Coupling a vascular graph model and the surrounding tissue to simulate flow processes in vascular networks Timo Koch 12 September 2014 Title Coupling a vascular graph model and the surrounding tissue to simulate flow processes in vascular networks Institutions Principle Supervisor Co-Supervisor Stuttgart Supervisor Oslo Co-Supervisor Oslo Student University of Stuttgart and Simula Research Laboratory Oslo Prof. Rainer Helmig, University of Stuttgart Dr. Natalie Schröder, University of Stuttgart Prof. Kent-Andre Mardal, University of Oslo / Simula André Massing, PhD, Simula Research Laboratory Oslo Timo Koch, B.Sc., University of Stuttgart Abstract Mathematical models of fluid exchange in the microcirculation can help to understand complex processes and may guide treatment of diseases in the future. To this end, a model with reduced computational demand is investigated making it possible to model large networks of vessels in interaction with the surrounding tissue. We derive a reduced model from a spatially resolved model and assess the error made with the model reduction. A two step reduction results in a first model with reduced vessel wall and finally in a second model with reduced vessel that couples a one-dimensional vessel graph with a three-dimensional tissue domain through line sources. Firstly, we construct a Darcy-Stokes coupled problem where the Darcy domain is separated from the Stokes domain by a thin membrane. For this problem a new set of interface conditions is derived. A locally conservative discontinuous Galerkin method is proposed to solve problems of this kind. Furthermore, it is shown that iterative Robin-Robin domain decomposition can be a more efficient alternative to direct solvers for Darcy-Stokes multi-compartment models. Secondly, it is shown that the reduced model is very accurate and efficient for geometrically symmetric problems in a wide range of physically relevant model parameters. Furthermore, it is shown that the error made by missing asymmetry features is smaller than that of model parameter uncertainty. The reduced model is also solved numerically for cases where the vascular graph can be chosen independently of the tissue grid. Deutsche Zusammenfassung Mathematische Modelle des Fluidaustausches in der Mikrozirkulation können zum Verständnis komplexer Vorgänge beitragen und in Zukunft die Krankheitstherapie begleiten. Reduzierte Modelle sind in der Lage große Netzwerke von Blutgefäßen, und die Interaktion mit dem umgebenden Gewebe, effizient zu berechnen. In dieser Arbeit wird in zwei Schritten ein voll aufgelöstes homogenisiertes Modell reduziert und die Fehler, die durch die Reduktion eingebracht werden beschrieben. Im ersten Schritt wird ein Modell mit reduzierter Gefäßwand entwickelt und neue Interfacebedingungen vorgeschlagen. Im zweiten Schritt wird ein Modell hergeleitet, bei dem ein eindimensionales Blutgefäßnetzerk mit einer dreidimensionalen Gewebeumgebung durch Linienquellen gekoppelt wird. Die Arbeit analysiert zunächst ein Darcy-Stokes Problem bei dem die beiden Gebiete durch eine dünne Membran getrennt sind. Ergebnis ist eine massenkonservative DiscontinuousGalerkin-Diskretisierung zur Lösung von Darcy-Stokes Problemen mit Drucksprung am Interface. Darüber hinaus zeigen die Ergebnisse das eine iterative Robin-Robin Gebietszerlegung bei solchen Mehrgebiets-Kopplungsproblem eine effiziente Alternative zu direkten Lösern ist. Das zweite reduzierte Modell zeigt sich akkurat und effizient in einem großen medizinisch relevanten Parameterbereich. Die Fehler durch Parameterunsicherheit übertreffen die Fehler durch die fehlende Abbildung von Asymmetrie im reduzierten Modell. Acknowledgment I thank my lovely parents for their support throughout my studies. I thank my girlfriend who had to borrow me for a while to the Norwegians. Speaking of those, a special thanks goes to André Massing who always took time to explain me the math and Kent-Andre Mardal for superb support and supervision and a great stay at the Simula Research Laboratory in Oslo. Further, I thank Rainer Helmig and Natalie Schröder for always having a friendly ear and great tips. I thank the German Research Foundation (DFG) for the funding within the international Research Training Group “Non-Linearities and Upscaling in Porous Media” (NUPUS). Contents 1 Introduction 1 2 Mathematical model 4 2.1 Fundamental balance equations in continuum mechanics . . . . . . . . . . . . . . 4 2.1.1 Balance of mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Balance of momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Modeling a blood vessel in the microcirculation . . . . . . . . . . . . . . . . . . . 7 2.3 Modeling the capillary bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Modeling transmural fluid exchange . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5 Model parameter values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.1 Viscosity of blood an interstitial fluid . . . . . . . . . . . . . . . . . . . . 14 2.5.2 Permeabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.5.3 Pressures and velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3 Coupling concepts 16 3.1 Interface conditions with a selective permeable membrane . . . . . . . . . . . . . 17 3.2 The coupled Darcy-Stokes system with selective permeable membrane . . . . . . . 18 3.3 A one-dimensional model for a blood vessel in the microcirculation . . . . . . . . . 19 3.4 A tissue model with source term on a line . . . . . . . . . . . . . . . . . . . . . . 23 3.5 The coupled 1D-3D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.6 The coupled 1D-2D model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4 5 The finite element method 26 4.1 The strong formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3 Essential and natural boundary conditions . . . . . . . . . . . . . . . . . . . . . . 28 4.4 The variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Finite element discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.6 Mixed variational formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.7 An interior penalty discontinuous Galerkin method for the Stokes problem . . . . . 32 Discretizing and solving coupled Darcy-Stokes systems 5.1 Unified mixed element formulation for the coupled Darcy-Stokes problem with selective permeable membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 5.3 39 39 Robin-Robin domain decomposition of the coupled Darcy-Stokes system with selective permeable membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Iterative domain decomposition of the 1D-2D reduced Darcy-Stokes problem . . . 44 5.3.1 45 Calculation of line sources . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Implementation 47 7 Comparison scenarios 50 7.A The reference scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.B Variations in geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 7.C Variations of model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8 Results and Discussion 57 9 Summary and Outlook 72 Nomenclature In this work, lower case symbols (p) denote scalar quantities and bold lowercase symbols (u) represent vectors or vector-valued functions. Bold uppercase symbols (T) denote second-order tensors or tensor-valued functions. Index notation of vectors or tensor operations uses Einstein notation. The operator ∇(·) denotes the gradient of a function with respect to the position vector x. So, ∇p = ∂p ∂xi = p,i and ∇u = ∂ui ∂xj = ui,j is the gradient of the scalar function p and the vector function u, respectively. The operator ∇ · (·) denotes the divergence of a function with respect to x with ∇·u = ∂ui ∂xi = ui,i and ∇ · T = Tij,j being the divergence of the vector function u and the tensor function T, respectively. The Laplace operator ∆(·) is equal to ∇ · ∇(·), the divergence of the gradient of a function. The determinant of the tensor T is denoted by det T. The trace of the tensor T is given by tr(T) = Tii . Furthermore, the following list of symbols is used. Symbol Description Unit General symbols Ω A physical domain ∂Ω The boundary of a domain Ω Ωp Darcy domain Ωf Stokes domain Γ Interface (·)f Physical quantity of the Stokes domain (·)p Physical quantity of the Darcy domain t Time s v Velocity m s p Pressure Pa % Density kg m3 µ Dynamic viscosity (of blood if not otherwise stated) P as ν Kinematic viscosity (of blood if not otherwise stated) m2 s Symbols introduced in Chapter 2 B Abstract physical body P Material point inside a physical body x Current position vector of a material point X Reference position vector of a material point ei Orthonormal basis of R3 Symbol Description O Origin of the coordinate system χ Lagrangian motion function −1 Unit χ Eulerian motion function F Deformation gradient u Deformation vector I Second-order Identity tensor f Body/volume force m s2 t Traction vector Pa T Stress tensor Pa a Acceleration field m s2 dv Volume integrand of the current configuration dV Volume integrand of the reference configuration M Mass kg I Momentum kg ms F External forces N D(v) Symmetric velocity gradient of velocity v 1 s ϕ Mixture ϕα Constituent α of the mixture ϕ (·)α Kinematic physical quantity of the constituent α (·) α m Non-kinematic physical quantity of the constituent α nα Volume fraction %α Partial density φ Porosity %̂ Density production term p̂ α Momentum production term kg sm3 kg s 2 m2 T Absolute temperature K Re Reynolds numer K Intrinsic permeability m2 K Scalar isotropic intrinsic permeability m2 Q Total flux (over the capillary wall) A Surface area m3 s 2 m π Oncotic pressure Pa Lp Filtration coefficient of the capillary wall m P as Symbol Description Unit KM Intrinsic permeability of the capillary wall m2 µi Viscosity of the interstitial fluid P as dM Thickness of the capillary wall m Symbols introduced in subsequent chapters KR Friction parameter m2 s p̄ Average pressure (see text for average operators) Pa δΓ Dirac delta distribution on Γ R Capillary radius γf , γp Acceleration parameters θ Relaxation parameter wi Weighting parameter for Gaussian quadrature rule xi Integration point for Gaussian quadrature rule p̄in Dirichlet boundary condition at Stokes inlet Pa p̄out Dirichlet boundary condition at Stokes outlet Pa p̄p Dirichlet boundary condition for Darcy domain Pa Function spaces Real numbers R L2 Square integrable functions n Functions with nth weak derivative ( H div) Functions with divergence in L2 C0 Continuous functions Cn Continuous functions n-times differentiable V Trial function space V̂ Test function space P1 Continuous linear functions P2 Continuous quadratic functions P0 Continuous constant functions P1 Discrete space of piecewise linear polynomials / P1 -element P2 Discrete space of piecewise quadratic polynomials / P2 -element DG0 Discrete space of piecewise constant functions / DG0 -elements H m 1 Introduction The microcirculation is the fundamental structure to provide cells with oxygen and nutrients and to distribute pharmaceuticals. Although geometries might be available through specialized imaging techniques, exact measurements of flow fields or distribution of a certain chemical are often too invasive and costly. Mathematical models of flow and transport processes in the microcirculation and the surrounding tissue help to understand the complex structure and processes and can guide treatment and therapy of diseases. Possible problems of interest include oxygen transport to the brain in case of a stroke, blood supply and growth of tumors (angiogenesis), treatment of tumors with therapeutic agents (e.g. nano particles), transport of antibiotics to biofilms on implants. Apart from diseases, mathematical models may contribute to understanding complicated whole-body processes like training effects on muscles, or regeneration of brain tissue during the sleep1 . Mathematical simulation can simulate system response to a wide range of parameters. The simulation can yield information even beyond the situation of the measurements it was calibrated with. The microcirculation is a complicated network that features extensive branching and looping or bypassing. A description from aterioles, or even arteries, down to thousands of tiny capillaries per cubic centimeter tissue [Formaggia et al., 2009a] is highly complex. A fully spatially resolved model of a network this size exceeds the limits of current computational power and time. These models usually do not go further than investigating a single capillary, e.g. the model by Baber [2014]. This demands reduced models which can be solved numerically at a fraction of the computational power required for solving fully resolved models. Two main ideas have been presented in the literature recently. The first kind are homogenized models of the microcirculation where the vessels are described as volume fractions in homogenized tissue control volumes [Erbertseder, 2012; Ehlers and Wagner, 2013; Chapman et al., 2008]. The second kind of models reduce the vessels to their centerlines, and the resulting one-dimensional flow in the microcirculation is coupled with the threedimensional tissue through line sources [D’Angelo, 2007; Cattaneo and Zunino, 2013; Sun and Wu, 2013; Secomb et al., 2004]. The reduced model in this thesis is in the latter category. Up to now, it has not been investigated which errors the model reduction introduces. 1 see recent study on Alzheimer’s: [Ju et al., 2013] 1 The objectives of this thesis are: ◦ Which assumptions are necessary to derive a reduced one-dimensional capillary flow model and surrounding three-dimensional tissue? ◦ In which situation do the assumptions hold, in which they do not? ◦ How much faster is the reduced model in comparison to fully resolved models? There are several approaches on how to derive the reduced model. However, a full derivation starting from a coupled Darcy-Stokes system with all necessary assumptions has not yet been published to the knowledge of the author. We perform a step by step reduction which allows us to compare models of different reduction levels. This work starts with a homogenized yet still fully spatially resolved model of a single capillary as proposed by Baber [2014] to study transport processes over the vessel wall in detail. In a first step, the vessel wall is reduced to a two-dimensional surface. This results in a coupled Darcy-Stokes system which is separated by a membrane on the vessel surface. DarcyStokes systems have been extensively studied in literature, we recommend the review by Discacciati and Quarteroni [2009]. However, the reduced vessel wall alters the well-known coupling conditions which results in a new set of conditions introducing a large pressure jump across the Darcy-Stokes interface. A locally conservative finite element discretization for this new problem is presented. Furthermore, the system is solved using a direct solver and an algorithm is presented in order to solve it iteratively following the idea of Discacciati et al. [2007]. A domain decomposition approach is highly flexible and accounts for the different physics of the subproblem. In a second step, the remaining three-dimensional vessel is reduced to its centerline. Quarteroni and Formaggia [2004] list three ways of deriving a one-dimensional model from the three-dimensional (Navier-)Stokes equations. In this work, we integrate the Stokes equations over a generic section and include the surrounding tissue. Furthermore, is questionable, whether the assumptions of the reduction hold in all imaginable, physical scenarios. With two models, i.e. a spatially resolved and a spatially reduced model, we can compare different cases and quantify model errors. An optimal result is achieved if the error introduced through the assumptions is small but the reduction in computational cost is large. The model reduction is visualized conceptionally in Figure 1.1. This thesis is structured as follows: In Chapter 2 the basic continuum mechanical framework is set up to derive the necessary model equations. The generally derived balance laws of mass and momentum are then adapted to the underlying physical problem. Medical knowledge is provided when needed for the model assumptions. With the mathematical equations for the subsystems vessel, tissue, and capillary wall at hand, coupling conditions are discussed in Chapter 3. Firstly, a new set of coupling conditions for a coupled Darcy-Stokes system is introduced by reducing the vessel wall. Secondly, the one-dimensional flow model is derived. For the second model, a different coupling strategy is needed than in the spatially resolved model. In a mathematical excursion, Chapter 4 presents the finite element method and the basic mathematical framework. Furthermore, the chapter explains 2 Figure 1.1 – Reducing a model. Starting from a fully spatially resolved tissue, vessel, and vessel wall (left) the wall is reduced first (middle). Then, the vessel is reduced to its centerline (right). more advanced finite element formulations. With these tools at hands the mathematical problems of Chapter 3 can be discretized and solved numerically. In Chapter 5, discretization methods for the coupled systems are presented. Additionally to a fully coupled approach, we discuss a domain decomposition method with the possibility to use specialized solvers in each subdomain suiting the prevalent physics. After introducing a few comparison scenarios in Chapter 7, results from all model are presented, discussed and compared in Chapter 8. Finally, Chapter 9 provides a summary of findings and future plans and research suggestions. 3 2 Mathematical model In this chapter the fundamental governing equations are derived. The balance of mass and the balance of momentum are introduced. Based on those, Section 2.2 develops a blood model governed by the incompressible Stokes equations. An introduction to the modeling of porous media flow is given in Section 2.3 and leads to Darcy’s law as a model for biological tissue. Section 2.4 explains how to model fluid flow across the vessel wall with Starling’s law. The chapter closes with remarks on model parameters and the primary variables. For a more detailed description of the continuum mechanical basis the interested reader is referred to [Ehlers and Bluhm, 2002; Boer, 2000]. Before mathematical models can be set up it is important to understand the structure of the underlying physical problem. For an extensive assertion of all relevant processes in a modeling context we refer to the excellent introduction of [Baber, 2009]. In this work, we only give a short introduction to the structure of capillaries and flow processes provided in place, when needed for model assumptions. 