Model reduction applied to finite-element techniques for the solution of porous-media problems

(1)

(2)

(3)

Model Reduction applied to

Finite-Element Techniques for the

Solution of Porous-Media Problems

Von der Fakult¨at Bau- und Umweltingenieurwissenschaften

der Universit¨at Stuttgart zur Erlangung der W¨

urde

eines Doktor-Ingenieurs (Dr.-Ing.)

genehmigte Abhandlung

vorgelegt von

Dipl.-Ing. Davina Fink

aus

Kassel

Hauptberichter:

Prof. Dr.-Ing. Dr. h. c. Wolfgang Ehlers

Mitberichter:

Prof. Dr. Bernard Haasdonk

Prof. Dr.-Ing. Stefanie Reese

Tag der m¨

undlichen Pr¨

ufung: 9. Juli 2019

Institut f¨

ur Mechanik (Bauwesen) der Universit¨at Stuttgart

Lehrstuhl f¨

ur Kontinuumsmechanik

Prof. Dr.-Ing. Dr. h. c. W. Ehlers

2019

(4)

Report No. II-37

Institut für Mechanik (Bauwesen) Lehrstuhl für Kontinuumsmechanik Universität Stuttgart, Germany, 2019

Editor:

Prof. Dr.-Ing. Dr. h. c. W. Ehlers

c

Davina Fink

Institut für Mechanik (Bauwesen) Lehrstuhl für Kontinuumsmechanik Universität Stuttgart

Pfaffenwaldring 7

70569 Stuttgart, Germany

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy-ing, recordphotocopy-ing, scanning or otherwise, without the permission in writing of the author.

ISBN 978 – 3 – 937399 – 37 – 9

(5)

Acknowledgements

The work presented in this doctoral thesis was developed during my profession as an assistant lecturer and research associate at the Institute of Applied Mechanics (Civil Engineering), Chair of Continuum Mechanics, at the University of Stuttgart. At this point, I want to take the opportunity to gratefully acknowledge numerous people who contributed in various ways to complete this work.

First of all, I would like to sincerely thank my supervisor Professor Wolfgang Ehlers for giving me the opportunity to prepare my thesis at his institute under his constant scientific support. I personally appreciate not only his structured way of working and his broad knowledge but also his well-balanced character. Furthermore, I would like to thank Professor Bernard Haasdonk, not only for evaluating my thesis, but also for most valuable discussions concerning the mathematical aspects of this work and for his time to extensively reading and discussing this thesis. I am also grateful to Professor Stefanie Reese for taking the third supervision in my dissertation procedure.

Moreover, I would like to thank all my former colleagues at the institute for the great and friendly atmosphere, creating a pleasant and efficient working environment. Thanks for all the warm conversations, for the constant willingness to share experiences and knowledge with me and for the time we spend apart from work. In particular, I would like to thank Arndt Wagner for his enthusiasm and his huge personal engagement as teaching assistant, with which he significantly inspired me to head into the field of mechanics. Furthermore, I am very thankful for all the discussions we had on various topics and for the proofreading of parts of this work. Equally, I would like to thank my proofreaders Sami Bidier, Lukas Eurich, Chenyi Luo and Alixa Sonntag, not only for the time to read parts of my thesis, but also for all the pleasurable moments. It was a fortune to have kind and reliable office-mates. When I wrote my diploma thesis at the institute, Nils Karajan supervised and taught me many important routine matters at the institute - many thanks for this. Beyond that, I would like to thank David Koch, Maik Schenke and Patrick Schr¨oder for their constant administrative support and for all the discussions about technical and many other issues, and Christian Bleiler and Mylena Mordhorst for the nice conversations and the cosy get-together at various conferences and workshops. This, of course, also applies to my former colleagues Kai H¨aberle, Said Jamei and Joffrey Mabuma, and all the others from the institute.

My special thanks go to my wonderful parents for always believing and trusting in me. Finally, I would like to thank my beloved husband for his limitless support and encour-agement.

(6)

(7)

B.1 Overall system of a quasi-static biphasic model of a porous material . . . . 127 B.2 Overall system of the dynamic porous-soil model . . . 130 B.3 Overall system of the simplified drug-infusion model for brain tissue . . . . 136 C Specific derivation of the reduced systems in abstract formulation 141

C.1 Reduced formulation of the quasi-static porous-soil model with linear sys-tem of equations . . . 141 C.2 Reduced system of the dynamic porous-soil model . . . 143 C.3 Reduced system of a nonlinear biphasic model of a porous material . . . . 145 C.4 Reduced system of the different drug-infusion models for brain tissue . . . 146 C.4.1 Simplified brain-tissue model with linear equation system . . . 146 C.4.2 General brain-tissue model with nonlinear equation system . . . 147

Bibliography 149

(10)

(11)

Abstract

Computational simulations have a tremendous impact on a growing number of scientific fields. In the course of time, increasingly precise predictions could be accomplished by means of simulation results. Furthermore, a wide variety of practical tests and exten-sive measurements could be substituted by simulations. In addition, simulations provide new possibilities in the modern medicine. In particular, they enable to gain a deeper understanding of the complex processes in the human body or even to contribute to the successful planning of a surgical intervention by providing supplementary process infor-mation. The increasing importance of computational simulations requires for a high level of trustworthiness of the simulation results. In order to meet these requirements, sim-ulations based on sophisticated models and with a sufficiently fine discretisation of the geometry are indispensable for the numerical realisation. Particularly when taking into account the high structural complexity of the underlying materials, a detailed knowledge about the inner structure and the composition of the materials of various components is essential for a sufficiently accurate modelling. A broad variety of materials cannot be described with classical continuum-mechanical models restricted to singlephasic (homo-geneous) materials. This applies in particular to materials with a porous micro-structure. The group of porous media, consisting of a porous soil whose pore space is filled with fluids and/or gases, includes, amongst others, partially saturated soils and biological tis-sue aggregates. Hence, a multiphasic and multicomponent modelling approach on the basis of the Theory of Porous Media (TPM) appears suitable in order to describe these complex materials. Concerning the numerical treatment of porous materials, the finite-element method (FEM) has been proven to be a well-suited technique for the solution of arbitrary initial-boundary-value problems. The resulting simulations are able to de-scribe the physical phenomena by repeatedly solving the descriptive set of coupled partial differential equations (PDE). However, the necessarily high accuracy of the approxima-tion results and the complexity of the underlying models result in many applicaapproxima-tions in an extremely high dimension of the resulting equation system. The consequences are time-consuming simulations, which are either too slow to satisfy the time constraints and to enable practical applications (such as an accompanying use in clinical practice) or which cannot be performed as often as needed due to the high computation time. In order to counteract these problems, to increase the solution speed and to reduce the computational expenses, model-reduction methods are increasingly important and are gaining a considerable scientific interest. While, on the one hand, the detailed theoreti-cal basis of the modelling approach needs to be maintained and, on the other hand, an efficient numerical computation should be provided, the available performance capacity of projection-based model-reduction techniques can be used to provide fast simulations whenever they are actually needed. Supported by the steadily growing potential of com-puting power and storage capacities, time-consuming simulations based on models with all their complexity can already be performed and all necessary information and data can be stored beforehand in more and more application areas. Following the concept of offline/online decomposition, a possibly time-consuming offline phase (including the

(12)

VI Abstract

ulations performed in advance and the reduction of the underlying system of equations) can be separated from a time-efficient online phase with fast simulations (performed using the reduced system). A computationally intensive offline phase pays off by the possibility to rapidly produce required simulation results in daily routines or by a sufficient num-ber of individual computationally inexpensive simulations with varying material and/or simulation parameters.

