Download Anisotropic Conduction in Electrocardiology

Anisotropic Conduction in
Electrocardiology
Diplomarbeit
zur Erlangung des akademischen Grades
Diplom-Mathematiker/in
Westfälische Wilhelms-Universität Münster
Fachbereich Mathematik und Informatik
Institut für Numerische und Angewandte Mathematik
Betreuung:
Prof. Dr. Martin Burger
Eingereicht von:
Stefanie Kälz
Münster, November 2012
i
Abstract
In this thesis we have applied the theory of periodic homogenization in order to calculate the macroscopic electrical conductivity values within the heart. Macroscopic
electrical conductivity values are an important factor in the modeling and simulation
of the electrical activity of the heart. The base for our calculations is a micro-CT image
of a pig heart. We solve the so-called cell problem of homogenization and calculate
the homogenized macroscopic conductivity values. Instead of periodic boundary conditions, we have used homogeneous Dirichlet and homogeneous Neumann boundary
conditions. Subsequently, we have compared the results.
ii
Acknowledgments
I would like to thank everybody who made my studies and this thesis possible,
especially:
Prof. Dr. Martin Burger for answering my questions, his guidance and support.
My family, especially my mother Martina Kälz, for always believing in me and supporting me.
Meinen Großeltern für ihre finanzielle Unterstützung.
iii
Contents
1 Introduction
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2 Physiological Background
2.1 The Human Heart . . . . . . . . . . . . . . . .
2.2 The Circulatory System of the Heart . . . . . .
2.3 The Electrical Conduction System of the Heart
2.4 The Structure of the Heart Muscle . . . . . . .
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3 Mathematical Background
3.1 Sobolev Spaces . . . . . . . . . . . .
3.2 Sobolev Spaces of Periodic Functions
3.3 The Homogeneous Dirichlet Problem
3.4 The Periodic Problem . . . . . . . .
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4 Homogenization
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Method of Two-Scale Asymptotic Expansion . . . .
4.2.1 Setting of the Problem . . . . . . . . . . . . . . .
4.2.2 Two-Scale Asymptotic Expansion . . . . . . . . .
4.3 The One-Dimensional Case . . . . . . . . . . . . . . . . .
4.4 Two-Scale Convergence . . . . . . . . . . . . . . . . . . .
4.5 Homogenization of Elliptic Partial Differential Equations
5 Numerical Realization
5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Computation of the Macroscopic Conductivity Values
5.3 MATLAB’s Partial Differential Equation Toolbox . .
5.4 Numerical Implementation . . . . . . . . . . . . . . .
6 Computational Results
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54
Contents
iv
7 Conclusion
60
List of Figures
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List of Tables
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8 Appendix
8.1 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
64
Bibliography
73
1
1 Introduction
Despite many advances in biomedical and cardiac research in the last decades, cardiovascular diseases remain the leading cause of death worldwide. In 2008, more than 17.3
million people worldwide died in consequence of cardiovascular diseases, representing
30% of all deaths. This number being expected to increase to more than 23.6 million
by 2030, according to the World Health Organization (WHO) [1].
In general, the treatment and prevention of cardiovascular diseases remains difficult
because there are still fundamental gaps in the knowledge of the complicated processes
that take place in the heart. Mathematical modeling and computer simulations of the
heart and its function are thus a useful and promising tool to fill these gaps in knowledge. They offer insights into the complex mechanisms of cardiac function in a way
that cannot be achieved experimentally. Hence, they are essential for a better understanding of the heart and it’s function in both, health and disease.
Of particular interest is the modeling and the simulation of the electrical activity of
the heart. In order to simulate realistic electrical impulse propagation, it is crucial to
incorporate on the one hand, an accurate description of the electrophysiological properties of the heart, and on the other hand, an appropriate and detailed representation of
the heart’s anatomy. A key factor thereby is the incorporation of the unique, complex
three-dimensional fiber and sheet architecture of the heart, which has been known to
influence the electrical as well as the mechanical properties of the heart.
As a result of this complex structural organization, the electrical conduction within the
heart is anisotropic. In cardiac electrophysiology, anisotropy means that cardiac tissue
has different electrical properties in different directions, in contrast to isotropy, which
means that the properties are the same in all directions. In other words, anisotropy
implies that the electrical conductivity changes depending on the orientation of the
fibers, with electrical conduction much faster in direction parallel to the long axis of
the fibers, than across the fiber direction. Thus anisotropy plays a key role in the
realistic propagation of the electrical excitation within the heart.
Since the heart consists of a few billions of cells, simulations that take into account
each individual cell are not possible from a computational point of view. Instead,
1
Introduction
2
macroscopic models are used in order to reduce the computational effort. Thus, a
widely used model to simulate the anisotropic electrical activity within the heart is the
so-called bidomain model, which represents macroscopic (average) tissue properties.
The macroscopic model can be derived from a microscopic model by a homogenization
process.
One of the most important factors that influences bidomain simulations are the electrical conductivity values. This influence was shown, for instance, by Johnston and
Kilpatrick [26]. In their study, they demonstrated that different electrical conductivity values result in significantly different epicardial potential distributions in otherwise
identical models. This underlines even more the need for accurate electrical conductivities in order to achive reliable results. However, the few conductivity values found
in the literature differ substantially from each other, see Table 8.1. In addition, there
exist no general agreement, which conductivity values are the ”right ones”. Often these
values are based on experimental data and reasons for the variations are for example
different experimental setups (in vivo or in vitro), different species experimented, or
different modeling assumptions to name but a few [18, 25, 59]. Thus, the experimental
determination of these conductivity values is very challenging.
In this thesis, we will use an alternative approach to derive the electrical conductivity
values. This approach is based on the homogenization of partial differential equations.
The thesis is organized as follows. Chapter 2 provides a short introduction to the
anatomy and physiology of the heart, followed by a more detailed description of the
heart’s structure. In Chapter 3, the mathematical background is presented as a base for
the homogenization theory, which is the content of chapter 4. Afterwards, in Chapter
5, we describe the used data and the numerical implementation. Finally, in Chapter
6, we present our computational results.
3
2 Physiological Background
The aim of this chapter is to provide the reader with some relevant background knowledge about the heart and it’s function. First the basic anatomy of the heart will be
described briefly. This will be followed by a short introduction into the circulatory
system and the electrical conduction system of the heart. The main focus of interest
will be the structure of the heart, in particular of the heart muscle. Thus, finally, we
are going to explain in more detail some structural features of the heart muscle.
2.1 The Human Heart
The human heart is a muscular organ and one of the most important organs in the
human body. It is responsible for circulating blood, oxygen and nutrient to all parts
of the body, in order to sustain life.
No larger than a size of a fist, the heart is located between the lungs in the middle of the
chest, behind and slightly to the left of breastbone. The heart, as illustrated in Figure
2.1, consists of four main chambers: two upper, smaller thin-walled chambers, called
the right and left atria, and two lower, larger thicker-walled chambers, called the right
and left ventricles. The ventricles have to fulfil the major part of the pumping action
of the heart and thus, have to generate much higher pressures. As a consequence, the
ventricles have ticker walls than the atria, especially the left ventricle, whose wall is
approximately three times thicker than the wall of right ventricle [28, 41].
The atria are completely isolated from the ventricles by the cardiac skeleton, also
referred to as the fibrous skeleton of the heart, which is an electrically non-conducting
sheet of connective tissue. At the base, the upper part of the heart, formed mainly
by the left atrium, the major blood vessels of the heart enter and leave. They include
the superior and inferior vena cava, the pulmonary artery and pulmonary vein as well
as the aorta. These vessels are responsible for the blood transport between the body,
the heart and the lungs. The opposite end of the heart is called the apex of the heart,
which is formed by the bottom tip of the left ventricle.
2
Physiological Background
4
A wall of muscular tissue, known as the septum, divides the heart into a left and
into a right half. Each half is composed of one atrium, the receiving chamber, and
one ventricle, the discharging chamber. The right half of the heart is responsible for
pumping blood to the lungs (pulmonary circulation), whereas the left half of the heart
is responsible for circulating blood to the entire body (systemic circulation).
Figure 2.1: Basic Anatomy of the Heart [2]
To ensure that the blood flows in the right direction through the heart, and to prevent
the backward flow of blood, the heart contains four heart valves, which are located in
the cardiac skeleton. Two of the heart valves, known as atrioventricular valves, are
located between the atria and the ventricles: on the right side the tricuspid valve and
on the left side the mitral (or bicuspid) valve. The two remaining valves, the pulmonary
and the aortic valve, also referred to as the semilunar valves, lie between the ventricles
and the great arteries. The pulmonary valve is found between the right ventricle and
the pulmonary artery, and the aortic valve between the left ventricle and the aorta.
The wall of the human heart, which surrounds all four chambers, consists of three
distinct layers: the endocardium, the myocardium and the epicardium.
The endocardium describes the inner layer of the heart wall, which covers the chambers
and the valves of the heart. The myocardium is the middle layer and forms the major
part of the heart wall. It consists primarily of heart muscle cells, also known as cardiac
2
Physiological Background
5
myocytes, and is responsible for the rhythmic contraction of the heart. The outermost
layer is referred to as the epicardium and is composed mostly of connective tissue.
Eventually, the entire heart is surrounded by the pericardium: a thin, double-walled
protecting membrane of connective tissue [10, 31].
2.2 The Circulatory System of the Heart
The human circulatory system, also called cardiovascular system, is extremely important in sustaining life. It is made of the heart, the blood, and the blood vessels, and
ensures the circulation of blood through the body.
There are three main circulatory processes that occur simultaneously in the body:
the coronary circulation, the (higher-pressurized) systemic circulation and the (lowerpressurized) pulmonary circulation.
In the first instance, the heart itself, as the hardest working muscle in the human body,
requires a steady supply of blood to function. For that reason, the heart has its own
blood supply, known as coronary circulation. The coronary circulation consists of two
main vessels, namely the left and right coronary arteries, which leave the aorta and
run along the epicardial surface of the heart [10, 28]. Each of these coronary arteries
branches off into smaller arteries (arterioles and capillaries), which transport the blood
to all parts of the heart muscle.
Beginning with the pulmonary circulation, deoxygenated blood enters the right atrium
via the superior and inferior vena cava, which carries deoxygenated blood from the
upper and lower part of the body, respectively [10]. Subsequently, this blood moves
through the tricuspid valve into the right ventricle. From there it is pumped through
the pulmonary valve into the pulmonary artery, which in turn carries it to the lungs. In
the lungs, the deoxygenated blood gets refreshed and oxygen-rich blood returns back
to the left atrium via the pulmonary veins, completing the pulmonary cycle.
Then, the systemic circulation starts. This blood then passes from the left atrium
through the mitral valve down into the left ventricle. From there, it reaches the aorta,
the largest artery in the body, through the aortic valve. Finally, the aorta distributes
this oxygen-rich blood through a network of arteries, arterioles and capillaries to all
cells of the body. Deoxygenated blood then returns back to the right ventricle via the
superior and inferior vena cava and the cycle starts again.
2
Physiological Background
6
2.3 The Electrical Conduction System of the Heart
The cardiac conduction system initiates and coordinates the electrical impulses, which
are responsible for the rhythmic and coordinated contraction of the heart called heartbeat. This system consists of the sinoatrial (SA) node, the atrioventricular (AV) node,
the bundle of His, the bundle branches, and the Purkinje fibers.
A schematic representation of the conduction system is given in Figure 2.2.
Figure 2.2: Electrical Conduction System of the Heart (adapted from [3])
The contraction of the heart is caused by an electrical signal, called action potential.
Under normal circumstances, the electrical signal starts in the sinoatrial (SA) node,
found within the upper wall of the right atrium. The SA node consists of specialized
cells, also known as pacemaker cells, which spontaneously initiate electrical impulses
at a rate of about 60-100 times per minute without any neural stimulation [61]. The
SA node is also known as the natural or primary pacemaker of the heart because it
sets the rate and the basic rhythm of the heart [28, 29]. Each electrical impulse that
enters the SA node, moves through the walls of both atria to the atrioventricular (AV)
node, which is located near the septum between the right atrium and the right ventricle
[30, 31]. Since the atria and the ventricles are usually electrically isolated from each
2
Physiological Background
7
other, the AV node presents the only electrical connection where electrical impulses
from the atria can be transmitted into the ventricles [20, 28, 29, 61, 55]. The AV node
slightly delays incoming electrical impulses from the atria to ensure that the atria contract before the signal reaches the ventricles [20, 29]. The AV node is also known as
secondary pacemaker [55] because it can take over the control of the rate and rhythm
of the heart, if the SA node fails. After passing the AV node, the electrical impulse
rapidly moves down to the atrioventricular bundle, also called the bundle of His, which
further splits into the right and left bundle branches. They transmit the impulse from
the AV node along the septum to the right and left ventricles, respectively. The bundle
branches finally end in an extensive network of tiny fibers, the Purkinje fibers situated
in the subendocardium of the left and right ventricle [31]. These specialized fast conducting fibers rapidly transmit the electrical impulse to all regions of the ventricular
myocardium, in the direction from the endocardium to the epicardium as well as from
the apex to the base of the heart [55, 57]. This coordinated spread of electrical current
causes the ventricles to contract and to pump blood to the lungs and to the rest of the
body. This process then repeats itself with each new heartbeat.