2.1 Fundamental balance equations in continuum mechanics In order to derive the fundamental balance equations, the following picture of a deforming body B should be kept in mind (Figure 2.1). Here, ei={1,...,n} is an orthonormal basis of Rn with origin O. The vectors x and X denote the current and the reference position vector, respectively. Furthermore, n is the outward pointing normal vector on ∂B, t is the traction vector, and %f represents a volume or body force acting on the whole body B, e.g. gravity. The motion of the deforming body can be described by a Lagrangian motion function, i.e. the current position vector x of a material point P is depending on the reference position vector X and the time t x = χ(X, t). (2.1) The basic kinematical quantity in a large strain setting is the deformation gradient F= ∂χ(X, t) ∂x ∂(X + u) ∂u = = =I+ , ∂X ∂X ∂X ∂X 4 (2.2) ∂B n x = χ(X, t) t B F= ∂x ∂X =I+ ∂u ∂X X x u=x−X e2 e3 %f e1 O Figure 2.1 – A deforming body B where u = x − X is the displacement vector and χ the motion function of a material point P ∈ B. In order to be unique, the motion function has be invertible, leading to the following constraint: X = χ−1 (x, t) if det F 6= 0.1 (2.3) It is then possible to describe the motion, velocity, and acceleration fields in a Lagrangian or material setting x = χ(X, t), ẋ = v = d χ(X, t), dt v̇ = a = d2 χ(X, t), dt 2 (2.4) or, using the inverse motion function, in an Eulerian or spatial setting v = v(x, t), where 2.1.1 d dt (·) ˙ = = (·) ∂ ∂t (·) a = a(x, t). (2.5) + v ∇(·) indicates the material time derivative of a physical quantitiy. Balance of mass The conservation of mass is a fundamental axiom in continuum mechanics Z dM % dv. = 0 with M = dt B (2.6) The density is denoted by %, and dv and dV are infinitesimal volume elements in the current and reference configuration, respectively. Then, using the identities dv = det F dV and 1 and det F > 0, to rule out interpenetration of matter. 5 d dt det F = det F∇·v 2 yields Z Z Z Z d % dv = (%̇ dv + % ḋv) = (%̇ dv + %(det˙ F)dV ) = (%̇ + %∇·v) dv = 0. dt B B B B Applying the localization theorem dv → 0 yields the local form of the mass balance %̇ + %∇·v = 0 ∂% + ∇·(%v) = 0 ∂t or (2.7) in its general form. For lots of applications in fluid mechanics the density of the fluid can be assumed constant, resulting in the incompressible mass balance ∇·v = 0. 2.1.2 (2.8) Balance of momentum In a similar manner as the mass balance one can derive the balance of momentum Z Z Z dI = F with I = %v dv and F = t ds + %f dv. dt B ∂B B (2.9) Applying Cauchy’s theorem (t(x, t, n) = T(x, t)n) and the Gauss-Green formula yields Z Z %v dv = B Z Tn ds + d dt Z ∇·T dv + %f dv = B ∂B Using the identities dv = det F dV, Z B %f dv. B det F = det F∇·v, and the balance of mass, the global form of the balance of momentum is obtained as Z Z Z Z v(%̇ + %∇·v) + %v̇ dv = %v̇ dv = ∇·T dv + %f dv. B B B B The localization theorem dv → 0 finally yields the local form of the balance of momentum dv %v̇ = % =% dt 2 d dt det F = ∂ det F ∂F ∂v + v · ∇v ∂t = ∇·T + %f. ·· Ḟ = det F(FT−1 ·· Ḟ) = det F(F−1 Ḟ ·· I) = det F tr( ∂X ∂x 6 ∂ ẋ ) ∂X = det F tr(∇v) = det F∇·v (2.10) 2.2 Modeling a blood vessel in the microcirculation The Navier-Stokes equations describe the motion of fluids. They are obtained from the general mass balance and general momentum balance by inserting the constitutive law for Newtonian fluids τ = 2µD(v), (2.11) where τ is the shear stress tensor and D(v) = 12 (∇v + ∇T v) the symmetric velocity gradient, via the relation T = τ + pI, (2.12) where T is the Cauchy stress tensor with respect to the current configuration and p the hydrostatic pressure. Thus, the incompressible Navier-Stokes equations read % ∂v + v · ∇v ∂t = 2µ∇·D(v) − ∇p + %f, (2.13) ∇·v = 0. Note that for incompressible fluids ∇·v = 0 (2.8) and thus ∇·∇T v = 0 inside the domain. Herein, blood is the considered fluid. Blood is a mixture of several components. Most prominently, it consists of red and white blood cells, blood platelets, plasma and plasma proteins [Formaggia et al., 2009b]. The stress behavior of the mixture is generally non-Newtonian. The blood viscosity depends on the plasma viscosity, the pressure, haematocrit, the deformation of red blood cells in small capillaries, the vessel diameter and the blood composition [Baber, 2009]. However, for simplicity and the reason that this work’s primary object is the verification of a model reduction to a one-dimensional model, blood is modeled as an incompressible Newtonian fluid with constant viscosity. More sophisticated viscosity models are easily implemented. Blood flow is mostly laminar, especially in the microcirculation. Reynolds numbers Re = vc Lc , ν (2.14) where we choose the characteristic length Lc as the vessel diameter, are very small (ca. 0.003 in capillaries according to Formaggia et al. [2009a]). For creeping flow (Re 1), the non-linear inertial term on the left-hand side can be omitted and the linear incompressible Stokes equations (2.15) are obtained −2µ∇·D(v) + ∇p = 0, (2.15) ∇·v = 0. Although gravity can have a noticeable influence on the flow field depending on the orientation of the vessel, we neglect the effects of gravity in this thesis. It is justifiable because we will compare 7 ∂B n χs (Xs , t) t B χf (Xf , t) Xs u = x − Xff x e2 γ Xf e3 e1 O Figure 2.2 – A deforming body B being a mixture of two constituents ϕS and ϕF model concepts rather than produce quantitative results or simulate experimental data. Gravity effects can be easily added later. 2.3 Modeling the capillary bed The capillary bed is a highly complex structure consisting of fibers, cells, amorphous ground substance, and interstitial fluid. To model flow processes, the system has to be simplified. To this end, we introduce the continuum mechanical framework for the modeling of porous media. For a more detailed description, we refer to [Ehlers and Blum, 2002]. Modeling biological tissue as a porous medium is common in literature, see [Erbertseder, 2012] as an example. Modeling porous media, one typically deals with a multiphase system where a mixture ϕ is constituted by several constituents α ϕ= [ ϕα . (2.16) α A porous medium is described given at least one solid phase ϕS constituting the porous solid matrix and one fluid phase ϕF , the pore fluid. Each constituent α of the mixture is described by an individual motion function χα , velocity and acceleration fields, vα , aα , respectively. It posseses, thus, also individual deformation gradients Fα = ∂x . ∂Xα (2.17) A deforming body with two constituents α ∈ {F, S} is depicted in Figure 2.2. The reduction of 8 a highly complex biological system to a simpler porous medium model is based on the concept of volume averaging introduced by Hassanizadeh and Gray [1979]. The domain is homogenized on the scale of a representative elementary volume (REV). The process of homogenization is shown in Figure 2.3. The size of an REV is defined at the point where further enlargement of the control volume does not change the value of a homogenized physical quantity, e.g. the porosity. Finding an REV can be challenging for highly heterogeneous materials. The capillary vessel wall is e.g. so thin that it is questionable if the REV concept is applicable [Baber, 2014]. The local composition of the mixture is described by partial volumes V α and volume fractions nα [Markert, 2005] Z V = dv = B X V α with V α Z = Z dv = B α α nα dv. (2.18) B The volume fractions nα are defined locally as nα := dvα . dv (2.19) In a biphasic model nS , nF are called solidity and porosity, respectively. When the solid matrix is assumed rigid, solidity and porosity become constant. The constant porosity is then, for simplicity, denoted by φ. It follows from (2.18) that no vacant space in the domain is allowed, thus X nα = 1. (2.20) α Furthermore, the concept of partial densities is introduced. Each constituent has a material realistic density %αR , but can be additionally associated with a partial density %α related to the density % of the mixture. They are defined as %αR := dmα , dvα %α := dmα , dv %= X %α , (2.21) α and further related via the volume fractions %α = nα %αR . (2.22) Note, that although the realistic density might be constant in case of material incompressibility, the density of the mixture can still change through the change of the volume fractions. For a rigid solid matrix, however, the density of the mixture remains constant as well. Balance equations can be formulated for a single constituent, as long as the action of the other constituents upon this constituent is considered. The mixture behaves like a single phase and its balance equations are obtained by adding up the balance equations of the constituents. These principles are known as Truesdell’s metaphysical principles [Truesdell, 1984]. Following the principles, 9 interstitial fluid ϕF cells ϕS dvF dvS microscale REV scale dv Figure 2.3 – Homogenization and the concept of volume fractions the mass balance of a constituent α is formulated analogously to the single phase mass balance (2.7) ∂%α + ∇·(%α vα ) = %̂α , ∂t (2.23) where %̂α is a production term that accounts for interaction with the other constituents. It can be visualized best for the two constituents ice and water, where %̂α quantifies how much ice melts into water and visa versa. For two immiscible constituents %̂α vanishes. From the above mentioned principles follow the constraints X %α = % X and α %̂α = 0. (2.24) α The balance of momentum for the constituent α reads ∂vα %α + vα · ∇vα = ∇·Tα + %α f α + p̂ α + %̂α vα , ∂t (2.25) where p̂ α accounts for the momentum production by interaction with other constituents, e.g. through friction, and %̂α vα is the momentum production resulting from a mass production, e.g. ice melts in water. Again from Truesdell’s metaphysical principles follow the constraints X α %α vα = %v , X [Tα − %α (vα − v)] = T α , X α %α f α = %f and X (p̂ α + %̂α vα ) = 0. α (2.26) The simplest multiphase model is called a biphasic model, or, when the solid phase is assumed to be rigid, it is also referred to one-phase fluid flow in a porous medium. In this work we will use a one-phase model to simplify the tissue domain. All solid constituents if the interstitial tissue are unified to a single solid phase perfused by the interstitial fluid. The interstitial fluid is generally a 10 mixture too. With all the solutes united in a single fluid phase it can be modeled as an incompressible Newtonian fluid. In order to derive the one-phase model used in this thesis we make the following assumptions: A1 All solid constituents are united in a single homogeneous, isotropic solid phase ϕS A2 The fluid phase ϕF and the solid phase are immiscible A3 Neglection of body forces f α = f = 0 A4 Solid and fluid are materially incompressible %αR = const. A5 Isothermal process at θ = 37◦ C A6 Creeping fluid flow Re 1 A7 Rigid solid skeleton vS = 0 Furthermore, the momentum production p̂ F is expressed by the following constitutive law, p̂ F = p∇φ − p̂µF = p∇φ − φ2 µF K−1 (vF − vS ), (2.27) where p is the fluid pressure, φ denotes the porosity, µF the dynamic viscosity of the interstitial fluid, and K the positive definite intrinsic permeability tensor of the porous medium. The production term p̂ F can be seen as the local momentum production through friction of the interstitial fluid with the solid matrix. The stress tensor TF for a general fluid can be expressed as TF = TFµ − φpI = 2µF DF + λ(DF · I)I − φpI (2.28) with the second Lamé constant λ. The mass balance of the interstitial fluid reduces to ∇·(φvF ) = ∇·vf = 0, (2.29) where vf is called filter or seepage velocity. Starting from the momentum balance for the interstitial fluid (2.25), A2, A3, and A6 yield 0 = ∇·TF + p̂ F . (2.30) A dimensional analysis [Ehlers et al., 1997] shows that TFµ p̂µF for small characteristic length, e.g. pore diameter scale. This results in 0 = −∇·(φpI) + p∇φ − φ2 µF K−1 vF 0 = −φ∇p − p∇φ + p∇φ − φ2 µF K−1 vF (2.31) K vf = − F ∇p µ Equation (2.31) is known as Darcy’s filter law and was found by Darcy [1856] as result of a sand 11 Figure 2.4 – The three different types of capillaries. Continuous capillary (left), fenestrated capillary (middle), discontinuous capillary (right). Figure from Baber [2014]. column experiment. Darcy’s law can be reformulated by substituting the velocity in the mass balance (2.29) with the momentum balance (2.31) K − ∇· ∇p µF = 0. (2.32) In a first approach, the porous medium is often assumed to be homogenous and isotropic, so the permeability can be substituted by a scalar K. In reality, however, porous materials are often highly heterogenous and anisotropic. 2.4 Modeling transmural fluid exchange The interface between Stokes and Darcy domain is given by the selective permeable vessel wall. The vessel wall can in fact itself be modeled as an additional Darcy domain, e.g. [Quarteroni and Formaggia, 2004]. However, it is questionable whether an REV really exists because of its small dimensions [Baber, 2014]. Section 2.4 shows the three types of capillaries and their capillary walls. The capillary wall consists of two layers. The inner one is formed by endothelial cells (pink), the outer one is a basement membrane or basal lamina (green) that consists of fibers like collagen. The endothelial cells are connected by tight junctions. Water can pass through pores where the tight junctions are defective. Few larger pores also permit the exchange of larger molecules like proteins. The number of pores and thickness of the two layers differs for different types of capillaries, so does the amount of fluid exchange. Larger pores are more numerous in discontinuous capillaries and the basement membrane is reduced to a minimum. They occur in liver, spleen and bone marrow and have the highest exchange rates. Continuous capillaries have the lowest fluid exchange and can be 12 found in muscles, skin, lungs, and the central nervous system [Formaggia et al., 2009a]. The fluid movement across the capillary wall is determined by Starling’s law Q = Lp A [(pf − pp ) − σ(πf − πp )] , (2.33) where Q is the flux across the vessel wall with the filtration coefficient Lp and the surface area A. Further, pf and pp denote the hydrostatic pressure in the vessel and the interstitium, respectively. The oncotic or colloid osmotic pressure π is an osmotic pressure exerted by proteins3 . It usually causes an osmotic drag of water inside the blood vessel and is therefore working against hydraulic pressure gradient. The reflection coefficient for plasma proteins σM says what fraction of proteins is retained by vessel through reflection at the capillary wall. It is close to 1 for macromolecules and close to 0 for micromolecules [Jain, 1987]. The oncotic pressure difference remains nearly constant along the capillary. In all the following models we therefore join the oncotic pressure and the fluid pressure to one new primary variable. From now on, p shall denote the effective pressure pαe := pα − σπα , α = {p, f }. (2.34) For the physiological informations in this paragraph [Hall, 2010] was consulted. Starling’s law can be also interpreted as a Darcy-type law where the tangential velocity component is neglected vM · n = KM [pf − pp ] , µi dM (2.35) where vM is the seepage velocity, n the normal vector on the surface of the vessel wall pointing towards the interstitium, and the filtration coefficient of the capillary wall in now expressed as Lp = KM , µi dM (2.36) with the intrinsic permeability of the wall KM , its thickness dM and the fluid viscosity µi . The fluid viscosity is that of water for very small pores but higher for bigger pores when loaded with heavy solutes. It is simply assumed to be equal to the viscosity of the interstitial fluid in this work. The flow then corresponds to a tube model, where water flow paths through the membrane are simplified as cylindrical pores. The effective pressure gradient must be interpreted discretized over the full vessel wall pf − pp . dM ∇p = (2.37) With this interpretation it is possible to integrate Starling’s law in a new set of Darcy-Stokes interface conditions (see Chapter 3). Literature values are available for both the intrinsic permeability of the wall KM and the filtration coefficient Lp . 3 http://en.wikipedia.org/wiki/Oncotic˙pressure 13 2.5 Model parameter values Each of the above presented models relies on empirical parameters that need to be determined by experiments. Ischinger [2013] has aggregated literature values for all necessary parameters in this work. We refer to his work for literature references. This section presents the key parameters and provides an estimated range within which the parameters can fall. We also calculated an average from the literature values obtained by Ischinger [2013]. Estimated averages are provided when literature values are given only for combinations of model parameters. It is sometimes not specified at which exact location or under which conditions a parameter was measured. Parameters can change even along a single capillary. However, we regard the range of parameters as legitimate range for testing our numerical models. The section starts out with the parameter of the blood model, the blood viscosity µ. It proceeds with the parameters of the tissue model, the viscosity of the interstitial fluid µi , the intrinsic permeability K, and the parameter of the transmural flow model, the filtration coefficient of the vessel wall Lp or its intrinsic permeability KM . The section concludes with pressures and velocities that are necessary to find meaningful boundary condition and to check numerical results to consistency. 