In the present contribution, the developments in the modelling and the simulation of porous materials in the framework of the well-founded TPM are combined with the cur-rent 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 de-veloped on the basis of detailed and thermodynamically consistent TPM models. Early studies concentrated on the multiphasic and multicomponent modelling of porous media with application of the TPM have shown the outstanding suitability of this modelling approach, cf., e. g., de Boer [15, 16] or Ehlers [39, 42, 43]. The present work makes use of a biphasic model for the simulation of a saturated porous soil (cf. Ehlers [40], Ehlers & Eipper [47], Eipper [54], Ellsiepen [55]), a multiphasic and multicomponent model for the simulation of drug-infusion processes in brain tissue (cf. Ehlers & Wagner [51, 52], Wagner [127], Wagner & Ehlers [128]) and an extended biphasic model for the description of an inhomogeneous and anisotropic intervertebral disc (cf. Ehlers et al. [49, 50], Karajan [81]). The continuum-mechanical fundamentals of the TPM, required for the description of these models, are outlined in Chapter 2. Therefore, the TPM is introduced, all necessary kine-matical 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 FEM, cf. Lewis & Schrefler [91], which is used in this contribution for the numerical treat-ment of the TPM models. Starting from the weak forms of the governing equations, the spatial and temporal discretisation strategies are described in Chapter 3. In this re-gard, 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 considera-tion of these circumstances, efficient reduced models for the simulaconsidera-tion 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 Chapter 4. In particular, projection-based model-reduction techniques are used to transform a high-dimensional system to a low-dimensional subspace. The advan-tage 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. The development of the POD method, also known as Karhunen-Lo `eve expansion, cf. Sirovich [115], traces back to fluid-dynamic applications including

(13)

turbu-Abstract VII

lence, cf. Berkooz et al. [10]. Beyond that, the POD method was successfully applied to various problems in fluid flow (cf. Kunisch & Volkwein [88], Rowley et al. [110]), optimal control (cf. Kunisch & Volkwein [86]), aerodynamics (cf. Bui-Thanh et al. [24], Hall et al. [73]), biomechanics (cf. Radermacher & Reese [102]) and structural mechanics (cf. Herkt et al. [76], Radermacher & Reese [103]). 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), which is the discrete variant of the empirical-interpolation method (EIM, cf. Barrault et al. [7]) and which was introduced by Chaturantabut & Sorensen [32], is used in combination with the POD method to reduce arising nonlinearities. In the works of Kellems et al. [85] for a model of spiking neurons, in Chaturantabut & Sorensen [34] for a model with application to non-linear miscible viscous fingering, in Nguyen et al. [98] for reacting flow applications, in Negri et al. [97] for parametrised systems or in Bonomi et al. [19] for the application to parametrised problems in cardiac mechanics, amongst others, it could be shown that the DEIM is able to significantly reduce the numerical effort of complex nonlinear processes. The high complexity of the underlying multiphasic and multicomponent modelling of the treated materials and the resultant strongly coupled equation systems require for indi-vidual 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 Chapter 5. 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. For example, it should be ensured that the block structure of the coupled equation systems is preserved while considering the different temporal (physical) behaviour of the primary variables. Since the outlined modifications might be of great interest for other applications, a generalised approach for an adaptation to other models is finally presented.

(14)

(15)

Deutschsprachige Zusammenfassung

Rechnerbasierte Simulationen sind heutzutage aus immer mehr Wissenschaftsfeldern nicht mehr wegzudenken. So konnten im Laufe der Zeit zunehmend präzisere Vorhersagen aus Simulationsergebnissen gewonnen und eine Vielzahl von praktisch durchgeführten Tests und aufwändigen Messungen durch numerische Simulationen ersetzt werden. Zudem eröffnen Simulationen auch in der modernen Medizin neue Möglichkeiten, ein erweitertes Verständnis der komplexen Vorgänge im menschlichen Körper zu erhalten oder gar me-dizinische Eingriffe in der Praxis durch Simulationen zu unterstützen. Die zunehmende Bedeutung rechnerbasierter Simulationen verlangt eine hohe Vertrauenswürdigkeit in die Simulationsergebnisse. Um diesen Anforderungen gerecht zu werden, sind komplexe Mo-delle als Basis der Simulationen sowie eine ausreichend feine Diskretisierung der Geometrie bei der numerischen Umsetzung unabdingbar. Insbesondere bei Materialien mit einer ho-hen struktureller Komplexität ist eine Kenntis über die vorliegende innere Struktur und die Zusammensetzung der Materialien aus verschiedenen Bestandteilen für eine ausrei-chend genaue Modellbildung unverzichtbar. Es existieren eine Vielzahl von Materialien, die sich nicht mit den klassischen Methoden der Kontinuumsmechanik für ein einzelnes homogenes Material beschreiben lassen. Dies gilt insbesondere für Materialien mit einer poröser Mikrostruktur. Zu der Gruppe der porösen Medien, bestehend aus einem porösen Festkörper, dessen Porenräume mit Flüssigkeiten und/oder Gasen gefüllt sind, gehören gesättigte Böden ebenso wie biologisches Gewebe. Zur Modellbildung dieser komplexen Materialien hat sich eine mehrphasige und mehrkomponentige Beschreibung auf Basis der Theorie Poröser Medien (TPM) bewährt. Des Weiteren konnte im Hinblick auf die nume-rische Simulation solcher poröser Materialien in einer Vielzahl von Arbeiten die Eignung der Finite-Elemente-Methode (FEM) als numerische Approximationsmethode dargelegt werden. Die sich daraus ergebenden Simulationen können physikalische Phänomene be-schreiben, indem wiederholt das beschreibenden Gleichungssystem aus gekoppelten par-tiellen Differentialgleichungen gelöst wird. Die Komplexität der zugrundeliegenden Mo-delle und die notwendig hohe Genauigkeit der Approximationsergebnisse führt jedoch in vielen Anwendungen zu einer extrem hohen Dimension des resultierenden diskreten Glei-chungssystems. Die Folge sind rechenintensive Simulationen, die in der Regel entweder zu hohe Rechenzeiten aufweisen um damit die zeitlichen Beschränkungen zu erfüllen und praktische Anwendungen (wie den unterstützenden Einsatz bei medizinischen Eingriffen im Klinikalltag) zu ermöglichen oder aufgrund der hohen Rechenkosten nicht ausreichend häufig durchgeführt werden können. Dieser Problematik kann man entgegenwirken, indem man durch den Einsatz geeigneter Reduktionsmethoden die sich ergebenden hohen Re-chenzeiten und Rechenkosten numerischer Simulationen deutlich verringert. Folglich ist es nicht verwunderlich, dass Methoden der Modellreduktion in den letzten Jahren eine stark zunehmende Bedeutung und ein bemerkenswertes wissenschaftliches Interesse gewonnen haben. Dabei ermöglichen projektionsbasierte Reduktionsmethoden, dass einerseits die komplexen theoretischen Grundlagen der Modellbildung beibehalten werden und ande-rerseits bei Bedarf zeiteffiziente numerische Berechnungen durchgeführt werden können. Gestützt durch das ständig wachsende Potential an Rechenleistungen und