The conduction velocity varies in different regions of the heart. Accordingly, the fastes
conduction velocity is found in the Purkinje fibers and the slowest in the AV and SA
node, see Table 2.1.
SA node
0.05 m/s
Atrial muscle
0.8 m/s
AV node
0.05 m/s
His bundle
1.75 m/s
Bundle branches
1.5 m/s
Purkinje fibers
Ventricular muscle
3 m/s
1-1.5 m/s
Table 2.1: Conduction velocities in different regions of the heart [55]
Abnormalities in the electrical conduction system can occur anywhere in this system,
and can result in an abnormal heartbeat also called arrhythmia. The heartbeat may
be too fast (tachycardia), or too slow (brachycardia), or irregularly (fibrillation). As
a consequence, the heart may not be able to pump blood efficiently to the rest of the
body, which may lead to a number of serious problems, including death. Thus, understanding the electrical processes that take place in the heart is therefore of fundamental
importance in order to understand and treat cardiac arrhythmias.
2
Physiological Background
8
2.4 The Structure of the Heart Muscle
The human heart muscle, the myocardium, is a complex structure of highly organized
and specialized tissue. It consists of a few billions of cardiac myocytes representing
the major constituent of the myocardium. Cardiac myocytes have roughly a cylindrical
shape with an average length of 50-100 µm and an average diameter of 10-25 µm [31].
The actual shape, size and alignment of the myocytes depends on their location and
function in the heart [12, 57]. For instance, atrial myocytes are smaller than ventricular
myocytes [40, 58].
Cardiac myocytes are enclosed by a cell membrane, the sacrolemma, which separates
the interior of the cells, the intracellular space, from the surrounding medium, the
extracellular space. Individual cardiac myocytes are connected end-to-end and/or sideto side to each other by so-called intercalated discs [22]. Intercalated discs comprise
gap junctions, which are non-selective channels. They connect the intracellular spaces
of neighboring cells, through which ions and other small molecules can readily pass
without crossing the cell membrane. These specialized cell-to-cell junctions ensure on
the one hand, the mechanical coupling between the connected cells, and on the other
hand, act as low-resistance pathways for the rapid propagation of electrical impulses
from cell to cell [28, 29]. In the ventricular myocardium, most of these gap junctions are
found at the ends of the cells, in longitudinal direction [29] and fewer along the length,
in transverse direction [17, 27]. As a consequence, electrical conduction in longitudinal direction, along the long axis of the fibers, is much greater (approximately three
times faster) than in transverse direction, across to the fiber direction [27, 50]. This
implies that the electrical conduction within the heart is anisotropic; a key feature of
the myocardium. Anisotropy means that the electrical properties of cardiac tissue vary
in different directions, depending on the orientation of the fibers. Furthermore, the
intracellular space is more anisotropic than the extracellular space [49]. This condition
is also known as ”unequal anisotropy ratios” [48].
There are several factors, which affect the anisotropic conduction within the heart.
On the one hand, electrical membrane properties, and on the other hand, structural
features including the size, shape and arrangement of the myocytes and the number,
density and distribution of gap junctions [32, 50, 55].
For instance, the density of gap junctions in the SA node is much less than in the
ventricular myocardium, resulting in an slower conduction velocity in the SA node
compared to the ventricular myocardium [27, 55].
From the end-to-end connections of the myocytes result the characteristic long fiber
structure of the heart, while the side-to-side contacts establish a connection between
2
Physiological Background
9
these fibers [63]. Figure 2.3 schematically illustrates the fiber structure of the heart.
Figure 2.3: Cardiac fiber structure [57]
This results in a complex, three-dimensional branching network of interconnected cardiac fibers [37]. Furthermore, these fibers are in turn arranged into muscle layers
(sheets), also known as laminar structure [47, 51]. Hence, it is possible to define three
principal directions at any point within the myocardium. The first one is along the fiber
direction (fiber axis), the second one is perpendicular to the fiber axis lying in the sheet
plane (sheet axis), and the third one is normal to the sheet plane (sheet normal) [24].
These sheets are several myocytes thick and are surrounded by collagenous connective
tissue. A schematic representation of this structure is shown in Figure 2.4.
Figure 2.4: Fiber and sheet structure of the left ventricle (modified from [64])
A transmural block from ventricular wall illustrating (1) change of muscle fiber direction
across the wall and (2) organization of muscle fibers into sheets. Vectors a,b,c define
the three principal directions within the heart, a: fiber axis, b: sheet axis, c: sheet
normal
2
Physiological Background
10
These sheets also branch and interconnect like the fibers [31, 62]. Between these sheets,
there exists gaps, called cleavage planes or interlaminar clefts [12, 33, 55].
Fibers within a sheet are not uniformly oriented across the ventricular wall, instead
the orientation of the fibers has been known to vary throughout the ventricular wall
[33, 24] with a change in fibre orientation of approximately l20◦ from endocardium to
epicardium [31, 60]. Within the epicardium and the endocardium, fibers are predominantly longitudinally oriented, whereas fibers within the midwall are mainly circumferentially oriented. Besides these two types of fibers, there are also oblique running
fibers, which wind helically around the chambers of the heart [37, 47]. Figure 2.5
presents a schematic illustration of the ventricular fiber orientation.
Figure 2.5: Cardiac fibre orientation [19]
More precise, when looking from the apex to the base of the heart, the orientation of
the left ventricular fibers changes smoothly from a left-handed (clockwise) helix in the
subepicardium to circumferential fibers in the midwall, and returns to a right-handed
(counterclockwise) helix in the subendocardium [7, 30, 47, 60].
Furthermore, there are also differences in fiber orientation between the base and the
apex. At the base, the fibers rotate clockwise from epicardium to endocardium, whereas
at the apex, the fibers rotate counterclockwise from epicardium to endocardium [37].
Accordingly, the anisotropy of the myocardium results on the one hand, from the sheet
structure and on the other hand, from the transmurally varying fiber orientation within
these sheets. An understanding of this complex structural arrangement of the heart is
therefore of fundamental importance for understanding the electrical and mechanical
properties of the heart. Moreover, it is also known that cardiac fiber architecture is
altered in several cardiac diseases such as myocardial infarction and ventricular hypertrophy [21, 53]. As a consequence, also the electrical anisotropy changes. From a
clinical point of view, a better understanding of this remodeling would improve the
diagnosis and the treatment of cardiac diseases.
11
3 Mathematical Background
The purpose of this chapter, which is based upon [42], is to give a brief introduction
to the mathematical background, which is useful in order to follow the subsequent
chapter. We first want to present the functional setting, essentially based on Sobolev
spaces, and some basics of functional analysis. This will be then used to study elliptic
boundary value problems with Dirichlet and periodic boundary conditions. Both form
the basis for the homogenization theory presented in chapter 4.
3.1 Sobolev Spaces
In the following Ω denotes a bounded open subset of Rn and C0∞ (Ω) defines the space
of infinitely differentiable functions f : Ω → R with compact support.
In the study of second-order elliptic boundary value problems with Dirichlet and periodic boundary conditions, we will basically focus on the Sobolev space H 1 (Ω) and
certain subspaces, which we will introduce now.
Definition 3.1.1 (Sobolev Space H 1 (Ω)).
The Sobolev space H 1 (Ω) consists of all square intergrable functions u : Ω → R whose
first-order weak derivatives exist and are square integrable:
1
H (Ω) =
∂u
∂u
2
,...,
∈ L (Ω) .
u | u ∈ L (Ω), ∇u =
∂x1
∂xn
2
The space H 1 (Ω) is a separable Hilbert space with corresponding norm
21
kukH 1 (Ω) = kuk2L2 (Ω) + k∇uk2L2 (Ω)
,
(3.1)
3
Mathematical Background
12
and inner product
(u, v)H 1 (Ω) = (u, v)L2 (Ω) + (∇u, ∇v)L2 (Ω)
∀u, v ∈ H 1 (Ω).
(3.2)
Since H 1 (Ω) is a Hilbert space, H 1 (Ω) is also reflexive. Thus, any bounded sequence in
H 1 (Ω) contains a weakly convergent subsequence. Moreover, the embedding of H 1 (Ω)
into L2 (Ω) is compact. This fact will be often used later and is the content of the
following theorem.
Theorem 3.1.1 (Rellich compactness theorem).
From every bounded sequence in H 1 (Ω) one can extract a subsequence which is strongly
convergent in L2 (Ω).
Furthermore, we note that the space C ∞ (Ω) of infinitely differentiable functions is dense
in H 1 (Ω). Since we are concerned with boundary value problems, the solutions we are
looking for have to fulfill the given boundary conditions in some appropriate sense.
Since functions in H 1 (Ω) are defined up to sets of measure zero and the boundary of
the domain Ω has measure zero in Rn , we have to clarify in which sense one can speak
about values on ∂Ω for functions in H 1 (Ω) 1 . This will be achieved by the definition of
the trace operator.
Theorem 3.1.2 (The trace theorem).
Assume that Ω is bounded and that ∂Ω is Lipschitz continuous. Then there exists a
bounded linear operator T : H 1 (Ω) → L2 (∂Ω) such that
1. T u = u|∂Ω
∀u ∈ H 1 (Ω) ∩ C(Ω).
2. kT ukL2 (∂Ω) ≤ C kukH 1 (Ω)
∀u ∈ H 1 Ω.
In the case of the Dirichlet problem with homogeneous boundary conditions, we are
interested in functions of H 1 (Ω), which vanish on the boundary ∂Ω of the domain.
To this end, we will consider the following subspace of H 1 (Ω).
Definition 3.1.2 (Sobolev Space H01 (Ω)).
The Sobolev space H01 (Ω) denotes the closure of C0∞ (Ω) with respect to the norm in
H 1 (Ω):
H01 (Ω) = u | u ∈ H 1 (Ω), T u = 0 .
1
For the case n = 1 we have the inclusion H 1 (Ω) ⊂ C 0 (Ω), and we can define the value on the
boundary in a classical sense. But this is not the case in higher dimensions [11].
3
Mathematical Background
13
The space H01 (Ω) is a subspace of H 1 (Ω) and consists of functions of H 1 (Ω), whose
trace vanishes on the boundary.
The next result is an important tool in order to ensure the existence and uniqueness
of solutions to partial differential equations.
Theorem 3.1.3 (Poincaré inequality).
Let Ω be a bounded open set in Rn . Then there exists a constant CΩ such that
kukL2 (Ω) ≤ CΩ k∇ukL2 (Ω)
∀u ∈ H01 (Ω).
The Poincaré inequality implies that
kukH 1 (Ω) = k∇ukL2 (Ω)
0
defines a norm on H01 (Ω), which is equivalent to the usual norm in the space H 1 (Ω).
This is the norm that we will later use in the study of Dirichlet problems with homogeneous boundary conditions. Another important space that we will need in our study
of elliptic partial differential equations is the dual space of H01 (Ω), which we introduce
now.
Definition 3.1.3 (Dual Space H −1 (Ω)).
H −1 (Ω) defines the dual space of H01 (Ω), i.e. the space of bounded, linear functionals
on H01 (Ω). H −1 (Ω) is a Banach space with the norm
n
o
kf kH −1 (Ω) := sup |hf, vi| | v ∈ H01 (Ω), kvkH 1 (Ω) ≤ 1 .
0
where h·, ·i denote the pairing between H −1 (Ω) and H01 (Ω). Additionally, the following
holds
(3.3)
hf, viH −1 ,H01 ≤ kf kH −1 kvkH 1 ∀f ∈ H −1 (Ω), ∀v ∈ H01 (Ω).
0
3
Mathematical Background
14
3.2 Sobolev Spaces of Periodic Functions
Since we want to study elliptic partial differential equations with periodic boundary
conditions, we will need some basic properties of Sobolev spaces of periodic functions.
First, we start with the definition of periodic functions.
Definition 3.2.1 (Periodic function).
A function f : Rn → R is called Y-periodic, if
f (x + k) = f (x)
∀x ∈ Rn , ∀k ∈ Zn .
In our case, where Y = (0, 1)n denotes the unit cube in Rn , the function f (x) is periodic
in each variable with period 1.
1
(Y )).
Definition 3.2.2 (Sobolev Space Hper
∞
(Y ) the space of infinitely differentiable functions in Rn that are
We denote by Cper
∞
1
(Y ) with respect
(Y ) is defined to be the closure of Cper
1-periodic. Then the space Hper
to the H 1 -norm.
1
(Y ). This is due to the fact,
The Poincaré inequality does not hold in the space Hper
that for constant functions the quantity in the Poincaré inequality will vanishes, since
the derivative of a constant function is zero. The inequality holds, however, if we add
an additional condition that eliminates constant functions.
The Poincaré inequality is important in the sense that it builds the framework, in which
the Lax-Milgram lemma is applied in order to ensure the existence and uniqueness of
solutions to our boundary value problem. Hence, we define the following space
1
Hper
(Y
Z
1
)/R := H = u | u ∈ Hper (Y )| u dy = 0 .
(3.4)
Y
1
By H we denote the subset of H 1 (Ω) of all functions u in Hper
(Y ) with mean value zero
over the unit cell Y . As a consequence, the Poincaré inequality now holds for elements
in H, i.e. there exists a constant Cp > 0 such that
kukL2 (Y ) ≤ Cp k∇ukL2 (Y )
∀u ∈ H.
3
Mathematical Background
15
This means that we can use
kukH = k∇ukL2 (Y )
∀u ∈ H ,
as the norm in H.