2.5.1 Viscosity of blood an interstitial fluid As mentioned above blood is a mixture of various components. However, it is legitimate to describe it with a constant viscosity parameter µ if the flow conditions and geometry of the vessel are invariant during the simulation. Large particles in the blood, in particular red blood cells, can not pass the vessel wall. The interstitial fluid therefore has equal properties as blood plasma and can be modeled as a Newtonian fluid with constant viscosity µi . The viscosity has the unit P as. According to the literature consulted by Ischinger [2013] the blood viscosity can be estimated ranging from 2 − 3.5 · 10−3 P as where a value of µ = 2.1 · 10−3 P as was conducted for small vessels. The viscosity of the interstitial fluid can be estimated ranging from 1.1 − 2 · 10−3 P as with an average of µi = 1.3 · 10−3 P as. 2.5.2 Permeabilities The intrinsic permeability K of the solid matrix quantifies the flow resistance these obstacles pose for the fluid. It is highly anisotropic in the interstitium and can be e.g. obtained by diffusion tensor imaging [Ehlers and Wagner, 2013]. Due to the lack of patient specific data and the general focus on model reduction of this work, the permeability is assumed isotropic and replaced by a scalar value. Some literature values are given only for the hydraulic conductivity K µi . The intrinsic permeability has the unit m . Ischinger [2013] found literature values in the range of 4.4 · 10−18 − 3 · 10−17 m2 2 14 for the intrinsic permeability and 2.3 · 10−15 − 6.6 · 10−15 estimated average of K = 6.5 · 10 −18 m P as for the hydraulic conductivity with an 2 m . The intrinsic permeability KM of the vessel wall contains several resistance mechanism due to the complex nature of the transmural flow. Often, literature values are only available for the effective parameter, the filtration coefficient Lp . The filtration coefficient includes the thickness of the capillary wall and thus has the unit 2.5 · 10 −12 −9 − 1.5 · 10 m P as m P as . Ischinger [2013] found literature values in the range of for the filtration coefficient. The value is highly dependent on the type of capillary. The highest literature values were obtained for fenestrated capillaries that have a high amount of large pores. Intrinsic permeability values ranged from 2.4 · 10−20 − 9.7 · 10−18 m2 . The estimated average is Lp = 3.0 · 10−11 2.5.3 m P as . Pressures and velocities Primary variables in all models are velocity field v and effective pressure field p. The primary variables are the solution of the numerical simulation. However, reasonable boundary values have to be provided beforehand to solve the numerical model. The capillary blood velocity in our model will be determined by pressure, geometry, and the above presented model parameters. For a reference the mean blood velocity in capillaries is estimated being |vf | < 10−3 m s [Quarteroni and Formaggia, 2004]. As introduced above, the effective pressure consists of parts form the hydrostatical pressure and the oncotic pressure. The oncotic pressure is nearly constant along the vessel wall. On the contrary the hydrostatic pressure exhibits large gradients from aterial to venous end of a capillary. At the arterial end one finds net filtration of fluid into the tissue, at the venous end fluid gets reabsorbed. We do not intent to vary the pressure values in the scope of this work. Therefore, the mean values obtained by Baber [2014] are used to construct a comparable model test. She estimated the hydrostatic pressure at the arterial end of a capillary to p̄in = 4000 P a and the hydrostatic pressure at p̄out = 2000 P a with respect to the interstitial hydrostatic pressure p̄i = 0 P a. The interstitial pressure was estimated to be close to atmospheric pressure. She further used πf = 3600 P a for the oncotic pressure in the vessel and πp = 933 P a for the oncotic pressure in the interstitium. In Chapter 2, the governing equations for modeling blood flow in small vessel, one-phase flow in biological tissue and flow across a selective permeable membrane were derived. Furthermore, model assumptions based on given geometry, processes, and composition of the real problem were presented and a values for parameters were obtained from the literature. However, modeling flow in one of the mentioned domains alone is not enough to solve the full flow field. The vessel is connected to the tissue and the two are separated by the vessel wall. The equations need to be coupled in a physical sensible manner in order to calculate the flow in the entire domain. The following Chapter 3 presents these coupling mechanisms. 15 3 Coupling concepts Modeling transport processes in blood vessels and tissue constitutes a multi-domain problem. One domain is the blood vessel with a pipe-like flow governed by the Navier-Stokes equations. The other domain is the connected tissue surrounding the vessel which can be modeled as porous medium governed by Darcy’s law. Both domains influence the behavior of the respective other domain, i.e. they are coupled. Considering the very different nature of the models in both domains, the coupled problem is also a multi-physics problem. To realize the coupling of the tissue and vessel domains, this work presents a new set of interface conditions coupling Darcy and Stokes flow separated by a thin membrane. The new interface conditions allow the description of the vessel wall without spatially resolving it. The new interface conditions are presented in Section 3.1. The subsequent sections present the two fundamental coupling concepts in two models. According to Helmig et al. [2013] the first model is classified as a multi-compartment model, the second model as a multi-dimensional model. The first model is derived by looking at two spatially resolved domains, a free-flow domain and a porous domain, the blood vessel and the surrounding tissue, respectively. The domains are coupled at a common interface with appropriate interface conditions (see Section 3.1). The vessel wall model is herein reduced to an interface condition. All domains are illustrated in Figure 3.3. The first model is introduced in Section 3.2. In the second model, the vessel domain is reduced to a one-dimensional domain placed inside a spatially fully resolved tissue domain. The two domains are coupled through (line) source terms. The second model can be obtained from the first model making further assumptions. A model reduction, starting from the spatially resolved first model, is presented in Section 3.3. The model problem for three and two dimensions is presented in Sections 3.5 and 3.6. 16 vessel wall M vessel wall M Ωf Ωp n n τ pp Ωf Ωp vM · τ ≈ 0 n τ τ pp pM,f pM,p pf pf Figure 3.1 – Reduction of the capillary wall to a line interface between the capillary and the surrounding tissue 3.1 Interface conditions with a selective permeable membrane We start by recalling that Starling’s law (2.35), describing fluid flow across the capillary wall can be interpreted as Darcy’s law assuming the flow in tangential direction τ is negligible. Thus, the capillary wall M is a Darcy domain where flow only occurs in direction of n. Further, let Γf = ∂Ωf ∩∂M denote the interface of the capillary wall with the vessel domain and Γp = ∂Ωp ∩∂M the its interface with the tissue domain. Figure 3.1 shows a part of the system tissue–capillary wall– capillary explaining the aforementioned symbols. The interface Γp requires interface conditions that couple a Darcy domain with another Darcy domain. These can be trivially formulated as the continuity of the pressure across the interface pM,p = pp , (3.1) and the continuity of the normal velocity (mass conservation) vM · n = vp · n. (3.2) The interface Γf requires interface condition that couple a Darcy domain with a Stokes domain. There is a vast number of literature on Darcy-Stokes coupling that all use the interface conditions comprehensively investigated e.g. in [Discacciati and Quarteroni, 2009]. Mass conserves across the interface. This interface condition can be written as the local mass balance as above, vf · n = vM · n 17 (3.3) For simplicity n = nf denotes the outward pointing normal on ∂Ωf . Another interface condition is obtained by balancing the normal stresses at the interface, − 2µD(vf )n · n + pf = pM,f . (3.4) A third interface condition is required for the tangential stresses. An interface condition introduced by Beavers and Joseph [1967] as an experimental result, simplified by Saffman [1971] and also justified later mathematically by Mikelic and Jäger [2000] is the Beavers-Joseph-Saffman condition µ − 2µD(vf )n · τ = α √ vf · τ K (3.5) We assume in this work that the slip velocity vf · τ |Γf is negligible. Thus, vf · τ = 0. (3.6) The tangential free-flow velocity gets in fact smaller the lower the permeability of the porous is. Such a no-slip condition is justifiable for the very low permeability, KM ≈ 10−20 m2 (see Section 2.5), of the vessel wall. In a second step, we reduce the capillary wall by one dimension (dM → 0). The interfaces Γf and Γp now fall on one single interface Γ. The new interface has modified interface conditions that are vf · n = (vM · n) = vp · n, (3.7) the mass balance across the interface, − 2µD(vf )n · n + pf = µi dM v f · n + pp KM (3.8) the balance of normal stresses, and the interface condition for the tangential velocity (3.6) that stays untouched. The three interface conditions (3.7), (3.8) and (3.6) couple the Darcy domain with the Stokes domain under consideration that the interface between them is actually constituted of a selective permeable membrane. 3.2 The coupled Darcy-Stokes system with selective permeable membrane The domain Ω is split into a free-flow domain Ωf representing the blood vessel and a porous domain Ωp representing the surrounding tissue separated by a selective permeable membrane Γ. It is illustrated by Figure 3.2. The Stokes equations govern the free-flow domain Ωf and Darcy’s law 18 Γ nf Ωp nf Ωp Ωf Figure 3.2 – The domain Ω consisting of the free-flow domain Ωf (vessel) and the porous domain Ωp (tissue). the porous domain Ωp . Problem 3.1 (Coupled Darcy-Stokes problem) Find (v, p) such that −2µ∇·D(vf ) + ∇pf = 0 in Ωf (3.9) −∇·vf = 0 µi vp + ∇pp = 0 K −∇·vp = 0 in Ωf (3.10) in Ωp (3.11) in Ωp (3.12) on Γ (3.13) on Γ (3.14) on Γ (3.15) The applied interface conditions on Γ = ∂Ωp ∩ ∂Ωf are vf · n = vp · n −2µD(vf )n · n + pf = µ i dM vf · n + pp KM vf · τ = 0 The system is closed by appropriate boundary conditions on ∂Ωf and ∂Ωp . For the applied boundary conditions see Chapter 8. The coupling concept is equivalently applicable for 3D-3D coupling and 2D-2D coupling. 3.3 A one-dimensional model for a blood vessel in the microcirculation The diameter of a small vessel is usually small in comparison to the characteristic length of the vessel. The flow in microcirculation is laminar with Reynolds numbers smaller than 1 resulting in rather simple velocity fields. This motivates the reduction of the vessel to a one-dimensional object 19 in order to reduce computational costs. This section presents the reduction of the Stokes equations from three dimensions to one. nr S Ωp z − S+ Ωf nz + nz − M ω dz Figure 3.3 – A part P of a blood vessel in the microcirculation surrounded by tissue Ωp To derive the one-dimensional Stokes equations we start from the incompressible full three-dimensional Stokes equations (2.15) in cylindrical coordinates (r, θ, z) and subsequently reduce the system, making the following assumptions: A1 Axial symmetry. The velocity profile is symmetric with respect to the axis ∂v ∂θ =0 A2 Rigid arterial wall. The displacement of the arterial wall can be neglected in the microcirculation. Thus, R = const. A3 Constant pressure. The pressure is assumed constant over a cross-section. p = p(z) A4 Negligible radial velocity. Inside the domain the radial velocity can be neglected in comparison to the axial velocity. This follows the derivation presented in [Quarteroni and Formaggia, 2004] for the full Navier-Stokes equations. We look at a part P of a capillary vessel Ωf surrounded by a tissue compartment Ωp . The vessel is depicted in Figure 3.3. Let S denote an axial section of a vessel with the measure A = 2πR2 . The axial component of the velocity field can be written as v · nz = vz (r, z) = v̄ (z)s(r ) (3.16) where s(r ) = h r γ i 1 (2 + γ) 1 − γ R (3.17) is a velocity profile of a power law type, yielding a parabolic profile for γ = 2. The mean velocity is given by 1 v̄ = A Z 1 v ds = A S (A4) 20 Z vz nz ds = v̄ (z). S (3.18) Note that therefore Z s ds = A. (3.19) S Let ω denote the wall of the part of the capillary vessel P, and S + and S − the outflow and the inflow cross section, respectively, so that ∂P = ω ∪ S + ∪ S − (see Figure 3.3). We will integrate the Stokes equations over P = {(r, θ, z) : r ∈ [0, R), θ ∈ [0, 2π), z ∈ (z − dz 2 ,z dz 2 )} + and then go to the limit dz → 0. An interface condition, modeling the behavior of the wall as a selective permeable membrane, is introduced as a Robin-type boundary condition on the vessel wall ω (see Section 2.4) v · nr = vr (r, z) = KM (p − pi ) µi dM The mass balance can then be reduced as follows Z Z Z Z Z 0= ∇·v dv = v·n ds = v·n ds− vz ds+ P ∂P S− ω on ω. (3.20) Z vz ds = S+ Z v·n ds− Z v̄ s ds+ S− ω v̄ s ds. S+ Note that the second fundamental theorem of calculus holds for z+ Z dz 2 dz dz ∂ v̄ − v̄ z − dz = A v̄ z + ∂z 2 2 A z− where we used R S dz 2 s ds = A. Applying the interface condition and recalling that ds = R dθ dz in cylindrical coordinates yields Z Z v · n ds = ω ω KM (p − pi )R dθ dz µi dM where 1 p̄i = 2πR dz→0 ≈ 2πR KM (p − p̄i ), µi dM (3.21) Z pi (z , θ)R dθ (3.22) θ is the interstitial pressure averaged over the surface of the vessel wall. As the vessel fluid pressure is assumed constant over a cross-section such an average operator is obsolete. The one-dimensional mass balance then reads −A ∂ v̄ KM = 2πR (p − p̄i ). ∂z µi dM (3.23) For the momentum balance, we follow the same procedure. The integration of the pressure term is straightforward 1 ρ For the viscous term Z Z ν∆v dv = P ∂P Z dz→0 ∇p dv = P Z ν∇vn ds = S− A ∂p nz . ρ ∂z ν∇vn− z ds + 21 Z S+ ν∇vn+ z ds + Z ν∇vnr ds ω we neglect the change of v with respect to z in comparison to the change in radial direction, ∇vnz = ∂v ≈0 ∂z and we split ∇vnr in its radial and its axial part, so that Z Z Z ν∇vnr ds = ν(nr ⊗ nr )∇vnr ds + ν(nz ⊗ nz )∇vnr ds ω ω ω Z Z ∂vz ∂vr nr ds + nz ds. ν = ν ∂r ∂r ω ω Recalling, that vz (r, z) = v̄ (z)s(r ) we get ∂vz ∂s ν nz ds = 2πRν v̄ nz = −KR v̄ nz . ∂r ∂r r =R ω Z where KR = −2πRν ∂s ∂r r =R is a friction parameter. The given power type law (3.17) for the axial velocity profile results in KR (γ) = 2πν(2 + γ). For the radial part of the velocity gradient, we get Z ∂vr ν nr ds = ∂r ω Z ∂vr ν ∂r z Z 2π nr dθ dz = 0. 0 This yields the one-dimensional momentum balance in a three-dimensional world A ∂p nz + KR v̄ nz = 0 ρ ∂z (3.24) where nz is a three-dimensional vector in axial direction of the reduced vessel. Finally, the full one-dimensional Stokes equations read A ∂p nz + KR v̄ nz = 0 ρ ∂z ∂ v̄ KM −A = 2πR (p − p̄i ) ∂z µi dM (3.25) Note that the velocity in the mass balance can be eliminated by inserting the momentum balance, resulting in A ∂p nz + KR v̄ nz = 0 ρ ∂z 2πR4 ∂ 2 p KM = 2πR (p − p̄i ) 2 µ(2 + γ) ∂z µi dM (3.26) where γ is the parameter for the power type axial velocity profile. The above derived model assumed that the vessel is surrounded by a three-dimensional tissue matrix. However, when looking at a two-dimensional model, the reduction to one dimension slightly differs. The measure for the cross-section S is then A2D = 2R. Integrals over the vessel wall ω are 22 calculated as R ω R dθ dz dz→0 = 2πR in three dimensions, but as R ω,2D 2 dz dz→0 = 2 in two dimensions. The reduced one-dimensional model then reads 2R ∂p nz + KR,2D v̄ nz = 0 ρ ∂z 2R3 ∂ 2 p KM =2 (p − p̄i,2D ) 2 µ(2 + γ) ∂z µi dM with KR,2D = 1 R 2ν(2 + γ) (3.27) and p̄i,2D = 12 (pi |R + pi |−R ). Consequently, nz is now a two-dimensional vector in direction of the reduced vessel. To this end, pi |R denotes and evaluation of the interstitial pressure at distance R from the vessel on one side of the vessel and pi |−R the evaluation at distance R on the opposite side. Note that the 2D formulation is then equivalent to the 3D formulation, except for the calculation of the source term average operator. 3.4 A tissue model with source term on a line The Darcy domain Ωp , the tissue, and the one-dimensional free-flow domain Γ, the vessel, are coupled via interface conditions on the vessel wall. The interaction can be modeled by including a source term f on a line in the mass balance (3.28). − ∇· K ∇pp = f δΓ µi in Ω, (3.28) where δΓ is the Dirac delta distribution with the following properties δΓ = Z 1 on Γ 0 elsewhere Z f δΓ dv = f ds. Ω (3.29) Γ It restricts the source term to a line representing the blood vessel. A comparison with (3.26) yields f = 2πR KM (pf − p̄p ) µ i dM (3.30) for a three-dimensional model and f =2 1 KM pf − (pp |R + pp |−R ) µ i dM 2 for a two dimensional model. 23 (3.31) 3.5 The coupled 1D-3D model Γ nz Ω Figure 3.4 – A 1D blood vessel Γ in the microcirculation surrounded by tissue Ωp The 1D-3D model features a one-dimensional vessel model inside a three-dimensional tissue model. The governing equations and coupling source term were introduced in the previous sections. The porous tissue domain Ωp is traversed by a line, the vessel domain Γ. The vessel domain has a null measure in R3 and we subsequently write Ωp as Ω. The domain is illustrated in Figure 3.4. In order to better identify the mathematical nature of the problem the coefficients in (3.26) are aggregated into one coefficient C= R3 µi dM µ(2 + γ)KM (3.32) The problem then reads Problem 3.2 (1D-3D coupled problem) Find (pf , pp ) such that ∂ 2 pf − pf = −p̄p ∂z 2 K KM −∇· ∇pp = (2πR (pf − p̄p ))δΓ µi µi dM C on Γ (3.33) in Ω The same model was also obtained by Cattaneo and Zunino [2013] using an immersed boundary method. The coupling is non-trivial since the formulation is a mixed integral differential formulation due to the pressure average operator. 