Preprint 2014/09 Non-linearities and Upscaling in Porous Media Coupling a vascular graph model and the surrounding tissue to simulate flow processes in vascular networks Koch, T. Supervisors: Prof. Dr.-Ing. Rainer Helmig Dr. Natalie Schröder Prof. Kent-Andre Mardal André Massing
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Preprint 2014/09
Non-linearities and Upscaling in Porous Media
Coupling a vascular graph model and the surrounding
tissue to simulate flow processes in vascular networks
In this work, lower case symbols (p) denote scalar quantities and bold lowercase symbols (u) repre-
sent 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 = Ti j,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) = Ti i .
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 ms
p Pressure Pa
% Density kgm3
µ Dynamic viscosity (of blood if not otherwise stated) Pas
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 Unit
O Origin of the coordinate system
χ Lagrangian motion function
χ−1 Eulerian motion function
F Deformation gradient
u Deformation vector m
I Second-order Identity tensor
f Body/volume force ms2
t Traction vector Pa
T Stress tensor Pa
a Acceleration field ms2
dv Volume integrand of the current configuration
dV Volume integrand of the reference configuration
M Mass kg
I Momentum kgms
F External forces N
D(v) Symmetric velocity gradient of velocity v 1s
ϕ Mixture
ϕα Constituent α of the mixture ϕ
(·)α Kinematic physical quantity of the constituent α
(·)α Non-kinematic physical quantity of the constituent α
nα Volume fraction
%α Partial density
φ Porosity
% Density production term kgsm3
pα Momentum production term kgs2m2
T Absolute temperature K
Re Reynolds numer
K Intrinsic permeability m2
K Scalar isotropic intrinsic permeability m2
Q Total flux (over the capillary wall) m3
s
A Surface area m2
π Oncotic pressure Pa
Lp Filtration coefficient of the capillary wall mPas
Symbol Description Unit
KM Intrinsic permeability of the capillary wall m2
µi Viscosity of the interstitial fluid Pas
dM Thickness of the capillary wall m
Symbols introduced in subsequent chapters
KR Friction parameter m2
s
p Average pressure (see text for average operators) Pa
δΓ Dirac delta distribution on Γ
R Capillary radius m
γf, γp Acceleration parameters
θ Relaxation parameter
wi Weighting parameter for Gaussian quadrature rule
xi Integration point for Gaussian quadrature rule
pin Dirichlet boundary condition at Stokes inlet Pa
pout Dirichlet boundary condition at Stokes outlet Pa
pp Dirichlet boundary condition for Darcy domain Pa
Function spaces
R Real numbers
L2 Square integrable functions
Hn 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
V 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
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 in-
vasive 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 treat-
ment 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 ther-
apeutic 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 three-
dimensional 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.
1see 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. Darcy-
Stokes 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 Bshould 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
∂X=∂(X + u)
∂X= I+
∂u
∂X, (2.2)
4
∂B
B
F = ∂x∂X = I+ ∂u
∂X
tn
%f
Xx
e1
e2
x = χ(X, t)
e3
u = x− X
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), x = v =d
dtχ(X, t), v = a =
d2
dt2χ(X, t), (2.4)
or, using the inverse motion function, in an Eulerian or spatial setting
v = v(x, t), a = a(x, t). (2.5)
where ddt (·) = ˙(·) = ∂
∂t (·) + v∇(·) indicates the material time derivative of a physical quantitiy.
2.1.1 Balance of mass
The conservation of mass is a fundamental axiom in continuum mechanics
dM
dt= 0 with M =
∫B% dv. (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 ddt det F =
1and det F > 0, to rule out interpenetration of matter.
5
det F∇·v 2 yields
d
dt
∫B% dv =
∫B
(% dv + % dv) =
∫B
(% dv + % ˙(det F)dV ) =
∫B
(%+ %∇·v) dv = 0.
Applying the localization theorem dv→ 0 yields the local form of the mass balance
%+ %∇·v = 0 or∂%
∂t+∇·(%v) = 0 (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. (2.8)
2.1.2 Balance of momentum
In a similar manner as the mass balance one can derive the balance of momentum
dI
dt= F with I =
∫B%v dv and F =
∫∂B
t ds +
∫B%f dv. (2.9)
Applying Cauchy’s theorem (t(x, t,n) = T(x, t)n) and the Gauss-Green formula yields∫B%v dv =
∫∂B
Tn ds +
∫B%f dv =
∫B∇·T dv +
∫B%f dv.
Using the identities dv = det F dV, ddt det F = det F∇·v, and the balance of mass, the global form
of the balance of momentum is obtained as∫B
v(%+ %∇·v) + %v dv =
∫B%v dv =
∫B∇·T dv +
∫B%f dv.
