Development of robust and efficient solution strategies for coupled problems

Det er mange modeller i moderne vitenskap hvor sammenkoblingen mellom forskjellige fysiske prosesser er svært viktig. Disse finner man for eksempel i forbindelse med lagring av karbondioksid i undervannsreservoarer, flyt i kroppsvev, kreftsvulstvekst og geotermisk energiutvinning. Denne avhandlingen har to fokusområder som er knyttet til sammenkoblede modeller. Det første er å utvikle pålitelige og effektive tilnærmingsmetoder, og det andre er utviklingen av en ny modell som tar for seg flyt i et porøst medium som består av to forskjellige materialer. For tilnærmingsmetodene har det vært et spesielt fokus på splittemetoder. Dette er metoder hvor hver av de sammenkoblede modellene håndteres separat, og så itererer man mellom dem. Dette gjøres i hovedsak fordi man kan utnytte tilgjengelig teori og programvare for å løse hver undermodell svært effektivt. Ulempen er at man kan ende opp med løsningsalgoritmer for den sammenkoblede modellen som er trege, eller ikke kommer frem til noen løsning i det hele tatt. I denne avhandlingen har tre forskjellige metoder for å forbedre splittemetoder blitt utviklet for tre forskjellige sammenkoblede modeller. Den første modellen beskriver flyt gjennom deformerbart porøst medium og er kjent som Biot ligningene. For å anvende en splittemetode på denne modellen har et stabiliseringsledd blitt tilført. Dette sikrer at metoden konvergerer (kommer frem til en løsning), men dersom man ikke skalerer stabiliseringsleddet riktig kan det ta veldig lang tid. Derfor har et intervall hvor den optimale skaleringen av stabiliseringsleddet befinner seg blitt identifisert, og utfra dette presenteres det en måte å praktisk velge den riktige skaleringen på. Den andre modellen er en fasefeltmodell for sprekkpropagering. Denne modellen løses vanligvis med en splittemetode som er veldig treg, men konvergent. For å forbedre dette har en ny akselerasjonsmetode har blitt utviklet. Denne anvendes som et postprosesseringssteg til den klassiske splittemetoden, og utnytter både overrelaksering og Anderson akselerasjon. Disse to forskjellige akselerasjonsmetodene har kompatible styrker i at overrelaksering akselererer når man er langt fra løsningen (som er tilfellet når sprekken propagerer), og Anderson akselerasjon fungerer bra når man er nærme løsningen. For å veksle mellom de to metodene har et kriterium basert på residualfeilen blitt brukt. Resultatet er en pålitelig akselerasjonsmetode som alltid akselererer og ofte er svært effektiv. Det siste modellen kalles Cahn-Larché ligningene og er også en fasefeltmodell, men denne beskriver elastisitet i et medium bestående av to elastiske materialer som kan bevege seg basert på overflatespenningen mellom dem. Dette problemet er spesielt utfordrende å løse da det verken er lineært eller konvekst. For å håndtere dette har en ny måte å behandle tidsavhengigheten til det underliggende koblede problemet på blitt utviklet. Dette leder til et diskret system som er ekvivalent med et konvekst minimeringsproblem, som derfor er velegnet til å løses med de fleste numeriske optimeringsmetoder, også splittemetoder. Den nye modellen som har blitt utviklet er en utvidelse av Cahn-Larché ligningene og har fått navnet Cahn-Hilliard-Biot. Dette er fordi ligningene utgjør en fasefelt modell som beskriver flyt i et deformerbart porøst medium med to poroelastiske materialer. Disse kan forflytte seg basert på overflatespenning, elastisk spenning, og poretrykk, og det er tenkt at modellen kan anvendes i forbindelse med kreftsvulstmodellering.There are many applications where the study of coupled physical processes is of great importance. These range from the life sciences with flow in deformable human tissue to structural engineering with fracture propagation in elastic solids. In this doctoral dissertation, there is a twofold focus on coupled problems. Firstly, robust and efficient solution strategies, with a focus on iterative decoupling methods, have been applied to several coupled systems of equations. Secondly, a new thermodynamically consistent coupled system of equations is proposed. Solution strategies are developed for three different coupled problems; the quasi-static linearized Biot equations that couples flow through porous materials and elastic deformation of the solid medium, variational phase-field models for brittle fracture that couple a phase-field equation for fracture evolution with linearized elasticity, and the Cahn-Larché equations that model elastic effects in a two-phase elastic material and couples an extended Cahn-Hilliard phase-field equation