3.6 The coupled 1D-2D model The 1D-2D model features a one-dimensional vessel model inside a two-dimensional tissue model. The governing equations and coupling source term were introduced in the previous sections. The domain is illustrated in Figure 3.5. The problem reads in analogy to Problem 3.2 24 Γ nz Ω Figure 3.5 – A 1D blood vessel Γ in the microcirculation surrounded by tissue Ωp Problem 3.3 (1D-2D coupled problem) Find (pf , pp ) such that ∂ 2 pf − pf = −p̄p,2D ∂z 2 K KM −∇· ∇pp = (2 (pf − p̄p,2D ))δΓ µi µi dM C on Γ (3.34) in Ω where the averaging operator is now the 2D averaging operator presented at the end of Section 3.3. In Chapter 3 we have presented two conceptionally different coupled models describing the flow field in and around a blood vessel in the microcirculation. In the first model, the vessel is fully spatially resolved. In the second model, the vessel is reduced to its centerline. Both models were derived for three and two dimensions. The following investigations are conducted with the two-dimensional model for sake of simplicity of implementation and solution. In order to solve the problems posed in this section using computers, we need to introduce numerical methods. Chapter 4 presents the finite element method. 25 4 The finite element method In this chapter, the numerical method used within this work is presented: the finite element method (FEM). The finite element method and its variations are versatile numerical methods to solve partial differential equations. This chapter provides the basic mathematical tools of FEM and introduces some numerical applications. Subsequent sections also introduce mixed finite element methods, discontinuous Galerkin methods, and stabilized FEM methods. For more comprehensive introductions to the finite element method, we refer to [Larson and Bengzon, 2013; Brenner and Scott, 2008; Logg et al., 2012a]. The finite element method is explained here by means of treating the Poisson equation numerically. − ∆u = f (4.1) The Poisson equation is an elliptic partial differential equation (PDE), i.e. information propagates equally in all directions. It can describe e.g. heat conduction, electrical conduction, diffusive transport or flow in porous media. In order to obtain a determined system to solve numerically we have to restrict it to a finite domain Ω and equip it with Dirichlet and Neumann boundary conditions. A Dirichlet boundary condition is of the form u = u0 and fixes the solution function u to a value u0 on the Dirichlet part of the boundary ∂ΩD . A Neumann condition boundary is of the form ∇u · n = g and fixes the normal derivative ∂u ∂n = ∇u · n of the solution function u to a value g on the Neumann part of the boundary ∂ΩN . 4.1 The strong formulation The Poisson problem (4.1) together with the boundary conditions is called strong formulation of the Poisson problem. Let the considered domain Ω ⊂ Rn , n ∈ {2, 3} be an open and bounded domain and let Ω̄ denote its closure. Problem 4.1 (Strong formulation) Find u ∈ C 2 (Ω̄) such that − ∆u = f in Ω, u = u0 on ∂ΩD , 26 ∇u · n = g on ∂ΩN , (4.2) where C k (Ω̄) = {u ∈ Ω̄ : u and its derivatives up to kth order are continuous}, ∂ΩD and ∂ΩN denote the boundary parts of Ω with Dirichlet and Neumann boundary conditions, respectively. Here, u ∈ C 2 (Ω̄) is called the strong or classical solution of the problem. The restriction for u ∈ C 2 is strong. In a numerical scheme we have to deal with discrete non-differentiable (in a classical sense) functions or even discontinuous functions. In what follows, we describe an alternative formulation of the problem called the variational or weak formulation. It is less restrictive towards u. The weak formulation employs function spaces making use of weak derivatives of the form Z 1 Z 1 f v 0 dx gv dx = − 0 ∀v (4.3) 0 where v is a test function satisfying v (0) = v (1) = 0 and g = f 0 is called the weak derivative of f . In order to continue the explanation a short introduction to finite element function spaces is required. 4.2 Function spaces Let us define two function spaces commonly encountered in a finite element setting. The function space Z 2 L (Ω) = {u ∈ Ω : 1 2 < ∞} u dv 2 (4.4) Ω is the space of functions where the squared function is bounded in a Lebesgue sense, or measurable, 1 R and kukL2 = Ω u 2 dv 2 its norm. In other words, a function u is in L2 (Ω) if kukL2 is smaller than infinity. The function space H1 (Ω) = {u ∈ L2 (Ω) : ∇u ∈ L2 (Ω)n } (4.5) is called Sobolev space (of first order). With the scalar product Z Z ∇u · ∇v dv + (u, v )H1 = Ω uv dv (4.6) Ω and the so induced norm kukH1 = p (u, u)H1 (4.7) H1 (Ω) is a Hilbert space. Or, in short, the space of L2 functions whose gradients are also L2 functions. Functions in L2 are only defined up to null sets. This enables weak differentiation of functions that would not be differentiable in a classical sense. As an example we can look at the 27 function f (x) = |x| on Ω = [−2, 2] shown in Figure 4.1. Z 2 f (x) = |x| ∈ L (Ω) because 1 2 |x| dv <∞ 2 (4.8) Ω Note that, e.g. f (x) ∈ / L2 (R), because the space of real numbers R is not bounded as a domain. The absolute function |x| is not differentiable in a classical sense because of its cusp at x = 0. However, in a weak sense we can derive f (x) = |x| and get the signum function. 1 if x > 0 sgn(x) = 0 if x = 0 −1 if x < 0 (4.9) We can choose the value at x = 0 arbitrarily because it is a null set and will not change the value of the integral. f 0 (x) = sgn(x) is an L2 (Ω) function and f (x) = |x| is therefore also a member of the Hilbert space H1 (Ω). The signum function itself can not be derived further with respect to x in a weak sense, f 0 (x) = sgn(x) ∈ / H1 (Ω). f (x) f (x) 1 x 1 −1 −1 x −1 1 1 Figure 4.1 – The functions f (x) = |x| and f 0 (x) = sgn(x) 4.3 Essential and natural boundary conditions The finite element theory distinguishes between essential and natural boundary conditions. Natural boundary conditions are enforced in a weak sense in the variational formulation, essential boundary conditions have to be included into the function space of solution and test function. In the following example the Dirichlet boundary condition will be an essential boundary condition and the Neumann boundary condition will be a natural boundary condition. This is not always the case, see e.g. Section 4.6 about mixed variational formulations. For the following example the Dirichlet boundary condition is incorporated in the function space. Choosing the solution or trial function u ∈ V and 28 the test function v ∈ V̂, where V(Ω̄) = {u ∈ H1 (Ω̄) : u = u0 on ∂ΩD } and V̂(Ω̄) = {u ∈ H1 (Ω̄) : u = 0 on ∂ΩD } (4.10) are spaces of functions satisfying the Dirichlet boundary condition and a shifted Dirichlet boundary condition, respectively, it is now possible to formulate the variational formulation. 4.4 The variational formulation Multiplying the strong form (4.2) with the test function v ∈ V̂ and integration over Ω, leads to Z Z −∆uv dv = f v dv. Ω (4.11) Ω Integration by parts of the left-hand side integral yields Z Z Z ∇u · ∇v dv = f v dv + Ω Ω gv ds, (4.12) ∂ΩN exploiting the fact that the test function vanishes on the Dirichlet boundary. The Neumann boundary condition is enforced weakly in the variational formulation. Now, the variational problem can be defined as Problem 4.2 (Variational formulation) Find u ∈ V(Ω̄) such that Z Z ∇u · ∇v dv = Ω Z f v dv + Ω gv ds ∀v ∈ V̂(Ω̄) (4.13) ∂ΩN The formulation in Problem 4.2 is called variational formulation of the Poisson problem. Herein, u ∈ V(Ω̄) is called the weak solution of the Poisson problem. The solution of the strong formulation is also a solution of the variational formulation. However, the variational integral formulation makes sense under less restrictive conditions. The weak solution of the Poisson problem exists, is unique, and changes continuously with the initial conditions. The problem is thus called well-posed (after Hadamard). 4.5 Finite element discretization After stating the mathematical foundation, we can now discretize the variational formulation. We split the domain Ω into smaller units, e.g. triangles in two dimension, or tetrahedrons in three 29 degree of freedom node P2 P1 Figure 4.2 – The P1 and the P2 Lagrange element dimension. We call T a mesh (or triangulation, in case of triangles) of Ω [Larson and Bengzon, 2013]. The mesh is (usually) a set of triangles {τ }, such that Ω= [ τ. (4.14) τ ∈T Depending on the type of the mesh and the dimension, triangles could be substituted by lines, squares, cubes, tetrahedrons, or even objects with round edges. Further, we have to choose a finite element type. A finite element is defined by an element domain τ ∈ Ω, a discrete function space Vh (Ω), and a basis φ of the dual space Vh0 [Brenner and Scott, 2008]. The dual space is the space of bounded linear functionals on Vh . φ is also called basis function or ansatz function. A common choice is the P1 Lagrange element [Logg et al., 2012a; Larson and Bengzon, 2013] τ ∈T Vh (T ) = {v ∈ C 0 (Ω) : v |τ ∈ P1 , ∀τ ∈ T } P1 (τ, Vh , φ) = 1 for i = j φ = φ (v ) = i , j = 1, 2, 3 j i j 0 for i 6= j (4.15) where C 0 is the space of continuous functions in Ω, and vi the nodal values of the function v . The basis functions are 1 on the node i and 0 elsewhere. The basis function are piecewise continuous linear functions. The degrees of freedom of the P1 element are situated on the nodes of the element. The next higher order Pk element is the P2 element. It has piecewise continuous quadratic basis functions. Three additional degrees of freedom are situated in the middle of each element edge. The P1 and the P2 element and it’s degrees of freedom are visualized in Figure 4.2. Since the the function v is continuous no jump over the interface of two triangles is possible. Additional types of elements used in this work will be discussed in Section 4.6. A so called discontinuous Galerkin 30 method allowing for jumps on element interfaces will be discussed in Section 4.7. With the previous definitions, we can approximate the function u in Problem 4.2 as uh = N X Uj φj , (4.16) j=1 where N is the number of degrees of freedom. We can now write the discrete formulation of the Poisson problem. Problem 4.3 (Discrete formulation) Find uh ∈ Vh (Ω̄) = {uh ∈ C 0 (Ω) : uh |τ ∈ P1 , ∀τ ∈ T Z Z and Z ∇uh · ∇v dv = f v dv + Ω uh = u0 on ∂ΩD } such that gv ds Ω ∀v ∈ Vˆh (Ω̄) (4.17) g∇φ̂i ds (4.18) ∂ΩN or, using (4.16) N X j=1 Z Z ∇φj · ∇φ̂i dv = Uj Ω Z f ∇φ̂i dv + Ω ∂ΩN This corresponds to solving the linear system AU = b (4.19) with the primary variable vector u and Z A= ZΩ b= ∇φj · ∇φ̂i dv Z f ∇φ̂i dv + Ω (4.20) g∇φ̂i ds ∂ΩN Note that the basis functions equal 1 on the node i and 0 on all other nodes. Thus, A has a sparse structure. 4.6 Mixed variational formulations Variational problems can also be formulated for more than one unknown. An example used in this work is Darcy’s law (3.28) with separate mass and momentum balance −∇·vf = 0, µF K−1 vf + ∇p = 0. 31 (4.21) The unknowns are the velocity field vf and pressure field p. The mixed variational formulation is obtained by multiplying the first equation with a test function q and the second equation with another test function w. After integration over the domain Ω the first and second equation are added. Problem 4.4 (Mixed variational formulation) Find (vf , p) ∈ V such that Z µF K−1 vf · w dv − Ω Z Z p∇·w dv − Ω Z pw · n dv = 0 q∇·vf dv + Ω ∀(w, q) ∈ V̂ (4.22) ∂Ω Note that in this formulation a Dirichlet boundary condition is a natural boundary condition. The Neumann boundary condition is essential and has to be enforced in the function space. The formulation holds for all test functions, which means it particularly holds if one of the test functions is zero. In that case we retrieve one of the original equations in variational form. The difficulty in mixed methods lies in finding suitable function spaces and finite elements. Not all combinations of finite elements produce stable schemes. A natural choice of function spaces for the Darcy case would be V = H(div) × L2 , (4.23) where H(div) is the space of L2 function that have a divergence in L2 . A stable discretization is a mixed formulation with BDM1 (Brezzi-Douglas-Marini elements) for the velocity and DG0 (Discontinuous Galerkin elements) for the pressure. The BDM element is suggested by Fortin and Brezzi [1991] as a H(div)-conforming element in the sense that the discrete function space is a subset of H(div). The degrees of freedom of this element are normal components evaluated on the edges of the element. The mixed Darcy formulation required the continuity of the normal component of the velocity. It has no restrictions for the tangential component. Therefore the BDM1 constitutes a natural element for the Darcy velocity. The DG-element of 0th order is an element with just one degree of freedom per element. It therefore has piecewise constant basis functions which are naturally discontinuous across element facets. The degrees of freedom of both the BDM1 and the DG0 are visualized in Figure 4.3. Note that this combination is e.g. not stable for the Stokes equations as the normal and tangential component must be continuous in the Stokes case. A stabilization technique will be presented subsequently. 4.7 An interior penalty discontinuous Galerkin method for the Stokes problem The spatially resolved coupled blood-tissue flow model features a pressure jump across the vessel wall. In order to resolve the jump, a discontinuous solution is mandatory. When coupling Darcy and 32 DOF node BDM1 DG0 Figure 4.3 – The BDM1 and the DG0 element with degrees of freedom (DOFs) Stokes flow, common finite elements in both domains have the advantage of simpler implementation. The following Stokes method can be discretized with a mixed BDM1 × DG0 -element. A DarcyStokes coupled problem can thus be treated with a single mixed element used in the whole domain and the scheme is additionally locally mass conservative. The method is based on the interior penalty method presented in [Rivière, 2008] for the Stokes equation. Rivière and Yotov [2005] extended the method to a Darcy-Stokes coupled problem with simple interface. The essential boundary conditions are weakly enforced using Nitsche’s method [Nitsche, 1971]. Some interior penalty methods can be in fact interpreted as a Nitsche type method weakly enforcing the continuity of the solution over interior facets [Arnold, 1982; Massing, 2012] . Massing et al. [2014] introduce a Nitsche method for the Stokes problem for interface conditions on overlapping meshes. Following this, we start by introducing the basics of Nitsche’s method and, after introducing helpful DG notation, end up with the desired scheme. Nitsche’s method allows to include boundary or interface conditions within the variational formulation of the problem instead of including the conditions in the solution’s function space. For a simple Poisson problem −∆u = f in Ω; u = u0 on ∂Ω the variational formulation is obtained by multiplying with a test function v and integration by parts. Z Z ∇u · ∇v dv − Ω Z (∇u · n)v ds = ∂Ω f v dv (4.24) Ω The boundary condition is now weakly enforced by penalizing (u − u0 ), yielding Z ∇u · ∇v dv − Ω Z Z (∇u · n)v ds + ∂Ω ∂Ω α (u − u0 )v ds = h Z f v dv, (4.25) Ω where h is the local mesh size and α > 0 a penalty parameter. Rendering (4.25) symmetric as the problem originally was is desirable to e.g. design efficient solvers. A consistent symmetrization can 33 be achieved by adding the term − Z R ∂Ω (∇v · n)(u − u0 ) ds, giving Z Z ∇u · ∇v dv − Ω | (∇u · n)v ds − ∂Ω {z }| Consistency Z Z (∇v · n)u ds ∂Ω {z } Symmetry + | Ω − | Penalty Z Z f v dv α uv ds = ∂Ω h {z } (∇v · n)u0 ds ∂Ω {z } Symmetry + | α u0 v ds . ∂Ω h {z } (4.26) Penalty The method is consistent in the sense that the original solution to the problem is also a solution to the altered problem and vice versa. The method can be applied analogously in a DG scheme to weakly enforce continuity of the solution across interior facets. A discontinuous Galerkin method features function spaces of discontinuous piecewise polynomials. Integrals of interior facets no longer vanish. It comes in handy to define the jump and average operators JvK = v+ − v− {v} = 1 + (v + v− ), 2 (4.27) respectively, and to introduce the following identity Jv · wK = JvK · {w} + {v} · JwK, (4.28) easily proven with the definitions in (4.27). We use ne to denote a fixed normal vector of a facet of two neighboring cells E + and E − , not affected by jump and average operators Jv · ne K = JvK · ne . Two neighboring cells are depicted in Figure 4.4. The choice of ne is arbitrary if consistent [Rivière, 2008]. A discontinuous formulation cannot be formulated in global integrals. Instead, we look at n− E− E+ n+ Γ∂Ω Figure 4.4 – Notation for discontinuous Galerkin techniques one element E of a triangulation E and sum over all elements, where e and Γ, Γ∂Ω here denote the set of facets, interior facets, and exterior facets, respectively. For the Poisson problem where u is 34 taken as piecewise linear on each element E we get Z Z Z ∇u · ∇v dv − (∇u · nE )v ds = ∇f · ∇v dv. E ∂E (4.29) E Then, summing over all elements, switching to the fixed normal vector ne between two neighboring elements, and adding penalty and symmetry term as above yields XZ E∈E ∇u · ∇v dv − E XZ e∈Γ − e {∇u} · ne Jv K ds X Z e∈Γ∂Ω ∇u · nv ds − e e∈Γ − e − f v dv E {∇v } · ne JuK ds X Z ∇v · nu ds e e∈Γ∂Ω XZ E∈E XZ X Z ∇v · nu0 ds e e∈Γ∂Ω XZ β JuKJv K ds + h e∈Γ e X Z α uv ds = + h e∈Γ∂Ω e X Z α u0 v ds, + e h e∈Γ∂Ω (4.30) where β is a second penalty parameter. The penalty parameters have to be chosen large enough to ensure stability but small enough to not worsen the condition number and emphasize numerical errors. Lower bound estimates can be obtained theoretically, e.g. [Epshteyn and Rivière, 2007]. The penalty parameter is dependent on the model parameters and nature of the problem and on the approximation degree of the numerical method. We now look at the Stokes problem for a tube shaped domain Ω and its wall ∂Ωω , inlet ∂Ωin , and outlet ∂Ωout . Problem 4.5 (Stokes) Find (u, p) such that −2µ∇·D(v) + ∇p = 0 in Ω ∇·v = 0 in Ω v=0 on ∂Ωω p = p̄in and ∇v n = 0 on ∂Ωin p = p̄out and ∇v n = 0 on ∂Ωout (4.31) with D = 12 (∇v +∇T v) denoting the symmetric velocity gradient as usual. We now have one vectorvalued and one scalar-valued equation and therefore choose a mixed variational formulation. Again, the variational formulation for an element E is obtained by multiplying with two test functions (w, q) and integration by parts. Z 2µD(v) ·· D(w) dv − E Z Z p∇·w dv − ZE − q∇·v dv Z pnE · w ds = 0 2µD(v)nE · w ds + E ∂E ∂E 35 (4.32) In this divergence formulation the pressure Dirichlet boundary condition is natural, while the velocity Dirichlet condition is essential. The essential boundary condition will be enforced with Nitsche’s method. In fact, when using BDM1 -elements the degrees of freedom are normal components and do not allow to set Dirichlet conditions strongly for the tangential velocity component. The mixed DG method is obtained by summing over all elements, symmetrization and penalization. XZ E∈E 2µD(v) ·· D(w) dv − E − XZ − 2µD(v)ne · w ds − e X e e∈Γ q∇·v dv E 2µ{D(w)}ne · JvK ds + XZ e∈Γ β 2µ JvK · JwK ds h e {q}JvK · ne ds e XZ X T µ∇ vne · w ds − X XZ e∈Γω e∈Γ∂Ω in − XZ e∈Γ XZ e∈Γω − {p}JwK · ne ds + e e∈Γ XZ E∈E 2µ{D(v)}ne · JwK ds − e XZ p∇·w dv − E E∈E e∈Γ + XZ XZ 2µD(w)ne · v ds + e e∈Γω α 2µ v · w ds h e (4.33) T µ∇ vne · w ds = e∈Γ∂Ω out p̄in (ne · w) ds − e∈Γ∂Ω in X p̄out (ne · w) ds e∈Γ∂Ω out Note that choosing BDMk -elements leads to JvK · ne = JwK · ne = 0. This method is similar to the one presented in [Rivière, 2008]. They show pressure and velocity convergence for mixed DGk × DGk elements. The mesh convergence of the velocity is shown by Wang et al. [2009] for H(div)-conforming elements. However, the convergence of the pressure was not investigated. We tested convergence of pressure and velocity for a domain Ω = [−0.2, 0.2] × [−1, 1], where we chose µ = 1, α = 10, and the boundary conditions so that the exact solution is vx = 0, vy = −2 + 50x 2 , p = 100(1 + y ). The error is calculated as Z e = ||u − ue ||L2 = (u − ue )2 dv 1/2 , Ω where ue is the respective exact solution. The rate of convergence is calculated as the experimental order of convergence r= ln e k+1 − ln e k , k+1 k ln hmax − ln hmax where hmax is the maximal element diameter of the mesh calculated as two times the circumradius1 and k the refinement step. The results of the grid convergence test is shown in Table 4.1. The above presented method is locally mass conservative. There are no constraints for functions concerning jumps over facets, so selecting an interior element E we choose q equals to 1 on E, and 1 http://www.wolframalpha.com/input/?i=circumradius+triangle 36 k hmax ||p k − pe ||L2 Rate r ||vk − ve ||L2 Rate r 1 0.141421 2.35427 0.978062 0.0736681 1.89229 2 0.0707107 1.18615 0.988995 0.0195418 1.91448 3 0.0353553 0.595311 0.994571 0.00526346 1.89248 4 0.0176777 0.298207 0.997327 0.00148557 1.82500 5 0.00883883 0.14924 0.998677 0.000456279 1.70303 Table 4.1 – Convergence rates and errors for the BDM1 × DG0 mixed Stokes discretization for k mesh refinements. 