The localization theorem dv→ 0 finally yields the local form of the balance of momentum
%v = %dv
dt= %
(∂v
∂t+ v · ∇v
)= ∇·T + %f. (2.10)
2 ddt
det F = ∂ det F∂F
·· F = det F(FT−1 ·· F) = det F(F−1F ·· I) = det F tr( ∂X∂x
∂x∂X
) = det F tr(∇v) = det F∇·v
6
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
∂t+ v · ∇v
)= 2µ∇·D(v)−∇p + %f,
∇·v = 0.
(2.13)
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 =vcLcν, (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,
∇·v = 0.(2.15)
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
Btn
γ
Xsx
e1
e2
χs(Xs, t)
e3
u = x− Xff
O
Xf
χf (Xf , t)
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 sub-
stance, 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 consti-
tuted by several constituents α
ϕ =⋃α
ϕα. (2.16)
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]
V =
∫B
dv =∑α
V α with V α =
∫B
dvα =
∫Bnα dv. (2.18)
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∑α
nα = 1. (2.20)
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, % =
∑α
%α, (2.21)
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
microscale REV scale
cells ϕS interstitial fluid ϕF
dv
dvF
dvS
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
∂t+∇·(%αvα) = %α, (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∑α
%α = % and∑α
%α = 0. (2.24)
The balance of momentum for the constituent α reads
%α(∂vα∂t
+ vα · ∇vα
)= ∇·Tα + %αfα + pα + %αvα, (2.25)
where pα accounts for the momentum production by interaction with other constituents, e.g.
through friction, and %αvα is the momentum production resulting from a mass production, e.g.
ice melts in water. Again from Truesdell’s metaphysical principles follow the constraints∑α
%αvα = %v ,∑α
[Tα − %α(vα − v)] = T ,∑α
%αfα = %f and∑α
(pα + %αvα) = 0.
(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 fα = f = 0
A4 Solid and fluid are materially incompressible %αR = const.
A5 Isothermal process at θ = 37C
A6 Creeping fluid flow Re 1
A7 Rigid solid skeleton vS = 0
Furthermore, the momentum production pF is expressed by the following constitutive law,
pF = p∇φ− pFµ = p∇φ− φ2µFK−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 pF 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 = 2µFDF + λ(DF · I)I− φpI (2.28)
with the second Lame 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 + pF . (2.30)
A dimensional analysis [Ehlers et al., 1997] shows that TFµ pFµ for small characteristic length,
e.g. pore diameter scale. This results in
0 = −∇·(φpI) + p∇φ− φ2µFK−1vF
0 = −φ∇p − p∇φ+ p∇φ− φ2µFK−1vF
vf = −K
µF∇p
(2.31)
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
µF∇p)
= 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 = LpA [(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
peα := 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 =KMµidM
[pf − pp] , (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 =KMµidM
, (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
∇p =pf − pp
dM. (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
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 Pas. According
to the literature consulted by Ischinger [2013] the blood viscosity can be estimated ranging from
2 − 3.5 · 10−3 Pas where a value of µ = 2.1 · 10−3 Pas was conducted for small vessels. The
viscosity of the interstitial fluid can be estimated ranging from 1.1− 2 · 10−3 Pas with an average
of µi = 1.3 · 10−3 Pas.
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 Kµi
. The intrinsic permeability
has the unit m2. Ischinger [2013] found literature values in the range of 4.4 · 10−18 − 3 · 10−17 m2
14
for the intrinsic permeability and 2.3 · 10−15− 6.6 · 10−15 mPas for the hydraulic conductivity with an
estimated average of K = 6.5 · 10−18 m2.
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 mPas . Ischinger [2013] found literature values in the range of
2.5 · 10−12 − 1.5 · 10−9 mPas 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 mPas .
2.5.3 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 ms [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 pin = 4000 Pa and the hydrostatic pressure
at pout = 2000 Pa with respect to the interstitial hydrostatic pressure pi = 0 Pa. The interstitial
pressure was estimated to be close to atmospheric pressure. She further used πf = 3600 Pa for
the oncotic pressure in the vessel and πp = 933 Pa 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
Ωf
vessel wall M
Ωp
n n
pM,p
pM,f
pf
pp
Ωf
vessel wall M
Ωp
n
pp
pf
vM · τ ≈ 0
τ τ τ
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∩∂Mthe 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 (3.3)
17
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 Jager [2000] is the Beavers-Joseph-Saffman condition
− 2µD(vf)n · τ = αµ√K
vf · τ (3.5)
We assume in this work that the slip velocity vf · τ |Γfis 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 =µidMKM
vf · n + pp (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 mem-
brane
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
Ωp
ΩfΩp
nf
nfΓ
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 in Ωf (3.10)
µiK
vp +∇pp = 0 in Ωp (3.11)
−∇·vp = 0 in Ωp (3.12)
The applied interface conditions on Γ = ∂Ωp ∩ ∂Ωf are
vf · n = vp · n on Γ (3.13)
−2µD(vf)n · n + pf =µidMKM
vf · n + pp on Γ (3.14)
vf · τ = 0 on Γ (3.15)
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 microcircula-
tion
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.