1
(Y ))∗ , which are orthogonal to
The dual space H ∗ of H contains all elements of (Hper
constants:
1
H ∗ = u ∈ (Hper
(Y ))∗ | hu, 1i = 0
(3.5)
1
1
(Y ).
(Y ))∗ and Hper
where h·, ·i denotes the pairing between (Hper
We will see that the space H 1 (Ω) and its subsets are appropriate spaces, in which we
will look for weak solutions of boundary value problems for second order elliptic partial
differential equations. The existence and uniqueness of such solutions is ensured by
the Lax-Milgram lemma, which we state now.
Definition 3.2.3.
Let H be a Hilbert space with norm k·k and inner product (·, ·). A bilinear form B : H×
H → R is called continuous (or bounded) if there exists a constant β ≥ 0 such that
|B(u, v)| ≤ β kuk kvk
∀u, v ∈ H,
and coercive if there exists a constant α ≥ 0 such that
B(u, u) ≥ α kuk2
∀u ∈ H.
Lemma 3.2.1 (Lax-Milgram Lemma).
Let B be a continuous, coercive bilinear form on a Hilbert space H. Then, for every
bounded linear functional f on H, there exists a unique element u ∈ H such that
B(u, v) = hf, vi
∀v ∈ H.
(3.6)
Furthermore, the following estimate holds
kukH ≤
1
kf kH∗
α
A proof of this result can be found in e.g. [15].
Now, we have the necessary ”tools” to study elliptic partial differential equations with
Dirichlet and periodic boundary conditions.
3
Mathematical Background
16
3.3 The Homogeneous Dirichlet Problem
We consider the following boundary value problem in divergence form with homogeneous Dirichlet boundary conditions
(
−∇ · (A∇u) = f
for x ∈ Ω
u = 0 for x ∈ ∂Ω
(3.7)
where Ω ⊂ Rn is a bounded open subset of Rn , A = A(x) is a positive definite matrix
and f = f (x) ∈ H −1 (Ω), where H −1 (Ω) denotes the dual space of H 1 (Ω).
Before we continue with the analysis of this boundary value problem, we will introduce
the class of coefficients A(x) that we will be concerned with.
Definition 3.3.1.
Let α, β ∈ R, such that 0 < α ≤ β < ∞. We denote by M (α, β, Ω) the set of n × n
matrices A ∈ L∞ (Ω)n×n such that for every ξ ∈ Rn and every x ∈ Ω
(i) hξ, A(x)ξi ≥ α |ξ|2
(ii) |A(x)ξ| ≤ β |ξ|
Similarly, Mper (α, β, Y ) is defined as the set of matrices in M (α, β, Y ) with Y-periodic
coefficients.
In the study of homogenization we have to deal with partial differential equations with
non-smooth coefficients. Thus, the definition of a classical solution, i.e. a function
u ∈ C 2 (Ω) ∩ C(Ω) that solves (3.7), is too strong and has to be weakend. This leads to
the concept of weak solutions.
To obtain the weak formulation of the Dirichlet problem (3.7), we multiply this equation
by a test function v ∈ H01 (Ω), integrate over Ω, and then use the integration by parts.
The corresponding weak formulation to (3.7) reads then as follows:
Find u ∈ H01 (Ω) such that
Z
Z
A(x)∇u(x)∇v(x) dx =
Ω
f (x)v(x) dx ∀u, v ∈ H01 (Ω) ,
Ω
or in a more concise form
B(u, v) = hf, vi ∀v ∈ H01 (Ω) ,
(3.8)
3
Mathematical Background
17
where B : H01 (Ω) × H01 (Ω) → R is a bilinear form defined by
Z
A(x)∇u(x)∇v(x) dx ∀u, v ∈ H01 (Ω) ,
B(u, v) =
Ω
and f : H01 (Ω) → R is a linear functional, and h·, ·i denotes the pairing between H −1 (Ω)
and H01 (Ω).
Definition 3.3.2 (Weak solution).
A function u ∈ H01 (Ω) is called weak solution of the Dirichlet problem (3.7) if
∀v ∈ H01 (Ω).
B(u, v) = hf, vi
Next, we have to prove the existence and uniqueness of solutions to the Dirichlet problem (3.7), or equivalently (3.8). Thus, we have the following theorem.
Theorem 3.3.1 (Existence and uniqueness of weak solutions).
Assume that A ∈ M (α, β, Ω) and f ∈ H −1 (Ω). Then the Dirichlet problem (3.7) has a
unique weak solution u ∈ H01 (Ω). Additionally, the following estimate holds:
kukH 1 (Ω) ≤
0
1
kf kH −1 (Ω) .
α
(3.9)
Proof. The proof is a straightforward application of the Lax-Milgram theorem.
First, we prove the coercivity condition. The positive definiteness of the matrix A
yields
Z
A∇u · ∇u dx
B(u, u) =
ΩZ
|∇u|2 dx
≥α
Ω
= α kuk2H 1 (Ω)
0
where we used the fact that k∇·kL2 (Ω) defines an equivalent norm in H01 (Ω).
To show continuity, we make use of the L∞ bound on A and the Cauchy-Schwarz
inequality to obtain
Z
|B(u, v)| = A∇u · ∇v dx
ZΩ
≤ β |∇u| |∇v| dx
Ω
≤ β k∇ukL2 (Ω) k∇vkL2 (Ω)
= β kukH 1 (Ω) kvkH 1 (Ω) .
0
0
3
Mathematical Background
18
Hence, the bilinear form B(u, v) is bounded and coercive so that the Lax-Milgram
lemma implies the existence of a unique solution u ∈ H01 (Ω) of equation (3.8).
Finally, we prove estimate (3.9). We have
α kuk2H 1 (Ω) ≤ B(u, u) = hf, uiH −1 ,H01 ≤ kf kH −1 (Ω) kukH 1 (Ω) ,
0
0
and thus
1
kf kH −1 (Ω) .
α
kukH 1 (Ω) ≤
0
Theorem 3.3.2.
Consider the Dirichlet problem
(
−∇ · (Aε ∇uε ) = f
for x ∈ Ω
uε = 0 for x ∈ ∂Ω
(3.10)
where f = f (x) ∈ H −1 (Ω) and Aε = Aε (x) is a sequence of matrices with Aε ∈ M (α, β, Ω)
for every ε > 0 2 . The following estimate holds
kuε kH 1 (Ω) ≤ C
0
where C is a constant independent of ε. This implies that the sequence uε is uniformly bounded in H01 (Ω). Since H01 (Ω) is reflexive, we can conclude that there exists
a subsequence uε0 and an element u0 ∈ H01 (Ω), such that
uε0 * u0
weakly in H01 (Ω).
Furthermore, from the Rellich compactness theorem we deduce that
uε0 → u0
strongly in L2 (Ω).
The above Dirichlet problem (3.10) will be our main object when studying homogenization for elliptic partial differential equations. Our goal will be to characterize this
limit u0 , i.e. to find the equation satisfied by the limit u0 , the so-called homogenized
equation.
2
Note, that the coefficients Aε are not assumed to be periodic.
3
Mathematical Background
19
3.4 The Periodic Problem
Now we consider the case of elliptic partial differential equations with periodic boundary conditions
(
−∇ · (A∇u) = f in Y
(3.11)
u is Y -periodic
where A = A(x) is a Y -periodic positive definite matrix and f = f (x) ∈ H ∗ , the dual
of H. It is obvious that the solution of the periodic problem (3.11) can be determined
1
(Y ) cannot be ensured.
only up to a constant. Thus, uniqueness in Hper
As we have already mentioned in section 3.2, the Poincaré inequality does not hold in
1
Hper
(Y ), which implies that the bilinear form associated to (3.11) is not coercive and
thus, the Lax-Milgram lemma does not apply.
In order to ensure uniqueness, we need to fix this constant. To this end we work in H,
1
(Y ) with mean value zero, see (3.4).
which consists of functions u ∈ Hper
Similarly to the previous section, we have the following definition.
Definition 3.4.1 (Weak solution).
A function u ∈ H is called a weak solution of the boundary value problem (3.11) if
B(u, v) = hf, vi
∀v ∈ H
(3.12)
where h·, ·i denotes the pairing between H ∗ and H.
We note that when f ∈ H ∗ , then the equation (3.11) has a unique solution because the
definition of H ∗ ensures that the function f has mean value zero over the unit cell Y .
This is a consequence of the Fredholm alternative, see e.g. [15]. Thus, we can state
the following theorem.
Theorem 3.4.1 (Existence and uniqueness of weak solutions).
Let A ∈ Mper (α, β, Y ) and f ∈ H ∗ . Then problem (3.11), or equivalently (3.12), has a
unique weak solution u ∈ H. Furthermore, the following estimate holds
kukH ≤
1
kf kH ∗ .
α
1
Proof. We show that for f ∈ (Hper
(Y ))∗ the problem
1
B(u, v) = hf, vi ∀v ∈ Hper
(Y )
(3.13)
3
Mathematical Background
20
has a unique solution u ∈ H if and only if
hf, 1i = 0.
(3.14)
The condition (3.14) is necessary, since
B(u, v) = 0 = hf, vi
1
for constant functions v ∈ Hper
(Y ). Similar to the proof of Theorem 3.3.1, we can
conclude that the bilinear form B is continuous and coercive. It follows, by the LaxMilgram lemma, that the problem
B(u, v) = hf, vi ∀v ∈ H
1
has a unique solution u. Next,
Z we have to show that this holds for all v ∈ Hper (Y ).
1
Let v ∈ Hper
(Y ). Then, v− v(y) dy = v̂ ∈ H and we obtain:
Y
B(u, v) = B(u, v̂) = hf, v̂i
Z
v(y) dy hf, 1i
= hf, vi −
Y
1
= hf, vi ∀v ∈ Hper
(Y ).
This proof can be found in [43].
21
4 Homogenization
The aim of this chapter is to introduce the reader in the theory of homogenization,
which will be the base for our numerical part in chapter 5.
First we are going to explain the main idea of homogenization with the focus on periodic
homogenization. Afterwards, we want to present the classical method in the theory of
homogenization, the two-scale asymptotic expansion method. Then, we are going to
show one special case where an analytical solution of the problem exists, which is in
general not the case. Since the two-scale asymptotic expansion method is only formal,
we are going to present another useful method to study homogenization problems,
the so-called two-scale convergence method. This method can also be used to justify
mathematically the two-scale asymptotic expansion method. Subsequently, we are
going to apply the two-scale convergence method to the homogenization of elliptic
partial differential equations. In this chapter, we go along the lines of [42].
4.1 Introduction
The theory of homogenization often deals with the analysis of partial differential equations with rapidly oscillating coefficients. Equations of this type describe a wide variety
of problems arising in many fields of natural science and engineering. Typical problems
are heat conduction, fluid flow in porous media or composite materials [23].
One common feature of these problems is that they involve multiple scales; usually a
macroscopic one describing the global behavior and a microscopic one describing the
heterogeneities of the underlying material. Due to these heterogeneities, the physical properties of the material, such as thermal or electrical conductivity, will oscillate
rapidly.
In order to explain the goal and the main ideas of homogenization, we consider the
following classical example of a linear second-order elliptic partial differential equation
4
Homogenization
22
with rapidly oscillating periodic coefficients
(
−∇ · (Aε ∇uε ) = f
in Ω
(4.1)
uε = 0 on ∂Ω .
Here, the coefficient matrix Aε , defined as Aε (x) := A xε , is assumed to be periodic,
i.e. A(y) is Y -periodic in y = xε with respect to the unit cell Y = (0, 1)n in Rn . The
coefficient matrix represents the physical properties of the underlying material, and
the small parameter ε > 0 describes the period of the oscillations, characterizing the
microscopic scale of the problem. Hence, the smaller ε gets, the more rapid are the
oscillations, and the finer the microstructure becomes.
Further, Ω ⊂ Rn denotes the domain with boundary ∂Ω, the function f is a given source
term, and uε is the unknown function to be solved for.
The actual difficulty is that if the characteristic size of the period ε is very small in
contrast to the size of the domain Ω of interest, a direct numerical analysis, even with
modern high-speed computers, will become extremely difficult, and is sometimes even
not possible.
This is due to the fact that solving equations like (4.1) requires a very fine discretization of the domain, fine enough (i.e. smaller than the period ε) to exactly resolve the
periodic microstructure in order to capture the microscopic variations in the underlying
structure. As a consequence, this in turn implies an enormous computational cost for
small ε.
To overcome this difficulty, alternative approaches are needed, which on the one hand
treat efficiently such problems, and on the other hand are easier to solve from a numerical point of view. And this is where homogenization theory comes into play. The
general idea of homogenization is to replace the original model problem (4.1) by a
simpler one that is independent of the small parameter ε, the so-called homogenized
problem
(
−∇ · A ∇u0 = f in Ω
(4.2)
u0 = 0 on ∂Ω ,
where A is a constant matrix characterized by the homogenized coefficients, also called
effective coefficients, which describe the global, macroscopic properties of the underlying structure. That means, instead of working with the original highly heterogeneous
model problem with rapidly oscillating coefficients, one works with an effective homogenized one with constant or slowly varying coefficients, whose macroscopic properties
are a good approximation of the original one [11].