0 elsewhere. The discretization of the mass balance then reduces to Z XZ 1 − JvK · ne ds = 0. ∇·v dv + E e 2 (4.34) e∈∂E With BDM1 -elements the normal velocity component is continuous, thus JvK · ne = 0. The method therefore exactly satisfies the mass balance for each element Z − ∇·v dv = 0. (4.35) E Locally conservative schemes are important, e.g. for coupled flow and transport problems in porous media. Newton solvers are observed to stop converging after a few time steps if the scheme is not locally conservative [Rivière, 2008]. In order to determine large enough penalty parameters, we set α = β and calculated the L2 -norm of the pressure and velocity error to the exact solution for various penalty parameters. The results are shown in Figure 4.5. The penalty term shifts the eigenvalues of the the stiffness matrix so that the matrix is positive definite which is a requirement for the stability. The penalty parameter has to be large enough to assure positive definiteness. Positive definite matrices can, however, also occur for small penalty parameters (local minima in Figure 4.5). Figure 4.5 shows that for ca. α > 2 the numerical method is stable and the error in comparison with the exact solution minimal. For values α 2 the error slightly increases due to numerical errors introduced by larger and larger condition numbers. 37 12 8 ||v − ve ||L2 ||p − pe ||L2 10 6 4 2 0 0 0.5 1 1.5 2 2.5 3 50 45 40 35 30 25 20 15 10 5 0 0 penalty parameter α 0.5 1 1.5 2 2.5 3 penalty parameter α Figure 4.5 – Error over penalty parameter α = β for the Stokes BDM1 −DG0 -method. Pressure (left) and velocity (right). In this chapter, several discretization techniques based on the finite element method were introduced. With those discretization techniques at hand, we can now discretize the problems presented in Chapter 3 subsequently in Chapter 5. In particular, the introduced discontinuous Galerkin discretization of the Stokes problem and the mixed variational formulation for the Darcy problem can be used in the coupled Darcy-Stokes system. The estimated penalty term also provides a first estimation for the penalty terms of the coupled problem. 38 5 Discretizing and solving coupled Darcy-Stokes systems This chapter presents formulations and solution algorithms for the introduced models. Two general concepts are the fully coupled method and iterative domain decomposition methods. Both methods decompose the domain into parts that use different physical models, namely a free-flow and a porous region. A fully coupled strategy solving all equations in a single linear system at once is presented in Section 5.1 for Problem 3.1. This method uses a direct solver for the Darcy-Stokes system with membrane since the systems generally have rather bad condition numbers. Iterative domain decomposition methods solve two separate system sequentially with suitable boundary conditions and source terms for each individual problem. The boundary conditions get updated every iteration step. A big advantage of iterative methods is the fact that well-known discretization methods, solvers and preconditioners are already available for the subproblems. Hanging nodes on the interface are possible. Considering time-dependent problems, different time step size can be used for each domain. The two systems can even be solved by different specialized code libraries, if a few data transfer mechanisms are available. Disadvantages are that for ill-conditioned problems the iterative solver is slow in comparison to direct solvers. Iterative algorithm are presented in Sections 5.2 and 5.3 for the Problems 3.1 and 3.3, respectively. 5.1 Unified mixed element formulation for the coupled Darcy-Stokes problem with selective permeable membrane The aim is a formulation that can be discretized with one type of finite element. This allows easy implementation. Because of the interface pressure jump, the described method has to allow for discontinuities. Starting point is Problem 3.1. The scheme features the mixed and DG techniques introduced in Section 4.6 and Section 4.7. The discrete mixed variational formulation of (3.9) can 39 be obtained as a(v, w) + b(p, w) + b(q, v) + c(v, w) + d(v, w) = L(w, q). (5.1) Here, Ωp denotes the porous domain, Ωf the free-flow domain. The symbols Γp , Γf denote interior facets and Γ = Γp ∩ Γf . The elements Ep , Ef denote elements of the triangulations Ep and Ef , and e element facets. The outer boundaries are written as ∂Ωp and ∂Ωf . Then, the symmetric bilinear form a(v, w) is defined as a(v, w) := XZ Ef Ef ∈Ef − XZ e∈Γf e X Z 2µ∇v ·· ∇w dv + Ep Ep ∈Ep 2µ{D(v)}ne · JwK ds − XZ e∈Γf µ v · w dv K 2µ{D(w)}ne · JvK ds + e XZ β 2µ JvK · JwK ds. h e e∈Γf (5.2) It includes the integrals over interior facets in the free-flow domain and penalizes velocity jumps. b(p, w) and b(q, v) are defined as XZ b(p, w) := − b(q, v) := − XZ Ef ∈Ef p∇·w dv − Ef Ef ∈Ef X Z q∇·v dv − Ef p∇·w dv + Ep Ep ∈Ep X Z q∇·v dv + (5.3) XZ e∈Γf {p}JwK · ne ds e e∈Γf Ep Ep ∈Ep XZ {q}JvK · ne ds e where the last term is consistent but vanishes when using BDM1 -elements. The bilinear form c(v, w) is the form of the interface conditions and is defined as XZ XZ 1 α c(v, w) := (vf · nf )(wf · nf ) ds + 2µ (vf · τ )(wf · τ ) ds L h p e∈Γ e e∈Γ e Z XZ X (2µD(wf )nf · τ )(vf · τ ) ds (2µD(vf )nf · τ )(wf · τ ) ds − − e∈Γ e e∈Γ (5.4) e All the quantities are restricted to the free-flow domain. The bilinear form d(v, w) consists of the weakly enforced boundary conditions at inlet and outlet of the free-flow domain and is defined by d(v, w) = − X e∈Γ∂Ωf µ∇T vnf · w ds − X e∈Γ∂Ωf in µ∇T vnf · w ds (5.5) X (5.6) out The right-hand side linear form reads L(w, q) := − X e∈Γ∂Ωf p̄in (nf · w) ds + in X e∈Γ∂Ωf p̄out (nf · w) ds − out p̄D (np · w) ds e∈Γ∂Ωp D Form c(v, w) requires some additional explanation. Let us look at the integral over one interface facet e ∈ Γ. Contributions from the free-flow and porous domain are marked as (·)f and (·)p , 40 respectively. Z Z − Z 2µD(vf )nf · wf ds + e∈Γ pf nf · wf ds + e∈Γ pp np · wp ds (5.7) e∈Γ Considering that np = −nf , and performing a split in tangential and normal component v = (v · nf )nf + (v · τ )τ yields Z − Z (2µD(vf )nf · τ )(wf · τ ) ds − e∈Γ Z (2µD(vf )nf · nf )(wf · nf ) ds + e∈Γ e∈Γ We now analyze the pressure jump term and see that Z Z Z Z (pf wf − pp wp )nf ds = JpwKnf ds = (JpK{w} + {p}JwK)nf ds = e∈Γ e∈Γ e∈Γ JpwKnf ds (5.8) (pf − pp )(w · nf ) ds e∈Γ (5.9) where we used JwK · nf = 0 and {w} · nf = wf · nf for BDM-elements. Recalling interface condition (3.8) resulting from the normal stress balance, we can then write Z Z − (2µD(vf )nf · τ )(wf · τ ) ds + e∈Γ e∈Γ 1 (vf · nf )(wf · nf ) ds Lp (5.10) Interface condition (3.6) is enforced weakly using Nitsche’s method Z Z 1 α (vf · nf )(wf · nf ) ds + µ (vf · τ )(wf · τ ) ds L e∈Γ p e∈Γ h Z Z − (2µD(vf )nf · τ )(wf · τ ) ds − (2µD(wf )nf · τ )(vf · τ ) ds e∈Γ (5.11) e∈Γ and summing over all interface facets we have form (5.4). All three interface conditions (3.7), (3.8), and (3.6) are thus satisfied. The continuity of normal velocities is incorporated in the function space. The normal stress balance can be seen as a natural boundary condition. The no-slip condition on the vessel wall, as an essential boundary condition, is enforced weakly using Nitsche’s method. The scheme is locally mass conservative as the mass balance is explicitly satisfied for all elements. The system is solved using a direct solver. 5.2 Robin-Robin domain decomposition of the coupled Darcy-Stokes system with selective permeable membrane The domain Ω is decomposed into two subdomains Ωf and Ωp . On each subdomain independent problems are solved. Each process transfers information from the other domain by boundary conditions on the original interface Γ. The solving process is iterative and serial. The easiest method for decomposing Darcy and Stokes domain is a Dirichlet-Neumann domain decomposition method. In each iteration step, a subproblems in Darcy and Stokes domain are solved with boundary con- 41 ditions calculated from the solution of the respective other domain. Such an algorithm for the Darcy-Stokes coupled problem was presented and mathematically analysed in [Discacciati, 2005]. However, Discacciati [2005] finds that the algorithm is impractically slow for high ratios of fluid viscosity to permeability as present in our case. A more advanced Robin-Robin domain decomposition algorithm for a Darcy-Stokes coupled system with simple interface (without considering a membrane on the interface) has been developed and thoroughly investigated by Discacciati et al. [2007]. A modified algorithm with the new interface conditions is presented here. The boundary conditions for the subproblem are of Robin type. This, third possibility of a boundary condition is of the form au + b∇u · n = au0 + bg and is a linear combination of Dirichlet and Neumann boundary conditions. For a → 0 the Neumann boundary condition is obtained, for b → 0 the Dirichlet boundary condition. The Darcy system in its variational form can be written as Z Z K K ∇pp · ∇ϕ dv + ∇pp · nf ϕ ds = 0 Γ µi Ωp µi (5.12) the Stokes system as Z 2µD(vf ) ·· D(w) dv − Ωf Z Z pf ∇·w dv − q∇·vf dv+ Z Z + [−2µD + pf I] nf · w ds − µ∇T vf nf · w ds Γ ∂Ωf in Z =− p0 nf · w ds. Ωf Ωf (5.13) ∂Ωf in The Darcy velocity is calculated in a decoupled step, solving the variational form Z Z K vf · ϕ dv = − ∇pp · ϕ dv µ Ωp Ωp i (5.14) The Darcy pressure is discretized with P2 -elements the velocity with P31 -elements. The Stokes is solved in the mixed formulation using P32 -P1 -elements (Taylor-Hood elements) for the pair (vf , pf ). ALGORITHM 1 — 1. Solve the Darcy problem Z Z Z K γp ∇pp k+1 · ∇ϕ dv + pp k+1 ϕ ds = Λk ϕ ds µi Ωp Γ Γ 42 (5.15) which corresponds to imposing the Robin boundary condition −γp K µi dM k ∇pp k+1 · nf + pp k+1 = γp vf k · nf − 2µDk n · nf + pf k − vf · nf µi KM (5.16) := Λk 2. Solve the Stokes problem Z 2µD(vf k+1 ) ·· D(w) dv − Z pf k+1 Z q∇·vf k+1 dv ∇·w dv − Ωf Ωf Ωf Z µi dM k+1 )(vf · nf )(nf · w) ds − µ∇T vf k+1 nf · w ds + (γf + KM ∂Ωf in Γ Z Z γf k γp + γf k+1 k+1 Λ − pp (nf · w) ds =− p0 nf · w ds + γp ∂Ωf in Γ γp (5.17) Z which corresponds to imposing the Robin boundary condition K µi dM k+1 ∇pp k+1 · nf − pp k+1 − vf · nf µi KM γf k γp + γf k+1 µi dM k+1 Λ − pp − vf · nf = γp γp KM (5.18) γf k γp + γf k+1 Λ + pp γp γp (5.19) K ∇pp k+1 · ϕ dv µi (5.20) 2µDk+1 n · nf − pf k+1 + γf vf k+1 · nf = −γf 3. Upadte Λ Λk+1 = (γp + γf ) vf k+1 · nf − (4.) Calculate the Darcy velocity field by solving Z vf k+1 · ϕ dv = − Z Ωp Ωp If the algorithm converges, the original interface conditions are retained. Let vf ∗ , vp ∗ , pf ∗ , and pp ∗ be the functions the primary variables vf , vp , pf , and pp converged to. Then, the Robin boundary conditions imposed in step 1 and 2 read µi dM ∗ K ∇pp ∗ · nf + pp ∗ = γp vf ∗ · nf − 2µD(vf ∗ )n · nf + pf ∗ − vf · nf µi KM µi dM ∗ K 2µD(vf ∗ )n · nf − pf ∗ + γf vf ∗ · nf = −γf ∇pp ∗ · nf − pp ∗ − vf · nf µi KM −γp (5.21) (5.22) Inserting (5.22) in (5.21) yields − (γp + γf ) K ∇pp ∗ · nf = (γp + γf )vf ∗ · nf . µi (5.23) Equation (5.23) is interface condition (3.7) for γp + γf 6= 0. With (5.23) inserted in (5.22) we 43 obtain 2µD(vf ∗ )n · nf − pf ∗ = −pp ∗ − µi dM ∗ vf · nf , KM (5.24) which is interface condition (3.8). The third interface condition (3.6) is an essential Dirichlet boundary condition for the Stokes domain on Γ. Discacciati et al. [2007] proves that the algorithm converges for γf > γp > 0, where γf , γp are chosen to be large enough to guarantee good convergence properties and small enough to keep the condition numbers of the subsystems low. 5.3 Iterative domain decomposition of the 1D-2D reduced DarcyStokes problem This section presents an iterative algorithm for solving the 1D-2D coupled problem (Problem 3.3). The domain as shown in Figure 3.5 is a porous tissue domain that is traversed by a vessel line. The leaky vessel model is formulated in one dimension. The mesh consists of intervals but lives in a two-dimensional world. Each node is associated with coordinates in two dimensions. The coupling is realized with the mutual source term. We can eliminate the velocity in both domains, thus, solving a problem with the effective pressure as only primary variable. The velocity field can be calculated in a second decoupled step. The 2D Darcy system in its variational form can be written as Z Z KM K ∇pp · ∇ϕ dv = (2 (p̄p,2D − pf ))ϕ ds (5.25) µi dM Γ Ωp µi the corresponding 1D Stokes or Hagen-Poiseuille system as Z C Γ ∂pf ∂q dz + ∂z ∂z Z Z pf q dz = Γ p̄p,2D q dz (5.26) Γ Both pressures are discretized with P1 -elements. ALGORITHM 2 — 1. Assemble the following Darcy system Z Ωp K ∇pp k+1 · ∇ϕ dv = 0 µi (5.27) 2. Calculate point sources for every integration point x (e.g. Gaussian quadrature with n = 1, associated interval Ix = [a, b]) fP (x) = (b − a)f ((a + b)/2), f (z) = 2 44 KM 1 (pf − (pp |R + pp |−R )) µi dM 2 (5.28) 3. Assemble point sources into right hand side vector of Darcy system and solve for pp k+1 4. Solve 1D Stokes system ∂pf k+1 ∂q dz + C ∂z ∂z Γ Z Z pf k+1 Z q dz = Γ Γ p̄p k+1 ,2D q dz (5.29) 5. Set pp k = (1 − θ)pp k+1 + θpp k (5.30) The relaxation parameter θ is used to accelerate convergence. The velocities can be obtained in a post processing step from the pressure solutions. The calculation of point sources for every integration point is a geometrically flexible method of realizing the coupling. 5.3.1 Calculation of line sources Ep integration point line integral over this line is approximated by quadrature rule intersection point Ef Ep Ef Figure 5.1 – Discrete approximation of a line source integral A source term on a line is easily written down mathematically but the implementation is more complex. Formally the source term is an integral over the entire vessel Z KM f (Ωp ) = 2 (pf − p̄p,2D ) dz . Γ µi dM (5.31) The discretized domain is triangulated. In a similar manner the source term gets discretized. Gaussian quadrature rules are a method of numerical integration. They only work well if the integrand is a polynomial. At element facets the pressure solution can have bends and the integral 45 needs to be split up. The discrete form of integral (5.31) can be written as X XZ fh (Ωp ) = Ep ∈Ep Ef ∈Ef 2 Γ∩Ep ∩Ef 1 KM (pf − (pp |R + pp |−R )) dz µi dM 2 (5.32) The remaining inner integral (the integration domain in marked red in Figure 5.1) is approximated by a quadrature rule. The discrete integral is illustrated in Figure 5.1. For the subsequent numerical examples a single integration point was used, corresponding to a Gaussian quadrature rule of degree n = 1. Thus, the integral over one (red) part of the facet Γ ∈ [a, b] is numerically approximated by Z a b n=1 b−a X b−a a+b f (z) dz ≈ wi f xi + , 2 2 2 (5.33) i=1 where f (z) = 2 µKi dMM (pf − 12 (pp |R + pp |−R )). For n = 1 the weighting function w1 = 2 and the only integration point xi = 0 [Abramowitz and Stegun, 1972]. Implementation-wise every addend of the discretized sum was applied to the right-hand side vector of the linear system as a point source. The point source affects the right-hand side of all degrees of freedom of the element the point falls in. This leads to a ”smearing” of the point source over the element that gets less the smaller the element is or the higher the polynomial degree of the basis functions of the element is. The intersection points and integration points are calculated in a preprocessing step after mesh generation. The intersection detection algorithm was implemented as a brute-force algorithm where all edges of a cell of the two-dimensional domain are tested for an intersection with all interval cells of the one-dimensional domain. If an intersection is found, the intersection points are calculated. Note that significantly faster algorithms exist, e.g. a bounding box hierarchy method1 . For an efficient intersection detection implementation for three-dimensional meshes see e.g. [Massing et al., 2013]. The herein employed meshes where relatively small so that fast implementation was more important than algorithm speed. For larger simulations, the intersection detection algorithm can consume a majority of the whole CPU time [Cattaneo and Zunino, 2013]. For the most part of this thesis a simplified line source algorithm was used that gives good results if the one-dimensional elements are around the same magnitude or the one-dimensional grid is small enough. Then it is enough to use one integration point in the middle of each one-dimensional element, regardless of the two-dimensional grid. Intersection do not have to be calculated at all. This only works for solution without large bends over element edges. Such solutions occur in our examples so that a difference between the above presented method and the simplified method was not visible. 