgart : Institut für Mechanik (Bauwesen), Lehrstuhl für Kontinuumsmechanik, Universität Stuttgart
metadata.ubs.publikation.typ: Dissertation
metadata.ubs.publikation.seiten: XX, 165
Series/Report no.: Report / Institut für Mechanik (Bauwesen), Lehrstuhl für Kontinuumsmechanik, Universität Stuttgart;37
URI: http://elib.uni-stuttgart.de/handle/11682/10706
http://nbn-resolving.de/urn:nbn:de:bsz:93-opus-ds-107069
http://dx.doi.org/10.18419/opus-10689
ISBN: 978-3-937399-37-9
Abstract: In the present contribution, the developments in the modelling and the simulation of porous materials in the framework of the well-founded Theorie of Porous Media (TPM) are combined with the current state of research in the field of model reduction using projection-based methods. In terms of the continuum-mechanical modelling, this work is focusing on problems developed on the basis of detailed and thermodynamically consistent TPM models. The present work makes use of a biphasic model for the simulation of a saturated porous soil, a multiphasic and multicomponent model for the simulation of drug-infusion processes in brain tissue and an extended biphasic model for the description of an inhomogeneous and anisotropic intervertebral disc. The continuum-mechanical fundamentals of the TPM, required for the description of these models, are outlined. Therefore, the TPM is introduced, all necessary kinematical relations are provided and the balance relations are presented. Furthermore, the general continuum-mechanical fundamentals are specified for the models used in this work. A convenient technique for the solution of arbitrary initial-boundary-value problems is the Finite Element Method (FEM), which is used in this contribution for the numerical treatment of the TPM models. Starting from the weak forms of the governing equations, the spatial and temporal discretisation strategies are described. In this regard, a reduction of the descriptive set of (strongly) coupled partial differential equations provides an enormous benefit to significantly reduce the dimension of these systems and, thus, the computation time and the numerical effort of the FE simulations. Particularly with regard to nonlinear systems, the computational effort is usually immense as high-dimensional equation systems need to be solved repeatedly for the determination of the nonlinearities. Following this, a suitable reduction of these systems essentially improves the efficiency by solving only a subset of equations of the original model. Under consideration of these circumstances, efficient reduced models for the simulation of different porous materials are provided in the present work by an application-driven approach. Thereby, only model-reduction techniques applied to the monolithic solution of the strongly coupled equation systems are considered. The applied model-reduction techniques are explained in detail. In particular, projection-based model-reduction techniques are used to transform a high-dimensional system to a low-dimensional subspace. The advantage of such an approach is to maintain the detailed theoretical basis of the modelling process while an efficient numerical computation is provided. In this contribution, the method of proper orthogonal decomposition (POD) is used as a starting point for the model reduction. However, since the POD-Galerkin approximation does in fact significantly reduce the dimension of the equation system but not the effort to evaluate the nonlinear terms, the computational effort of nonlinear problems cannot be (sufficiently) reduced when exclusively using the POD method. This drawback motivates the application of additional methods for the reduction of the nonlinear terms. Within the scope of this work, the discrete-empirical-interpolation method (DEIM) is used in combination with the POD method to reduce arising nonlinearities. The high complexity of the underlying multiphasic and multicomponent modelling of the treated materials and the resultant strongly coupled equation systems require for individual adaptations and modifications of the used reduction methods to achieve satisfying results. Therefore, the scope of this monograph is the development of an application-driven approach for providing reduced models, which are capable of simulating specific porous materials in a time-efficient manner. The necessary modifications are discussed in detail in this work and are additionally illustrated with examples. In this regard, an in-depth knowledge of the form and the characteristics of the underlying equation system is essential and is therefore treated intensively. Since the outlined modifications might be of great interest for other applications, a generalised approach for an adaptation to other models is finally presented.
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