(16)

X Deutschsprachige Zusammenfassung

pazitäten können in immer mehr Anwendungsbereichen rechenaufwändige Simulationen, basierend auf Modellen mit all ihrer Komplexität, bereits im Vorfeld durchgeführt und alle notwendigen Informationen und Datensätze abgelegt werden. Die Basis für die Ef-fizienz des Reduktionsprozesses stellt eine Offline/Online-Zerlegung dar. Bei dieser kann eine meist rechenintensive Offline-Phase, bestehend aus vorab durchgeführte Simulatio-nen und der Reduktion des zugrundeliegenden Gleichungssystems, von einer zeiteffizienten Online-Phase (Simulationen basierend auf dem reduzierten Gleichungssystem) entkoppelt werden. Eine teure Offline-Phase zahlt sich dadurch aus, dass man die Möglichkeit hat, im Alltag schnell Simulationsergebnisse erzeugen zu können. Zudem ergibt sich bei einer genügend hohen Anzahl an Einzelsimulationen, die mit wechselnden Parametern durch-geführt werden, eine deutliche Reduktion der Rechenzeiten und Rechenkosten.

Ziel dieser Arbeit ist es, die Fortschritte in der Modellbildung und Simulation poröser Ma-terialien auf Basis der TPM mit dem aktuellen Stand der Forschung zur Modellreduktion unter Verwendung projektionsbasierter Methoden zu kombinieren. Hinsichtlich der konti-nuumsmechanischen Modellbildung werden im Rahmen dieser Arbeit Probleme behandelt, die auf Grundlage detaillierter thermodynamisch konsistenter Modelle unter Anwendung der TPM entwickelt wurden. Frühe Arbeiten zur mehrphasigen und mehrkomponentigen Beschreibung poröser Medien auf Basis der TPM zeigen die Eignung dieses Modellan-satzes (de Boer [15, 16], Ehlers [39, 42, 43]). Die vorliegende Arbeit macht Gebrauch von einem Zweiphasenmodell zur Beschreibung gesättigter Böden (Ehlers [40], Ehlers & Eipper [47], Eipper [54]), einem mehrphasigen und mehrkomponentigen Modell zur Simulation von Infusionsprozessen in Gehirngewebe (Ehlers & Wagner [51, 52], Wagner [127], Wagner & Ehlers [128]) und einem erweiterten Zweiphasenmodell zur Beschreibung inhomogener und anisotroper Bandscheiben (Ehlers et al. [49, 50], Karajan [81]). Die zur Beschrei-bung der zugrundeliegenden Modelle notwendigen kontinuumsmechanischen Grundlagen der TPM werden in Kapitel 2 erläutert. Dabei werden neben einer Einführung der TPM die notwendigen kinematischen Beziehungen eingeführt und die benötigten Bilanzglei-chungen zusammengestellt. Darüber hinaus werden die allgemeinen kontinuumsmechani-schen Grundlagen für die in dieser Arbeit verwendeten Modelle spezifiziert. Eine effiziente Möglichkeit zur Behandlung beliebiger Anfangs-Randwertprobleme bietet die FEM (Lewis & Schrefler [91]), mit deren Hilfe die TPM-Modelle in dieser Arbeit numerisch diskretisiert werden. Ausgehend von den schwachen Formen der Bilanzgleichungen folgt in Kapitel 3 eine Orts- und Zeitdiskretisierung der Bestimmungsgleichungen. Eine Reduktion der be-schreibenden Gleichungssysteme aus (stark) gekoppelten partiellen Differentialgleichung-en ist von großem NutzDifferentialgleichung-en um die DimDifferentialgleichung-ension der Differentialgleichung-entsprechDifferentialgleichung-endDifferentialgleichung-en diskretisiertDifferentialgleichung-en Systeme und damit einhergehend die Rechenzeiten und Rechenkosten der numerischen Simulatio-nen signifikant zu reduzieren. Insbesondere bei nichtlinearen Systemen ist der Rechenauf-wand in der Regel immens, da zur Bestimmung der Nichtlinearitäten wiederholt hoch-dimensionale Gleichungssysteme gelöst werden müssen. Eine geeignete Reduktion dieser Systeme ermöglicht es, dass nur eine kleine Teilmenge von Gleichungen gelöst werden muss. Unter Berücksichtigung der oben genannten Sachverhalte werden in der vorliegen-den Arbeit durch ein anwendungsorientiertes Vorgehen effiziente reduzierte Modelle der verschiedenen porösen Materialien erarbeitet. Dabei wird sich auf Reduktionsmethoden beschränkt, die auf die monolithische Lösung des stark gekoppelten Gleichungssystems der TPM-Modelle angewandt werden. Diese werden in Kapitel 4 näher erläutert.

(17)

Kon-Deutschsprachige Zusammenfassung XI

kret werden projektions-basierte Reduktionsmethoden genutzt, um ein hochdimensionales System auf einen niedrigdimensionalen Unterraum zu transformieren. Während auf diese Weise die detaillierten theoretischen Grundlagen des Systems erhalten bleiben, ist es in vielen Anwendungen möglich, den Rechenaufwand der numerischen Simulationen deutlich zu reduzieren. Die Reduktion der hochdimensionalen Gleichungssysteme erfolgt in dieser Arbeit mittels der POD-Methode (method of proper orthogonal decomposition). Erste Ar-beiten zur Entwicklung dieser Methode (auch bekannt als Karhunen-Lo ève-Zerlegung, siehe Sirovich [115]) sind bei fluiddynamischen Anwendungen unter Einbeziehung von Turbulenzen zu finden, siehe Berkooz et al. [10]. Des Weiteren wurde die POD-Methode erfolgreich bei einer Vielzahl von Problemen im Bereich von Fluidströmungen (Kunisch & Volkwein [88], Rowley et al. [110]), optimaler Steuerung (Kunisch & Volkwein [86]), Aerodynamik (Bui-Thanh et al. [24], Hall et al. [73]), Biomechanik (Radermacher & Reese [102]) und Strukturmechanik (Herkt et al. [76], Radermacher & Reese [103]) angewandt. Da die POD-Galerkin-Approximation zwar die Dimension des zu lösenden Gleichungs-systems deutlich reduziert, nicht jedoch den Aufwand zur Bestimmung der nichtlinearen Anteile, kann der Rechenaufwand bei nichtlinearen Gleichungssystemen bei alleiniger Ver-wendung der POD-Methode in der Regel (wenn überhaupt) nicht ausreichend reduziert werden. Dies motiviert bei nichtlinearen Problemen den Einsatz ergänzender Methoden. Im Rahmen dieser Arbeit wird zur Reduktion solcher nichtlinearer Terme die DEIM (discrete-empirical-interpolation method, siehe Chaturantabut & Sorensen [32]), die ei-ne diskrete Variante der EIM (empirical-interpolation method, siehe Barrault et al. [7]) darstellt, verwendet. So konnte unter anderem in den Arbeiten von Kellems et al. [85] bei der Anwendung auf ein Neuronen-Modell, in Chaturantabut & Sorensen [34] bei der Be-schreibung spezieller Effekte (viscous fingering) bei der Modellierung nichtlinearer viskoser Strömungen, in Nguyen et al. [98] bei der Anwendung auf ein Modell zur Beschreibung reaktiver Strömungen, in Negri et al. [97] bei der Reduktion von parametrisierten Syste-men oder in Bonomi et al. [19] bei der Anwendung auf parametrisierte Probleme bei der Modellierung von Kontraktionen des Herzens gezeigt werden, dass die DEIM in der Lage ist, den numerischen Aufwand komplexer nichtlinearer Prozesse deutlich zu reduzieren. Die hohe Komplexität der zugrundeliegenden mehrphasigen und mehrkomponentigen Be-schreibung der betrachteten Materialien und der sich daraus ergebenden stark gekoppelten Gleichungssysteme erfordert individuelle Anpassungen und Modifikationen der verwende-ten Reduktionsmethoden, um zufriedenstellende Ergebnisse zu erhalverwende-ten. Daraus resultiert ein anwendungsorientiertes Vorgehen zur Bereitstellung reduzierter Modelle, welches eine zeiteffiziente Simulation der spezifischen Anwendungsbeispiele ermöglicht. Die notwendi-gen Modifikationen werden in dieser Arbeit ausführlich erläutert und in Kapitel 5 mit entsprechenden numerischen Beispielen unterlegt. Da für ein solches Vorgehen die Form und die Eigenschaften der zugrundeliegenden Gleichungssysteme eine wesentliche Rolle spielen, wird zudem umfassend auf diese eingegangen. So sollte zum einen die sich erge-bende Block-Struktur der gekoppelten Gleichungssysteme beibehalten werden, und zum anderen das physikalische (zeitlich stark differenzierte) Verhalten der Primärvariablen berücksichtigt werden um numerisch effiziente Simulationsergebnisse zu erhalten. Da die in dieser Monographie erarbeiteten Modifikationen auch für andere Anwendungen von ho-hem Interesse sein können, wird außerdem auf ein generalisiertes Vorgehen zur Adaption auf anderer Modelle eingegangen.