and linearized elasticity. Finally, the new system of equations that is proposed models flow through a two-phase deformable porous material where the solid phase evolution is governed by interfacial forces as well as effects from both the fluid and elastic properties of the material. In the work that concerns the quasi-static linearized Biot equations, the focus is on the fixed-stress splitting scheme, which is a popular method for sequentially solving the flow and elasticity subsystems of the full model. Using such a method is beneficial as it allows for the use of readily available solvers for the subproblems; however, a stabilizing term is required for the scheme to converge. It is well known that the convergence properties of the method strongly depend on how this term is chosen, and here, the optimal choice of it is addressed both theoretically and practically. An interval where the optimal stabilization parameter lies is provided, depending on the material parameters. In addition, two different ways of optimizing the parameter are proposed. The first is a brute-force method that relies on the mesh independence of the scheme's optimal stabilization parameter, and the second is valid for low-permeable media and utilizes an equivalence between the fixed-stress splitting scheme and the modified Richardson iteration. Regarding the variational phase-field model for brittle fracture propagation, the focus is on improving the convergence properties of the most commonly used solution strategy with an acceleration method. This solution strategy relies on a staggered scheme that alternates between solving the elasticity and phase-field subproblems in an iterative way. This is known to be a robust method compared to the monolithic Newton method. However, the staggered scheme often requires many iterations to converge to satisfactory precision. The contribution of this work is to accelerate the solver through a new acceleration method that combines Anderson acceleration and over-relaxation, dynamically switching back and forth between them depending on a criterion that takes the residual evolution into account. The acceleration scheme takes advantage of the strengths of both Anderson acceleration and over-relaxation, and the fact that they are complementary when applied to this problem, resulting in a significant speed-up of the convergence. Moreover, the method is applied as a post-processing technique to the increments of the solver, and can thus be implemented with minor modifications to readily available software. The final contribution toward solution strategies for coupled problems focuses on the Cahn-Larché equations. This is a model for linearized elasticity in a medium with two elastic phases that evolve with respect to interfacial forces and elastic effects. The system couples linearized elasticity and an extended Cahn-Hilliard phase-field equation. There are several challenging features with regards to solution strategies for this system including nonlinear coupling terms, and the fourth-order term that comes from the Cahn-Hilliard subsystem. Moreover, the system is nonlinear and non-convex with respect to both the phase-field and the displacement. In this work, a new semi-implicit time discretization that extends the standard convex-concave splitting method applied to the double-well potential from the Cahn-Hilliard subsystem is proposed. The extension includes special treatment for the elastic energy, and it is shown that the resulting discrete system is equivalent to a convex minimization problem. Furthermore, an alternating minimization solver is proposed for the fully discrete system, together with a convergence proof that includes convergence rates. Through numerical experiments, it becomes evident that the newly proposed discretization method leads to a system that is far better conditioned for linearization methods than standard time discretizations. Finally, a new model for flow through a two-phase deformable porous material is proposed. The two poroelastic phases have distinct material properties, and their interface evolves according to a generalized Ginzburg–Landau energy functional. As a result, a model that extends the Cahn-Larché equations to poroelasticity is proposed, and essential coupling terms for several applications are highlighted. These include solid tumor growth, biogrout, and wood growth. Moreover, the coupled set of equations is shown to be a generalized gradient flow. This implies that the system is thermodynamically consistent and makes a toolbox of analysis and solvers available for further study of the model.Doktorgradsavhandlin