1 http://en.wikipedia.org/wiki/Bounding˙volume˙hierarchy 46 In this chapter the solution strategies for the Problems 3.1 and 3.3 were presented. The spatially resolved model can be solved in a unified approach in a single linear system, or, decomposed in two subsystems that communicate via appropriate interface conditions. For the spatially reduced second model and iterative solver is presented accounting for the different geometrical and mathematical nature of the subsystems. The solvers can now be implemented. Remarks on the implementation are found in Chapter 6. Testing scenarios and results are presented subsequently. 47 6 Implementation The implementation was accomplished in FEniCS (http://fenicsproject.org/ ). The FEniCS project [Logg et al., 2012a] is a collection of open source software to solve differential equations. Its heart is the C++ (and Python) library DOLFIN [Logg et al., 2012b]. The form compiler FFC compiles variational forms, finite elements, and functionals written in UFL (Unified Form Language) to basic C++–code. UFL allows to write variational forms in a close to paper notation. The following presents how to solve a Poisson problem with discontinuous elements in FEniCS using DOLFIN’s C++–interface. We solve a simple poisson problem ∇u = f in Ω ; u = 0 on ∂Ω using FEniCS. We choose f = f (x, y ) = 500 ∗ exp −((x − 0.5)2 + (y − 0.5)2 )/0.02 . First we obtain the variational formulation by multiplying with a test function v and integration by parts. Z Z Z ∇u · ∇v dv + ∇u · nv ds = f v dv Ω ∂Ω (6.1) Ω The domain is dicretized using DG1 -elements. Therefore we penalize the jump of u with Nitsche’s method. The essential boundary condition u = 0 on ∂Ω is also enforced with Nitsche’s method as introduced in Section 4.7. We obtain the discrete variational problem as XZ E∈E E XZ α JuKJv K ds h e∈Γ e e∈Γ e e∈Γ e X Z X Z X Z β XZ + ∇une v ds + ∇v ne u ds + uv ds = f v dv e e e h E ∇u · ∇v dv + XZ {∇u} · Jv Kne ds + e∈Γ∂Ω XZ e∈Γ∂Ω {∇v } · JuKne ds + e∈Γ∂Ω (6.2) E∈E This can be easily translated into the following UFL code. This code and the following code snippets in this section are taken and slightly altered from the undocumented dg-poisson-demo included in the newest DOLFIN 1.4 release. 48 UFL code # Elements element = FiniteElement ( ” DG ” , triangle , 1 ) # Trial and test functions u = TrialFunction ( element ) v = TestFunction ( element ) f = Coefficient ( element ) # Normal component , cell size and right - hand side h = 2 . 0 * triangle . circumradius h˙avg = ( h ( ’+ ’) + h ( ’ - ’) ) / 2 n = element . cell () . n # Parameters alpha = 4 . 0 beta = 8 . 0 # Bilinear form and Linear Form a = inner ( grad ( u ) , grad ( v ) ) * dx “ - inner ( jump (u , n ) , avg ( grad ( v ) ) ) * dS “ - inner ( avg ( grad ( u ) ) , jump (v , n ) ) * dS “ + ( alpha / h˙avg ) * jump ( u ) * jump ( v ) * dS “ - inner ( u *n , grad ( v ) ) * ds - inner ( grad ( u ) , v * n ) * ds “ + ( beta / h ) * u * v * ds L = f * v * dx FFC compiles UFL code to C++ code. To do this we call Bash code ffc -l dolfin Poisson . ufl from a terminal. This generates the header file Poisson.h with all the classes we need to solve the Poisson problem in our C++ code. The corresponding DOLFIN C++ code looks like this. The code is explained via comments inside the code. C++ code # include ¡ dolfin . h ¿ # include ” Poisson . h ” using namespace dolfin ; int main () – // Source term 49 class Source : public Expression – public : Source () : Expression () –˝ void eval ( Array ¡ double ¿ & values , const Array ¡ double ¿ & x ) const – const double dx = x [ 0 ] - 0 . 5 ; const double dy = x [ 1 ] - 0 . 5 ; values [ 0 ] = 500 . 0 * exp ( - ( dx * dx + dy * dy ) / 0 . 02 ) ; ˝ ˝; // Create mesh UnitSquareMesh mesh ( 24 , 24 ) ; // Create functions Source f ; // Create function space Poisson : : FunctionSpace V ( mesh ) ; // Define variational problem Poisson : : BilinearForm a (V , V ) ; Poisson : : LinearForm L ( V ) ; L.f = f; // Compute solution Function u ( V ) ; solve ( a = = L , u ) ; // Save solution in VTK format File file ( ” poisson . pvd ” ) ; file ¡ ¡ u ; // Plot solution plot ( u ) ; interactive () ; return 0 ; ˝ In this chapter an exemplary implementation of a DG Poisson problem was given. It can be seen that the UFL code is very close to the mathematical description and notation of the problem. The C++–Interface to DOLFIN provides comprehensible classes for the implementation of the solving process. Further implementation code is attached in the appendix. 50 7 Comparison scenarios In this chapter numerical scenarios are developed to compare the aforementioned models and solution strategies in efficiency, generality and robustness. To this end, we construct a reference case (Section 7.A). The reference case is developed in analogy to the one presented in [Baber, 2014] in order to allow for comparison. The subsequently presented cases are alterations of the reference case in order to test specific behavior of the models. The focus of the comparison scenarios is testing the effect of the model reduction to a one-dimensional vessel. Therefore, we compare the behavior of the two models presented in Chapter 3 to changes of parameters and geometry. In order to simplify reference [2D-FC] shall refer to the spatially resolved model with direct solver (Section 5.1), [2D-IT] to the spatially resolved model with iterative solver (Section 5.2), and [1D] to the reduced model with the one-dimensional vessel geometry (Section 5.3). In order to compare model results, meaningful indicators have to be define. We calculate the total net flux over the interface. For the [2D-FC] and the [2D-IT] model this refers to calculating the functional Z vf · n dv. Q2D = (7.1) Γ For the [1D] model the total net flux is equal to the source term Z Q1D = 2 Γ KM 1 pf − (pp |R + pp |−R ) dv. µi dM 2 (7.2) A indicator is the shape of the plot-over-line curve obtained by plotting the velocity normal to the vessel wall in a distance of 50 µm from the vessel wall. A indicator is the effective pressure profile obtained as a plot-over-line curve on a characteristic cross-section. The meshes used in this work are triangulations of the model domain. The meshes were generated with the software Gmsh1 (GNU General Public License). The meshes for the spatially resolved models [2D-FC] and [2D-IT] are refined towards the vessel in order to resolve the geometry of the capillary. For a mesh refinement study the mesh is additionally uniformly refined using a build-in mesh refinement function in FEniCS. Figure 7.1 shows a cutout of the reference case mesh (refined 1 http://www.geuz.org/gmsh/ 51 Figure 7.1 – Cutout of the mesh of the reference case showing mesh refinement towards the vessel. Figure 7.2 – Mesh of the reference case for the [1D] model. Mesh turned by 90◦ . once) that displays the mesh refinement towards the capillary. Figure 7.2 shows the reference case mesh (unrefined) used for the [1D] model. 52 Symbol lcap Name Value Length of the capillary Unit 1 · 10 −3 m −6 m R Radius of the capillary 4.3 · 10 dp Half the estimated intercapillary distance 100 · 10−6 m µ Blood viscosity 2.8 · 10−3 P as µi Viscosity of interstitial fluid 1.3 · 10−3 P as −18 m2 2.34 · 10−20 m2 6.5 · 10 K Intrinsic permeability of the tissue KM Intrinsic permeability of the vessel wall dM Thickness of the vessel wall 6.0 · 10−7 3.0 · 10 m −11 m P as Lp Hydraulic conductivity of the vessel wall p̄f,in Effective pressure at aterial end 400 Pa p̄f,out Effective pressure at venous end −1600 Pa p̄p Effective pressure at distance dp from vessel wall −933 Pa γf Acceleration parameter [2D-IT] 1.0 · 1011 – γp Acceleration parameter [2D-IT] 10 – θ Relaxation parameter [1D] 0.26 – 3.3333 · 10 Table 7.1 – Parameters used in the reference case. 7.A The reference scenario The reference scenario describes a single capillary of length lcap = 1 mm, surrounded by the tissue of the capillary bed. The setup is symmetric with blood flowing in the capillary from top to bottom, i.e. from the arterial to the venous end. The free-flow domain Ωf , the surrounding tissue Ωp and the applied boundary conditions are shown in Figure 7.3. In case of reduction of the vessel to one dimension the vessel domain Ωf is reduced to its centerline. The same boundary conditions are applied. To give an overview over the parameters associated with the reference scenario all parameters are listed in Table 7.1. 7.B Variations in geometry Blood vessels in living tissue exhibit a large variety of geometrical shapes. Vessels split into two vessels at bifurcations, rejoin, and even bypasses and loops are often encountered. This scenario is designed to find how the model behaves when the geometry is altered from the symmetrical 53 Symbol µ Name Lower bound Upper bound −3 −3 P as −14 2.0 · 10 Blood viscosity 3.5 · 10 Unit K µi Hydraulic conductivity of tissue 1.2 · 10 Lp Filtration coefficient of the vessel wall 2.5 · 10−12 1.5 · 10−9 m2 P as m P as R Radius of the capillary lumen 1.5 · 10−6 5.0 · 10−6 m −15 2.7 · 10 Table 7.2 – Parameters and their range for case C obtained from the literature study in Ischinger [2013]. Remarks to the parameters can be found in Section 2.5 reference case. Furthermore, it is to be determined if the [2D-FC] model responds differently from the [1D] model. To this end, the geometry is altered to an arc and a bifurcation. In both cases the geometry of the tissue follows the geometry of the vessel in order to have comparable boundary effects to the reference case. Both scenarios use the same model parameters as the reference case (Table 7.1). Comparisons exclude the [2D-IT] model for time reasons. The Arc. — For the arc geometry the capillary makes a 90◦ turn. The length of the capillary is still lcap = 1 mm. The boundary conditions are chosen in analogy to the reference case. The arc introduces asymmetry as the interface is longer and the tissue area greater on the outer, left side of the vessel, and the interface is shorter and the tissue area smaller on the inside. The geometry and the applied boundary conditions are shown in Figure 7.4. The Bifurcation. — In the bifurcation geometry the capillary splits into two vessel of each lcap,2 = 0.5 mm after a distance of lcap,1 = 0.5 mm from the inlet. For simplicity the two child branches have the same radius R = 4.3 · 10−6 m as the mother branch. The geometry and the applied boundary conditions are shown in Figure 7.5. The geometry has the property that the problem is close to symmetric (with respect to the vessel centerline) towards the end of a branch and asymmetric around the bifurcation. 7.C Variations of model parameters The current model is far from a real, in-vivo scenario. Testing the model with a wide range of parameters makes it possible to foresee model behavior for different parameters. A range of parameters for capillaries was obtained in a literature study by Ischinger [2013]. Model response to parameter change and the comparison of the [2D-FC] model with the [1D] model is the purpose of this scenario. The geometry is chosen to be the same as in the reference case. One parameter is altered while the others stay fixed. The parameters in question and the tested range are listed in Table 7.2. 54 Neumann no-flow boundary condition (vp · n = 0) Dirichlet boundary condition for effective pressure pf = 400 P a Centerline Interface Γ Ωf Ωp Ωp y x n 1 mm n Dirichlet boundary condition for effective pressure pp = −933 P a Dirichlet boundary condition for effective pressure pf = −1600 P a 100 µm 8. 6 100 µm µm Figure 7.3 – Reference scenario (case A) and applied boundary conditions. 55 Neumann no-flow boundary condition (vp · n = 0) Dirichlet boundary condition for effective pressure pf = 400 P a Dirichlet boundary condition for effective pressure pp = −933 P a Centerline lcap = 1 mm Interface Γ Ωp Ωf Dirichlet boundary condition for effective pressure pf = −1600 P a Ωp n n 100 µm y x 8.6 µm Dirichlet boundary condition for effective pressure pp = −933 P a 100 µm Figure 7.4 – Arc scenario (case B.1) and applied boundary conditions. 56 m 0µ 10 m µ 8.6 100 µm Neumann no-flow boundary condition (vp · n = 0) Dirichlet boundary condition for effective pressure pf = 400 P a n n Centerline Interface Γ Dirichlet boundary condition for effective pressure pp = −933 P a Ωf Dirichlet boundary condition for effective pressure pf = −1600 P a y x Ωp Ωp Ωp 100 µm 100 µm 8.6 µm 8.6 µm 100 µm Dirichlet boundary condition for effective pressure pp = −933 P a 100 µm Figure 7.5 – Bifurcation scenario (case B.2) and applied boundary conditions. 57 8 Results and Discussion Reference case A. — The reference case shows excellent compliance of all three models. Differ- ences in pressure, velocity, and total flux over the interface are less than 1 %. The reference case is characterized by its symmetric geometry with respect to the vessel centerline. The radial symmetry of the vessel was an integral assumption in the derivation of the [1D] model (see Section 3.3). This assumption holds true for the reference case. In fact, the major difference between the fully spatially resolved models and the [1D] model is the negligence of the velocity in radial direction and the resulting pressure gradient in radial direction. However, the results show that the neglect is justified for the reference case as the differences between the spatially resolved models and the reduced vessel model are small. A surface plot of the pressures is shown in Figure 8.1. One notices the reduced geometry of the vessel. Visually, the pressure plots are equal. Figure 8.2 shows a plotover-line for the pressure for both the [2D-FC] model and the [1D] model. Looking closely it can be seen that the altitude of the pressure jump differs by less than 1 %. The pressure jump across the interface in around 200 P a, which corresponds to fluid filtration into the tissue. The pressure jump at the inlet is as high as 800 P a. At the outlet the pressure jump is around −400 P a and results in reabsorption of fluid into the capillary. Figure 8.3 shows the velocity plotted over a line through the tissue domain. The normal component of the velocity shows an almost linear profile, while at the top and bottom of the domain the slope is 0 due to the no-flow boundary condition. In close inspection, one can see that the functions differ by approximately 1 % on the upper and lower boundary, while both models show the same zero-crossing. This shows that the sign of the pressure jump over the vessel wall is identical whereas the altitude slightly differs. The total net fluxes over the interface as an integral measure, shown in Table 8.1, match well. Model Geometry Total net flux [2D-FC] Reference 1.24516 · 10−11 [2D-IT] Reference 1.24528 · 10−11 [1D] Reference 1.24603 · 10−11 Unit m2 s m2 s m2 s Table 8.1 – Total net flux across the capillary wall for the [2D-FC], [2D-IT], and [1D] models for the reference case. 58 Figure 8.1 – Effective pressure solution for the [2D-FC]/[2D-IT] model (right) and for the [1D-IT] model (left). The differences between the model cannot be traced back to a specific source. The neglect of the radial pressure gradient can be one of them. Also numerical error can lead to small differences. Figure 8.1 shows the pressure in the middle of the vessel. Grid convergence. — Usually, the solution is expected to converge to the exact solution when the mesh gets finer and finer, also called grid convergence. For the coupled Darcy-Stokes coupled problem with the new interface conditions the author could not identify a manufactured exact solution. Yet, it is possible to obtain the experimental range of convergence calculated as rk = ln e k − ln e k−1 , k−1 k ln hmax − ln hmax where e k is the error with respect to the solution u N calculated for the finest grid in the L2 -norm for the considered primary variable e k = ||u k − u N ||L2 . The refinement step is denoted by k, where the coarsest grid is k = 0, and hmax is the maximum element diameter calculated as the length for (one-dimensional) intervals and twice the circumradius for triangles. The results are shown in Tables 8.2 to 8.5. The convergence orders are, as expected, close to (p + 1), where p is the 59 p in P a −600 [2D-FC] [1D-IT] −650 −700 −750 −800 −850 −900 −950 −1 0 1 ×10−4 x in m Figure 8.2 – Plot-over-line (y = 0 m) for the effective pressure. Also the [2D-IT] model was in excellent accordance but is omitted in this plot for the sake of clarity. polynomial degree of the finite element [Knabner and Angermann, 2000]. For the [2D-IT] model it was difficult to construct a coarse enough mesh so that the norms do not fall under the numerical threshold of double precision and then are distorted by numerical errors. This shows, given that the algorithm converges against the exact solution, that the approximation is already excellent even for coarse grids. The order of polynomials in the Stokes domain is one degree higher that for the [2D-FC] model with BDM1 × DG0 -elements. On the other hand, the approximation degree in the Darcy domain is one degree higher for the pressure but one degree lower for the velocity compared to the [2D-FC] model. This results in a difference in performance as the velocity is a vector-valued function. k hmax ||p k − p N ||L2 Rate r k ||vk − vN ||L2 Rate r k 0 2.40511 · 10−5 0.00256983 − 1.41953 · 10−8 − 1 1.20256 · 10 −5 6.01278 · 10 −6 0.00056086 3.00639 · 10 −6 − 2 3=N 0.00125404 3.47647 · 10 −9 2.02972 1.16088 7.95118 · 10 −10 2.12838 − − 1.03509 − Table 8.2 – Convergence rates and errors for the [2D-FC] model with BDM1 × DG0 -elements for k mesh refinements (reference case). 