(18)

(19)

Nomenclature

As far as possible, the notation in this monograph follows the common conventions of modern tensor calculus, such as in de Boer [14] or Ehlers [41]. The particular symbols used in the context of porous-media theories are chosen according to the established nomenclature given by de Boer [16] and Ehlers [39, 43].

Conventions

General conventions

( · ) placeholder for arbitrary quantities

δ( · ) test functions of primary unknowns

d( · ) or ∂(· ) differential or partial derivative operator a, b, . . . or φ, ψ, . . . scalars (zero-order tensors)

a, b, . . . or φ, ψ, . . . vectors (first-order tensors) A, B, . . . or Φ, Ψ, . . . second-order tensors

n

A, B, . . . orn Φ,n Ψ, . . . nn th- or higher-order tensors

a, b, . . . or A, B, . . . general column vectors (n× 1) and matrices (n × m)

Index and suffix conventions i, j, k, l, . . . indices

( · )α subscripts indicate kinematical quantities of a constituent ϕα

within porous-media or mixture theories

( · )α _{superscripts indicate non-kinematical quantities of a}

con-stituent ϕα _{within porous-media or mixture theories}

( · )(·)0α initial values of non-kinematical quantities with respect to

the referential configuration of a constituent ϕα

( · )k( · )k =Pk( · )k( · )k Einstein’s summation convention yields a summation over

indices that appear twice unless stated otherwise ·

( · ) = d( · )/dt total time derivatives with respect to the overall aggregate ϕ ( · )′

α = dα( · )/dt material time derivatives following the motion of a

con-stituent ϕα

(20)

XIV Nomenclature

Symbols

Greek letters

Symbol Unit Description

α constituent identifier in super- and subscript, e. g., α =_{{S, F }} ˜

αB _{[ N/m}2_] _{material parameter of a blood constituent ϕ}B

β identifier for the pore liquids in super- and subscript

˜

βB _{[ - ]} _{material parameter of a blood constituent ϕ}B

γβR _{[ N/m}3_] _{effective weight of a liquid constituent ϕ}β

˜ γS

1 [ - ] parameter of the anisotropic part of the solid strain energy

Γ , Γ(·) domain boundary and Dirichlet and Neumann boundaries

∆π [ N/m2_] _{osmotic pressure difference}

ε, εα _{[ J/kg ]} _{mass-specific internal energy of ϕ and ϕ}α

ˆ

εα _{[ J/m}3_{s ]} _{volume-specific direct energy production of ϕ}α

εbound [ - ] (estimated) bound for the error

εNRMS [ - ] normalised root-mean-square error

εtol pre-defined tolerance

ˆ

ζα _{[ J/K m}3_{s ] volume-specific direct entropy production of ϕ}α

η, ηα _{[ J/K kg ]} _{mass-specific entropy of ϕ and ϕ}α

ˆ

η, ˆηα _{[ J/K m}3_{s ] volume-specific total entropy production of ϕ and ϕ}α

θ, θα _{[ K ]} _{absolute Kelvin’s temperature of ϕ and ϕ}α

ϑs identifier for the primary variables

κ [ - ] exponent governing the deformation dependency of KS

λr eigenvalues of the Gramian matrix C

λS

0 [ N/m2] first Lam´e constant of ϕS

µβ_{, µ}βR _{[ N s/m}2_] _{partial and effective dynamic viscosity of ϕ}β

µS

0 [ N/m2] second Lam´e constant of ϕS

˜ µS

1 [ N/m2] parameter of the anisotropic part of the solid strain energy

ρ [ kg/m3_] _{density of the overall aggregate ϕ}

ρα_{, ρ}αR _{[ kg/m}3_] _{partial and effective (realistic) density of ϕ}α

ˆ

ρα _{[ kg/m}3_{s ]} _{volume-specific mass production term of ϕ}α

σ, σα _{scalar-valued supply terms of mechanical quantities}

ση, σαη volume-specific external entropy supply of ϕ and ϕα

ςi identifier for a group of similar primary variables

ϕ, ϕα _{entire aggregate model and particular constituent}

φS

0 [◦] half of the fibre angle between two fibres

(21)

Nomenclature XV

ˆ

Ψ , ˆΨα _[_·/m3_] _{volume-specific productions of scalar mechanical quantities}

Ω , ∂Ω spatial domain and boundary of the aggregate body_B

Ωe, Ωh finite element and discretised finite-element domain

ε vector-valued error between the full and the reduced system

µ parameter configuration

ξi local coordinate system of a referential finite element

σ, σα _{vector-valued supply terms of mechanical quantities}

ϕr basis vectors of the subspace Vl

φ, φα _{vector-valued efflux terms of mechanical quantities}

φη, φαη [ J/K m2s ] entropy efflux vector of ϕ and ϕα

χα, χ−1α motion and inverse motion functions of the constituents ϕα

ψr basis vectors of the subspace Vk

Ψ, Ψα _[_·/m3_] _{volume-specific densities of vectorial mechanical quantities}

ˆ

Ψ, ˆΨα _[_·/m3_] _{volume-specific productions of vectorial mechanical quantities}