nz− nz
+
S+Ωp
Ωf
S−
nr
z
dzω
M
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, mak-
ing 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 microcircu-
lation. 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 = πR2. The axial component of the velocity field can be written as
v · nz = vz(r, z) = v(z)s(r) (3.16)
where
s(r) =1
γ(2 + γ)
[1−
( rR
)γ](3.17)
is a velocity profile of a power law type, yielding a parabolic profile for γ = 2. The mean velocity is
given by
v =1
A
∫S
v ds(A4)=
1
A
∫S
vznz ds = v(z). (3.18)
20
Note that therefore ∫S
s ds = A. (3.19)
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 − dz2 , 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) =KMµidM
(p − pi) on ω. (3.20)
The mass balance can then be reduced as follows
0 =
∫P∇·v dv =
∫∂P
v·n ds =
∫ω
v·n ds−∫S−vz ds+
∫S+
vz ds =
∫ω
v·n ds−∫S−v s ds+
∫S+
v s ds.
Note that the second fundamental theorem of calculus holds for
A
∫ z+ dz2
z− dz2
∂v
∂zdz = A
[v
(z +
dz
2
)− v
(z −
dz
2
)]
where we used∫S s ds = A. Applying the interface condition and recalling that ds = R dθ dz in
cylindrical coordinates yields∫ω
v · n ds =
∫ω
KMµidM
(p − pi)Rdθdzdz→0≈ 2πR
KMµidM
(p − pi), (3.21)
where
pi =1
2πR
∫θ
pi(z, θ)R dθ (3.22)
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∂v
∂z= 2πR
KMµidM
(p − pi). (3.23)
For the momentum balance, we follow the same procedure. The integration of the pressure term
is straightforward1
ρ
∫P∇p dv
dz→0=
A
ρ
∂p
∂znz.
For the viscous term∫Pν∆v dv =
∫∂Pν∇vn ds =
∫S−ν∇vn−z ds +
∫S+
ν∇vn+z ds +
∫ω
ν∇vnr ds
21
we neglect the change of v with respect to z in comparison to the change in radial direction,
∇vnz =∂v
∂z≈ 0
and we split ∇vnr in its radial and its axial part, so that∫ω
ν∇vnr ds =
∫ω
ν(nr ⊗ nr)∇vnr ds +
∫ω
ν(nz ⊗ nz)∇vnr ds
=
∫ω
ν∂vr∂r
nr ds +
∫ω
ν∂vz∂r
nz ds.
Recalling, that vz(r, z) = v(z)s(r) we get∫ω
ν∂vz∂r
nz ds = 2πRνv∂s
∂r
∣∣∣∣r=R
nz = −KRvnz.
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∫ω
ν∂vr∂r
nr ds =
∫z
ν∂vr∂r
(∫ 2π
0
nr dθ
)dz = 0.
This yields the one-dimensional momentum balance in a three-dimensional world
A
ρ
∂p
∂znz +KRvnz = 0 (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
ρ
∂p
∂znz +KRvnz = 0
−A∂v
∂z= 2πR
KMµidM
(p − pi)(3.25)
Note that the velocity in the mass balance can be eliminated by inserting the momentum balance,
resulting inA
ρ
∂p
∂znz +KRvnz = 0
πR4
2µ(2 + γ)
∂2p
∂z2= 2πR
KMµidM
(p − pi)(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 = R. Integrals over the vessel wall ω are calculated
22
as∫ω R dθ dz
dz→0= 2πR in three dimensions, but as
∫ω,2D 2 dz
dz→0= 2 in two dimensions. The
reduced one-dimensional model then reads
R
ρ
∂p
∂znz +KR,2D vnz = 0
R3
2µ(2 + γ)
∂2p
∂z2= 2
KMµidM
(p − pi ,2D)
(3.27)
with KR,2D = 1R2ν(2 + γ) and pi ,2D = 1
2 (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
µi∇pp = f δΓ in Ω, (3.28)
where δΓ is the Dirac delta distribution with the following properties
δΓ =
1 on Γ
0 elsewhere∫Ω
f δΓ dv =
∫Γ
f ds.
