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

Robust preconditioners for a new stabilized discretization of the poroelastic equations

In this paper, we present block preconditioners for a stabilized discretization of the poroelastic equations developed in [C. Rodrigo, X. Hu, P. Ohm, J. Adler, F. Gaspar, and L. Zikatanov, Comput. Methods Appl. Mech. Engrg., 341 (2018), pp. 467-484]. The discretization is proved to be well-posed with respect to the physical and discretization parameters and thus provides a framework to develop preconditioners that are robust with respect to such parameters as well. We construct both norm-equivalent (diagonal) and field-of-value-equivalent (triangular) preconditioners for both the stabilized discretization and a perturbation of the stabilized discretization, which leads to a smaller overall problem after static condensation. Numerical tests for both two-and three-dimensional problems confirm the robustness of the block preconditioners with respect to the physical and discretization parameters

Domain Decomposition And Time-Splitting Methods For The Biot System Of Poroelasticity

In this thesis, we develop efficient mixed finite element methods to solve the Biot system of poroelasticity, which models the flow of a viscous fluid through a porous medium along with the deformation of the medium. We study non-overlapping domain decomposition techniques and sequential splitting methods to reduce the computational complexity of the problem. The solid deformation is modeled with a mixed three-field formulation with weak stress symmetry. The fluid flow is modeled with a mixed Darcy formulation. We introduce displacement and pressure Lagrange multipliers on the subdomain interfaces to impose weakly the continuity of normal stress and normal velocity, respectively. The global problem is reduced to an interface problem for the Lagrange multipliers, which is solved by a Krylov space iterative method. We study both monolithic and split methods. For the monolithic method, the cases of matching and non-matching subdomain grid interfaces are analyzed separately. For both cases, a coupled displacement-pressure interface problem is solved, with each iteration requiring the solution of local Biot problems. For the case of matching subdomain grids, we show that the resulting interface operator is positive definite and analyze the convergence of the iteration. For the non-matching subdomain grid case, we use a multiscale mortar mixed finite element (MMMFE) approach. We further study drained split and fixed stress Biot splittings, in which case we solve separate interface problems requiring elasticity and Darcy solves. We analyze the stability of the split formulations. We also use numerical experiments to illustrate the convergence of the domain decomposition methods and compare their accuracy and efficiency in the monolithic and time-splitting settings. Finally, we present a novel space-time domain decomposition technique for the mixed finite element formulation of a parabolic equation. This method is motivated by the MMMFE method, where we split the space-time domain into multiple subdomains with space-time grids of different sizes. Scalar Lagrange multiplier (mortar) functions are introduced to enforce weakly the continuity of the normal component of the mixed finite element flux variable over the space-time interfaces. We analyze the new method and numerical experiments are developed to illustrate and confirm the theoretical results

Rainfall-induced landslides: simulation and validation through case studies with a multiphase porous media model in conjunction with the second order work criterion

Rainfall is one of the most common triggering factors of landslides. Due to the usual large extension of rainfall events, hydrologically-driven instability can be triggered over large areas and frequently results in a diffuse, flow type of failure which occurs abruptly, involving a shallow soil deposit of different grading and origin. Considering the destructiveness of this type of landslides, the forecast of this risk consists a fundamental issue. This PhD thesis is primarily motivated by the need for a better understanding of the slope's mechanical response to rainfall infiltration and for a more accurate and realistic prediction of the location and time of the slope failure occurrence with regards to physical, mathematical and numerical modelling. To this end, the modelling of rainfall induced landslides is considered as a coupled variably saturated hydro-mechanical problem. For the numerical simulations the geometrically linear finite element code Comes-Geo is used, in which soils are considered as non-isothermal elasto-plastic multiphase solid porous materials. Furthermore, this PhD manuscript is dedicated to the concept of material instability in multiphase geomaterials, with particular reference to rainfall-induced landslides. A recently proposed criterion, the second order work criterion, based on Hill's sufficient condition of stability (Hill, 1958) has been implemented on the abovementioned code. It consists in studying the sign of the second order work at the material point level and it is used for the detection of the onset of the local failure. The definition of the second order work criterion is reviewed and three different expressions are presented additionally, which could be used in the case of variably saturated porous materials; the second order work is expressed in terms of effective stress, of total stress and thirdly by taking into account the hydraulic energy contribution for the case of partially saturated soils. The abovementioned modelling framework in conjunction with Hill's criterion is applied for the finite element analysis of three initial boundary value problems: a plane strain compression test on dense sand and isochoric granular material where strain localisation is observed; the failure initiation of a well-documented flowslide (Sarno-Quidinci event, southern Italy 1998); a large scale slope stability experimental test subjected to rainfall infiltration (University of Padua, 2014)