εS [ - ] linear solid strain tensor

ξ coefficient matrix used in the DEIM approach

τ, τα _{[ N/m}2_] _{Kirchhoff stress tensor of ϕ and ϕ}α

Υ subspace matrix used for a P etrov-Galerkin projection

Φ, Φα _{general tensor-valued mechanical quantities}

Φu subspace matrix used in the POD approach

Ψw subspace matrix used in the DEIM approach

Latin letters

cD

m [ mol/m3] molar concentration of a therapeutic agent ϕD

da, daα _{[ m}2_] _{actual area element of ϕ and ϕ}α

dmα _{[ kg ]} _{local mass element of ϕ}α

dt [ s ] time increment

dv, dvα _{[ m}3_] _{actual volume element of ϕ and ϕ}α

dVα [ m3] reference volume element of ϕα

ˆ

eα _{[ J/m}3_{s ]} _{volume-specific total energy production of ϕ}α

¯

D _{[ mol/m}2_{s ] area-specific therapeutic efflux of ϕ}D _{over the boundary Γ} ¯ D

Jα [ - ] Jacobian determinant of ϕα

k [ - ] number of DEIM (or magic) points

kF _{[ m/s ]} _{Darcy flow coefficient (hydraulic conductivity)}

KS _{[ m}2_] _{intrinsic (deformation-dependent) permeability}

(22)

XVI Nomenclature

m [ - ] number of snapshots of the state variables

n, ne [ - ] number of nodal points for Ωh and Ωe

nα _{[ - ]} _{volume fractions of a constituent ϕ}α

N [ - ] number of degrees of freedom

N_(·)j [ - ] global basis function of a degree of freedom p, pαR _{[ N/m}2_] _{overall pore pressure and liquid pore pressures}

pdif [ N/m2] differential pressure of the pore liquids

pr [ - ] interpolation indices

P E [ - ] projection error

¯

q [ m/s ] volume efflux of the fluid over the boundary Γq

Q [ m3_{/s ]} _{application rate during the CED application}

r, rα _{[ J/kg s ]} _{mass-specific external heat supply (radiation) of ϕ and ϕ}α

sβ _{[ - ]} _{saturation function of the pore liquids ϕ}β

t, tn−1, tn [ s ] actual time and temporally discretised time steps

T [ s ] specific (final) time

¯

vβ _{[ m}3_/m2_{s ] area-specific volume efflux of ϕ}β _{over the boundary Γ} ¯ vβ

V , Vα [ m3] overall volume of B and partial volume of Bα

w [ - ] number of Newton steps

111 vector of all ones

a0S [ - ] unit vector pointing in the fibre direction

aS [ - ] fibre direction within the actual configuration

b, bα _{[ m/s}2_] _{mass-specific body force vector}

dα [ m/s ] diffusion velocity vector of ϕα

da [ m2_] _{oriented actual area element}

dAα [ m2] oriented reference area element of ϕα

dx [ m ] actual line element

dXα [ m ] reference line element of the constituent ϕα

ei [ - ] (Cartesian) basis of orthonormal vectors

f [ N ] total volume force vector

fα _{[ N ]} _{volume force vector acting on} _Pα _{from a distance}

g [ m/s2_] _{constant gravitation vector}

ˆ

hα _{[ N/m}2_] _{volume-specific total angular momentum production of ϕ}α

kα_{, k}α

c, kαv [ N ] total, contact and volume force element of ϕα

ˆ

mα _{[ N/m}2_] _{volume-specific direct angular momentum production of ϕ}α

n [ - ] outward-oriented unit surface normal vector

ˆ

pα _{[ N/m}3_] _{volume-specific direct momentum production of ϕ}α

(23)

Nomenclature XVII

¯r [ m ] level arm of the surface traction with regard to the COG ˆsα _{[ N/m}3_] _{volume-specific total momentum production of ϕ}α

tα _{[ N/m}2_] _{surface traction vector of ϕ}α

¯t [ N/m2_] _{external load vector acting on the boundary Γ} t

¯tF _{[ N/m}2_] _{external load vector of the fluid acting on the boundary Γ} tF

u vector containing the primary variables

uS [ m ] solid displacement vector

vF, vS [ m/s ] fluid and solid velocity vector

wβ [ m/s ] seepage velocity vector of ϕβ

x [ m ] actual position vector of ϕ

Xα = x0α [ m ] reference position vector of Pα

˙x, x′

α [ m/s ] velocity vector of the aggregate ϕ and the constituent ϕα

¨ x, x′′

α [ m/s2] acceleration vector of the aggregate ϕ and the constituent ϕα

Aα [ - ] Almansian strain tensor of ϕα

Bα [ - ] left Cauchy-Green deformation tensor of ϕα

Cα [ - ] right Cauchy-Green deformation tensor of ϕα

Dα [·/s ] symmetric deformation velocity tensor of ϕα

DD _{[ m}2_{/s ]} _{diffusion tensor of a therapeutic agent ϕ}D

Eα [ - ] Green-Lagrangean strain tensor of ϕα

Fα [ - ] material deformation gradient of ϕα

I [ - ] identity tensor (second-order fundamental tensor) ˜I [ - ] matrix consisting of identity matrices

Kα [ - ] Karni-Reiner strain tensor of ϕα

Kβ _{[ m/s ]} _{Darcy permeability (or hydraulic conductivity) tensor of ϕ}β

Kβ

spec [ m4/N s ] specific permeability tensor of ϕβ

KSβ _{[ m}2_] _{intrinsic permeability tensor of ϕ}β

L, Lα [·/s ] spatial velocity gradient of ϕ and ϕα

M [ Nm ] moment vector (mechanical reaction to the surface traction) N

NN_ϑ_s abstract matrix representing the global basis functions of ϑs

P, Pα _{[ N/m}2_] _{first P iola-Kirchhoff (or nominal) stress tensor of ϕ and ϕ}α

R [ N ] force vector (mechanical reaction to the surface traction) Rα [ - ] proper orthogonal rotation tensor of the polar decomp. of Fα

S, Sα _{[ N/m}2_] _{second P iola-Kirchhoff stress tensor of ϕ and ϕ}α

T, Tα _{[ N/m}2_] _{overall and partial Cauchy (true) stress tensor of ϕ and ϕ}α

Tα

E [ N/m2] partial Cauchy extra stress tensor of ϕα

TS

E, aniso [ N/m2] anisotropic contribution to the solid’s extra stress TES

TS

(24)

XVIII Nomenclature

TS

E, mech [ N/m2] purely mechanical part of the solid’s extra stress TES

TS

osm [ N/m2] osmotic part of the solid’s extra stress TES

Uα, Vα [ - ] right and left stretch tensors of the polar decomposition of Fα

Wα [·/s ] skew-symmetric spin tensor of ϕα 4

De [ N/m2] fourth-order elasticity tensor

Calligraphic letters

B, Bα _{aggregate body and body of the constituent ϕ}α

O origin of a coordinate system

P [ N/m2] hydraulic pore pressure

P, Pα _{material points of ϕ and ϕ}α

S, Sα _{surface of the overall and the constituent body}

Su trial spaces of the primary variables

Tu test spaces of the primary variables

V, Vl _{Hilbert space and l-dimensional subspace}

b vector containing the body-force terms

c coefficient vector used in the DEIM approach

∆yk

n vector of stage increments at the current Newton step k

f, fext generalised force vector and generalised external force vector

ˆ

f reduced force vector

F vector containing the global and local system of equations

GGG(·),GGGh(·) abstract function vectors containing the weak forms

k generalised stiffness vector

ˆ

k reduced stiffness vector

L L

Lq,LLLhq vector-valued operators containing the evolution equations

q discrete vector containing the history variables

r residual vector

rev vector containing the non-differential terms of LLLhq

u discrete vector containing the nodal unknowns of each dof

¯

u approximation of the vector of unknowns u

ured reduced vector of unknowns

vr eigenvectors of the Gramian matrix C

w vector containing the nonlinear terms of the equation system

¯

w approximation of the nonlinear-terms vector w

y abstract vector containing the all variables of q and u

(25)