(3.29)
It restricts the source term to a line representing the blood vessel. A comparison with (3.26) yields
f = 2πRKMµidM
(pf − pp) (3.30)
for a three-dimensional model and
f = 2KMµidM
[pf −
1
2(pp|R + pp|−R)
](3.31)
for a two dimensional model.
23
3.5 The coupled 1D-3D model
Ω
nz
Γ
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 =R3µidM
4µ(2 + γ)KM(3.32)
The problem then reads
Problem 3.2 (1D-3D coupled problem)
Find (pf, pp) such that
C∂2pf∂z2
− pf = −pp on Γ
−∇·K
µi∇pp = (2πR
KMµidM
(pf − pp))δΓ in Ω
(3.33)
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
Ω
nz
Γ
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
C∂2pf∂z2
− pf = −pp,2D on Γ
−∇·K
µi∇pp = (2
KMµidM
(pf − pp,2D))δΓ in Ω
(3.34)
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 Ω denote its closure.
Problem 4.1 (Strong formulation) Find u ∈ C2(Ω) such that
− ∆u = f in Ω, u = u0 on ∂ΩD, ∇u · n = g on ∂ΩN , (4.2)
26
where Ck(Ω) = u ∈ Ω : 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 ∈ C2(Ω) is called the strong or classical solution of the problem. The restriction for u ∈ C2
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∫ 1
0
gv dx = −∫ 1
0
f v ′ dx ∀v (4.3)
where v is a test function satisfying v(0) = v(1) = 0 and g = f ′ 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
L2(Ω) = u ∈ Ω :
(∫Ω
u2 dv
) 12
<∞ (4.4)
is the space of functions where the squared function is bounded in a Lebesgue sense, or measurable,
and ‖u‖L2 =(∫
Ω u2 dv
) 12 its norm. In other words, a function u is in L2(Ω) if ‖u‖L2 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
(u, v)H1 =
∫Ω
∇u · ∇v dv +
∫Ω
uv dv (4.6)
and the so induced norm
‖u‖H1 =√
(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.
f (x) = |x | ∈ L2(Ω) because
(∫Ω
|x |2 dv
) 12
<∞ (4.8)
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.
sgn(x) =
1 if x > 0
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 ′(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 ′(x) = sgn(x) /∈ H1(Ω).
1
1−1
1−1
1
−1x
f (x) f (x)
x
Figure 4.1 – The functions f (x) = |x | and f ′(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 ∈ V, where
V(Ω) = u ∈ H1(Ω) : u = u0 on ∂ΩD and V(Ω) = u ∈ H1(Ω) : 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 ∈ V and integration over Ω, leads to∫Ω
−∆uv dv =
∫Ω
f v dv. (4.11)
Integration by parts of the left-hand side integral yields∫Ω
∇u · ∇v dv =
∫Ω
f v dv +
∫∂ΩN
gv ds, (4.12)
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(Ω) such that∫Ω
∇u · ∇v dv =
∫Ω
f v dv +
∫∂ΩN
gv ds ∀v ∈ V(Ω) (4.13)
The formulation in Problem 4.2 is called variational formulation of the Poisson problem. Herein,
u ∈ V(Ω) 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
P1 P2
node
degree of freedom
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∈T
τ. (4.14)
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 V ′h [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]
P1(τ,Vh, φ) =
τ ∈ T
Vh(T ) = v ∈ C0(Ω) : v |τ ∈ P1,∀τ ∈ T
φj = φj(vi) =
1 for i = j
0 for i 6= ji , j = 1, 2, 3
(4.15)
where C0 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∑j=1
Ujφj , (4.16)
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(Ω) = uh ∈ C0(Ω) : uh|τ ∈ P1,∀τ ∈ T and uh = u0 on ∂ΩD such that∫Ω
∇uh · ∇v dv =
∫Ω
f v dv +
∫∂ΩN
gv ds ∀v ∈ Vh(Ω) (4.17)
or, using (4.16)N∑j=1
Uj
∫Ω
∇φj · ∇φi dv =
∫Ω
f∇φi dv +
∫∂ΩN
g∇φi ds (4.18)
This corresponds to solving the linear system
AU = b (4.19)
with the primary variable vector u and