Model development for efficient simulation of CO2 storage

Carbon capture and storage (CCS) is an important component of several initiatives to reduce global greenhouse gas emissions by injecting and storing CO2 in underground reservoirs. Simulation technology plays an important role in providing storage capacity estimates and analyzing long-term safety and risk factors of leakage to the surface. Two of several important questions that need to be answered before a storage project may be approved is how fast and how much CO2 can be injected without compromising the integrity of the sealing caprock, which stops it from migrating to the surface. To evaluate the integrity we rely on mathematical models, but due to the large extent of the area that needs to be considered and the many processes that are involved the calculations can quickly become large and complicated and very time consuming to solve. For screening purposes of potential storage sites, or investigation of potential storage sites when little data is available, many model realizations are needed, thus fast and robust yet accurate numerical techniques are not only tractable but also essential. To understand what happens to the CO2, formation water and rock during injection and storage, we have thoroughly reviewed the main processes that are relevant to the integrity of the reservoir and sealing formations. These main processes are fluid flow, stress change and temperature change and they are all coupled where for instance a change in pore pressure and temperature due to CO2 injection causes deformations and stress alterations that can affect the integrity of the injection reservoir and caprock. Considering the low solubility of CO2 in formation water under typical storage con- ditions (depth, temperature, pressure and salinity) we have illustrated that it is a good approximation to treat the injected CO2 and formation water as two separate fluids. Miscibility is therefore not an important process to consider in relation to long-term mechanical integrity and this simplifies the mathematical description of fluid flow. Whether the thermo-hydro-mechanical coupling, where the temperature change is also considered, is important to evaluate is less obvious. Through examples we show that the in situ temperature is important to consider when estimating material proper- ties, but the effect of the cold (CO2) injection, relative to the storage formation, is very local and mainly affects the near-field of the injector. The cooling effect reduces the spreading of the CO2, but has little effect on the pore pressure. In general, cold injec- tion (relative to formation temperature) lowers the fracture pressure of the rock and the limit for maximum sustainable injection rate, and therefore, ignoring non-isothermal effects can underestimate the risk of failure, and vice versa for hot injection. A risk analysis of reactivation of faults in the sealing formation in the CO2 storage project at In Salah, Algeria, revealed that the thermal effect can make the difference between safe and risky storage. To achieve a procedure for faster numerical evaluation, the layering structure and high aspect ratio of typical storage reservoirs can be used to simplify the mathematical description of the internal physical processes using a method of dimensional reduction. This has previously been found particularly attractive in simulating the migration of CO2 in the context of CCS. Since hydro-mechanical coupling is particularly essential to consider when evaluating the integrity of the caprock, we have extended this concept to also include the geomechanical processes. The underlying assumptions of negligible vertical flow compared to horizontal flow (Vertical Equilibrium, VE, assumption) and linearly varying displacement across the thickness of the reservoir (Linear Vertical Deflection, LVD, assumption) has proved promising in providing significant savings in computational time and effort with up to more than ten times faster calculations compared to a full-dimensional model. It has also been demonstrated that such models can retain a high accuracy when applied to realistic field data, such as the conditions at the CO2 storage plant at In Salah, Algeria. Also, the range of applicability of the dimensionally reduced model is to a leading order the thickness of the reduced domain and accurate solutions in the order of 0.1 % and less difference in solution compared to a full-dimensional formulation for aquifers up to 100 meters thick has been achieved

Preconditioners for Soil-Structure Interaction Problems with Significant Material Stiffness Contrast

Ph.DDOCTOR OF PHILOSOPH

A Computational model for fluid-porous structure interaction

This work utilizes numerical models to investigate the importance of poroelasticity in Fluid- Structure Interaction, and to establish a connection between the apparent viscoelastic behavior of the structure part and the intramural filtration flow. We discuss a loosely coupled computational framework for modeling multiphysics systems of coupled flow and mechanics via finite element method. Fluid is modeled as an incompressible, viscous, Newtonian fluid using the Navier-Stokes equations and the structure domain consists of a thick poroelastic material, which is modeled by the Biot system. Physically meaningful interface conditions are imposed on the discrete level via mortar finite elements or Nitsche's coupling. We further discuss the use of our loosely coupled non-iterative time-split formulation as a preconditioner for the monolithic scheme. We further investigate the interaction of an incompressible fluid with a poroelastic structure featuring possibly large deformations, where the assumption of large deformations is taken into account by including the full strain tensor. We use this model to study the influence of different parameters on energy dissipation in a poroelastic medium. The numerical results investigate the effects of poroelastic parameters on the pressure wave propagation, filtration of the incompressible fluid through the porous media, and the structure displacement