On the optimization of iterative schemes for solving non-linear and/or coupled PDEs

In this thesis we study the optimization of iterative schemes as both linearization methods, and as splitting methods for solving non-linear and coupled partial differential equations (PDEs). We consider two equations that are describing processes in porous media; Richards’ equation, a possibly degenerate, non-linear and elliptic/parabolic equation that models flow of water in saturated/unsaturated porous media, and Biot’s equations, a coupled system of equations that models flow in deformable porous media. For Richards’ equation we compare the numerical properties of several linearization schemes, including the Newton-Raphson method, the modified Picard method and the L-scheme. Additionally, we prove convergence of the linearly and globally convergent L-scheme and discuss theoretically and practically how to choose its stabilization parameter optimally in the sense that convergence is obtained in the least amount of iterations. The second aim of the thesis is to effectively solve the quasi-static, linear Biot model. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot’s equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization parameter. We propose a new approach to optimize this parameter, and show theoretically that it does not only depend on the mechanical properties and the coupling coefficient, but also on the fluid’s flow properties. The type of analysis presented in this thesis is not restricted to a particular spatial discretization, but we require it to be inf-sup stable. The convergence proof also applies to low-compressible or incompressible fluids, and low-permeable porous media. We perform illustrative numerical examples, including a well-known benchmark problem, Mandel’s problem. The results largely agree with the theoretical findings. Furthermore, we show numerically that for conditionally inf-sup stable discretizations, the performance of the fixed-stress splitting scheme behaves in a manner which contradicts the theory provided for inf-sup stable discretizations.Masteroppgave i anvendt og beregningsorientert matematikkMAMN-MABMAB39

The distribution of small consignments from a production facility remote to the markets: The case of OMYA Hustadmarmor AS

Confidential until 20. May 201

Health, Social, and Lifestyle Factors Related to Combined Alcohol and Sleep Problems: Investigating Risks of Chronic Pain, Welfare Benefits, Divorce, Smoking, and Low Physical Activity in Relation to Alcohol and Sleep Problems

The purpose of the study was to uncover whether there are increased risks associated to having combined alcohol and sleep problems in association to health, social, and lifestyle factors. This in comparison to having either alcohol and sleep problems, as opposed to having neither alcohol nor sleep problems. This study investigates potential risks in association to chronic pain, reception of welfare benefits, divorce, smoking, and physical activity. This was investigated by using data collected from the sixth Tromsø study, Tromsø 6. Results show that combined alcohol and sleep problems are associated to risks in chronic pain, welfare benefits, and smoking among men, and only smoking among women. For the same factors it seems that sleep is the decisive factor among women. This study has recognised some associations in combined problems and other factors, future studies are needed to decide the causality of the associations

Segmentation and Unsupervised Adversarial Domain Adaptation Between Medical Imaging Modalities

Segmenting and labelling tumors in multimodal medical imaging are often vital parts of diagnostics and can in many cases be very labor intensive for clinicians. The effort in advancing time-saving methods in the medical health sector might be of great help for busy clinicians and can maybe even save lives. Furthermore, creating methods that generically, accurately and successfully process unlabelled data would be a major breakthrough in deep learning. This thesis aims to address both these challenges by exploring and improving current methods involving adversarial discriminative domain adaptation (ADDA) on multimodal imaging, and address weaknesses, not only in ADDA, but also in the general adversarial discriminative cases. More specifically, this thesis - applies convolutional neural networks to segment soft tissue sarcomas in PET, CT and MRI modalities, and to the author's best knowledge achieves state-of-the-art results, - explores unsupervised adversarial discriminative domain adaptation on segmentation of soft tissue sarcoma tumors between permutations of PET, CT and MRI and - demonstrates weaknesses in state-of-the-art adversarial discriminative training, and finally - improves and provides groundwork for further research on said techniques. Additionally, the thesis will also provide strong fundamental background for applying ADDA for use in medical modalities, including a solid introduction to deep learning in medical imaging, both from a theoretical and practical aspect

Development of an Experimental Model to Quantify Lumbar Spine Kinematics during Military Seat Ejection

The initial phase of a military ejection sequence exerts substantial axial loads on the spinal column. Eccentric inertial loading on the thoracolumbar spine can lead to injury. Most serious injuries due to ejection are in the form of a vertebral fracture, most commonly occurring at the thoracolumbar junction. The objective of the current study was to understand characteristics of a military seat ejection by employing an experimental model designed to simulate the boost or in-rail phase. The model incorporates realistic boundary conditions and is capable of quantifying metrics associated with injury tolerance such as applied accelerations and resultant loads and spinal kinematics. A total of four human cadaveric spine specimens (T12-L5) were tested. The test matrix consisted of two parts. The first part subjected specimens to sub-failure loading to outline spinal kinematics during dynamic vertical acceleration. The second part of the test matrix consisted of acceleration tests designed to induce compression and/or burst fractures as sustained by military aviators during ejection. The developed experimental model is the first to simulate realistic inertial loading during ejection-type accelerations using isolated osteoligamentous spines and may provide imperative injury mechanism data for future safety design considerations

Hvordan forbedre egen praksis for yrkesfaglærere i det digitale klasserommet?