From a mathematical point of view, the aim of homogenization is to describe the limit
4
Homogenization
23
process of uε when the length of the period ε tends to zero. More precisely, one assumes that the solutions uε of (4.1) converge in some appropriate sense to some limit
u0 , where u0 is the solution of the corresponding homogenized equation (4.2) that does
not depend on the small scale ε. Thus, by letting the parameter ε tend to zero, the
heterogeneities become infinitesimally small, so that the underlying structure appears
to be homogeneous; hence the name homogenization.
Figure 4.1 schematically illustrates the described principle of homogenization.
Figure 4.1: The material for successively smaller ε
Several methods have been developed in order to study homogenization problems such
as the method of asymptotic expansion, G-convergence, H-convergence, Γ-convergence,
the oscillating test function method of Tartar, or the two-scale convergence method.
One classical method in the homogenization theory is the method of two-scale asymptotic expansion, which we are going to introduce in more detail in Section 4.2.
This method is based on the assumption that the solution uε to a given problem like
(4.1) admits a two-scale asymptotic expansion of the form
uε (x) = u0
x
x
x
2
+ εu1 x,
+ ε u2 x,
+ ...,
x,
ε
ε
ε
(4.3)
where the functions ui (x, y) are Y -periodic in y.
The idea behind the method is to insert the expansion (4.3) into the given problem
(4.1) and to equate terms of the same power in ε, which leads to a sequence of equations
for the functions ui (x, y). Finally, we derive the corresponding homogenized equation
(4.2) as well as the so-called cell problem (see Section 4.2).
Even though the method is simple and powerful, and it yields heuristically the homogenized equation, it has the disadvantage that one cannot be sure a priori that the
solution uε actually admits an expansion like (4.3).
However, the method can be used to guess the form of the homogenized equation [6].
As a consequence, a second step is needed to prove the validity of this result, i.e. the
actual convergence of the sequence uε to u0 .
4
Homogenization
24
In the case of periodic homogenization, the two-scale convergence method has proven
to be particularly suited. This method was first introduced by Nguentseng [39] in 1989
and further developed by Allaire [6] and others.
It has the advantage, in contrast to other methods, that it combines the above two
steps in just one step, i.e. we obtain the homogenized equation, and simultaneously
prove the convergence [6].
However, the two-scale convergence is suited only for periodic homogenization problems. This method uses oscillating test functions of the form ϕ(x, xε ), where ϕ(x, xε ) is
Y -periodic in y. One characteristic property of this method is that the two-scale limit
of a sequence of functions uε (x) consists of two variables, incorporating the two scales
of the problem. Namely, the macroscopic variable x, and the microscopic variable y.
Thus, two-scale convergence can be regarded as a special kind of weak convergence in
the sense that the informations of the microscopic oscillations of the function, which
are lost in the weak limit are captured by the two-scale limit in an additional variable
[16]. Furthermore, the concept of two-scale convergence can be used to justify mathematically the formal two-scale asymptotic expansion method [43].
That is why the two-scale convergence became an important and powerful tool in periodic homogenization theory, and we are going to describe this method in more detail
in Section 4.4.
There exists a large literature dealing with periodic homogenization, see e.g. [8, 11, 52].
Finally, it should be mentioned that homogenization is not restricted to the periodic
case, and can also be applied to random media.
4.2 The Method of Two-Scale Asymptotic Expansion
In this section, we briefly present the method of two-scale asymptotic expansion applied to a classical elliptic boundary value problem. We are going to show that the
corresponding homogenized equation to this boundary value problem can be formally
obtained with this method. A rigorous mathematical justification will be given in
section 4.4 by means of the two-scale convergence method.
4
Homogenization
25
4.2.1 Setting of the Problem
We consider the following second-order elliptic boundary value problem in divergence
form with homogeneous Dirichlet boundary conditions:
(
−∇ · (Aε ∇uε ) = f
in Ω
(4.4)
uε = 0 on ∂Ω ,
where Ω is an open and bounded subset of Rn with smooth boundary ∂Ω and the matrix
Aε is defined as Aε = A xε . We will assume that A(y), y = xε , is smooth, uniformly
positive definite, and Y -periodic in y, where Y = (0, 1)n denotes the unit cube in Rn .
Furthermore, the function f = f (x) is also assumed to be smooth and independent of
ε. To sum up, we have the following assumptions:
f ∈ C ∞ (Rn ) ,
∞
A ∈ Cper
(Y ) ,
∃α > 0 : hξ, A(y)ξi ≥ α |ξ|2 , ∀ y ∈ Y, ∀ ξ ∈ Rn .
The assumptions on A imply on the one hand that Aε ∈ Mper (α, β, Y ), see Definition
3.3.1, and on the other hand that the differential operator Aε := −∇ · (Aε ∇) on the
left hand side of (4.4) is uniformly elliptic.
At this point it should be mentioned that the regularity assumptions are more restrictive than needed in order to perform the formal calculations. Later we will see that
much less regularity is required. That is an important aspect, since in many applications the coefficients A(y) are not smooth functions [43].
Our goal is now to derive the following homogenized equation to the problem (4.4)
above
(
−∇ · A ∇u0 = f in Ω
(4.5)
u0 = 0 on ∂Ω ,
where the constant effective homogenized coefficients A = (Aij )1≤i,j≤n are given by
Z
A(y) + A(y)∇y χ(y)T dy ,
A=
Y
or, equivalently
Z
A(y) (ei + ∇y χi (y)) · ej dy ,
Aij =
Y
(4.6)
4
Homogenization
26
where {ei }ni=1 and {ej }nj=1 denotes the i-th and j-th unit vector on Rn , respectively.
The vector field χ : Y → Rn satisfies the so-called cell problem


−∇y · ∇y χ(y)AT (y) = ∇y · AT (y) in Y




χ(y) is Y -periodic
Z




χ(y) dy = 0.

(4.7)
Y
Alternatively, the cell problem (4.7) can be written component wise as


−∇y · (A(y)∇y χi (y)) = ∇y · (A(y) ei )




χi (y) is Y -periodic
Z




χi (y) dy = 0

in Y
(4.8)
Y
for i = 1, . . . , n and {ei }ni=1 denotes the i-th unit vector on Rn .
4.2.2 Two-Scale Asymptotic Expansion
The method of two-scale asymptotic expansion, as stated earlier, is based on the assumption that the solution uε to the problem (4.4) can be expressed in form of the
following ansatz
x
x
x
+ εu1 x,
+ ε2 u2 x,
+ ...,
uε (x) = u0 x,
ε
ε
ε
(4.9)
where the functions ui (x, y) are Y -periodic in y = xε , i.e. the functions ui (x, y) take
equal values on the opposite faces of Y [14].
This ansatz is legitimate in the sense that it takes into account the two different length
scales arising in our problem. The variable x denotes the macroscopic scale, which
represents slow variations, and the variable y = xε characterizes the microscopic scale,
which describes the rapid oscillations.
We will treat x and y to be independent variables, since for sufficiently small ε the
variable y changes more rapidly than x. In view of y = xε the gradient operator, i.e.
the partial differential derivatives with respect to x become
1
∇ = ∇x + ∇y .
ε
4
Homogenization
27
Using this, the differential operator
Aε := −∇ · (A(y)∇)
can be rewritten as
1
1
A = − ∇x + ∇y · A(y)(∇x + ∇y )
ε
ε
1
1
A1 + A2
=
A
+
0
ε2
ε
ε
(4.10)
where
A0 = −∇y · (A(y)∇y ) ,
A1 = −∇y · (A(y)∇x ) − ∇x · (A(y)∇y ) ,
A2 = −∇x · (A(y)∇x ) .
In virtue of (4.10), equation (4.4) becomes
1
1
A0 + A1 + A2 uε = f
ε2
ε
in Ω × Y ,
(4.11)
uε = 0 on ∂Ω × Y .
Then, inserting the ansatz (4.3) into (4.11) and rearranging terms yields
Aε uε − f =
1
1
A
u
+
(A0 u1 + A1 u0 ) + (A0 u2 + A1 u1 + A2 u0 )+
0
0
ε2
ε
ε(A1 u2 + A2 u1 ) + ε2 A2 u2 + . . . − f.
The next step consists in equating terms, which have the same order in ε and disregarding all terms of order higher than 1. That leads to the following sequence of
equations:
A0 u0 = 0 ,
u0 (x, y) is Y -periodic ,
(4.12)
A0 u1 = −A1 u0 ,
u1 (x, y) is Y -periodic ,
(4.13)
A0 u2 = −A1 u1 − A2 u0 + f , u2 (x, y) is Y -periodic ,
(4.14)
1 1
where the equations (4.12), (4.13), (4.14) correspond to the 2 , and ε terms, respecε ε
tively. We note that A0 is a uniform elliptic partial differential operator with respect to
y, and we may treat x merely as a parameter. Furthermore, the periodicity condition
4
Homogenization
28
can be regarded as boundary condition [14].
Before we continue with the analysis of these equations, we remark the following important result, which ensures the existence and uniqueness of solutions to our problem.
The equations (4.12)-(4.14) are of the form
(
A0 u = F
in Y
(4.15)
u is Y -periodic .
This equation admits a solution if and only if the right hand side of (4.15) averages to
zero over the unit cell Y , i.e. if the following solvability condition is satisfied
Z
F (y) dy = 0 .
(4.16)
Y
If a solution exists, it is unique up to an additive constant, and among all solutions of
(4.15) we will choose the unique solution whose integral over Y vanishes:
Z
u(y) dy = 0 .
Y
The solvability condition (4.16) is a consequence of the Fredholm alternative.
For more details we refer e.g. to [15], [42], or [43].
However, the above results will be sufficient for our purposes in this section.
Thus, the equations (4.12), (4.13), (4.14) have a unique solution, up to an additive
constant, if and only if, the right hand side averages to zero over the unit cell Y .
With this in mind, we turn now to the analysis of the above equations.
Let us consider equation (4.12), which reads
−∇y · (A(y)∇y u0 (x, y)) = 0.
It turns out that the solutions of the homogenized equation (4.12) depend only on the
macroscopic variable x, i.e. u0 (x, y) = u0 (x) for each x ∈ Ω.
As a consequence, the first term u0 in the two-scale asymptotic expansion (4.9) does
not depend on the microscopic variable y.
Next, we proceed with equation (4.13). Since u0 is independent of y, it follows that
∇y u0 (x) = 0. Thus, equation (4.13) reduces to
− ∇ · (A(y)∇y u1 (x, y)) = ∇y · (A(y)∇x u0 (x)).
(4.17)
4
Homogenization
29
We have to verify that the solvability condition is satisfied
Z
Z
∇y · (A(y)∇x u0 (x)) dy = ∇x u0 (x)
Y
∇y · A(y) dy = 0 ,
Y
where we used the divergence theorem and the periodicity of A.
Since the solvability condition is satisfied, equation (4.13) has a unique solution up to
an additive constant. In order to solve this equation, we are going to use the separation
of variables.
Thus, the solution u1 (x, y) can be represented in the form
u1 (x, y) = χ(y) · ∇x u0 (x) ,
(4.18)
where χ(y) = (χ1 (y), . . . , χn (y))T is a Y -periodic vector field, also known as first-order
corrector. It is the unique solution of the so-called cell problem
−∇y · (A(y)∇y χi (y)) = ∇y · (A(y) ei ) in Y ,
(4.19)
χi (y) is Y -periodic ,
Z
χi (y) dy = 0 ,
(4.20)
Y
which we obtain by substituting (4.18) into (4.17).
We note that due to the periodicity of the coefficients the right hand side of (4.19)
averages to zero over the unit cell Y , i.e. the necessary condition for the existence of a
solution is satisfied. The uniqueness of solutions to (4.19) is ensured by the condition
(4.20).
Finally, we turn to equation (4.14). Applying the solvability condition, we obtain for
the left hand side of (4.14) by periodicity of u2
Z
Z
A0 u2 dy =
Y
∇y · (A(y)∇y u2 (x, y)) dy
ZY
A(y)∇y u2 (x, y) · n dσ(y)
=
∂Y
= 0,
4
Homogenization
30
where n is the normal vector over ∂Ω. This implies that the right hand side of equation
(4.14) must have a zero average over Y , i.e.
Z
Z
(A2 u0 + A1 u1 ) dy =
Y
f (x) dy = f (x)
(4.21)
Y
since the function f was assumed to be independent of y.
For the first term on the left hand side of (4.21), we obtain
Z
Z
A2 u0 dy =
−∇x · (A(y)∇x u0 (x)) dy
Z
= −∇x ·
A(y) dy ∇x u0 (x) ,
Y
Y
(4.22)
Y
and for the second term
Z
Z
Z
A1 u1 dy =
−∇y · (A(y)∇x u1 (x, y) dy − ∇x · (A(y)∇y u1 (x, y)) dy
Y
Y
Y
{z
}
|
=0 by divergence theorem and periodicity
Z
= − ∇x · (A(y)∇y u1 (x, y)) dy
(4.23)
ZY
= − A(y)∇x · ∇y (χ(y) · ∇x u0 (x)) dy
Y
Z
A(y)∇y χ(y) dy ∇x u0 (x) .
= −∇x ·
Y
Substituting (4.22) and (4.23) into (4.21) finally yields the homogenized equation
(
−∇ · A ∇u0 = f
in Ω
(4.24)
u0 = 0 on ∂Ω
with the homogenized coefficient matrix A given by (4.6).
Furthermore, it can be shown that the homogenized matrix A is symmetric and positive
definite if the matrix A(y) is symmetric and positive definite. Hence, the homogenization procedure preserves these two properties, but it does not preserve isotropy and
therefore, can create anisotropies. For more details, we refer to e.g. [42, 43].
To summarize, if we postulate an asymptotic expansion of the form (4.9), then the
first term u0 of this expansion can be identified with the solution of the homogenized
equation (4.5). The solution of the homogenized equation can be obtained in the
4
Homogenization
31
following way: In a first step, one has to solve the cell problem (4.7). The next step
consists in inserting the solution of the cell problem into the equation (4.8) and to
calculate the homogenized coefficient matrix A. Finally, the homogenized equation
can be solved.
In general, it is not possible to solve the cell problem analytically so that numerical
methods are required. One exception is the one-dimensional case, which we are going
to present know.
4.3 The One-Dimensional Case
Let n = 1 and Ω = [0, L]. Then the Dirichlet problem (4.4) reduces to a two-point
boundary value problem:
x duε
(x) = f (x) for x ∈ (0, L),
a
ε dx
d
−
dx
uε (0) = uε (L) = 0.
Similarly, the cell problem becomes a boundary value problem for an ordinary differential equation
d
−
dy
dχ(y)
a(y)
dy
=
da(y)
,
dy
for y ∈ (0, 1),
(4.25)
χ(y) is Y -periodic
Z 1
χ(y) dy = 0.
0
In the one dimensional case, we only have one homogenized coefficient given by
Z
a=
0
1
dχ(y)
a(y) + a(y)
dy
dy.
(4.26)
We assume that a(y) is smooth and Y -periodic. Further, we assume that f is smooth
and that there exist constants α, β such that
0 ≤ α ≤ a(y) ≤ β < ∞,
∀y ∈ [0, 1] .
4
Homogenization
32
Now, we can solve equation (4.25). Integration from 0 to y yields
a(y)
dχ(y)
= −a(y) + c1 .
dy
(4.27)
where the constant c1 is undetermined at this point. The inequality (4.3) allows us to
divide (4.27) by a(y) since it implies that a is strictly positive. Then, integrating once
again from 0 to y gives
Z
χ(y) = −y + c1
0
y
1
dy + c2 .
a(y)
whereas the constant c2 is not required for the calculation of the a. In order to determine
the constant c1 we use the fact that χ(y) is a periodic function:
Z
χ(0) = χ(1)
⇒ 0 = 1 − c1
0
1
1
dy
a(y)
1
.
⇒ c1 = Z 1
1
dy
0 a(y)
Thus, from (4.27)
dχ(y)
1+
=Z
dy
1
1
a(y)−1 dy
Z
0
y
1
dy.
a(y)
0
Finally, inserting this expression into equation (4.26) yields the following homogenized
coefficient
1
(4.28)
a= Z 1
−1
a(y) dy
0
Thus, one can see that the homogenized coefficient is not only the average of the
unhomogenized coefficients over a period of the microstructure.In this case it is the
harmonic average.
4.4 Two-Scale Convergence
Since the method of two-scale convergence has proven to be a useful method in periodic
homogenization, we first want to present in this section some basic facts about twoscale convergence. Subsequently, we are going to show how this concept can be used
4
Homogenization
33
in the homogenization of second-order elliptic partial differential equations.
For more information concerning two-scale convergence, the reader is referred to [6]
and [35].
In the following let Ω be an open bounded subset of Rn with smooth boundary ∂Ω and
let Y = (0, 1)n denote the unit cube in Rn .
Before we start with the definition of the two-scale convergence, we will recall two
important function spaces in this context.
The first of them is the space L2 (Ω; L2 (Y )) := L2 (Ω×Y ), which is a Hilbert space with
inner product
Z Z
u(x, y)v(x, y) dy dx
(u, v)L2 (Ω×Y ) =
Ω
and norm
kuk2L2 (Ω×Y )
Y
Z Z
|u(x, y)|2 dy dx.
=
Ω
Y
The second one is space L2 (Ω; Cper (Y )), which is the set of all measurable functions
u : Ω → Cper (Y ) with ku(x)kCper (Y ) ∈ L2 (Ω). This is a separable Banach space, which
is dense in L2 (Ω × Y ) with the corresponding norm
kuk2L2 (Ω;Cper (Y ))
2
Z =
sup |u(x, y)| dy.
y∈Y
Ω
Definition 4.4.1 (Two-scale convergence).
A sequence of functions uε in L2 (Ω) two-scale converges to a limit u0 (x, y) ∈ L2 (Ω×Y ),
2−s
denoted by uε * u0 , if
Z
x
dx =
lim uε (x)ϕ x,
ε→0 Ω
ε
Z Z
u0 (x, y)ϕ(x, y) dy dx
Ω
(4.29)
Y
for every test function ϕ(x, y) ∈ L2 (Ω; Cper (Y ). If, in addition,
Z
Z Z
|uε (x)| dx =
lim
ε→0
2
Ω
Ω
|u0 (x, y)|2 dy dx
Y
2−s
holds, then uε two-scale converges strongly to u0 (x, y) ∈ L2 (Ω×Y ), denoted by uε → u0 .
At this point it should be mentioned that an appropriate choice of test functions is
important in the concept of two-scale convergence and its applications. A detailed
discussion on this subject can be found e.g. in [6], [35] or [65].
In many cases it is necessary to extend the class of test functions. For our purposes,
4
Homogenization
34
especially with regard to the homogenization, it will be sufficient to make use of the
following type of test functions.
Lemma 4.4.1.
Let uε be a sequence in L2 (Ω) that two-scale converges to u0 . Then the definition of
two-scale convergence (4.29) holds true for all test functions of the form ϕ(x, y) =
ϕ1 (y)ϕ2 (x, y) with ϕ1 (y) ∈ L∞ (Y ) and ϕ2 (x, y) ∈ L2 (Ω; Cper (Y )).
Theorem 4.4.1.
The two-scale limit is unique.
Proof. See Section 4.1, Theorem 49 in [16] or Section 3 in [35].
Next, we want to recall one of the most important results in the theory of two-scale
convergence, the so-called compactness theorem.
Theorem 4.4.2 (Two-scale compactness).
For each bounded sequence uε in L2 (Ω) there exists a subsequence, still denoted by uε ,
and a function u0 (x, y) ∈ L2 (Ω × Y ), such that this subsequence two-scale converges to
u0 .
In order to prove this compactness theorem, we will need the following lemma.
Lemma 4.4.2.
The space L2 (Ω; Cper (Y )) has the following properties:
1. L2 (Ω; Cper (Y )) is a separable Banach space.
2. L2 (Ω; Cper (Y )) is dense in L2 (Ω × Y ).
3. Let u ∈ L2 (Ω; Cper (Y )), then u(x, xε ) ∈ L2 (Ω) and
x ≤ ku(x, y)kL2 (Ω;Cper (Y )) .
u x,
ε L2 (Ω)
4. Let u ∈ L2 (Ω; Cper (Y )), then
x → ku(x, y)kL2 (Ω×Y ) .
u x,
ε L2 (Ω)
5. For every u ∈ L2 (Ω; Cper (Y )), it holds that
x
u x,
*
ε
Z
u(x, y) dy
Y
weakly in L2 (Ω).
4
Homogenization
35
A proof of this result is found in e.g. Section 5 in [6] or Section 2 in [35].
Proof of Theorem 4.4.2. Here we give an outline of the proof and refer to [6], [35].
The boundedness of uε , Hölder inequality and Lemma 4.4.2 yield
Z
x uε (x)ϕ x, x dx ≤ kuε k 2 ≤ C kϕ(x, y)kL2 (Ω;Cper (Y )) . (4.30)
L (Ω) ϕ x,
ε
ε L2 (Ω)
Ω
This implies that the left hand side of (4.30) is a continuous linear form on L2 (Ω; Cper (Y )),
which can be identified with an element Uε in the dual space L2 (Ω; Cper (Y ))0 of L2 (Ω; Cper (Y ))
through the formula
Z
x
hUε , ϕi =
uε (x)ϕ x,
dx.
ε
Ω
In view of the inequalities (4.30), Uε is bounded and since L2 (Ω; Cper (Y )) is separable,
we can extract a weak-∗ convergent subsequence, still denoted by Uε , such that
∗
hUε , ϕi * hU0 , ϕi
(4.31)
for any ϕ ∈ L2 (Ω; Cper (Y )) and some U0 ∈ L2 (Ω; Cper (Y ))0 . Using this relation and
Lemma 4.4.2, we deduce that
Z
x x dx ≤ C lim ϕ x,
≤ C kϕ(x, y)kL2 (Ω×Y ) .
|hU0 , ϕi| = lim uε (x)ϕ x,
ε→0
ε→0
ε
ε L2 (Ω)
Ω
(4.32)
2
Thus, U0 can be extended to become a bounded linear form on L (Ω × Y ) by density of
L2 (Ω; Cper (Y )) in L2 (Ω × Y ). Finally, according to the Riesz representation theorem,
there exist an element u0 ∈ L2 (Ω × Y ) such that
Z Z
hU0 , ϕi =
Ω
u0 (x, y)ϕ(x, y) dy dx ∀ϕ ∈ L2 (Ω × Y ).
(4.33)
Y
Combining (4.31) and (4.33) yields the desired result.
In the next section we will show how two-scale convergence can be used to study
periodic homogenization problems. Moreover, it provides a mathematical rigorous
justification of the formal two-scale asymptotic expansion presented in Section 4.2.
Thus, we have the following useful lemma.
Lemma 4.4.3.
Consider a function uε (x) ∈ L2 (Ω) which admits the following two-scale asymptotic
4
Homogenization
36
expansion
x
x
+ εu1 x,
+ ...
uε (x) = u0 x,
ε
ε
2−s
where uj (x, y) ∈ L2 (Ω; Cper (Y )), j = 0, 1, . . . , d. Then uε * u0 .
Proof. See Section 5.2, Lemma 5.7 in [43].
Hence, we obtain that the first term u0 (x, y) in the asymptotic expansion (4.9) is the
two-scale limit of the sequence uε . Next, we want to show that two-scale convergence is
also related to more classical types of convergence such as strong and weak convergence.
Theorem 4.4.3 (Strong and two-scale convergence).
If a sequence of functions uε converges strongly to u0 in L2 (Ω), then it two-scale converges to the same limit u0 .
Proof. See Section 3, Theorem 9 in [35] or Theorem 1.16 in [38].
Theorem 4.4.4 (Weak and two-scale convergence).
Let uε be a sequence in L2 (Ω) that two-scale converges to u0 (x, y) ∈ L2 (Ω × Y ). Then
Z
u0 (x, y) dy
uε * u(x) :=
weakly in L2 (Ω).
Y
Furthermore, we have
lim inf kuε kL2 (Ω) ≥ ku0 |kL2 (Ω×Y ) ≥ kukL2 (Ω) .
ε→0
(4.34)
Proof. See Section 1, Proposition 1.6 in [6] or Section 3, Theorem 10 in [35].
Hence, a sequence uε in L2 (Ω) that two-scale converges to a limit u0 ∈ L2 (Ω), also
converges weakly to its integral mean value over Y in L2 (Ω) [56]. Moreover, if the
two-scale limit does not depend on the microscopic variable y, then the two-scale limit
and the weak limit coincide. In particular, a sequence uε that two-scale converges in
L2 (Ω), is also bounded in L2 (Ω). This is a consequence of the fact that every weakly
convergent sequence is bounded.
Furthermore, from the second part of Theorem 4.4.4 one can see that there is more
information in the two-scale limit u0 than in the weak limit u about the oscillations
of the sequence uε . However, the two-scale limit captures only the oscillations of the
sequence uε that have the same frequency as the test functions [6].
All in all, Theorem 4.4.3 and Theorem 4.4.4 state that strong convergence implies
4
Homogenization
37
two-scale convergence and two-scale convergence in turn implies weak convergence. In
other words, two-scale convergence is weaker than strong convergence, but stronger
than weak convergence. The converse is not true [44]. For for more details we refer to
[38].
Up to now, we have only discussed bounded sequences in L2 (Ω). The next theorem
gives further information on the behavior of gradients of sequences uε that are bounded
in H 1 (Ω). It is important for the two-scale convergence, in particular for its applications
to homogenization, as we will see in the section 4.5.
Theorem 4.4.5.
1. Let uε be a bounded sequence in H 1 (Ω), which converges weakly to a limit u ∈ H 1 (Ω).
Then, there exist a function u1 (x, y) ∈ L2 (Ω; H) 1 such that, up to a subsequence, uε
two-scale converges to u(x) and ∇uε two-scale converges to ∇x u(x) + ∇y u1 (x, y).
2. Let uε and ε∇uε be bounded sequence in L2 (Ω). Then, there exist a function
u0 (x, y) ∈ L2 (Ω; H) such that, up to a subsequence, uε two-scale converges to u0 (x, y)
and ε∇uε two-scale converges to ∇y u0 (x, y).
Proof. See Theorem 1.30 in [38] or [6].
Let us conclude this section with a few remarks on the above Theorem, found in [44].
Remark 4.4.1. 1. Theorem 4.4.5 remains true if we replace H 1 (Ω) by the subspace
H01 (Ω), which consists of functions in H 1 (Ω) that vanish on the boundary of the domain
∂Ω. In the next section we will study problems, where the solutions are assumed to
vanish on the boundary, thus it will be convenient at this point to consider H01 (Ω) as
the relevant function space rather than H 1 (Ω).
2. Certainly, in virtue of the gradient characterization, the function u1 (x, y) above may
1
more generally be assumed to belong to L2 (Ω; Hper
(Y )), i.e. that its mean value over
Y is not necessary zero.
3. Notice that the boundedness of uε in H 1 (Ω) ensures, up to a subsequence, that uε
converges strongly to u in L2 (Ω) by the Rellich compactness theorem ??.
1
R
1
H = u | u ∈ Hper
(Y )| Y u dy = 0 , see (3.4)
4
Homogenization
38
4.5 Homogenization of Elliptic Partial Differential
Equations
In this section we are going to demonstrate how two-scale convergence can be used for
the homogenization of linear second-order uniformly elliptic partial differential equations with periodic coefficients and Dirichlet boundary conditions. Our goal will be to
prove the following homogenization theorem.
Theorem 4.5.1.
Let uε be the weak solution of
−∇ · (Aε ∇uε ) = f
on Ω
(4.35)
uε = 0 on ∂Ω
with f = f (x) ∈ L2 (Ω) ⊂ H −1 (Ω), Ω ⊂ Rn bounded and Aε (x) = A( xε ), A(y) ∈
Mper (α, β, Y ). Further, let u0 be the weak solution of the homogenized equation
−∇ · A ∇u0 = f
on Ω
(4.36)
u0 = 0 on ∂Ω
with the homogenized coefficients A given by
Z
A=
A(y) + A(y)∇y χ(y)T dy,
(4.37)
Y
where the vector field χ(y) is the weak solution of the cell problem
−∇y · (∇y χ(y) AT (y)) = ∇y · AT (y) in Y
(4.38)
χ(y) is Y -periodic.
(4.39)
Then
uε * u0
weakly in H01 (Ω)
and
uε → u0
strongly in L2 (Ω).
The proof of this theorem is divided into several steps. First, we are going to show
that uε as well as ∇uε have two-scale convergent subsequences, defined by a pair of
functions {u0 , u1 }. In a second step we use a test function of the form
x
vε (x) = v0 (x) + εv1 x,
ε
4
Homogenization
39
in order to pass to the two-scale limit. Thereby, we receive a coupled system of equations for the functions {u0 , u1 }; the so-called two-scale system, see Lemma 4.5.1. This
is an alternative formulation of the limit problem consisting of the usual homogenized
and cell equation [6]. The third step consists in proving the existence and uniqueness
of this system by application of the Lax-Milgram lemma, see Lemma 4.5.2. The last
step is to decouple this system of equations in order to obtain the homogenized and
the cell equation, see Lemma 4.5.3.
Lemma 4.5.1.
Let uε (x) be the solution of (4.35) with the assumptions of Theorem 4.5.1. Then, there
exist functions u0 (x) ∈ H01 (Ω), u1 (x, y) ∈ L2 (Ω; H) such that, up to a subsequence, uε
two-scale converges to u0 (x) and ∇uε two-scale converges to ∇x u0 (x) + ∇y u1 (x, y).
Moreover, {u0 , u1 } solve the following two-scale system
− ∇y · A(y)(∇x u0 (x) + ∇y u1 (x, y)) = 0
Z
− ∇x ·
A(y)(∇x u0 (x) + ∇y u1 (x, y)) dy = f
in Ω × Y
(4.40a)
in Ω
(4.40b)
Y
u0 = 0
on ∂Ω
(4.40c)
u1 (x, y)
is Y-periodic in y
(4.40d)
Proof. As we have seen in section 3.3, our problem (4.35) has a unique solution uε ∈
H01 (Ω), where uε is uniformly bounded in H01 (Ω), i.e.
kuε kH 1 (Ω) ≤ C.
0
The uniform boundedness of uε implies, since H01 (Ω) is reflexive, that, up to a subsequence, uε converges weakly to a limit u0 in H01 (Ω). Furthermore, due to fact that
H01 (Ω) is compactly embedded in L2 (Ω), we deduce by the Rellich compactness theorem ?? that, up to a subsequence, uε converges strongly to u0 in L2 (Ω). According
to Theorem 4.4.5, there exist u0 ∈ H01 (Ω) and u1 ∈ L2 (Ω; H) such that, up to a subsequence, uε two-scale converges to u0 and ∇uε two-scale converges to ∇x u0 + ∇y u1 .
In order to show the second part of the lemma, we introduce the corresponding weak
formulation of (4.35), which reads
Z
A
Ω
x
ε
Z
∇uε (x) · ∇vε (x) dx =
f (x)vε (x) dx ∀vε ∈ H01 (Ω).
(4.41)
Ω
Now, we assume that the solution of (4.35) takes the form uε ≈ u0 (x) + εu1 (x, xε ) + . . ..
This suggest to multiply (4.41) by a test function of the form vε (x) = v0 (x) + εv1 (x, xε ),
4
Homogenization
40
∞
where v0 ∈ C0∞ (Ω) and v1 ∈ C0∞ (Ω; Cper
(Y )). This is possible since vε ∈ H01 (Ω) by
construction. Then, equation (4.41) becomes
Z
x
x x,
dx + ε
ε
Z
∇uε (x) · ∇v0 (x) + ∇y v1
ε
Ω
Z
Z
x
dx.
= f (x)v0 (x) dx + ε f (x)v1 x,
ε
Ω
Ω
A
x
x ∇uε (x) · ∇x v1 x,
dx
ε
ε
A
Ω
which can be rewritten as follows
Z
Z
x x x x T
∇uε (x) · A
∇v0 (x) + ∇y v1 x,
dx + ε ∇uε (x) · AT
∇x v1 x,
dx
ε
ε
ε
ε
Ω
Ω
Z
Z
(4.42)
x
= f (x)v0 (x) dx + ε f (x)v1 x,
dx.
ε
Ω
Ω
Next, we want to pass to the two-scale limit as ε → 0. For the first term on the left hand
side of (4.42), this is possible, according to Lemma 4.4.1, since AT xε ∈ L∞ (Y ) and
∇v0 (x)+∇y v1 (x, xε ) ∈ L2 (Ω, Cper (Y )). Thus, the function AT xε ∇v0 (x) + ∇y v1 x, xε
can be used as a test function, and we obtain
Z
∇uε (x) · AT
x
Ω
ε
x ∇v0 (x) + ∇y v1 x,
dx
ε
Z Z
A(y)T ∇u0 (x) + ∇y u1 (x, y) · ∇v0 (x) + ∇y v1 (x, y) dy dx
→
Ω
Y
Since the function AT xε ∇x v1 (x, y) is also an admissible test function, we can also
pass to the two-scale limit in the second term of the left hand side of (4.42), which
yields
Z
x x T
ε ∇uε (x) · A
∇x v1 x,
dx → 0.
ε
ε
Ω
For the term on the right hand side of (4.42), we have
Z
Z
f (x)v0 (x) dx + ε
Ω
f (x)v1
Ω
x
x,
dx →
ε
Z
f (x)v0 (x) dx.
Ω
since v0 (·) + εv1 (·, ε· ) * v0 weakly in H01 (Ω). Combining these results, we obtain the
limiting equation
Z Z
Ω
A(y)(∇u0 (x) + ∇y u1 (x, y) · ∇v0 (x) + ∇y v1 (x, y)) dy dx =
Y
Z
f (x)v0 (x) dx
Ω
(v0 , v1 ) ∈ H01 (Ω)×L2 (Ω; H)
which holds for all
in H01 (Ω)×L2 (Ω; H).
since
∞
C0∞ (Ω)×C0∞ (Ω; Cper
(Y
(4.43)
)) is dense
4
Homogenization
41
We note that equation (4.43) is the weak formulation of the two-scale system (4.40).
First choose v0 = 0 and we obtain the weak formulation of equation (4.40a)
Z Z
A(y) (∇x u0 (x) + ∇y u1 (x, y)) · ∇y v1 (x, y) dy dx = 0.
Ω
Y
Choosing then v1 = 0 yields
Z Z
Z
A(y) (∇x u0 (x) + ∇y u1 (x, y)) · ∇x v0 (x) dy dx =
Ω
Y
f (x)v0 (x) dx
Y
which is the weak formulation of equation (4.40b). The boundary conditions (4.40c)
and (4.40d) result from the fact that u0 ∈ H01 (Ω) and u1 ∈ L2 (Ω; H).
Thus far, we have proved the result for a subsequence. If we show that the two-scale
system has a unique solution {u0 , u1 }, then the above result holds true for the entire
sequence.
Lemma 4.5.2.
Under the assumptions of Theorem 4.5.1 the two-scale system (4.40) has a unique
solution {u0 , u1 } ∈ H01 (Ω) × L2 (Ω; H).
Proof. The proof follows by an application of the Lax-Milgram lemma in the space
H01 (Ω) × L2 (Ω; H). To ease the notation, we will denote this space by X. This is a
Hilbert space with inner product
(ũ, ṽ)X = (∇u0 , ∇v0 )L2 (Ω) + (∇y u1 , ∇y v1 )L2 (Ω×Y )
and norm
kũk2X = k∇u0 k2L2 (Ω) + k∇y u1 k2L2 (Ω×Y )
where ũ = (u0 , u1 ) and ve = (v0 , v1 ). Furthermore, let us introduce the bilinear form
Z Z
B(ũ, ṽ) =
Ω
Y
A(y)(∇u0 (x) + ∇y u1 (x, y) · ∇v0 (x) + ∇y v1 (x, y)) dy dx.
4
Homogenization
42
We first prove that B is coercive. From the positive definiteness of A it follows
Z Z
A(y)(∇u0 (x) + ∇y u1 (x, y)) · (∇u0 (x) + ∇y u1 (x, y))dy dx
B(ũ, ũ) =
ΩZ YZ
|∇u0 (x) + ∇y u1 (x, y)|2 dy dx
ZΩ ZY
Z Z
Z Z
2
=α
|∇u0 | dy dx + 2
∇u0 (x) · ∇y u1 (x, y)dy dx +
|∇y u1 |2 dy dx
Ω Y
Ω Y
| Ω Y
{z
}
≥α
=0 by divergence theorem and periodicity
= α kũk2X
Next, we prove that B is bounded. The Cauchy Schwartz inequality and the L∞ bound
on A(y) implies that
Z Z
|B(ũ, ṽ)| = A(y)(∇u0 (x) + ∇y u1 (x, y)) · (∇v0 (x) + ∇y v1 (x, y))dy dx
ZΩ ZY
(4.44)
|∇u0 (x) + ∇y u1 (x, y)| |∇v0 (x) + ∇y v1 (x, y)| dy dx
≤β
Ω
Y
≤ β kũkX kṽkX .
Thus, the assumptions of the Lax-Milgram lemma are satisfied so that there exists
a unique solution of the two-scale system in X. Consequently, the whole sequence
converges.
Finally, we want to show the relation between the homogenized equation and the twoscale system.
Lemma 4.5.3.
Consider the unique solution {u0 , u1 } ∈ X of the two-scale system (4.40). Then, u0 is
the unique solution of the homogenized equation (4.36), and u1 (x, y) is of the form
u1 (x, y) = χ(y) · ∇u0 (x)
(4.45)
where χ(y) is the solution of the cell problem (4.38).
Proof. Inserting (4.45) into equation (4.40a) yields
−∇y · (∇y χ(y)AT (y)) · ∇x u0 = (∇y · AT (y)) · ∇x u0 .
This equation is satisfied given that χ ∈ H is the unique solution of the cell problem
4
Homogenization
43
(4.38). Inserting (4.45) into equation (4.40b) yields
Z
A(y)(∇x u0 + (∇y χT (y))∇x u0 ) dy = f
−∇x ·
Y
respectively
Z
−∇x ·
A(y)(I + ∇y χT (y)) dy ∇x u0 ) = f
Y
and thus
−∇x · A ∇x u0 = f
which is the homogenized equation with the homogenized coefficients given by (4.37).
At this point, it should be mentioned that the decoupling of the two-scale system is not
always possible, and can lead to very complicated forms of the homogenized equation
[6]. In order to decouple the homogenized equation from that system, we need an H 1 estimate on the solution uε , i.e. a good control regarding the oscillations of the solution
[43]. In this case, it follows that the two-scale limit is independent of the microscopic
scale and thus, a homogenized equation actually exists.
Finally, we want to shows that we can obtain improved approximations by adding extra
terms, also called correctors, in the multiscale expansion. The theorem below states
that we can get strong convergence in H 1 (Ω) if we take the first-order corrector field
into account. Furthermore, the theorem rigorously validates the two first terms in the
two-scale asymptotic expansion (4.9).
Theorem 4.5.2.
Let uε (x) and u0 (x) be defined as in Theorem 4.5.1 and let u1 (x, y) be given by (4.45).
Assume that f ∈ L2 (Ω) and that the coefficient matrix A is sufficiently smooth such
1
that χ ∈ Cper
(Y ). Then
x lim uε (x) − u0 (x) + εu1 x,
1 = 0.
ε→0
ε
H (Ω)
Proof. Since we already know that uε converges strongly to u0 in L2 (Ω), it is enough
to prove that
x lim ∇uε (x) − ∇ u0 (x) + εu1 x,
2 n=0
ε→0
ε
L (Ω)
4
Homogenization
44
or, equivalently
x x
+ ∇y u1 x,
lim ∇uε (x) − ∇u0 (x) + ε∇x u1 x,
2 n = 0.
ε→0
ε
ε
L (Ω)
1
(Y ), we have
Since f ∈ L2 (Ω) it follows that u0 (x) ∈ H 2 (Ω) ∩ H01 (Ω), and since χ ∈ Cper
1
(Y )). From Theorem 4.4.2 we deduce that
that u1 (x, y) = χ(y) · ∇u0 (x) ∈ H 1 (Ω; Cper
x ∇
u
≤C
x 1 x,
ε L2 (Ω)n
and thus
x → 0.
ε∇x u1 x,
ε L2 (Ω)n
Finally, we prove the following
x = 0.
lim ∇uε (x) − ∇u0 (x) − ∇y u1 x,
ε→0
ε L2 (Ω)n
From the uniform ellipticity of A it follows that
x 2
α ∇uε (x) − ∇u0 (x) − ∇y u1 x,
ε L2 (Ω)n
Z x 2
= α ∇uε (x) − ∇u0 (x) − ∇y u1 x,
dx
ε
Z Ω x x x
A
∇uε (x)−∇u0 (x)−∇y u1 x,
· ∇uε (x)−∇u0 (x)−∇y u1 x,
dx
≤
ε
ε
ε
ZΩ x
=
A
∇uε (x) · ∇uε (x) dx
(4.46)
ε
Ω
Z
x x − ∇uε (x) · (A + AT )
∇u0 (x) + ∇y u1 x,
dx
ε
ε
Ω
Z
x x x ∇u0 (x) + ∇y u1 x,
· ∇u0 (x) + ∇y u1 x,
dx.
+ A
ε
ε
ε
Ω
From Theorem 4.5.1 we know that uε converges strongly to u0 in L2 (Ω). Since strong
convergence implies weak convergence, and in view of (4.41), we obtain for the first
term of the right hand side of (4.46)
Z
A
Ω
x
ε
Z
∇uε (x) · ∇uε (x) dx =
Z
f (x)uε (x) dx →
Ω
f (x)u0 (x) dx as ε → 0.
Ω
From Lemma 4.4.1 it follows that (A+AT )( xε )(∇u0 (x)+∇y u1 (x, xε )) and A( xε )(∇u0 (x)+
4
Homogenization
45
n
2
1
∇y u1 (x, xε )) are suitable test functions, since A ∈ L∞
per (Y ) , u0 ∈ H (Ω) ∩ H0 (Ω) and
1
χ ∈ Cper
(Y ). Hence, we can pass to the two-scale limit in the last two terms on the
right hand side of (4.46). For the second term, we obtain
Z
x x ∇u0 (x) + ∇y u1 x,
dx
∇uε (x) · (A + AT )
ε
ε
Ω
Z Z
→−
(∇u0 (x) + ∇y u1 (x, y)) · (A + AT )(y)(∇u0 (x) + ∇y u1 (x, y)) dy dx
Ω
Z YZ
=−2
A(y)(∇u0 (x) + ∇y u1 (x, y)) · (∇u0 (x) + ∇y u1 (x, y)) dy dx.
−
Ω
Y
and for the third term, we have
Z
x x x A
∇u0 (x) + ∇y u1 x,
· ∇u0 (x) + ∇y u1 x,
dx
ε
ε
ε
Ω
Z Z
→
A(y)(∇u0 (x) + ∇y u1 (x, y)) · (∇u0 (x) + ∇y u1 (x, y)) dy dx.
Ω
Y
Finally, by combining the above relations, we get that
x 2
lim α ∇uε (x) − ∇u0 (x) − ∇y u1 x,
ε→0
ε L2 (Ω)n
Z x 2
= lim α ∇uε (x) − ∇u0 (x) − ∇y u1 x,
dx
ε→0
ε
Ω
Z
Z Z
≤ f (x)u0 (x) dx−
A(y)(∇u0 (x) + ∇y u1 (x, y))·(∇u0 (x) + ∇y u1 (x, y)) dy dx = 0
Ω
Ω Y
|
{z
}
Z
f (x)u0 (x) dx
=
Ω
Consequently,
x lim ∇uε (x) − ∇u0 (x) − ∇u1 x,
=0
ε→0
ε L2 (Ω)n
46
5 Numerical Realization
In the present chapter we want to apply the theory of homogenization, introduced in
Chapter 4, in order to calculate the macroscopic homogenized electrical conductivity
values within the heart. First, we want to briefly describe the data used for our
calculations, and the idea behind. This will be followed by a short description of
the homogenization steps. Since our calculations are based on MATLAB’s Partial
Differential Equation Toolbox, we subsequently want to give a short overview of this
Toolbox. Finally, we are going to briefly explain our implementation steps.
5.1 Data
The data we are going to use for our calculations are high resolution 2D pneumographic
micro-CT data of a pig heart, provided by Lunkenheimer et al. [4].
A slice of the given CT data set is shown in Figure 5.1.
Figure 5.1: Slice of the Micro-CT data
5
Numerical Realization
47
The CT scans were taken during coronary pneumatic distension. Pneumography is
related to the fact that the heart muscle primarily consists of water, and that we
can contrast it with air [9]. In order to do this, the coronary arteries of excised pig
hearts were catheterized, and subsequently inflated by compressed air at a pressure of
1.5 atmospheres. Then, micro-computed tomography (micro-CT) was performed from
cardiac base to apex [36].
As indicated in Figure 5.1, this technique enables a high contrast differentiation of the
heart’s microstructure with a clearly visible fiber structure.
For our calculations, we will focus only on a part of this CT slice, where the fiber
structure is clearly visible. The Region of Interest is depicted in Figure 5.2 (b), and
has a size of 376 × 376 pixels.
(a) Slice
(b) ROI
Figure 5.2: CT slice and the Region of Interest (ROI)
The idea behind is that we are going to assume that the intensity, the brightness, of
the fibers, given by the grayscale values, is correlated with the electrical conductivity.
More precisely, we are going to assume that the intensity value of each pixel represents
a microscopic electrical conductivity value.
Since our image is of class double, these intensity values range from 0 to 1, where
a value of 0 indicates black (lowest intensity), a value of 1 indicates white (highest
intensity), and values in between represents shades of gray.
In other words, in our case, a value of 0 means no electrical conductivity and a value
of 1 means high electrical conductivity.
Our aim will be to calculate the macroscopic conductivity values on the base of these
microscopic conductivity values using homogenization. Thus, these microscopic values
will be used as input for our implementation.
5
Numerical Realization
48
5.2 Computation of the Macroscopic Conductivity
Values
In this section we are going to briefly describe how the homogenization theory can be
applied to calculate the homogenized macroscopic conductivity values.
In this work, we restrict our attention to the two-dimensional case.
In order to calculate the macroscopic conductivity values using the theory of homogenization, we have to perform the following two steps.
In a first step, we have to solve the so-called cell problem in the unit cell Y = (0, 1)2
introduced in Subsection 4.2.1.
The cell problem is given by: Find u1 and u2 , solutions of
−∇y · (A(y)∇y ui (y)) = ∇y · (A(y) ei ) in Y
(5.1)
ui (y) is Y -periodic
Z
ui (y) dy = 0
i = 1, 2 .
(5.2)
Y
The coefficient matrix A(y) in the cell problem (5.1) represents the microscopic conductivity values, i.e. the intensity values of the micro-CT data.
As already mentioned in Section 3.4, the solutions of the cell problem can be determined only up to a constant. Thus, in order to ensure uniqueness we have to fix this
constant. This will be achieved by the condition (5.2).
Finally, we have to impose boundary conditions. Usually, we have to deal with periodic
boundary conditions. But periodic boundary conditions are quite difficult to realize in
practice. For that reason, we are going to consider two different sets of boundary conditions: homogeneous Dirichlet boundary conditions ui = 0 on ∂Y , and homogeneous
Neumann boundary conditions n·(A ∇ui ) = 0 on ∂Y .
Subsequently, we are going to compare the results.
The second and last step is to insert the solution of the cell problem (5.1) into the
equation of the homogenized coefficients
Z
A(y)(ei + ∇y ui (y)) · ej dy ,
Aij =
Y
i = 1, 2 ,
(5.3)
5
Numerical Realization
49
which represents the macroscopic homogenized conductivity values.
Thus, for the two-dimensional case, we obtain the following matrix
 Z ∂u1
A(y) + A(y)

∂y1
Z Y
Aij = 

∂u1
A(y)
dy
∂y2
Y
Z ∂u2
dy
A(y)
dy
∂y1
Z Y
∂u2
A(y) + A(y)
dy
∂y2
Y


.

5
Numerical Realization
50
5.3 MATLAB’s Partial Differential Equation Toolbox
Since we are going to use MATLAB’s Partial Differential Equation Toolbox for our
calculations, we want to give a short overview of the Toolbox.
MATLAB’s Partial Differential Equation (PDE) Toolbox is a tool for studying and
solving partial differential equations in two dimensions and time.
It can be started by typing pdetool at the MATLAB command line.
The PDE Toolbox provides a set of command line functions as well as an intuitive
graphical user interface (GUI) for preprocessing, solving, and postprocessing different
types of of partial differential equations, including elliptic, hyperbolic, parabolic, eigenvalue, nonlinear elliptic, and systems of partial differential equations.
The PDE Toolbox allows to
• define the 2D geometry of the problem
• define the boundary conditions
• define the coefficients of the partial differential equation
• create a triangular mesh with the option to refine the mesh
• solve the problem
• visualize the solution.
To create the geometry of the problem, the Toolbox provides a set of solid objects
(rectangle, circle, ellipse, and polygon), which can be combined in a flexible way.
The boundary conditions that can be used are of Dirichlet, generalized Neumann, and
mixed type. The PDE Toolbox automatically generates an unstructured triangular
mesh using the Delaunay triangulation algorithm, but the user has also the possibility
to refine the mesh. The partial differential equations are numerically solved by means
of the Finite Element Method (FEM).
The process of modeling and solving a particular problem can be done in several ways.
The problem can be solved completely within the graphical user interface, or, one uses
the command line functions. A combination of both is also possible, i.e. to perform
only a few of the above steps in the graphical user interface, which can be then made
available in the MATLAB workspace through the data export facilities of the GUI.
In this way it is possible to define your own application-specific problems.
For more information about MATLAB’s PDE Toolbox, we refer to [5].
5
Numerical Realization
51
5.4 Numerical Implementation
In this section we are going to briefly describe the main steps of our implementation
using MATLAB’s Partial Differential Equation Toolbox.
The first step of our implementation consists in defining the domain, the geometry,
of the problem. In our case, the geometry of the problem is the unit cell Y = (0, 1)2 ,
depicted in Figure 5.3.
Subsequently, we have to specify the boundary conditions. This can be done either by
a boundary file, or by a boundary matrix. We use the latter. To this end we draw the
geometry and specify the boundary conditions within the graphical user interface, and
subsequently export the boundary conditions as boundary matrix.
Figure 5.3: Unit cell
The next step is to define the partial differential equation we want to solve.
The PDE Toolbox cannot solve the cell problem (5.1) directly, it can only solve elliptic
partial differential equations of the following form
− ∇ · (c∇u) + au = f.
(5.4)
Thus, we have to specify the coefficients a, c, and the right hand side f for our purposes.
In our case, the coefficient c in equation (5.4) respresents the microscopic conductivity
values, i.e. the intensity values of the micro-CT data.
In order to use these values as input, we have to interpolate these values from node
data to triangle midpoint data. The same applies to the right hand side f .
5
Numerical Realization
52
The coefficient a is usually equal to zero in our setting. That also holds for the case of
Dirichlet boundary conditions. However, in the case of Neumann boundary conditions,
there arises a problem. For a = 0 we have a non-empty null space consisting of constant
functions. Thus, in order to ensure the uniqueness of the solution we have to fix this
constant. In order to do so, we set a = 10−6 , which also holds in the Dirichlet case.
The next step consists in generating the triangular mesh. This will be automatically
done by the PDE Toolbox. Figure 5.4 shows an initial mesh.
Figure 5.4: Initial mesh consiting of 175 nodes and 308 triangles
The PDE Toolbox also offers the possibility to refine the triangular mesh.
Figure 5.5 depicts the initial mesh after one refinement.
The number of triangles within a mesh influences the accuracy of the solution. Thus,
a mesh with more and smaller triangles will increase the accuracy of the results but
at the same time it will also increase the computation time. Thus, one has to find the
right balance between the accuracy and the computation time.
As we will see in the next Chapter, in our case, two mesh refinements are the ”right
choice”. Finally, we can solve the cell problem (5.1).
Subsequently, with the solutions u1 and u2 of the cell problem, we are able to compute
the macroscopic conductivity values given by the formula
Z
A(y)(ei + ∇y ui (y)) · ej dy ,
Aij =
Y
i = 1, 2 ,
(5.5)
5
Numerical Realization
53
where ei and ej denote the i-th and j-th unit vector, respectively.
This integral can then be approximated as the sum of the areas of the triangles, i.e.
Z
Fij dy ≈
Aij =
Y
X
|T | Fij ,
ij
where Fij = A(y)(ei + ∇y ui (y)) · ej and |T | denotes the area of the triangles.
Figure 5.5: Mesh after 1 refinement consiting of 657 nodes and 1232 triangles
54
6 Computational Results
The aim of this Chapter is to present some of our computational results. We are going
to present the computed macroscopic conductivity values for different image sections;
once for Dirichlet boundary conditions, and once for Neumann boundary conditions.
Subsequently, the results will be discussed.
Our computations are based on 6 different sections of the CT image depicted in Figure
5.2 (b). Four of the six images have a size of 50 × 50 pixels, and the other two images
have a size of 100 × 100 pixels. Each of these images is illustrated as a grayscale image
as well as a colored image.
Some of these images show a clearly visible fiber structure of the heart, i.e. a preferential
alignment of the fibers. Thus, we would expect that the electrical conduction along
this fiber direction is greater than across it, which means that the electrical conduction
is anisotropic.
In contrast to this, the other images do not show such a preferential alignment of the
fibers. In this case, we would expect that the electrical conductivity is almost identical
in all directions, which means that the electrical conduction is isotropic.
Befor we take a closer look at the images, and discuss the results, it is important to
mention that the coordinate axes of our images are reversed, i.e. in our case, x is the
vertical axis, and y is the horizontal axis.
Furthermore, it should be mentioned that we have used two mesh refinements for the
computation of the conductivity values, since more refinements do not show a real
improvment, and thus, only increase the computation time. An example can be found
in the Appendix.
We also note that the conductivity values were symmetrized, in order to eliminate
numerical errors.
With this in mind, let us sonsider the first image depicted in Figure 6.1.
As we can see, there exists a visible fiber structure with a preferential fiber direction
in x-direction. Thus, we would expect that the electrical conduction along the fiber
direction is greater than across it.
6
Computational Results
55
Figure 6.1: Image 1, 50×50 pixels
One look at the computed macroscopic conductivity values A supports that view:
Adir =
!
0.2050 −0.0095
−0.0095 0.1983
Aneu =
!
0.2234 −0.0012
−0.0012 0.2003
where Adir and Aneu are the macroscopic conductivity values for Dirichlet and for
Neumann boundary conditions, respectively. In both cases, the first diagonal element
is greater than the second one, which reflect the higher conduction in the x-direction.
In the Dirichlet case, the difference between the two elements is not as significant as
in the Neumann case.
Furthermore, it might be also helpful to consider the eigenvalues of A, since the largest
eigenvalue represents the main direction:
Edir =
!
0.1919
0
0
0.2117
Eneu =
!
0.2003
0
0
0.2335
As we can see, in both cases, the first eigenvector is much smaller than the second one,
which contradicts our oberservations. At this point, we have to keep in mind that the
coordinate axes of the images are reversed, as mentioned before.
Thus, in our case, the second eigenvalue will represent the main fiber direction in
x-direction, and not the first one as usual. Hence, also the eigenvalues indicate a
preferential direction of the electrical conduction in the x-direction, i.e. anisotropy.
The second image, depicted in Figure 6.2, also indicates a preferential fiber direction
along the x-axis.
6
Computational Results
56
Figure 6.2: Image 2, 50×50 pixels
In the Dirichlet case, the situation is similar to image 1. The macroscopic conductivity
values as well as the eigenvalues indicate a higher electrical conductivity in the xdirection as we would expect. But in the Neumann case, there are some contradictions.
Adir =
!
0.1927 0.0002
0.0002 0.1756
Edir =
!
0.1756
0
0
0.1927
Aneu =
!
0.1834 0.0139
0.0139 0.2247
Eneu =
!
0.1792
0
0
0.2289
Looking at the macroscopic conductivity values, then one would assume that the electrical conduction in the y-direction is greater than in the x-direction, contrary to our
expectations. However, if we look at the eigenvalues, one might expect the opposite,
i.e. that the electrical conduction is higher in the x-direction.
Image 3, see Figure 6.3, also shows a clearly visible fiber structure in x-direction.
Figure 6.3: Image 3, 100×100 pixels
6
Computational Results
Adir =
57
!
0.1957 −0.0077
−0.0077 0.1893
Edir =
!
0.1842
0
0
0.2008
Aneu =
!
0.1914 −0.0102
−0.0102 0.1894
Eneu =
!
0.1802
0
0
0.2007
In the Dirichlet as well as in the Neumann case, both the conductivity values, and
the eigenvalues, support that view. It is noticeable that the macroscopic conductivity
values as well as the eigenvalues in the Dirichlet and the Neumann case are quite
similar. Furthermore, the anisotropy can be seen more clearly in the eigenvalues, than
in the conductivity values.
In Image 4, see Figure 6.4, a preferential fiber direction is not clearly observable, in
contrast to the images before. One might expect that there is maybe a preferential
direction along the x-axis.
Figure 6.4: Image 4, 100×100 pixels
In the Dirichlet case, the conductivity values as well as the eigenvalues are quite similar
in both directions, indicating isotropy. However, in the Neumann case, the conductivity
Adir =
Edir =
!
0.2261 0.000
0.0000 0.2239
!
0.2239
0
0
0.2261
A4 =
!
0.2212 −0.0102
−0.0102 0.2092
Eneu =
!
0.2000
0
0
0.2205
values as well as the eigenvalues indicate a higher conductivity in the x-direction.
6
Computational Results
58
In Image 5, depicted in Figure 6.5, for us, there is almost nothig to recognize, i.e. there
is no real preferential direction oberservable.
Figure 6.5: Image 5, 50×50 pixels
In the Dirichlet case, the conductivty values as well as the eigenvalues indicate a higher
conductivity in the x-direction. In the Neumann case, however, it is totally different.
The conductivty values indicate a higher conductivity along the y-axis, whereas the
eigenvalues show a higher conduction in the x-direction.
Adir =
!
0.1992 −0.0067
−0.0067 0.1934
Edir =
!
0.1890
0
0
0.2037
Aneu =
!
0.2070 −0.0012
−0.0012 0.2256
Eneu =
!
0.2069
0
0
0.2257
In Image 6, depicted in Figure 6.6, there is also no real preferntial fiber direction
visible. In the case of Dirichlet boundary conditions as well as in the case of Neumann
boundary conditions, both the conductivity values and the eigenvalues show a higher
conductivty in the y direction.
A6 =
!
0.2247 −0.0003
−0.0003 0.2276
E6 =
!
0.2277
0
0
0.2247
A6 =
!
0.1935 −0.0039
−0.0039 0.2215
E6 =
!
0.2220
0
0
0.1930
6
Computational Results
59
Figure 6.6: Image 6, 50×50 pixels
(a) Dirichlet
(b) Neumann
Figure 6.7: Quiver plot of the CT image depicted in Figure 5.2 (b
60
7 Conclusion
In this thesis we have applied the theory of periodic homogenization in order to calculate the macroscopic electrical conductivity values within the heart.
In order to do so, we have first solved the so-called cell problem, and subsequently, we
have used the solutions of the cell problem to calculate the macroscopic homogenized
conductivity values.
Usually, the cell problem is solved by imposing periodic boundary condition. Since periodic boundary conditions are quite difficult to realize in practice, we have used instead
two different sets of boundary conditions, namely homogeneous Dirichlet boundary
conditions, and homogeneous Neumann boundary conditions.
The base for our calculations was a CT image of a pig heart. We have assumed that
the intensities of the cardiac fibers in the CT-image, given by the grayscale values, are
correlated with the microscopic conductivity values. More precisely, we have assumed
that the intensity value of each pixel in the image represents a microscopic conductivity
value. On this basis we have performed our calculations for different image sections
of this CT image. We have chosen images with a clearly visible fiber structure, and a
preferential direction as well as images, where no preferential direction of the fibers was
observable. Subsequently, we have compared the results for Dirichlet and Neumann
boundary conditions.
We have seen, especially in the case where the fiber structure, and the preferential direction of the fibers was visible, that our expectations and the computed conductivity
values coincide.
Furthermore, we have seen that the eigenvalues are an important factor for the interpretation of the results. In most of the cases they show a more clearly preferential
direction of the conduction as the macroscopic conductivity values. Hence, they should
be taken into account to validate the results.
We also found that in some cases the results of the Neumann problem differ substantially from the results of the Dirichlet problem. That underlines the importance of the
boundary conditions.
61
List of Figures
2.1
2.2
2.3
2.4
2.5
Basic Anatomy of the Heart . . . . . . . . .
Electrical Conduction System of the Heart .
Cardiac fiber structure . . . . . . . . . . . .
Fiber and sheet structure of the left ventricle
Cardiac fibre orientation . . . . . . . . . . .
.
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4
6
9
9
10
4.1
The material for successively smaller ε . . . . . . . . . . . . . . . . . .
23
5.1
5.2
5.3
5.4
5.5
Slice of the Micro-CT data
The Region of Interest . .
Unit cell . . . . . . . . . .
Initial mesh . . . . . . . .
Refined mesh . . . . . . .
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46
47
51
52
53
6.1
6.2
6.3
6.4
6.5
6.6
6.7
Image 1 . .
Image 2 . .
Image 3 . .
Image 4 . .
Image 5 . .
Image 6 . .
Quiver plot
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55
56
56
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59
59
8.1
Data Structures in the PDE Toolbox . . . . . . . . . . . . . . . . . . .
64
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62
List of Tables
2.1
Conduction velocities in different regions of the heart . . . . . . . . . .
7
8.1
Macroscopic effective conductivities . . . . . . . . . . . . . . . . . . . .
63
63
8 Appendix
Reference
σil
σit
σel
σet
σil /σel
Clerc[13]
0.17
0.019
0.62
0.24
0.27
Roberts et al. [45]
0.28
0.026
0.22
0.13
1.27
Roberts and Scher [46]
0.34
0.06
0.12
0.08
2.83
LeGuyader [34]
0.20
0.024
0.3
0.20
0.66
Table 8.1: Macroscopic effective conductivities (ventricular muscle) (S/m) [54]
The macroscopic conductivity values for Image 5, after 1, 2, and 3 mesh refinements.
A1 =
!
0.2028 −0.0047
−0.0043 0.1992
A2 =
!
0.1992 −0.0069
−0.0066 0.1934
A3 =
!
0.1975 −0.0079
−0.0075 0.1911
8
Appendix
64
8.1 Figures
The folowing Figure shows an overview of the data structures in the PDE Toolbox.
The3rectangles
are functions, and the ellipses are data represented either by matrices,
Solving PDEs
or by M-files. The arrows indicate data necessary for the functions.
Geometry
Description
matrix
decsg
Decomposed
Geometry
matrix
Geometry
M-file
initmesh
Boundary
Condition
matrix
Boundary
M-file
Mesh
data
refinemesh
Coefficient
matrix
assempde
Solution
data
pdeplot
3-110
Figure 8.1: Data Structures in the PDE Toolbox
Coefficient
M-file
8
Appendix
65
(a)
(b)
Figure 8.2: Image 1, solution u1, Neumann
(a)
(b)
Figure 8.3: Image 2, solution u1, Neumann
(a)
(b)
Figure 8.4: Image 3, solution u1, Neumann
8
Appendix
66
(a)
(b)
Figure 8.5: Image 4, solution u1, Neumann
(a)
(b)
Figure 8.6: Image 5, solution u1, Neumann
(a)
(b)
Figure 8.7: Image 6, solution u1, Neumann
8
Appendix
67
(a)
(b)
Figure 8.8: Image 1, solution u2, Neumann
(a)
(b)
Figure 8.9: Image 2, solution u2, Neumann
(a)
(b)
Figure 8.10: Image 3, solution u2, Neumann
8
Appendix
68
(a)
(b)
Figure 8.11: Image 4, solution u2, Neumann
(a)
(b)
Figure 8.12: Image 5, solution u2, Neumann
(a)
(b)
Figure 8.13: Image 6, solution u2, Neumann
8
Appendix
69
(a)
(b)
Figure 8.14: Image 1, solution u1, Dirichlet
(a)
(b)
Figure 8.15: Image 2, solution u1, Dirichlet
(a)
(b)
Figure 8.16: Image 3,, solution u1, Dirichlet
8
Appendix
70
(a)
(b)
Figure 8.17: Image 4, solution u1, Dirichlet
(a)
(b)
Figure 8.18: Image 5, solution u1, Dirichlet
(a)
(b)
Figure 8.19: Image 6, solution u1, Dirichlet
8
Appendix
71
(a)
(b)
Figure 8.20: Image 1, solution u2, Dirichlet
(a)
(b)
Figure 8.21: Image 2, solution u2, Dirichlet
(a)
(b)
Figure 8.22: Image 3,, solution u2, Dirichlet
8
Appendix
72
(a)
(b)
Figure 8.23: Image 4, solution u2, Dirichlet
(a)
(b)
Figure 8.24: Image 5, solution u2, Dirichlet
(a)
(b)
Figure 8.25: Image 6, solution u2, Dirichlet
73
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Eidesstattliche Erklärung
Hiermit versichere ich, Stefanie Kälz, dass ich die vorliegende Arbeit selbstständig
verfasst und keine anderen als die angegebenen Quellen und Hilfsmittel verwendet
habe. Gedanklich, inhaltlich oder wörtlich Übernommenes habe ich durch Angabe
von Herkunft und Text oder Anmerkung belegt bzw. kenntlich gemacht. Dies gilt in
gleicher Weise für Bilder, Tabellen, Zeichnungen und Skizzen, die nicht von mir selbst
erstellt wurden.
Alle auf der CD beigefügten Programme sind von mir selbst programmiert worden.
Münster, 14. November 2012
Stefanie Kälz