A =
∫Ω
∇φj · ∇φi dv
b =
∫Ω
f∇φi dv +
∫∂ΩN
g∇φi ds
(4.20)
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,
µFK−1vf +∇p = 0.(4.21)
31
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∫Ω
µFK−1vf ·w dv−∫
Ω
p∇·w dv−∫
Ω
q∇·vf dv +
∫∂Ω
pw · n dv = 0 ∀(w, q) ∈ V (4.22)
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 for-
mulation 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
BDM1
DG0
node
DOF
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 Darcy-
Stokes 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 [Riviere, 2008] for the Stokes equation. Riviere 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. ∫Ω
∇u · ∇v dv−∫∂Ω
(∇u · n)v ds =
∫Ω
f v dv (4.24)
The boundary condition is now weakly enforced by penalizing (u − u0), yielding∫Ω
∇u · ∇v dv−∫∂Ω
(∇u · n)v ds +
∫∂Ω
α
h(u − u0)v ds =
∫Ω
f v dv, (4.25)
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 −∫∂Ω(∇v · n)(u − u0) ds, giving∫
Ω
∇u · ∇v dv−∫∂Ω
(∇u · n)v ds︸ ︷︷ ︸Consistency
−∫∂Ω
(∇v · n)u ds︸ ︷︷ ︸Symmetry
+
∫∂Ω
α
huv ds︸ ︷︷ ︸
Penalty
=
∫Ω
f v dv −∫∂Ω
(∇v · n)u0 ds︸ ︷︷ ︸Symmetry
+
∫∂Ω
α
hu0v ds︸ ︷︷ ︸
Penalty
.
(4.26)
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
2(v+ + v−), (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 · neK = JvK · ne .
Two neighboring cells are depicted in Figure 4.4. The choice of ne is arbitrary if consistent [Riviere,
2008]. A discontinuous formulation cannot be formulated in global integrals. Instead, we look at
n+
n−
Γ∂Ω
E−E+
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∫E
∇u · ∇v dv−∫∂E
(∇u · nE)v ds =
∫E
∇f · ∇v dv. (4.29)
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
∑E∈E
∫E
∇u · ∇v dv−∑e∈Γ
∫e
∇u · neJvK ds −∑e∈Γ
∫e
∇v · neJuK ds +∑e∈Γ
∫e
β
hJuKJvK ds
−∑e∈Γ∂Ω
∫e
∇u · nv ds −∑e∈Γ∂Ω
∫e
∇v · nu ds +∑e∈Γ∂Ω
∫e
α
huv ds =
∑E∈E
∫E
f v dv −∑e∈Γ∂Ω
∫e
∇v · nu0 ds +∑e∈Γ∂Ω
∫e
α
hu0v ds,
(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 Riviere, 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 = pin and ∇v n = 0 on ∂Ωin
p = pout 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 vector-
valued 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.∫E
2µD(v) ·· D(w) dv−∫E
p∇·w dv−∫E
q∇·v dv
−∫∂E
2µD(v)nE ·w ds +
∫∂E
pnE ·w ds = 0
(4.32)
35
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.
∑E∈E
∫E
2µD(v) ·· D(w) dv−∑E∈E
∫E
p∇·w dv−∑E∈E
∫E
q∇·v dv
−∑e∈Γ
∫e
2µD(v)ne · JwK ds−∑e∈Γ
∫e
2µD(w)ne · JvK ds +∑e∈Γ
∫e
2µβ
hJvK · JwK ds
+∑e∈Γ
∫e
pJwK · ne ds +∑e∈Γ
∫e
qJvK · ne ds
−∑e∈Γω
∫e
2µD(v)ne ·w ds−∑e∈Γω
∫e
2µD(w)ne · v ds +∑e∈Γω
∫e
2µα
hv ·w ds
−∑
e∈Γ∂Ωin
µ∇T vne ·w ds−∑
e∈Γ∂Ωout
µ∇T vne ·w ds =
−∑
e∈Γ∂Ωin
pin(ne ·w) ds−∑
e∈Γ∂Ωout
pout(ne ·w) ds
(4.33)
Note that choosing BDMk -elements leads to JvK · ne = JwK · ne = 0. This method is similar
to the one presented in [Riviere, 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 + 50x2,
p = 100(1 + y). The error is calculated as
e = ||u − ue ||L2 =
(∫Ω
(u − ue)2 dv
)1/2
,
where ue is the respective exact solution. The rate of convergence is calculated as the experimental
order of convergence
r =ln ek+1 − ln ek
ln hk+1max − ln hkmax
,
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
http://www.nupus.uni-stuttgart.de 2007/1 Cao, Y. / Eikemo, B. / Helmig, R.: Fractional flow formulation for two-phase flow in porous media 2007/2 Korteland, S.-A., The average equilibrium capillary pressure-saturation relationship
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