60 vx = v · n in m s ×10−8 2.5 [2D-FC] [1D-IT] 2 1.5 1 0.5 0 −0.5 −1 −1.5 −5 −4 −3 −2 0 −1 1 2 3 4 5 ×10−4 y in m Figure 8.3 – Plot-over-line (x = 50 · 10−6 m) for the x-velocity and reference geometry. Also the [2D-IT] model was in excellent accordance but is omitted in this plot for the sake of clarity. k hmax 0 0.00017506 1 2 3 4 5 6 7=N ||pp k − pp N ||L2 Rate r k ||pf k − pf N ||L2 Rate r k 0.000747285 − 0.000159071 − 8.75298 · 10 −5 4.37649 · 10 −5 2.18824 · 10 −5 1.09412 · 10 −5 5.47061 · 10 −6 2.73531 · 10 −6 1.82932 · 10 2.1678 5.01169 · 10 1.36765 · 10 −6 − − − 0.000206575 −5 5.36662 · 10 −5 1.35698 · 10 −6 3.38178 · 10 −7 8.21981 · 10 −7 1.85499 1.94458 1.98361 2.00455 2.04061 3.99714 · 10 −5 1.99263 1.00007 · 10 −5 1.99886 2.49555 · 10 −6 2.00268 6.18063 · 10 −7 2.01353 1.60748 · 10 −7 1.94296 −8 1.68143 − Table 8.3 – Convergence rates and errors for the [1D] model for k mesh refinements (reference case). The maximal element diameter hmax is related to the mesh of the tissue domain. Performance. — The models are not easy to compare performance-wise. Due to different discretization schemes that arise from the applied discretization techniques, the [2D-FC] and the [2D-IT] model can have the same approximation qualities in the Stokes domain. The velocity can be approximated with 2nd order polynomials and the pressure with 1st order polynomials. However then, in the Darcy domain, the velocity is approximated with 2nd order polynomials and the pressure 61 k hmax ||pf k − pf N ||L2 Rate r k ||vf k − vf N ||L2 Rate r k 0 0.000106216 1.11083 · 10−7 − 3.24379 · 10−12 − 1 5.3108 · 10 −5 −8 2.75876 · 10 2 2.6554 · 10−5 3 4=N 2.00955 4.0964 · 10 −13 2.98525 6.62825 · 10−9 2.05732 5.2472 · 10−14 2.96474 1.3277 · 10−5 1.37215 · 10−9 2.27219 7.52902 · 10−15 2.80101 6.6385 · 10−6 − − − − Table 8.4 – Convergence rates and errors for the [2D-IT] model (Stokes domain) for k mesh refinements (reference case). The maximal element diameter hmax is the maximal diameter of both meshes. ||pp k − pp N ||L2 Rate r k 0.000106216 0.000987003 − 5.3108 · 10 −5 0.000248515 1.98972 2.6554 · 10 −5 3 1.3277 · 10 −5 4=N 6.6385 · 10−6 k hmax 0 1 2 −5 2.01591 −5 1.45899 · 10 2.07438 − − 6.14473 · 10 Table 8.5 – Convergence rates and errors for the [2D-IT] model (Darcy domain) for k mesh refinements (reference case). The maximum element diameter hmax is the maximum diameter of both meshes. with 1st order polynomials in the [2D-FC] model, but in the [2D-IT] model the velocity is calculated in a post processing step from the 1st order polynomial pressure and is thus constant on each element. The pressure is the only primary variable in the Darcy domain. The velocity can be interpolated to a higher degree space. This leads to a speed up of the [2D-IT] model in comparison to the [2D-FC] at the expense of approximation quality. The [2D-IT] model has the advantage of a non-matching grid being possible on the vessel-tissue interface. Thus, the degrees of freedom in the Darcy domain can be reduced with almost the same approximation quality. All linear sub systems of the [2D-IT] were solved with direct solvers. The LU-factorization only needs to be calculated in the first iteration step and can be reused for all following steps. The resulting CPU time for different orders of approximation and grid refinements is listed in Table 8.7 for comparison. With BDM1 × DG0 -elements the [2D-FC] model has a lower order of approximation for velocity and pressure in the Stokes domain than the [2D-IT] model. In the Darcy domain the pressure is then approximated with piecewise constant functions whereas the [2D-IT] model uses continuous piecewise linear polynomials. With BDM2 × DG1 -elements the [2D-FC] model has the same order of approximation in the Stokes domain, but the velocity in the Darcy domain is approximated with piecewise quadratic polynomials whereas the [2D-IT] model approximates the Darcy velocity with piecewise constant functions. Comparison shows a small speed-up when using the iterative method 62 a a Model Stokes Darcy [2D-FC] BDM1 × DG0 mixed BDM1 × DG0 mixed [2D-FC] BDM1 × DG0 mixed [2D-FC] aa Refined CPU time 62464 0× 3.13686 s BDM1 × DG0 mixed 249536 1× 12.0146 s BDM1 × DG0 mixed BDM1 × DG0 mixed 997504 2× 52.8845 s [2D-FC] BDM1 × DG0 mixed BDM1 × DG0 mixed 3988736 3× 801.189 s [2D-FC] BDM2 × DG1 mixed BDM2 × DG1 mixed 163788 0× 7.11062 s [2D-FC] BDM2 × DG1 mixed BDM2 × DG1 mixed 654672 1× 33.4671 s [2D-IT] P32 -P1 P32 -P1 mixed P1 (pressure only) 24907 0× mixed P1 (pressure only) 19217 0× mixed P1 (pressure only) 69229 1× 8.76692 s [2D-IT] P32 -P1 P32 -P1 mixed P1 (pressure only) 365965 2× 37.173 s [2D-IT] P32 -P1 mixed P1 (pressure only) 1054939 3× 275.892 s [2D-IT] [2D-IT] DOFs 3.0776 s aaa 1.91084 s Table 8.6 – Comparison of performance measured in CPU time on the same meshes. Total CPU time includes solver time, all iteration steps, and all necessary pre- and post-processing steps. All calculation were executed on a MacBookPro (2.53 GHz Intel Core 2 Duo / 4GB Memory). One refinement step divides every triangle into four triangles.a a a Discretization method of the primary variables pressure and velocity. Number of degrees of freedom; for the [2D-IT] the number is calculated from the size of all linear systems to be solvedaina a one iteration step, the Darcy velocity is only calculated once in a post-processing step. The mesh of the Darcy domain was coarsened resulting in hanging nodes on the aaa interface. Refinement of Darcy mesh towards interface was reduced. of model [2D-IT] due to a reduction of the degrees of freedom of the system. The speed-up is more significant the finer the mesh. This is easily explained as degrees of freedom on a vertex are shared with several elements but degrees of freedom on edges are only shared with the neighboring element (for continuous function spaces). Due to discretization schemes fitted to the subproblem and the mesh flexibility on the interface, the [2D-IT] is faster than the [2D-FC] model for the reference case. With the reduction of the Stokes domain to one dimension the number of degrees of freedom can be reduced by 98 % in the Stokes domain. This leads to a significant speed-up. The velocity is calculated as a post-processing step for both domains. In each iteration step ca. 10 % of the time is used to calculate the Stokes step, 90 % are used for the Darcy step. From the latter, 50 % of the time is consumed by pressure evaluation, point-source calculation, and point source application to the Darcy system right-hand-side vector, and 50 % is consumed by solving the actual Darcy linear system. The post-processing, namely, output VTK files and calculation of the velocities takes about 45 % of the total CPU time. Because of the small total time of the algorithm the time is overproportionally distorted by simple operation, e.g. file output, that get insignificant for larger linear systems. It can be concluded that the model reduction leads to a speed-up of at least 90 % in comparison with the [2D-FC] model. 63 a a Model Stokes Darcy DOFs [1D] P1 (pressure only) P1 (pressure only) [1D] P1 (pressure only) [1D] [1D] aa Refined CPU time 1162 0× 0.466947 s P1 (pressure only) 4322 1× 0.773448 s P1 (pressure only) P1 (pressure only) 16642 2× 2.59135 s P1 (pressure only) P1 (pressure only) 65282 3× 10.1879 s Table 8.7 – CPU time for several refined meshes. Total CPU time includes solver time, all iteration steps, and all necessary pre- and post-processing steps. All calculation were executed on a MacBookPro (2.53 GHz Intel Core 2 Duo / 4GB Memory). One refinement step divides every triangle into four triangles and every interval into two intervals. a a a Discretization method of the primary variables pressure and velocity. Number of degrees of freedom; for the [2D-IT] the number is calculated from the size of all linear systems to be solved in one iteration step, the Darcy and Stokes velocity are only calculated once in a post-processing step. Acceleration and relaxation parameters. — The performance of the iterative algorithms of model [2D-IT] and [1D] is highly dependent on the number of iterations until convergence. Errors for both models were calculated as the sum of the ||u k − u k−1 ||L2 -norms of all primary variables, where k is the current iteration step. Primary variables for the [2D-IT] model are velocity and effective pressure in the Stokes domain and pressure in the Darcy domain. Primary variables for the [1D] model is effective pressure in both domains. The method is here considered to be converged when the error in the L2 -norm drops below the tolerance of 10−10 . Figure 8.4 shows the number of iterations with respect to the acceleration parameter γp . The least iterations (5) are needed if γp = 1 Lp . is dominated by the filtration coefficient. parameter remained γp = 1 Lp . This shows that the relation between the subsystems Also for parameter changes the optimal acceleration The algorithm is surprisingly robust towards parameter changes with the number of iterations being always under 10 for the tested literature value range (see Section 7.C). Figure 8.5 shows the number of iteration with respect to the relaxation parameter θ for the [1D] model. The least iterations (20) in the reference case are needed when θ = 0.26. The optimal choice of θ is highly dependent on the set of parameters. For higher values of the filtration coefficient, e.g., the algorithm might even diverge for θ = 0.26. The right graph in Figure 8.5 shows the iteration number with respect to θ for several filtration coefficients Lp . The respective optimal choices for θ seem to fall on one exponential curve. For the highest filtration coefficient Lp = 1.5 · 10−9 m P as the algorithm does not converge within 500 iteration steps. The optimal choice of θ is beyond 0.95. This means in particular that the [1D] model has a bad performance for high filtration coefficients and a very good performance for low filtration coefficient. One could say that if there is a high flow resistance between the free-flow and the porous domain the algorithm converges faster. For all 64 number of iterations, tolerance = 10−10 100 90 80 70 60 50 40 30 20 10 0 1 2 3 4 5 6 7 ×1010 γp Figure 8.4 – Acceleration parameter γp and number of iteration until convergence. The other parameter γf fixed to γf = 3γp . The filtration coefficient Lp = 3.0 · 10−11 is that of the reference case. Maximum iteration number was set to 100. filtration coefficients and permeabilities a θ can be found so that the algorithm converges. However, for very high filtration coefficients θ can be so high that the algorithm does not converge within a reasonable time. On the other hand, modeling capillary systems, the filtration coefficient is limited by values observed in experiments and it can be estimated from the literature values that it is seldom higher than the values experimented with in this work. It can be thus stated that the presented [1D] algorithm converges for all parameter ranges of interest. Penalty parameter for the coupled problem [2D-FC]. — The [2D-FC] discretization features a stabilization term on the interface. It is supposed to assure stability on the interface and enforce the Dirichlet boundary condition for the tangential velocity (vf · τ = 0 on Γ). An estimation of the penalty parameter was obtained for the Stokes problem in Section 4.7. A comparison with an exact solution was not available for the coupled problem. However, as the pressure and the velocity converge, possible oscillations get smaller by reducing grid size. To test the parameter on the interface we set α = 0.5β and calculated the normal velocity field on the interface for the [2D-FC] and the [1D] model. The [1D] velocity on the interface was then assumed as the reference value [1D] vref . We calculated the norm ||(v · n)[2D-FC] − vref ||l 2 and plotted it over the penalty parameter α in Figure 8.6 (left). In the right plot of Figure 8.6 a plot of the normal velocity (normal to the interface) over the interface is plotted for two different penalty parameters. It is visible that for α = 1.0 velocity oscillations occur while they are not visible for α = 3.4. As for the Stokes 65 number of iterations number of iterations 500 450 400 350 300 250 200 150 100 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 500 450 1.5 · 10−12 400 3.0 · 10−11 350 1.5 · 10−10 300 1.5 · 10−9 250 200 150 100 50 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 θ θ Figure 8.5 – Number of iterations of the [1D] model until convergence with respect to the relaxation parameter θ. Maximum number of iterations was set to 500. Reference case (left) and for different filtration coefficients Lp (right). m s −8 ||v · n − vref ||l 2 v · n in ×10−7 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ×10 3 α = 1.0 2.5 α = 3.4 2 1.5 1 0.5 0 −0.5 −1 −1.5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −4 ×10 y in m penalty parameter α Figure 8.6 – Error between [1D] and [2D-FC] model for different penalty parameters (left). Plot-over-line on the vessel-tissue interface (x = 4.3 · 10−6 m) for the x-velocity and reference case (right). discretization of Section 4.7 the error shrinks until an optimal penalty parameter (here α = 3.5) and then steadily but slowly rises (for values α > 3.4). Geometry tests (case B). — The geometry test features two altered geometries, an arc and a bifurcation. Both mimic shapes of capillaries as they could occur in mammals. The arc geometry has a larger exchange surface (vessel wall) on the outside of the bending than on the inside. 66 Furthermore, the surrounding tissue has a larger volume outside than on the inner side of the bending. This introduces asymmetrical features. The bifurcation has asymmetric features around the bifurcation and symmetric features at the end of the respective branches. Figures 8.7 to 8.9 show the pressure field and plot-over-line graphs for pressure and velocity for the arc geometry. Figures 8.10 and 8.11 display analogous results for the bifurcation geometry. At the points were we have an asymmetrical flow field with respect to the vessel centerline, e.g. where the vessel bifurcates, the pressure distinctly differs in the two models. A difference of 10 % is observed at the most extreme points. Regarding the interpretation it is unclear which of the models would be actually closer to an in-vivo measurement. The reason is the dimension of the models used for the results of this paper. In a 3D-3D model fluid has the opportunity to flow around the vessel. In a 2D-2D model fluid has to go inside the vessel first and leave the vessel on the other side in order to cross. A 2D-2D model behaves similar to a 3D-3D model in a radial symmetric case where it represent a slice through the centerline of the vessel. In an asymmetric case such as the arc the 2D-2D model corresponds more to a 3D-3D model where the vessel is a vertical cleft separating a left and a right tissue domain. Therefore, the model reduction to two dimension does not represent the original geometry anymore. In contrast, the set-up of the [1D] model still allows for flow in the porous domain without obstacles. It is thus still similar to a 1D-3D model or a 3D-3D model where fluid can pass the vessel by flowing around. Unfortunately this conceptual mistake does not allow for comparison of accuracy between the spatially resolved and the reduced model. The comparison must be done with a full 3D-3D model. The implementation was unfortunately not possible within the time framework of this thesis and is an interesting research question for future works. Nevertheless it can be estimated that the difference with a 3D-3D model will be smaller than the differences observed with the 2D-2D model. For the sake of completeness we continue the analysis with the available results. The surface pressure plots Figures 8.7 and 8.10 at first show no significant differences between the two models. Differences become visible when plotting over characteristic lines. In Figure 8.8 the [2D-FC] model shows different sized jumps on the inner and outer interface, whereas the [1D] model pressure field is almost symmetrical. Two effects occur. The inner interface is slightly shorter than the outer interface, and, the inner tissue area is smaller than the outer tissue area leading to a ”damming” effect on the inside. In the [1D] model these pressure differences can be evened out by flow exchange not hindered by the vessel. Figure 8.9 shows higher velocities on the inside due to a reduced tissue area in comparison with the outer tissue domain. The [1D] model shows only little asymmetry which is most likely a boundary effect. Parameter variations (case C). — We varied the model parameters Lp , µ, K µi , and R in a wide range (see Table 7.2). For all parameter sets and the symmetric reference geometry, the [1D] and the [2D-FC]/[2D-IT] models are in excellence accordance. The most sensitive parameter was found to be Lp which is partly due to its large uncertainty in literature but also because it dominates 67 Figure 8.7 – Effective pressure solution for the [2D-FC] model (right) and for the [1D-IT] model (left). p in P a −600 [2D-FC] [1D-IT] −650 −700 −750 −800 −850 −900 −950 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 −4 ×10 x in m Figure 8.8 – Plot-over-line (y = x m) for the effective pressure. the flow exchange of vessel and tissue and therefore the whole flow field. The pressure is shown as a plot-over-line in Figure 8.12 for four different filtration coefficients. The filtration coefficient 68 v · n in m s ×10−8 1.5 [2D-FC] left [2D-FC] right [1D-IT] left [1D-IT] right 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 0 10 20 30 40 50 60 70 80 90 θ in degree Figure 8.9 – Plot-over-line at d = 50 · 10−6 m perpendicular to the vessel centerline for the normal velocity v · n. The normal vector n is the unit normal on the vessel-tissue interface. Model Geometry Total net flux [2D-FC] Reference 1.24516 · 10−11 [2D-IT] Reference 1.24528 · 10−11 [1D] Reference 1.24603 · 10−11 [2D-FC] Arc 1.23978 · 10−11 [1D] Arc 1.24487 · 10−11 [2D-FC] Bifurcation 1.39925 · 10−13 [1D] Bifurcation −1.42522 · 10−13 Unit m2 s m2 s m2 s m2 s m2 s m2 s m2 s Table 8.8 – Total net flux across the capillary wall for the [2D-FC] model and the [1D] model for different geometries. determines the height of the pressure jump and correlated, the velocity across the vessel wall. The dashed lines ([1D]) and the drawn through lines ([2D-FC]) align excellently. The quality of the result was the same for all parameters, while the hydraulic conductivity of the tissue had a bigger influence than the viscosity or the radius that had a small impact on pressure and velocity fields. The parameters also have an influence on the numerical parameters. The impact of the filtration coefficient on the acceleration parameters of the [2D-IT] have already been discussed. With the 69 Figure 8.10 – Effective pressure solution for the [2D-FC] model (right) and for the [1D-IT] model (left). p in P a −920 [2D-FC] [1D-IT] −940 −960 −980 −1000 −1020 −1040 −1060 −1080 −1100 −1120 −2 0 −1 1 2 ×10−4 x in m Figure 8.11 – Plot-over-line (y = −0.1 · 10−3 m) for the effective pressure. 