Nomenclature XIX

A matrix made up of unit matrices resulting from the terms in q

C Gramian matrix

D generalised system (damping) matrix

ˆ

D reduced system (damping) matrix

G weighting matrix

J_nk global residual tangent at the current Newton step k

K generalised stiffness matrix

ˆ

K reduced stiffness matrix

Nuj global basis functions at a nodal pointPj in a finite element

P matrix containing information on the DEIM (or magic) points

Rk_n nonlinear functional at the current Newton step k

U snapshot matrix containing snapshots of the state variables

Uϑs part of the snapshot matrix U allocated to ϑs

W snapshot matrix containing snapshots of the nonlinear terms

Wϑs part of the snapshot matrix W allocated to ϑs

Selected acronyms

Symbol Description

2-d two-dimensional

3-d three-dimensional

dof degree of freedom

AF anulus fibrosus

CED convection-enhanced (drug) delivery

COG centre of gravity

DAE differential-algebraic equations

DEIM discrete-empirical-interpolation method

DIRK diagonally implicit Runge-Kutta

DOF degrees of freedom

DDS discrete deformation states

ECM extracellular matrix

EIM empirical-interpolation method

FE finite element

FEM finite-element method

GNAT Gauss-Newton with approximated tensors

HR hyper reduction

(26)

XX Nomenclature

IVD intervertebral disc

LBB Ladyshenskaya-Babuˇska-Brezzi

LDEIM localised discrete-empirical-interpolation method

MBS multi-body system

MDEIM matrix-discrete-empirical-interpolation method

MPE missing-point estimation

NP nucleus pulposus

NRMS normalised root mean square

ODE ordinary differential equations

PDE partial differential equations

POD proper orthogonal decomposition

PVL Pad´e via Lanczos

REV representative elementary volume

SVD singular-value decomposition

TM Theory of Mixtures

TPM Theory of Porous Media

TPWL trajectory piecewise linear

(27)

Chapter 1:

Introduction and overview

1.1 Motivation

Due to progressive technological development, computational simulations have a tremen-dous impact on an increasing number of scientific fields. In the course of time, increasingly precise predictions could be accomplished by means of simulation results. Furthermore, a wide variety of practical tests and extensive measurements could be substituted by simulations. Typical established applications can be found in the material modelling in engineering sciences, in crash simulations in the automotive industry, in weather forecasts in meteorology or in earthquake simulations in computer-aided seismology. But also in the modern medicine, simulations provide new possibilities to gain a deeper understanding of the complex processes in the human body or even to contribute to the successful planning of a surgical intervention by providing supplementary process information. The increasing importance of computational simulations requires for a high level of trustworthiness of the simulation results. In order to meet these requirements, simulations with a sufficiently fine discretisation of the geometry (especially for complex and/or irregular geometries of the considered materials) that are based on sophisticated models are indispensable for the numerical realisation.

Particularly when taking into account the high structural complexity of the underlying materials, a detailed knowledge about the inner structure and the composition of the materials of various components is essential for a sufficiently accurate modelling. A broad variety of materials cannot be described with classical continuum-mechanical models re-stricted to singlephasic (homogeneous) materials. This applies in particular to materials with a porous micro-structure, which, for example, can be found in the scientific fields of geotechnical engineering or biomechanics. The group of porous media, consisting of a porous solid whose pore space is filled with fluids and/or gases, includes, amongst oth-ers, partially saturated soils and biological tissue aggregates. Hence, a multiphasic and multicomponent modelling approach on the basis of the Theory of Porous Media (TPM) appears suitable in order to describe these complex materials. Concerning the numeri-cal treatment of porous materials, the finite-element method (FEM) has been proven to be a well-suited technique for the solution of arbitrary initial-boundary-value problems. The resulting simulations are able to describe the physical phenomena by repeatedly solving the descriptive set of coupled partial differential equations (PDE). However, the necessarily high accuracy of the approximation results and the complexity of the under-lying models result in many applications in an extremely high dimension (in terms of the system’s degrees of freedom) of the resulting equation system. The consequences are time-consuming simulations, which are either to slow to satisfy the time constraints and to enable practical applications (such as an accompanying use in clinical practice) or which cannot be performed as often as needed due to the high computation time.

(28)

2 1 Introduction and overview

In order to counteract these problems, to increase the solution speed and to reduce the computational expenses, model-reduction methods are increasingly important and are gaining a considerable scientific interest. While, on the one hand, the detailed theoreti-cal basis of the modelling approach needs to be maintained and, on the other hand, an efficient numerical computation should be provided, the available performance capacity of projection-based model-reduction techniques can be used to provide fast simulations whenever they are actually needed. Supported by the steadily growing potential of com-puting power and storage capacities, time-consuming simulations based on models with all their complexity can already be performed and all necessary information and data can be stored beforehand in more and more application areas. Following the concept of offline/on-line decomposition, a potentially time-consuming offoffline/on-line phase (including the simulations performed in advance and the reduction of the underlying system of equations) can be separated from a time-efficient online phase with fast simulations (performed using the reduced system). A computationally intensive offline phase pays off by the possibility to rapidly produce required simulation results in daily routines or by a sufficient number of individual computationally inexpensive simulations with varying material and/or simula-tion parameters.

1.2 State of the art, scope and aims

In the present contribution, the developments in the modelling and the simulation of porous materials in the framework of the well-founded TPM, cf. de Boer [16], Bowen [21] or Ehlers [39, 42, 43], with use of the FEM are combined with the current state of research in the field of model reduction using projection-based methods. 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 well-known 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 (with sufficient accuracy of the considered quantities). In this regard, the computation time and the crucial simulation results of simulations using the original unreduced system (hereinafter referred to as “full” or “full-order” systems) or the reduced system are compared to each other in order to demonstrate the efficiency of the considered reduction methods. Moreover, possibilities for an adaptation of the evolved modifications to other models are presented.

Currently, there are several models available, which are used to describe the mechanical behaviour of porous materials. For a proper description, the developed models should be, on the one hand, as simple as possible and, on the other hand, complex enough to capture the relevant properties of the materials. Thereby, the requirements of the in-tended application have an important part in the appropriate description of the model. In terms of the continuum-mechanical modelling, this work is focusing on problems de-veloped on the basis of detailed and thermodynamically consistent TPM models. Early studies concentrated on the multiphasic and multicomponent modelling of porous media with application of the TPM have shown the outstanding suitability of this modelling

(29)

1.2 State of the art, scope and aims 3

approach, cf., e. g., de Boer [15, 16] or Ehlers [39, 42, 43]. In recent years, a wide field of geomechanical and biomechanical applications of the TPM have been successfully derived, cf., e. g., Graf [61] or Ehlers [44] for flow and transport processes in unsaturated soils or Acart¨urk [1], Ehlers [44], Ehlers et al. [49], Ehlers & Wagner [52] or Ricken & Bluhm [108] for biomechanical problems. The present work makes use of a biphasic model for the simulation of a saturated porous soil (cf. Ehlers [40], Ehlers & Eipper [47], Eipper [54], Ellsiepen [55]), a multiphasic and multicomponent model for the simulation of drug-infusion processes in brain tissue (cf. Ehlers & Wagner [51, 52], Wagner [127], Wagner & Ehlers [128]) and an extended biphasic model for the description of an inhomogeneous and anisotropic intervertebral disc (cf. Ehlers et al. [49, 50], Karajan [81]).

Concerning the numerical treatment of the developed TPM models, the FEM is a con-venient technique for the solution of arbitrary initial-boundary-value problems, cf. Lewis & Schrefler [91]. Thereby, the spatial semi-discretisation of an initial-boundary-value problem leads to a system of differential-algebraic equations (DAE) in the time domain. Furthermore, suitable time-integration schemes, such as diagonally implicit Runge-Kutta (DIRK) methods (cf. Diebels et al. [37], Ellsiepen [55]), can be used for the numerical integration of the semi-discrete systems. 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. The numerical implementation is realised with the finite-element solver PANDAS1, which is going back to Ehlers & Ellsiepen [48] and Ellsiepen [55] and is continuously maintained and further developed at the Institute of Applied Mechanics (Continuum Mechanics) at the University of Stuttgart. All computations were performed on a single core of an Intel i5-4590 with 32 GB of memory running at clock speed of 3.30 GHz.

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. Herein, only model-reduction techniques applied to the monolithic solution of the strongly coupled TPM model are considered. In particular, projection-based model-reduction techniques are used to transform a high-dimensional system to a low-dimensional subspace. The advantage of these approaches is to maintain the detailed theoretical basis of the modelling approach while an efficient numerical computation is provided. Besides model-reduction techniques, parallelised solution methods or decou-pled solution strategies may also be considered as alternatives to increase the solution speed and, thus, reducing the computational expenses. On the one hand, parallelised solution methods can be applied to the full system (which is also used to perform the pre-computations required for the determination of the reduced basis) as well as to the reduced one. However, since it is out of the scope of this work, the combination of parallelised 1_{Porous media Adaptive Nonlinear finite element solver based on Differential Algebraic Systems, cf.}

(30)

solution methods with model-reduction techniques is not intended here. On the other hand, decoupled solution strategies can break the problem down to smaller subproblems, which can be subsequently integrated in a staggered manner, cf. Zinatbakhsh [132] or Markert et al. [93]. Nevertheless, this process may generate non-dissipative subproblems and thus render the problem just conditionally stable, cf. Ehlers et al. [53].

Restricting to projection-based model-order-reduction methods, Antoulas & Sorensen [5] provide an overview of different techniques. In this regard, the so-called Krylov-based methods and the singular-value-decomposition(SVD)-based methods can be identified as central approaches. The Krylov-based methods are approximation methods, which are based on the matching of the so-called moments (coefficients of series expansions) of the full and the reduced model, cf. Grimme [64] or Freund [59]. Herein, the response func-tion of the model is interpolated by comparing the coefficients of the T aylor expansion, whereby the first moments of the transfer function of the reduced model need to match with the ones from the full system. Widely used Krylov-based methods are the Padé via Lanczos (PVL), the Arnoldi procedures and the multipoint rational interpolation. Using SVD-based methods, a matrix is broken down into the product of three matrices by means of a singular-value decomposition for the benefit that the singular values can be read off directly. Typical SVD-based methods are the balanced truncation, cf. Moore [96], and the method of proper orthogonal decomposition (POD), which is used as a starting point for the model reduction in this contribution. The development of the POD method, also known as Karhunen-Loève expansion, cf. Sirovich [115], traces back to fluid-dynamic ap-plications including turbulence, cf. Berkooz et al. [10]. Beyond that, the POD method was successfully applied to various problems in fluid flow (cf. Kunisch & Volkwein [88], Rowley et al. [110]), optimal control (cf. Kunisch & Volkwein [86]), aerodynamics (cf. Bui-Thanh et al. [24], Hall et al. [73]), biomechanics (cf. Radermacher & Reese [102]) and structural mechanics (cf. Herkt et al. [76], Radermacher & Reese [103]). The error bounds for POD-Galerkin approximations of linear and nonlinear parabolic equations have been proven by Kunisch and Volkwein [87, 88]. One great advantage of the POD method is that this method is independent from the type of the model and can be used for nonlinear systems as well as for systems of second order. Its flexibility in application is based on analysing a given data set to determine a reduced basis. In order to generate this “training” data, pre-computations are performed in a time-consuming offline phase using the full model and the state solutions (so-called snapshots) of the system are collected. Specifically, a set of snapshots consists of discrete samples of state variables at certain time instances and associated with specific material parameters and particular initial and boundary con-ditions. After a reduced basis is determined, time-efficient simulations can be performed in the online phase using the reduced model. However, since the POD-Galerkin approxi-mation does in fact significantly reduce the dimension of the equation system but not the effort to determine the nonlinear terms, the computational effort of nonlinear problems cannot be (sufficiently) reduced when exclusively using the POD method. This draw-back motivates the application of additional methods for the reduction of the nonlinear terms. In combination with the POD method, adaptive sub-structuring as presented in Radermacher & Reese [103], the trajectory-piecewise-linear (TPWL) method, analysed in Rewieński & White [107], the lookup-table approach, cf. Herkt et al. [76] or Herkt [75], the Gauss-Newton-with-approximated-tensors (GNAT) method, cf. Carlberg et al. [29],

(31)

Carl-1.3 Outline of the thesis 5

berg [30], the approach of hyper-reduction (HR), cf. Ryckelynck [111] and Ryckelynck & Missoum-Benziane [112], and the discrete-empirical-interpolation method (DEIM), which is the discrete variant of the empirical-interpolation method (EIM, cf. Barrault et al. [7]) and which was introduced by Chaturantabut & Sorensen [32], are commonly used tech-niques. At this point, it should be mentioned that the DEIM is similar to the empirical operator interpolation, which was presented in Haasdonk et al. [71] and extended to non-linear problems in Haasdonk & Ohlberger [68]. In the works of Kellems et al. [85] for a model of spiking neurons, in Chaturantabut & Sorensen [34] for a model with application to non-linear miscible viscous fingering, in Nguyen et al. [98] for reacting flow applications, in Negri et al. [97] for parametrised systems or in Bonomi et al. [19] for the application to parametrised problems in cardiac mechanics, amongst others, it could be shown that the DEIM is able to significantly reduce the numerical effort of complex nonlinear processes. Hence, the POD method is used in this work in combination with the DEIM as additional method for the reduction of nonlinear terms. In this regard, problem-dependent modifi-cations of the classical model-reduction approaches are realised exemplarily for the three above-mentioned applications.

To sum it up, the desired goal of this monograph is neither the development of a finite-element model for a specific multiphasic and multicomponent material on the basis of a sophisticated continuum-mechanical modelling approach nor the derivation and formu-lation of an entirely new model-reduction technique. Instead, existing model-reduction approaches are taken as a basis to enable fast simulations of already developed continuum-mechanical models of porous materials. However, it can be shown within this work that the used model-reduction methods in their original form cannot be applied straightfor-ward to the utilised complex material models. To provide time-efficient but also accurate simulations in daily routines, specific, problem-dependent modifications of the existing model-reduction approaches are necessary and are therefore performed within this work. In this regard, an in-depth knowledge of the form and the characteristics of the under-lying equation systems is essential and is therefore treated intensively. For example, it should be ensured that the block structure of the (strongly) coupled equation systems is preserved while considering the different temporal (physical) behaviour of the primary variables. Moreover, the characteristics of the treated DAE systems make it impossible to use already established error estimators with the consequence that alternative error bounds need to be formulated. Ultimately, the derived modifications can be transferred to various other problems where physical phenomena occur on different time-scales. Fol-lowing this, the present work delivers a generalised approach for an adaptation of the evolved modifications to other models.

1.3 Outline of the thesis

Starting with Chapter 2, the continuum-mechanical fundamentals, required for the de-scription of porous media, are summarised on the basis of a general biphasic solid-fluid mixture. Therefore, the basic concept of the TPM is introduced, all necessary kinematical relations, such as the motion of an overall aggregate body, are provided and the balance relations for porous media are presented for the particular constituents and for the entire

(32)

aggregate. Afterwards, the general continuum-mechanical fundamentals are specified for three models of porous materials, namely a biphasic standard problem of a saturated porous soil, a multiphasic and multicomponent description of human brain tissue with application to drug-infusion processes and an extended biphasic model for the description of an inhomogeneous and anisotropic intervertebral disc.

The numerical treatment of the derived governing equations, in the framework of the mixed finite-element method in space and the finite-difference method in time, is shown in Chapter 3. In this regard, the weak formulations of the governing balance equations are presented for the multi-phasic modelling approaches of the previously described examples. This is done in a detailed way for the biphasic standard problem of the porous soil, distinguishing between quasi-static and dynamic initial-boundary-value problems, and briefly for the remaining examples.

Chapter 4 is concerned with the theoretical basis of the considered projection-based model-reduction techniques. After an overview of the different methods is given, all nec-essary mathematical fundamentals of the reduction processes, applied for both linear and nonlinear problems, are shown. In this context, the main focus lies on an application of the POD method, either individually or, for the treatment of nonlinearities, in com-bination with the DEIM. In this regard, application-driven modifications of the classical reduction processes are pointed out.

The application of the model-reduction methods for the three specified TPM models is carried out in Chapter 5. Therein, the utilisation of the POD method for linear and for nonlinear porous-media problems is discussed, as well as the application of the POD-DEIM for nonlinear problems. Starting with the relatively simple biphasic porous-soil model under quasi-static and linear-elastic material behaviour, more complex phenomena, such as dynamic or Neo-Hookean material behaviour, are investigated. Subsequently, a generalised approach for an adaptation of the evolved modifications to other models is presented.

Finally, a summary and an outlook are given in Chapter 6, discussing the presented work and showing possible further potentials of model-reduction implementations. For a better understanding of the discussed topics, additional information regarding the mathematical fundamentals is given in Appendix A and Appendix B. Furthermore, Appendix C provides a detailed presentation of the reduced systems of equations used in this monograph.

(33)

Chapter 2:

Continuum-mechanical fundamentals for the

modelling of porous media

The purpose of this chapter is to review the theoretical fundamentals for the continuum-mechanical description of multiphasic and multicomponent materials using the framework of the well-founded Theory of Porous Media (TPM). Therefore, the concept of volume fractions is presented, followed by a brief overview of the kinematical relations of su-perimposed constituents, providing the introduction of relevant deformation and strain measures. In addition, general balance relations for the overall aggregate as well as for the particular constituents are derived. Finally, this chapter is closed by a specification of the continuum-mechanical fundamentals for the modelling of three particular porous materials.

2.1 The concept of the Theory of Porous Media

With regard to a continuum-mechanical description of multiphasic porous materials, the earliest approaches trace back to the work of Biot [12, 13], describing the consolidation problem of biphasic geomaterials. The two major milestones for a further extension to the current understanding of the TPM (cf. de Boer & Ehlers [17], Ehlers [39, 40, 42, 43]) have been the development of the Theory of Mixtures (TM), cf. Truesdell & Toupin [124] and Bowen [20], and the enhancement by the concept of volume fractions (cf. Mills [94] and Bowen [21, 22]). An excellent overview of the historical evolution of the TPM is given in de Boer [15], de Boer & Ehlers [18] and Ehlers [39, 45].

2.1.1 Macroscopic modelling approach

When regarding the complex inner structure of porous media by means of the TPM, a homogenisation process over a locally defined representative elementary volume (REV), whose constituents are assumed to be in a state of ideal disarrangement, needs to be performed and yields a model on the macroscale. Due to this, the real microstructure of porous materials can remain unknown and the information about the underlying mi-crostructure is assured by the concept of volume fractions. In a multiphasic and mul-ticomponent approach of an entire aggregate model ϕ, multiple constituents ϕα _{can be}

identified, yielding ϕ = S

α

ϕα_{. In this regard, the union symbol is used to characterise}

that the aggregate consists of all the identified consituents (and is therefore not used in a strictly mathematical notation). In order to avoid any confusion, the terms component, phase and constituent are clarified here. Regarding the real composition of a material, several interacting or independent components are recognised and together form the in-tegrated whole of the described material. The term phase is used in this work to describe

(34)

8 2 Continuum-mechanical fundamentals for the modelling of porous media

the chemical state of aggregation, such as solid, liquid or gaseous. In general, each com-ponent can exist in different phase states. Moreover, the term constituent is used in the context of the modelling process. In this regard, all constituents add up to the entire theoretical model of the specified material.

2.1.2 Volume fractions and density functions

In order to account for the local composition of the aggregate, local structure parameters are introduced. The volume V of the overall aggregate bodyB is divided into the sum of its partial volumes Vα _{of the constituent bodies} _Bα_{. This yields}

V = Z B dv = X α Vα. (2.1)

Therein, the partial volume Vα_{can be described by the volume fraction n}α_{, which allocates}

for each spatial point of the actual configuration the local part of the volume Vα _{of the}

constituent ϕα _{on the volume V of the overall aggregate ϕ, yielding}

Vα = Z Bα dv = Z B dvα = Z B nαdv , with nα := dv α dv . (2.2)

Proceeding from equations (2.1) and (2.2), the so-called saturation constraint for a mul-tiphasic approach can be introduced as

X α

nα = 1 , (2.3)

assuming fully saturated conditions. Due to the applied homogenisation, two different densities can be defined for each constituent ϕα_{, the partial density ρ}α _{and the material}

or realistic density ραR_{. Following the concept of volume fractions yields the relations}

ρα := dm

α

dv and ρ

αR _:= dmα

dvα , (2.4)

with the local mass element dmα _{of the constituent ϕ}α_{. The densities are related to each}

other by

ρα = nαραR. (2.5)

Following this, the partial density ρα _{changes with a change of the volume fraction n}α _or

with a change of the realistic density ραR_{. Consequently, the property of material}

incom-pressibility (ραR _{=const.) will not necessarily cause bulk incompressibility (ρ}α _=const.).

Finally, the overall density ρ of the aggregate body _{B can be obtained as the sum of all} partial densities ρ = X α ρα = X α nαραR. (2.6)

Model reduction applied to finite-element techniques for the solution of porous-media problems