I denne mastergradsoppgaven har vi gjennomført et aksjonsforskningsprosjekt med følgende problemstilling: Hvordan forbedre egen praksis for yrkesfaglærere i det digitale klasserommet? Ved å gjennomføre til sammen syv ulike aksjoner i to ulike team ved en videregående skole i Østfold har vi forsøkt ut en modell for praktisk gjennomføring av et lokalt pedagogisk utviklingsarbeid, med det formål å utvikle digital kompetanse hos yrkesfaglærerne. Prosjektet tar utgangspunkt i en grunnleggende undersøkelse om den digitale kompetansen blant deltakerne. Utviklingsprosjektet har utviklet en god kollegabasert opplæring knyttet til praktisk anvendelse av digital kompetanse i yrkesfaglærernes pedagogiske og didaktiske tilnærming til undervisningen. Deltakerne får gjennom refleksive samtaler satt fokus på egen praksis.Prosjektet belyser en rekke av de vanskeligheter både lærere, skoleleder og skoleeier står ovenfor når den digitale kompetansen skal utvides. Prosjektet viser at solid forankring i ledelse er en forutsetning for å kunne heve den digitale kompetansen i lærerkollegiet. Prosjektet viser at utviklingen av den digitale kompetansen for yrkesfaglærere er kontekstavhengig og gir best resultat når den er knyttet til fag og kan utvikles som en del av den faglige kompetansen.Prosjektet problematiserer de begrepene som lærere og elever møter i skolen. Hva er digital kompetanse, hva er digitale verktøy og hva er digitale ferdigheter. Å utvikle god pedagogisk praksis med digital kompetanse og IKT tar tid, og prosjektet setter fokus på viktigheten av gode rammebetingelser. Etter hvert som aksjonene forløp observerte vi at deltakerne ble mer bevisste og aktive i sin anvendelse av sin digitale kompetanse versus elevene. Praksisendring skjer i møtet med elevene og relevansen til elevenes læringssituasjon og utbytte bør være tydelig, slik at yrkesfaglærere ser en praktisk nytteverdi i forhold til sitt daglige arbeid.For å få bekreftet vår problemstilling har vi anvendt ulike metoder, teoretisk tilnærming og praktisk utprøving.In this master assignment, we have performed an action research project with the following question: How to improve vocational teachers own practice in the digital classroom? By performing a total of seven different actions in two different teams working in a secondary school in Østfold county, Norway, we have tried out a model of practical implementation of a local, educational development, to improve own practice in the classroom with respect to digital literacy. Our project is based on a basic study on digital competence among our participants. The project has developed a good, colleague-based training related to the practical application of digital competence in vocational teachers pedagogical and didactic approach to teaching. Participants have developed their experience, through reflective conversations, based on their own practice. The project highlights a number of difficulties that both teachers, school leaders and school owners face when the digital competence is expanded. The project shows that a solid management grounding is a prerequisite to improve the digital competence among teachers.The project shows that the development of digital competence for vocational educators is context dependent and provides the best results when it is linked to subjects and can be developed as part of their professional competence. The project discusses the concepts many teachers and students face in school. What is digital literacy, what are digital tools and what are digital skills. To develop good teaching practices with digital literacy and ICT takes time. As the action progressed, we observed that participants were more aware and active in their use of their digital skills versus students. Development of good practice evolves in the meeting with the students and relevance to pupils' learning situation and the outcomes should be clear, so that vocational teachers see a practical value in relation to their daily work.In order to verify our approach, we have applied various methods, theories and practical testing.Master i yrkespedagogik