70 p in P a −500 1.5 · 10−12 −550 3.0 · 10−11 −600 1.5 · 10−10 −650 1.5 · 10−9 −700 −750 −800 −850 −900 −950 −1 0 1 ×10−4 x in m Figure 8.12 – Plot-over-line (x = 0 m) of the effective pressure for different filtration coefficients Lp . Dashed lines show the results of the [1D] model and the results of the [2D-FC] model are drawn through. Also the [2D-IT] model was in excellent accordance to the other results. The graph is not plotted for reasons of clarity. optimal γp = 1 Lp the iterative algorithm was very robust with respect to all parameter variations with ±2 iteration steps. Less robust was the iterative algorithm for the [1D] model. The algorithm converges but the number of iteration rises significantly with higher filtration coefficients. Non-matching grids for the [1D] model. — The [1D] model can also handle non-matching grids. All previous grids were matching in the sense that intervals always aligned with edges. Vertices do not necessarily match. For the purpose of demonstration a case was constructed where the one-dimensional vessel grid does not match the two-dimensional tissue grid. This introduces errors that get smaller with shrinking grid size. However, when thinking of large vessel networks it is a great advantage of the tissue and vessel mesh can be constructed totally independent of each other. The grid is shown in Figure 8.14 and the pressure field in Figure 8.13. The plot clearly shows the pressure jump between free-flow and porous domain. It can be seen that directly underneath the vessel the porous pressure is not symmetric due to the shifted one-dimensional grid. The error introduced with the non-matching grid shrink with the grid size or the approximation degree of the pressure used in the numerical method. 71 Figure 8.13 – Pressure in free-flow pf and porous domain pp at the example of a non-matching grid. The plot is warped perpendicular to the x-y-plane and scaled with the pressure value at each grid point. Figure 8.14 – Non-matching grid for the [1D] model. The one-dimensional domain is a bifurcation, with Dirichlet boundary conditions at inlet (top branch) p̄in = 400 P a and outlet (both bottom branches) p̄out = −1600 P a. The porous domain is rectangular with Dirichlet boundary conditions p̄ = −933 P a on the whole boundary. 72 9 Summary and Outlook In this work, a spatially resolved model of a capillary bed with three compartments, vessel, tissue, and capillary wall, was reduced in two steps. First, the vessel wall was reduced to a two-dimensional surface of the vessel resulting in a new set of interface conditions for a Darcy-Stokes coupled problem with selective permeable membrane. Secondly, the vessel was reduced to its centerline resulting in a one-dimensional vessel coupled with the surrounding full-dimensional tissue through line sources. For the latter step and the immediate step, numerical discretization and solution methods were proposed and discussed. A locally conservative discontinuous Galerkin discretization using BDM-elements was designed and evaluated for a 2D-2D coupled Darcy-Stokes system with new interface conditions. An alternative continuous Galerkin discretization was introduced in order to decompose the system into two subsystems that can be solved in a sequential iterative algorithm. Finally, an iterative algorithm for solving a system of a one-dimensional vascular graph placed inside a two-dimensional (porous) tissue domain and coupled through line sources was proposed and tested. The model assumptions were challenged by constructing several test cases. Numerical and physical model parameters were varied over a wide range obtained from literature. It was shown for model parameter variations (µ, K µi , Lp , R) that ◦ spatially resolved and spatially reduced models produce (almost) identical results for a wide range of parameters (for symmetrical geometries) ◦ the parameter with the highest influence is the filtration coefficient Lp ◦ for a high filtration coefficients the relaxation parameter is so high that the [1D] converges significantly slower whereas the [2D-IT] iterative algorithm is robust to parameter changes. Two geometrical test cases were introduced to analyze the model response to asymmetry. The comparison shows that a model reduction and the corresponding assumptions are valid in a wide range of parameters and geometries. The [1D] model can not resolve highly asymmetric situation as one assumption in its derivation is that of radial symmetry of the vessel and the close surrounding. Concerning the geometry, an open questions remains if a 2D-2D model can be compared with the presented 1D-2D model, or, if the only valid comparison would be 3D-3D to 1D-3D. This is to be determined in future works. Looking into the future and envisioning larger problem domains a fast 73 intersection detection algorithm can become a relevant time saver for the reduced model. Further reduction could include the homogenization and upscaling of the smallest capillaries, combining the two methods of model reduction. The model is yet to be verified with flow measurements and related geometries from experiments. The discretization method introduced for the [2D-FC] if further improved to avoid any oscillation could be of great interest modeling flow processes over membranes locally. The solvers have been evaluated with respect to CPU time consumed, convergence order, and for numerical parameter variations. The [2D-IT] showed that an iterative solver can be faster if advantages of domain decomposition like possibility of non-matching grids and possibility of reduced polynomial approximation degree are exploited. The iterative solution algorithm of the [1D] model can be a fast and easy to implement alternative to direct solution methods, e.g. as in Cattaneo and Zunino [2013], if the exchange between free-flow and porous domain is not too large. Chapter 8 showed in detail for the numerical parameters that ◦ the acceleration parameters of the [2D-IT] algorithm are highly dependent on the filtration coefficient ◦ the optimal acceleration parameter is found to be γp = 1/Lp ◦ the relaxation parameter of the [1D] model is highly parameter dependent and sets a lower bound for the number of iterations until convergence ◦ the convergence of the [1D] algorithm is fast if the exchange between vessel and tissue is low ◦ the penalty parameters of the [2D-FC] model could not be chosen perfectly in order to avoid any oscillations on the interface. The results show optimal convergence order of all introduced methods. With respect to CPU the following results were obtained: ◦ the [2D-FC] is the slowest but not necessarily the one with the highest approximation order ◦ the [2D-IT] can save some degrees of freedom with suitable functions spaces chosen for each subproblem and the possibility of non-matching grids (ca. −50 %) ◦ the [1D] model exhibits significant speed-up with at least −90 % in comparison with the [2D-FC] model. Finally, it was additionally shown that an SIPG method can stabilize a BDM-DG mixed variational formulation of the Stokes problem and that this leads to optimal grid convergence of velocity and pressure in the L2 -norm and the penalty parameter has a lower bound. This makes it possible to treat Darcy-Stokes coupled problems with a unified discretization that allows for jumps inside the domain. 74 For the future, it is still undetermined what restrictions apply to the penalty parameter for the DarcyStokes coupled problem with the new interface conditions. The system, using a BDMk /DGk−1 discretization is yet to be mathematically analysed. Further, concerning computational time, a comparison between a preconditioned [2D-FC] model and the [2D-IT] iterative method where each subsystem is preconditioned would yield more insight into which model is computationally more efficient in order to calculated localized problems involving a single blood vessel. The computationally by far most efficient [1D] model has to be improved with respect to parameter sensitivity. Alternatively, a (preconditioned) fully coupled approach is worth investigating and can be compared with the iterative solver. Most importantly, a 3D-3D vessel-tissue model has to implemented to be compared with the [1D] model in order to correctly determine the influence of geometrical effects on the solution of the spatially resolved and the spatially reduced model. The developed model for coupling the flow in blood vessels with the flow in the surrounding tissue can be applied to specific topics of medical interest in the future. To this end, not only the flow fields have to be modeled but in particular transport and reaction processes are essential. Furthermore, the model can be combined with a vascular growth model to simulate flow field of the vascular network is growing. This way, it may be possible to make prediction for treatment of tumors with therapeutic agents (e.g. nano particles), blood supply and growth of tumors (angiogenesis), or oxygen transport to the brain in case of a stroke. Finally, contributions to predicting transport of antibiotics to biofilms on artificial implants or to understanding of regeneration of brain tissue during the sleep become possible. 75 Bibliography Abramowitz, M. and Stegun, I. A. (1972). Handbook of mathematical functions: with formulas, graphs, and mathematical tables, pages 887–888. Number 55. Courier Dover Publications. available here: http://people.math.sfu.ca/˜cbm/aands/abramowitz˙and˙stegun.pdf. Arnold, D. N. (1982). An interior penalty finite element method with discontinuous elements. SIAM journal on numerical analysis, 19(4):742–760. Baber, K. (2009). Modeling the transfer of therapeutic agents from the vascular space to the tissue compartment (a continuum approach). Master’s thesis, Universität Stuttgart. Baber, K. (2014). Coupling fuelcells and bloodflow. PhD thesis, University of Stuttgart. Beavers, G. S. and Joseph, D. D. (1967). Boundary conditions at a naturally permeable wall. Journal of fluid mechanics, 30(01):197–207. Boer, R. (2000). Theory of porous media: highlights in historical development and current state. Springer New York. Brenner, S. C. and Scott, R. (2008). The mathematical theory of finite element methods, volume 15. Springer. Cattaneo, L. and Zunino, P. (2013). Computational models for fluid exchange between microcirculation and tissue interstitium. Networks and Heterogeneous Media. Chapman, S., Shipley, R., and Jawad, R. (2008). Multiscale modeling of fluid transport in tumors. Bulletin of Mathematical Biology, 70(8):2334–2357. Darcy, H. (1856). Les fontaines publiques de la ville de Dijon. Discacciati, M. (2005). Iterative methods for Stokes/Darcy coupling. In Barth, T., Griebel, M., Keyes, D., Nieminen, R., Roose, D., Schlick, T., Kornhuber, R., Hoppe, R., Périaux, J., Pironneau, O., Widlund, O., and Xu, J., editors, Domain Decomposition Methods in Science and Engineering, volume 40 of Lecture Notes in Computational Science and Engineering, pages 563– 570. Springer Berlin Heidelberg. 76 Discacciati, M. and Quarteroni, A. (2009). Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation. Revista Matematica Complutense, 22(2):315–426. Discacciati, M., Quarteroni, A., and Valli, A. (2007). Robin-Robin domain decomposition methods for the Stokes-Darcy coupling. SIAM Journal on Numerical Analysis, 45(3):1246–1268. D’Angelo, C. (2007). Multiscale modelling of metabolism and transport phenomena in living tissues. Bibliotheque de l’EPFL, Lausanne. Ehlers, W. and Bluhm, J. (2002). Porous media: theory, experiments and numerical applications. Springer. Ehlers, W. and Blum, J. (2002). Porous media: theory, experiments and numerical applications, chapter Foundations of multiphasic and porous materials, pages 3–86. Berlin: Springer-Verlag. Ehlers, W., Ellsiepen, P., Blome, P., Mahnkopf, D., and Markert, B. (1997). Theoretische und numerische Studien zur Lösung von Rand-und Anfangswertproblemen in der Theorie poröser Medien. Abschlußbericht zum DFG-Forschungsvorhaben Eh, 107:6–2. Ehlers, W. and Wagner, A. (2013). Multi-component modelling of human brain tissue: a contribution to the constitutive and computational description of deformation, flow and diffusion processes with application to the invasive drug-delivery problem. Computer methods in biomechanics and biomedical engineering, (ahead-of-print):1–19. Epshteyn, Y. and Rivière, B. (2007). Estimation of penalty parameters for symmetric interior penalty Galerkin methods. Journal of Computational and Applied Mathematics, 206(2):843 – 872. Erbertseder, K. M. (2012). A multi-scale model for describing cancer-therapeutic transport in the human lung. PhD thesis, University of Stuttgart. Formaggia, L., Quarteroni, A. M., and Veneziani, A. (2009a). Cardiovascular mathematics. Number CMCS-BOOK-2009-001. Springer. Formaggia, L., Quarteroni, A. M., and Veneziani, A. (2009b). Cardiovascular mathematics, pages 22–24. Number CMCS-BOOK-2009-001. Springer. Fortin, M. and Brezzi, F. (1991). Mixed and hybrid finite element methods. Springer. Hall, J. E. (2010). Guyton and Hall Textbook of Medical Physiology: Enhanced E-book, chapter 16. Elsevier Health Sciences. Hassanizadeh, M. and Gray, W. G. (1979). General conservation equations for multi-phase systems: 1. Averaging procedure. Advances in Water Resources, 2:131–144. 77 Helmig, R., Flemisch, B., Wolff, M., Ebigbo, A., and Class, H. (2013). Model coupling for multiphase flow in porous media. Advances in Water Resources, 51:52–66. Ischinger, F. (2013). Modeling transvascular exchanges of therapeutic agents - sensitivity analysis. http://www.hydrosys.uni-stuttgart.de/institut/hydrosys/publikationen/ paper/2013/ProjektArbeit˙FelixIschinger˙2013.pdf. Independent Study, University of Stuttgart. Jain, R. (1987). Transport of molecules across tumor vasculature. Cancer and Metastasis Reviews, 6(4):559–593. Ju, Y., McLeland, J., and Toedebusch, C. (2013). Sleep quality and preclinical alzheimer disease. JAMA Neurology, 70(5):587–593. Knabner, P. and Angermann, L. (2000). Numerik partieller Differentialgleichungen: eine anwendungsorientierte Einführung, page 121ff. Larson, M. G. and Bengzon, F. (2013). The Finite Element Method: Theory, Implementation, and Applications: Theory, Implementation, and Applications, volume 10. Springer. Logg, A., Mardal, K.-A., and Wells, G. (2012a). Automated solution of differential equations by the finite element method: The fenics book, volume 84. Springer. Logg, A., Wells, G. N., and Hake, J. (2012b). DOLFIN: A C++/Python finite element library. Springer. Markert, B. (2005). Porous media viscoelasticity with application to polymeric foams. PhD thesis, University of Stuttgart. Massing, A. (2012). Analysis and implementation of finite element methods on overlapping and fictitious domains. PhD thesis, University of Oslo. Massing, A., Larson, M. G., and Logg, A. (2013). Efficient implementation of finite element methods on nonmatching and overlapping meshes in three dimensions. SIAM Journal on Scientific Computing, 35(1):C23–C47. Massing, A., Larson, M. G., Logg, A., and Rognes, M. E. (2014). A stabilized Nitsche overlapping mesh method for the Stokes problem. Numerische Mathematik, pages 1–29. Mikelic, A. and Jäger, W. (2000). On the interface boundary condition of Beavers, Joseph, and Saffman. SIAM Journal on Applied Mathematics, 60(4):1111–1127. Nitsche, J. (1971). Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36(1):9–15. 78 Quarteroni, A. and Formaggia, L. (2004). Mathematical modelling and numerical simulation of the cardiovascular system. Handbook of numerical analysis, 12:3–127. Rivière, B. (2008). Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation. Society for Industrial and Applied Mathematics. Rivière, B. and Yotov, I. (2005). Locally conservative coupling of Stokes and Darcy flows. SIAM Journal on Numerical Analysis, 42(5):1959–1977. Saffman, P. G. (1971). Boundary condition at surface of a porous medium. Studies in Applied Mathematics, 50(2):93. Secomb, T. W., Hsu, R., Park, E. Y., and Dewhirst, M. W. (2004). Green’s function methods for analysis of oxygen delivery to tissue by microvascular networks. Annals of Biomedical Engineering, 32(11):1519–1529. Sun, Q. and Wu, G. X. (2013). Coupled finite difference and boundary element methods for fluid flow through a vessel with multibranches in tumours. International journal for numerical methods in biomedical engineering, 29(3):309—331. Truesdell, C. (1984). Thermodynamics of diffusion. In Rational thermodynamics, pages 219–236. Springer. Wang, J., Wang, Y., and Ye, X. (2009). A robust numerical method for Stokes equations based on divergence-free H(div) finite element methods. 31(4):2784–2802. 79 SIAM Journal on Scientific Computing, Appendix: Programming code The working procedure in FEniCS was outlined in Chapter 6. The code used to produce the results of this thesis is available under https://bitbucket.org/timokoch/koch2014˙masterthesis. The requirements to execute are FEniCS 1.4 (http://fenicsproject.org/) configured with the MUMPS solver library (http://mumps.enseeiht.fr/). 80

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement