A Numerical Investigation of the Stability of Different Schemes for a
Simplified Fluid-Structure Interaction Scheme with Application to
Physiological Problems
Bachelor Thesis
by
Jan Helmig
Supervisors:
Prof. Marek Behr
Prof. Kent-Andre Mardal
Dipl.-Ing. Lutz Pauli
Aachen, September 2, 2013
CHAIR FOR
COMPUTATIONAL
ANALYSIS OF
TECHNICAL
SYSTEMS
Affidavit:
I hereby declare that I wrote this thesis on my own and without the use of any other than the cited
sources and tools and all explanations that I copied directly or in their sense are marked as such, as well
as that the thesis has not been available to any audit authority yet.
Eidesstattliche Erklärung:
Ich versichere, dass ich die Bachelorarbeit selbstständig und ohne Benutzung anderer als der angegebenen Quellen und Hilfsmittel angefertigt habe und alle Ausführungen, die wörtlich oder sinngemäß
übernommen wurden, als solche gekennzeichnet sind, sowie dass die Bachelorarbeit in gleicher oder
ähnlicher Form noch keiner anderen Prüfungsbehörde vorgelegt wurde.
Aachen, September 2, 2013
Jan Helmig
Contents
List of Figures
I
List of Tables
II
Glossary
III
Abbreviations
IV
1. Introduction
1
2. General Mathematical and Numerical Model
2.1. Governing Equations . . . . . . . . . . . .
2.1.1. Fluid . . . . . . . . . . . . . . . . .
2.1.2. Wall . . . . . . . . . . . . . . . . . .
2.2. Finite Element Method . . . . . . . . . . .
2.3. Coupling at the Variational Level . . . . .
2.3.1. Weak Form of the Fluid . . . . . .
2.3.2. Weak Form Wall . . . . . . . . . .
2.3.3. Fluid-Solid Coupling . . . . . . . .
3
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3. Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
3.1. FEniCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2. SUPG stabilized FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3. Incremental Pressure Correction Scheme (IPCS) . . . . . . . . . . . . . . . . . . . . . . . .
3.4. Wall Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1. Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2. Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3. Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
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11
13
16
16
17
18
4. Numerical Behaviour and Results
4.1. Tube Flow as an Example for an Idealized Artery . . . . . . . . . . . . . . . .
4.1.1. Steady Run and Discussion about Initialization Steps . . . . . . . . .
4.1.2. Comparison of the Membrane Approach in Contrast to Hooke’s Law
4.1.3. Pulsatile Flow Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Analysis of IPCS Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Application in CSF flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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29
5. Summary and Discussion
5.1. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
34
6. Acknowledgments
35
References
36
A. Appendix
A.1. Computation of modified stress tensor for a membrane approach . . . . . . . . . . . . . .
38
38
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I
List of Figures
1.1. The CSF spaces in a healthy individual (left) and in a patient with Chiari Malformation
(right). (Taken from http://www.chiariinstitute.com) . . . . . . . . . . . . . . . . . . . . .
2.1. Fluid-Structure domain decomposition (taken from [2]) . . . . . . . . . . . . . . . . . . . .
3.1. Stresses of a membrane patch with transverse shear (taken from [2]) . . . . . . . . . . . . .
4.1. Tube flow model with boundary domains . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Hydrostatic displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. Displacement field after convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4. Comparison of pressure and velocity field of the rigid and deformable solution after 120
time steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5. Steady displacement field with Hooke’s law for the wall stress tensor after 100 iterations .
4.6. Inflow pulse for the idealized artery model . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7. Maximum and minimum velocity and displacement fields in the unsteady idealized artery
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8. Phase leg between rigid and deformable model at t = 0.3 . . . . . . . . . . . . . . . . . . .
4.9. Phase leg between rigid and deformable model at t = 0.7 . . . . . . . . . . . . . . . . . . .
4.10. First deformable time step with hydrostatic initial displacement . . . . . . . . . . . . . . .
4.11. First deforamble timestep with zero initial displacement . . . . . . . . . . . . . . . . . . .
4.12. Idealized subarachnoid space model with bounday domains . . . . . . . . . . . . . . . . .
4.13. Inflow pulse for the idealized subarachnoid space model . . . . . . . . . . . . . . . . . . .
4.14. Velocity field in idealized subarachnoid space model with glyphs prescibing the wall velocity scaled by a factor of 3 in the third periodic cycle. . . . . . . . . . . . . . . . . . . . .
4.15. Radial displacement field in idealized subarachnoid space model in the third periodic cycle.
4.16. Pressure field in idealized subarachnoid space model in the third periodic cycle. . . . . .
1
3
16
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30
31
32
33
II
List of Tables
4.1. Wall properties for the idealized artery model. . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. Fluid properties for the idealized artery model. . . . . . . . . . . . . . . . . . . . . . . . . .
4.3. CSF and brain tissue properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
21
29
III
Glossary
Symbol
Definition
Variables
Ω
b
g
h
ul , v l and wl
ρ
u (x, t)
f
t
Γ
H 1h
I
ν
λ and µs
M
D
C
n
P (w)
νs
p
h
n
σ (x, t)
ε (v)
τ (v)
w, q
v (x, t)
µ
ζ
E
ζ
computational domain
body force
essential boundary condition
natural boundary condition
components of the local displacement vector
density
displacement field
force
fluid traction
boundary interface
Hilbert space
identity matrix
kinematic viscosity
Lame’s first and second parameters
mass
material constants
fourth-order tensor of material constants
normal vector
operator applied to the test function
Poisson’s ratio
pressure
local mesh size
number of spatial dimension
Cauchy stress tensor
rate-of-strain tensor
viscous stress tensor
test functions
velocity vector
dynamic viscosity
wall thickness
Young’s modulus
wall thickness
Subscripts
t
e
0
n
l
g
s
t
T
h
time derivative
element
initial
iteration step
local
inflow
structure
time
time level
outflow
IV
Abbreviations
Abbreviation
Definition
ALE
CFD
CMM-FSI
CST
UFLACS
UFL
FEM
FSI
CSF
GLS
SUPG
IPCS
PSPG
Arbitrary Lagrangian-Eulerian
Computational Fluid Dynamics
Coupled Momentum Method - Fluid-Structure Interaction
Constant strain triangle
UFL Analyser and Compiler System
Unified Form Language
Finite Element Method
Fluid-Structure Interaction
Cerebrospinal fluid
Galerkin/Least-Squares
Streamline-Upwind/Petrov-Galerkin
Incremental Pressure Correction Scheme
Pressure-Stabilizing/Petrov-Galerkin
Introduction
1
1. Introduction
Mathematical and numerical models have become more and more important in the fields of medicine
and biology. They enable a more successful treatment of issues in these fields. Mathematical models
offer a new and interesting way of describing phenomena like the regulatory mechanisms of the heart
[6, 3] or blood flow in the vascular system or Cerebrospinal fluid (CSF) flow [14]. By means of these
non-invasive methods, physiological processes in the human body can be examined and understood
with less effort and less danger for the patient. As a consequence, scientists can gain more insight into
the inherent mechanisms and improve their diagnosis techniques. With the continuous development
of Computer Science, it is possible to reach a new understanding of the complex physical processes
affecting human health and to improve treatment.
Two examples are aneurysm development and rupture and its association with abnormal blood flow or
the development of syringomyelia in association with the Chiari I malformation [14, 7].
Chiari I malformation is a serious neurological condition that affects regions of the brain and spinal cord.
It is characterized by herniation of the cerebellum through the large opening in the base of the skull
(foramen magnum) into the spinal canal. The herniated cerebellum alters the flow of the cerebrospinal
fluid (CSF) that surrounds the brain and the spinal cord, what can lead to a cavity within in the spinal
cord (see Fig. 1.1). Possible symptoms reach from headache to more serious neurological disorders, like
visual disturbances and dizzines among others.
Figure 1.1: The CSF spaces in a healthy individual (left) and in a patient with Chiari Malformation
(right). (Taken from http://www.chiariinstitute.com)
In most of the simulations of this two applications, the Chiari I malformation and aneurysm development, pure CFD is used [4, 13], which means that rigid walls are assumed. However, this does not
properly reflect the behaviour since vessels or brain tissue may deform under the action of flow forces
and thus alter the flow. For the modeling to be more realistic, coupled fluid-structure interacton (FSI)
modeling must be employed. A standard technique for FSI is the Arbitrary Lagrangian-Eulerian (ALE)
method. Using this method for large, realistic anatomical and physiological models is still problematical
and simplified or reduced models have to be applied [2]. This is due to the fact that the ALE formulations must continually update the geometry of the fluid and the structural elements, making the method
computationally expensive and susceptible to instabilities. However, these problems should be solved
for patient-specific geometries that yield large problems. Furthermore, the flow fields are very complex.
Reference [10] proposes that turbulence might occur in these applications. In order to resolve this phenomenon properly, very fine meshes have to be used which makes the problem even larger. Therefore,
a more efficient and robust FSI method is required.
A possible solution is the coupled momentum FSI method (CMM-FSI), introduced by [2]. It couples
Introduction
2
the fluid and structure equations at the variational level as a boundary condition for the fluid domain.
Thus, the mesh can be kept fixed and only some additional terms have to be added at the boundary; this
promises to be an efficient scheme because the equations do not differ much from those already solved
by efficient flow solvers.
This thesis will use two popular fluid solver methods, a Streamline-Upwind/Petrov-Galerkin (SUPG)
stabilized method with mixed elements and an operator splitting scheme, the Incremental Pressure Correction Scheme (IPCS). The different approaches will be implemented with FEniCS, a FEM software
package library. Simplified geometries will be used to analyze and test the implemented approaches.
Based on these tests, feasibility, efficiency and stability of the methods will be examined.
General Mathematical and Numerical Model
3
2. General Mathematical and Numerical Model
This section is intended to introduce the basic idea of the CMM-FSI model. The governing equations
are derived and proper boundary conditions are stated, followed by a short introduction of the FiniteElement Method (FEM). The last part of this section describes the coupling of the fluid with the solid,
based on a weak formulation, where the solid domain is only imposed as a boundary condition.
Figure 2.1 gives a schematic description of the model domain. A thin wall structure is used for the solid.
This means that the domain can be expressed as a surface times a thickness. Therefore the solid domain
Ωs can be topologically related to the fluid boundary interface with the structure Γs s.t. Ωs = Γs × ζ
where ζ denotes the wall thickness. Due to this assumption and the introduction of a membrane model
for the wall, which will be discussed in Section 3.4, a strong coupling of degrees-of-freedom of the two
domains will enable a strong coupling at the variational level.
Figure 2.1: Fluid-Structure domain decomposition (taken from [2])
2.1. Governing Equations
2.1.1. Fluid
Both blood and Cerebrospinal fluid (CSF) can be considered as Newtonian fluids and thus described
by the Navier-Stokes equations for incompressible fluids. These equations form a system of partial
differential equations. They underly the basic concepts of conservation of both mass and momentum.
Conservation of Mass
Consider a varying material volume Ωt ∈ Rnsd , with nsd being the number of spatial dimension, occupied by the fluid at an instant time t ∈ [0, T ], that is bounded by ∂Ωt then a fundamental law in
Newtonian mechanics states that the mass M contained in this material volume is conserved
0=
dM
d
=
dt
dt
Z
ρ dx,
Ωt
General Mathematical and Numerical Model
4
where ρ is the fluid density. Applying Reynold’s transport theorem and Green’s theorem, we get
dM
0=
=
dt
Z
Ωt
∂ρ
dx +
∂t
Z
Z
ρv · n dx =
∂Ωt
Ωt
∂ρ
+ ∇ · (ρv)
∂t
dx,
where v (x, t) is the velocity vector. This relation must hold for every control volume and thus the
expression inside the integral must be zero. Furthermore, we are assuming incompressibility s.t. the
density is constant and we obtain the continuity equation for an incompressible fluid:
∇ · v = 0.
(2.1)
Conservation of Momentum
Newton’s second law states that the change of momentum of a point mass is equal to the forces acting
on it:
d
dt
Z
ρv dx =
X
F.
(2.2)
Ωt
Applying Reynold’s theorem and Gauss’s theorem under the assumption of incompressibility we obtain
Z
Ωt
∂ (ρv)
dx +
∂t
Z
Z
(ρv) v · n ds =
ρ
∂Ωt
Ωt
X
∂v
+ ρ∇ · (vv) dx =
F.
∂t
With help of the continuity equation 2.1, the expression can be reformulated to
Z
∂v
+ ρv · ∇v ds.
∂t
ρ
Ωt
(2.3)
The forces acting on a portion of fluid are volume forces (e.g. gravity) and surface forces (e.g. pressure
and viscous forces). We denote the volume force by f (x, t) and therefore the total volume force acting
R
on a portion of fluid is Ωt f dV . The force exerted on the surface of the portion of fluid can be expressed
R
with the Cauchy stress tensor σ (x, t) as ∂Ωt σ · n ds. With the use of Green’s theorem we get
Z
Z
σ · n ds =
∂Ωt
∇ · σ dx.
Ωt
For Newtonian fluids the Cauchy stress tensor is defined by following constitutive equation [15]
σ (v, p) = 2µε (v) + pI,
where
1
T
∇v + (∇v) ,
2
ε (v) =
denotes the rate-of-strain tensor.
Thus the total force acting on a portion of fluid is
X
Z
∇ · τ − ∇p + f dx,
F =
Ωt
with τ being defined as
(2.4)
General Mathematical and Numerical Model
5
T
τ = µ ∇v + (∇v) .
Under the assumption that Eq. (2.2) holds for a an arbitrary control volume the integrands have to be
equal s.t. this yields together with Eq. (2.1) to
ρ
∂v
+ ρ (v · ∇) v =∇ · τ − ∇p
∂t
∇ · v =0
(2.5)
which is known as the Navier-Stokes equation for incompressible fluids.
2.1.1.1. Boundary Conditions
To get a mathematical closed system for Eq. (2.5) proper initial and boundary conditions have to be
specified.
Considering Figure 2.1 the boundary conditions split up into three parts ∂Ω = Γg ∪ Γh ∪ Γs and Γg ∩
Γg ∩ Γs = ∅.
v (x, t) = g on Γg ,
σ · n = h on Γh ,
(2.6)
σ · n = tf on Γs .
Γg is the Dirichlet boundary where all degrees of freedom are described, except the pressure. Usually
it represents the inflow boundary condition. Γh and Γs both describe Neumann boundaries where the
tractions h and tf are prescribed. Γh typically describes the outfow boundary and is called pressure
boundary since h can be described with a given pressure and the assumption that the velocity gradients
are zero. Γs is the boundary that describes the interface of the fluid with the solid domain. In case
of rigid walls, a no-slip Dirichlet boundary condition would be imposed. In order to allow non-zero
velocities at the wall it is replaced by a traction f f , that occurs due to the interaction of the fluid with
the wall. It is unknown for the moment but will be described in more detail later on because it plays an
important role in the coupling process.
The divergence free initial condition is given by:
v (x, 0) = v 0 (x)
on Ω0 .
(2.7)
2.1.2. Wall
The wall mechanics are modeled with the classic elastodynamics equations. Based on Newton’s second
law the motion of the wall in a bounded domain Ωst ∈ Rnsd states as follows
ρs
∂2u
= ∇ · σ s + bs ,
∂t2
(2.8)
where u (x, t) is the displacement field, ρs is the density of the wall, b is the prescribed body force per
unit volume and σ s is the wall stress tensor. Under the assumption of small deformations, isotropy and
non-yielding material, linear elasticity can be used to obtain the constitutive equation:
General Mathematical and Numerical Model
6
σ s = Cs : (u) ,
(2.9)
where Cs is a fourth-order tensor of material constants. Due to the membrane approach special care has
to be taken on C. The standard approach with Hooke’s law; σ s = µs + λ tr () I, where λ and µs are
Lame’s first and second parameters, does not yield stable results. This will be discussed in Section 3.4
and 4.1.2.
2.1.2.1. Boundary Conditions
Similar to Section 2.1.1.1 boundary and initial conditions have to be stated in order to close the system
of equations (2.8)
u (x, t) = g s on Γsg ,
σ s · n = hs on Γsh ,
(2.10)
u (x, 0) = u0 (x) on Ωs0 ,
ut (x, 0) = u0t (x) on Ωs0 .
Γsg and Γsh are the parts of the boundary of Ωs where essential (g s ) and natural (hs ) boundary conditions
are described. u0 (x) and ust (x) denote the given initial conditions for the displacement and velocity
field, respectively.
2.2. Finite Element Method
The Finite Element Method (FEM) is a popular solution strategy to solve partial differential equations.
Its success is due its feasibility to all kind of differential equations. Furthermore, it is very flexible with
respect to complex geometries and it is easy to construct higher order approximations. The basic ideas
will be only shortly presented within this thesis. For more detailed information the reader should refer
to e.g., [5] and [1].
Based on the strong form of a boundary value problem, e.g., Eqs (2.5) and (2.6) for the fluid domain or
Eqs (2.8) and (2.10) for the solid domain, a weak form can be derived. This is done by multiplying the
equations with test functions and integrating over the domain. In order to get a discrete setting local
approximations for the unknown variables as well as for the test functions have to be found.
2.3. Coupling at the Variational Level
2.3.1. Weak Form of the Fluid
As mentioned in Section 2.2 the strong form of the boundary value problem of the fluid is
ρv ,t + ρ (v · ∇) v = ∇ · τ − ∇p on Ωt ,
∇·v =0
on Ωt ,
v (x, t) = g
on Γg ,
σ·n=h
on Γh ,
f
on Γs .
σ·n=f
(2.11)
General Mathematical and Numerical Model
7
To obtain the weak form, the momentum equation has to be multiplied with the velocity test function
w as well as the continuity equation with the pressure test function q and both equations are integrated
over Ω. Furthermore, the stress term (w (∇ · τ − ∇p)) is integrated by parts and thereby a natural
boundary condition is created. In the next step the integral equations have to be discretized. The
Galerkin formulation is used to discretize the derived integral equations. This means, that local approximations for velocity v h and pressure ph , the trial functions are introduced, as well as the associated
test functions wh and q h . A requirement on the trial functions is that they have to fulfill the Dirichlet
boundary condition whereas the test functions have to be zero.
Within the context of this thesis only first order finite elements are used. This implies that the same
spaces are prescribed for the pressure as well as the velocity variables. This requires stabilization terms
for the fully coupled scheme that is used in the following for the derivation of the FSI-coupling whereas
same order elements work with IPCS (Section 3.3). Due to integration by part only first derivatives appear in the variational problem. This means, that test and trial functions are required to be continuous
and their first derivative has to be square integrable. A first order Sobolev space like the first Hilbert
space H 1 meets this conditions. H 1h is the space that consits of piecewise continuous functions and it is
well-known that H 1h ⊂ H 1 . All in all this yields the following function spaces:
.
, v h = g h on Γhg },
.
n
Ṽvh = { wh | wh ∈ H 1h (Ω) sd , wh = 0 on Γhg },
S̃vh = { v h | v h ∈ H 1h (Ω)
nsd
(2.12)
Ṽph = S̃ph = { ph | ph ∈ H 1h (Ω) }.
With the help of these spaces we get the semi-discrete Galerkin finite element formulation for the fluid
that results in the following weak form:
Find v h ∈ S̃vh and ph ∈ Ṽph such that
Z
F (w, q; v, p) =
Ωh
t
wh · ρv h,t + ρv h · ∇v h − f + ∇wh : −ph I + τ v h + q h ∇ · v h dx
Z
h
Z
h
−w · h ds +
+
Γh
h
(2.13)
−wh · tf,h ds = 0
Γh
s
holds for all wh ∈ Ṽvh and q h ∈ Ṽph .
The integral containing the fluid traction tf,h still is not specified yet and will be determined on the basis
of the solid equations weak form under several assumptions.
2.3.2. Weak Form Wall
Similarly to Section 2.3.2 the weak form of the wall equations is obtained on the basis of the strong form
of the boundary value problem. Collecting Eq. (2.8) and (2.10) one obtains:
ρs u,tt = ∇ · σ s + bs
on Ωst ,
u (x, t) = g s
on Γsg ,
s
Γsh .
s
σ ·n=h
on
(2.14)
Based on Equation 2.14 the finite element formulation is derived. Once again appropriate function
spaces have to be found. A strong coupling between the degrees-of-freedom of the fluid and the solid
domain is assumed. Therefore, displacement, velocity, and acceleration are identical on the fluid-solid
interface. Additionally, the elastodynamic equation has the same requirements of differentiability on
General Mathematical and Numerical Model
8
the function spaces as the fluid equations. Therefore, we can use the same type of function spaces for
the wall weak form as for the fluid. They state as:
.
, uh = g sh on Γsh
g },
1h
h
h
s nsd
h .
sh
= { w | w ∈ H (Ω ) , w = 0 on Γg }.
nsd
S̃ush = { uh | uh ∈ H 1h (Ωs )
Ṽush
(2.15)
Furthermore the acceleration term of the displacement field is expressed in terms of the time derivative
of the velocity, because we want to express the wall equations in terms of the fluid unknowns for further coupling. Considering this and the function spaces stated above, the semi-discrete Galerkin finite
element formulation yields in the following weak form.
Find v h ∈ S̃ush such that
Z
s
h
ρ w ·
Ωh
s
v h,t
Z
h
∇w : σ
dx +
s,h
Z
h
s
Z
wh · hs,h ds.
w · b dx +
dx =
Ωh
s
(2.16)
Γsh
Ωh
s
holds for all wh ∈ Ṽush .
This equation is the basis for the coupling as a boundary condition, what is discussed in the next section.
2.3.3. Fluid-Solid Coupling
Based on the shown weak formulations for the fluid and the wall we can now develop the fluid-wall
coupling conditions. The first step is to map the structural domain Ωs onto the fluid-solid-interface
boundary Γs . Once again this is done under the assumption of a thin-walled structure. With a given
wall thickness ζ, this mapping can be done with the following relations
Z
Z
(·) dx = ζ
(·) ds,
Ωs
Γs
Z
(2.17)
Z
(·) ds = ζ
Γsh
(·) dl.
∂Γh
Thus Eq. 2.16 can be rewritten to
Z
ζ
Γh
s
ρs wh · v h,t ds + ζ
Z
∇wh : σ s,h ds = ζ
Z
Γh
s
wh · bs ds + ζ
Γh
s
Z
wh · hs,h dl.
(2.18)
∂Γsh
With the principle of Actio and Reactio the forces acting onto the wall due to the fluid are equal and opposite to those acting onto the fluid due to the wall. In this context these “forces” have to be interpreted
as the surface tractions
tf = −ts .
(2.19)
Considering once again that the wall is thin-structured it can be assumed that the traction onto the wall
is felt uniformly through the thickness ζ. Therefore it can be interpreted as the body force bs acting on
the solid domain. This yields to
bs = −
tf
.
ζ
We insert Eq. (2.20) into Eq. (2.18), resort the terms and obtain
(2.20)
General Mathematical and Numerical Model
Z
−
h
f,h
w ·t
Z
s
ρ w ·
ds = ζ
Γh
s
h
Γh
s
v h,t
9
Z
h
∇w : σ
ds + ζ
s,h
Z
ds − ζ
wh · hs,h dl,
(2.21)
∂Γh
h
Γh
s
that is an expression for the fluid surface traction tf . Therefore, Eq. (2.21) is the piece of the puzzle to
close the up to now unknown part of Eq. (2.13) as a function of the solid internal stresses and inertial
forces. Therefore, the final weak Galerkin formulation for the coupled method is:
Find v h ∈ S̃vh and ph ∈ Ṽph such that
Z
F (w, q; v, p) =
Ωh
t
h
w · ρv h,t + ρv h · ∇v h − f + ∇wh : −ph I + τ v h + q h ∇ · v h dx −
Z
s
h
ρ w ·
+ζ
Γh
s
v h,t
Z
h
∇w : σ
ds + ζ
Γh
s
s,h
Z
ds − ζ
Z
wh · hh ds
Γh
h
wh · hs,h dl = 0
∂Γh
h
(2.22)
holds for all wh ∈ Ṽvh and q h ∈ Ṽph .
Remarks
Until now Eq (2.22) is only a semi-discrete form. Therefore, the time integration still has to be discussed.
Important to note is that it does not yield stable results if first order elements are used. Therefore it is
only used as a basis for more advanced solution schemes for Navier-Stokes equations, like stabilized
finite elements or IPCS. A further look has to be taken onto the wall stress tensor σ s as well.
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
10
3. Special Discretization Schemes for the Fluid and Solid Model
and Implementation Aspects
The coupled scheme has been derived for the simplest semi-discrete finite element approach. To take
stability and efficiency into account popular solution schemes for the Navier-Stokes equations are used
as a basis for the coupling. The first scheme (SUPG stabilized FEM) is a further developed method of
(2.22), that takes stability into account. The other one is a more advanced operator splitting scheme
(IPCS) that is promising with respect to computational effort. Both schemes are derived within this
section and it is shown how the extra coupling terms are included.
Furthermore an appropriate model for the wall stress tensor σ s has to be described in order to close the
coupled system.
All the schemes presented have been implemented in FEnICS and a short introduction to FEnICS will
be given in the following section.
3.1. FEniCS
FEniCS is a programm package to solve partial differential equations (PDEs) based on the finite element
method [1]. This includes features for automated error control and adaptivity, a comprehensive library
of finite elements, high performance linear algebra and many more. All kind of variational forms can be
solved, what makes FEniCS are very general tool. FEniCS programs can be programmed in Python or
C++ but within this thesis we use Python. Programming a FEniCS programm is quite straight forward
since variational forms can be specified in a near-mathematical notation.
In order to get a better understanding of this we solve the Poisson problem as an example program [1].
It’s variational form after partial integration reads as follows:
Z
Z
|Ω
f vdx
∇u · ∇vdx =
{z
} | Ω {z }
a(u,v)
∀v ∈ V.
L(v)
The programm starts with,
from d o l f i n import ∗
what imports all key classes from the DOLFIN library. DOLFIN is a software library of efficient C++
classes for finite element computing and DOLFIN provides access to these classes from Python programs.
In the next step we create a mesh and define the FEM function space over this mesh.
# Mesh
mesh = UnitSphere ( 4 , 4 )
# Function spaces
V = FunctionSpace ( mesh , " Lagrange " , 1 )
It is possible to use predefined mesh classes for simple geometries like the unit square in this example
or use other preprocessing programms to create more complicated meshes.
Furthermore we define the discrete function spaces. In this example, a first order Lagrange element is
used but in general it is possible to use a whole bunch of finite element function spaces. Note that it
is not necessary to define separate spaces for test and trial functions since boundary conditions are not
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
11
specified here.
Furthermore we can describe boundary conditions as follows:
# Boundary c o n d i t i o n s
def u_boundary_domain ( x , on_boundary ) :
r e t u r n on_boundary
u_boundary = E x p r e s s i o n ( " x [ 0 ] + x [ 1 ] " )
bc = D i r i c h l e t B C (V, u_boundary , u_boundary_domain )
First of all we mark the corresponding part of the boundary domain where we want to set Dirichlet
boundary conditions. Additionally we define the values that have to be set by an expression. In the end
we combine these two information in the DOLFIN class that holds the information about the Dirichlet
boundary condition.
Finally the variational problem itself has to be specified and solved:
# V a r i a t i o n a l problem
u = T r i a l F u n c t i o n (V)
v = T e s t F u n c t i o n (V)
f = Constant ( 1 . 0 )
a = i n n e r ( grad ( u ) , grad ( v ) ) ∗dx
L = f ∗v∗dx
# Compute S o l u t i o n
u = Function (V)
s o l v e ( a == L , u , bc )
Test- and trial function u and v are defined on the finite element space introduced above. The right hand
side term f is set as constant value. Now we have all ingridients for a(u, v) and L(u, v). As mentioned
above this is done in a near-mathematical way such that the mathematical ”pen-written” variational
from can be transformed into finite element code without taking care of element matrix formulations.
Using the solve approach as shown here, the resulting linear system is solved with an LU decomposition. Since this is only practical for small problems there are several options for iterative methods and
preconditioning as well.
Remarks
This short introduction to FEniCS is intended to show that variational forms can be easily translated
into FEniCS code. Therefore it is sufficient to derive the variational form of the schemes presented in the
following section, since no element level matrix formulations are needed.
3.2. SUPG stabilized FEM
In this Section we derive a stabilized method for the semi-discrete formulation of Eq. (2.22). It has already been mentioned that the first order element approximations for the pressure as well as the velocity
are not stable. This is due to the fact that they do not fulfill the famous LBB (Ladyzhenskaya, Babuska
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
12
and Brezzi) condition. This condition ensures the stability of mixed methods when using continous or
discrete spaces.
In order to make the method stable, additional stabilization terms have to be added to the standard
Galerkin approximation of Eq. (2.22). Furthermore, this method is not stable for convection dominated
flows. This also requieres additional stabilization terms. The first two of these additional terms underly
the the same idea [5]: Add
XZ
P (w) τ R (v) dx
(3.1)
Ωe
e
to the standart Galerkin formulation where P (w) is a certain operator applied to the test function, τ is
the stabilization parameter and
R (v) = ρv t + ρ (v · ∇) v − ∇ · τ + ∇p
(3.2)
is the residual of the momentum equation. By multiplying with R (v) the method is ensured to be
consistent. This means that a solution of the differential equation is also a solution of the weak form.
Note that (∇ · τ ) can be neglected from R (v) because of the use of first order elements.
Pressure-Stabilizing/Petrov-Galerkin
The additional term presented in the following adds a certain amount of compressibility to the system
and allows the use of mixed elements with equal-order interpolations for velocity and pressure. This
implies that the LBB condition is fulfilled. It states as follows:
nel Z
X
e=1
τPSPG ∇q · R (v) dx,
(3.3)
Ωe
with τPSPG being evaluated element whise based on the local mesh size he :
τPSPG =
h2e
.
2
(3.4)
Streamline-Upwind/Petrov-Galerkin
Streamline-Upwind/Petrov Galerkin (SUPG) stabilization is needed for convection dominated flow.
Problematic with the Galerkin formulation is that it produces negative diffusion under this flow conditions. A first approach to balance it, is to add artificial diffusion. The SUPG stabilization uses this idea as
a starting point but ensures that artificial diffusion is only added in streamline direction. Thus excessive
over diffusion is avoided. The extra term is defined as:
nel Z
X
e=1
(v · ∇) w · τSUPG R (v) dx,
(3.5)
Ωe
where τSUPG is based on element level matrices and vectors to automatically account for local mesh sizes,
advection fields and the fluid viscosity. [8] propopes
τSUPG =
1
0.5∆t
2
+
2v
he
2
+9
4ν
h2e
2 !− 12
.
(3.6)
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
13
Artificial Diffusion
Artifical diffusion provides additional stability for convection dominated flows. It is basically a leastsquare stabilization on the incompressibility constraint. It states as follows:
nel Z
X
e=1
τLSIC ∇ · w : ∇ · v dx
(3.7)
Ωe
where a typical choice of τLSIC is
τLSIC = ||v||
he
2
(3.8)
according to [17]
Adding the PSPG and SUPG stabilization terms to Eq. (2.22) yields to the overall stabilized formulation:
Find v ∈ S̃vh and p ∈ Ṽph such that
Z
F (w, q; v, p) =
Z
{w · (ρv ,t + ρv · ∇v − f ) + ∇w : (−p I + τ (v)) + q∇ · v} dx −
Ωt
Z
Z
Z
+ζ
ρs w · v ,t ds + ζ
∇w : σ s ds − ζ
w · hs dl
Γs
+
+
Γs
nel Z
X
e
e=1 Ω
nel Z
X
e=1
(v · ∇) w · τSU P G R (v) dx +
w · h ds
Γh
∂Γh
nel Z
X
e=1
(3.9)
τP SP G ∇q · R (v) dx
Ωe
τLSIC ∇ · w : ∇ · v dx = 0
Ωe
holds for all w ∈ Ṽvh and q ∈ Ṽph .
Note that all superscripts h have been dropped for simplicity.
The computation of the element level stabilization parameters is quite costly when implemented in
Python. To efficiently calculate these expressions it is done as C++-implementations of the “Expression”
class. A similar code already excits in [9]. It is used as a basis and modified for the paramters within this
thesis.
Eq. (3.9) is still a semi-discrete formulation. Therefore a discretization in time has to be specified.In or-
n
v n−1 .
der to have less restrictions on time stepping in terms of stability, we use a Backward Euler scheme v t ≈ v −∆t
Furthermore we have to linearize the convective term (v · ∇) v. This is done in a semi-implicit manner
v n−1 · ∇ v n .
3.3. Incremental Pressure Correction Scheme (IPCS)
As mentioned in the previous section, one way to make equal first order approximations work is to
add stabilization terms. Still pressure and velocity are coupled, what makes these schemes expensive to
solve. Therefore an operator splitting scheme is presented that breaks the coupling of the pressure and
velocity. According to [9] the Incremental Pressure Correction Scheme (IPCS) is quite a good approach
for these kind of methods.
Derivation of the strong Form
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
14
We reconsider Eq. (2.5) and want to uncouple pressure and velocity. Therefore, we use an approximation for the pressure by taking the pressure of the previous timestep pn−1 instead of the actual one pn
[9]. Additionally, the time derivate ∂∂tv has to be approximated. This is done for similar reasons with
Backward Euler like in the stabilized method in the previous section. The convective term is linearized
as well in a semi-implicit approach yielding to v n−1 · ∇ v n . Keeping the unknown pressure pn for a
moment, it yields following system of equations, known as the Oseen equations:
v n + ∆tv n−1 · ∇v n − ∆t∇ · 2νε (v n ) +
∆t
∇pn =v n−1 ,
ρ
(3.10)
n
∇ · v =0,
where ν = µρ is the kinematic viscosity.
In the next step we use the approximation for the pressure und thus get an approximated value for the
velocity, v ∗ ≈ v n , which gives the simplified equation
v ∗ + ∆tv n−1 · ∇v ∗ − ∆t∇ · 2νε (v ∗ ) +
∆t
∇pn−1 = v n−1 .
ρ
(3.11)
Note that v ∗ does not have to be divergence-free. We would still like v n to fulfill this condition. Thus
we define a velocity correction v c = v n − v ∗ . With ∇ · v n = 0 we get ∇v c = −∇ · v ∗ . By substracting
Eq. 3.11 form Eq. 3.10 we obtain the new set of equations:
v c + ∆tv n−1 · ∇v c − ∆t∇ · 2νε (v c ) +
∆t
∇Φn = 0,
ρ
(3.12)
∇ · v c = −∇ · v ∗ ,
with Φn = pn − pn−1 . Since we replace pn with pn−1 we are only first order accurate O (∆t). Thus, we
can simplify Eq 3.12 without decreasing the order of approximation:
vc +
∆t
∇Φn = 0,
ρ
(3.13)
∗
c
∇ · v = −∇ · v .
Taking the divergence on the first equation and inserting the second one into it we obtain a Poisson
problem for pn :
∆pn = ∆pn−1 +
ρ
∇ · v∗ .
∆t
Knowing pn , v n can be computed with the definition of v c .
Therefore IPCS can be summarized into three steps:
1. Compute the tentative velocity v ∗ with Eq. (3.11).
2. Compute the corrected pressure pn with Eq. (3.14).
3. Compute the corrected velocity v n with v n = v ∗ −
∆t
n
ρ ∆Φ .
(3.14)
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
15
Derivation of the weak Form (FE Discretization)
To obtain a weak formulation we once again have to multiply the tentative velocity step (Eq. (3.11)) and
the velocity correction with the test function w as well as Eq. (3.14) with q. In the next step we have to
find appropriate approximations for both of them as well as for v ∗ , v n and pn . As described in Section
2.3.2 we use a first order Galerkin approach. Therefore, we can use the test and trial function described
in Eq. (2.12). For simplicity the superscripts (h ) are dropped in the following.
IPCS in a Galerkin formulation can be rewritten to:
1. Starting with the equation for the tentative velocity and after integrating by parts, we obtain the
following Galerkin finite element formulation:
Find v ∗ ∈ S̃vh such that
Z
Z
Z
ρ
w · v ∗ − v n−1 dx +
ρ w · v n−1 · ∇v ∗ dx +
∇w −pn−1 I + τ (u∗ ) dx
Ω ∆t
Ω
Z
Z Ω
(3.15)
∗ n−1
f
∗ n−1
−
w · h v ,p
ds −
w · t v ,p
ds = 0.
Γh
Γs
holds for all w ∈ Ṽvh .
Similar to Section 2.3.3, we insert Eq. (2.21) for the fluid traction tf to get a fully coupled system.
Thus, Eq. (3.15) changes to
Z
ρ w · v n−1 · ∇v ∗ dx +
∇w −pn−1 I + τ (u∗ ) dx
Ω
Ω
s
(3.16)
ρ
w · v ∗ − v n−1 + w : σ (u∗ ) ds = 0.
∆t
Γs
Z
ρ
w · v ∗ − v n−1 dx +
Ω ∆t
Z
Z
−
w · h v ∗ , pn−1 ds + ζ
Γh
Z
2. Multiplying the pressure correction equation with q and integrating by parts we obtain: Find
pn ∈ Ṽph such that
Z
Z
n
n−1
∇q · ∇p dx −
Ω
∇q · ∇p
Ω
Z
dx +
Ω
ρ
q∇ · v ∗ dx = 0
∆t
(3.17)
holds for all q ∈ Ṽph .
3. Similar to the previous steps the velocity correction equation is multiplied with w and integrated
over the domain Ω and integrated by parts s.t. we find:
Find v n ∈ S̃vh such that
Z
Z
∆t
w · v dx −
w · v dx +
ρ
Ω
Ω
n
∗
Z
∇q · ∇pn − ∇pn−1 dx = 0
(3.18)
Ω
holds for all w ∈ Ṽvh .
The implementation of this scheme is straight forward. There exists an IPCS solver for rigid wall problems in [9]. This can be used as a basis where only the additional forms have to be added.
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
16
3.4. Wall Discretization
3.4.1. Spatial Discretization
In section 2.1.2 the stress tensor σ s has not been further specified. This will be done within this section
together with its discretization within the finite element model according to [2]. According to [20],
experiments have shown that a linear model for the wall is reasonable for the pressure ranges occuring
in the studied physiological processes. Therefore Hooke’s law seems to be an appropriate first model
for σ s :
σ s = µs ε (u) + λ tr (ε (u)) I,
(3.19)
where λ and µs are Lame’s first and second parameters.
In 4.1.2 it is shown and explained that this model is not sufficient to describe the wall in a correct and
stable manner. Therefore an alternative model is presented in the following.
Recalling that we want to have a strong coupling of degrees-of-freedom of wall and fluid at the boundary, we use membrane elements with only translational degrees-of-freedom. This represents a simple
and convenient choice. The mesh of the vessel wall is defined by linear triangles since the internal fluid
mesh consists of tetrahedral elements. This yields the problem that constant strain triangles (CST), representing the membrane modes, are not working for three-dimensional geometries with transverse loads
[18]. To add additional stiffness to the linear membrane element we extend the system with transverse
shear (Fig. 3.1).
Figure 3.1: Stresses of a membrane patch with transverse shear (taken from [2])
This approach requires a coordinate transformation between the local plane of a given triangular element and the global reference system, in order to correctly represent the membrane behavior in the
global frame of the fluid domain.
Recall that we consider a thin membrane and thus, it is possible to make an assumption similar to Eq.
2.20. Since the forces are felt uniformly across the thickness, variations can be neglected in this direction
l
= 0 . Making use of the fact that C, and σ s are symmetric, it is possible to use a reduced vector
σzz
form of these tensors and reduce C to a second-order tensor of material contant D. Therefore local stress
and strain vectors are defined as follows:
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects

 l 
σxx
σ l 
 yy 
σl 


s,l
,
{σ } =  zz
l 

 τxy
 l 
 τxz 
l
τyz
∂ul
∂xl
∂v l
∂y l
17









0


l
{ } =  ∂ul
l.
∂v 
 ∂yl + ∂x
l


 ∂wl 
l
 ∂x

(3.20)
∂wl
∂y l
Note that ul , v l and wl are the three components of the displacement vector ul and that xl and y l are
the local nodal coordinates. In the next step, the tensor of material constant for a plane stress state of an
isotropic, incompressible solid extended with transverse shear stresses is defined:

1
ν s

0
E

D=

s
2
(1 − (ν ) )  0

0
0
νs
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0 0.5 (1 − ν s )
0
0
0
0.5k (1 − ν s )
0
0
0

0

0


0

,

0


0
s
0.5k (1 − ν )
(3.21)
where E and ν s are the Young’s modulus and Poisson’s ratio. In order to account for a parabolic variation of transverse shear stress through the membrane, the parameter k is added. Reference [18] proposes
that it takes the value 5/6 for a solid, homogenous plate.
Recall that we want to get a formulation for the following integral expression:
Z
ζ
∇w : σ s ds.
Γs
Due to the symmetry of σ s it can be rewritten as
Z
ζ
(w) : σ s ds = ζ
Z
(w) : C : (u) ds,
(3.22)
Γs
Γs
and thus, we obtain a discrete formulation for the wall with help of the index-collapsed vectors and
tensors defined by Eqs (3.20) and (3.21).
What is left to do is the local to global mapping of Eq. (3.20). We define the local normal vector nl of the
plane triangle and the two tangential vectors tl1 and tl2 that span the triangle plane. Then the local strain
vector {l } mapped into the global reference domain is defined as

tl1
T
∇u tl1



T


tl2 ∇u tl2




0


{ (u)} =  l T
.
 t1 ∇u tl2 + tl2 T ∇u tl1 




l
l T
n
∇u
t


1
T
nl ∇u tl2
(3.23)
This formulation have been implement with the Unified Form Language (UFL) [12] that is one of the
core components of FEniCS and it is shown in Appendix A.1
3.4.2. Time Discretization
In Eq. 2.22 we do not solve explicitly for the displacement field. Therefore it has to be related with the
velocity and acceleration of the fluid on Γs . This is done using Newmark’s formula:
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
un+1 = un + ∆tv n +
∆t2
(1 − 2β) v ,tn + 2βv ,tn+t .
2
18
(3.24)
According to [2], β = 1 is used within this thesis.
3.4.3. Initialization
The only thing missing to have the coupled method complete is a discussion about the initialization of
the algorithm. In order to get a stable flow field, a rigid wall simulation is performed. The results are
used as an initial condition for the velocity field. In contrast to a rigid wall problem, initial conditions
have to be specified for the displacement and velocity field on the fluid-solid interface as well, as stated
in Eq. (2.10). Based on the rigid wall run the velocity is set to zero. A more detailed look has to be taken
to the displacement field.
Zero Displacement
The simplest idea that comes to mind is to be to start from a zero displacement field as well. Most physiological applications are non-steady problems. In order to obtain stable results a steady deformable
problem is run until convergence before starting with varying inflow and outflow conditions. This
yields the following algorithm for initialization:
1. Solve the steady rigid wall problem
2. Initialize u0 , v 0 as zero on the fluid-solid interface
3. Solve the steady deformable wall problem
Hydrostatic Pressure
Another possibility is to start with a non-zero displacement field. Therefore we want that the wall and
fluid traction balance each other to start from an equilibrium state, Eq. (2.19). Reference [2] proposes
that based on (2.20) the wall is loaded with the average pressure P of the rigid wall problem. This load
is equivilant to the one in the hydrostatic case (v = v t = 0, τ = 0 ). Considering this, Eq. (2.22) reduces
to
Z
ζ
∇ : σ s (u) ds =
Γs
Z
wP · n dx.
(3.25)
Γs
For simplicity the displacement field is defined on whole domain Ω within the FEniCS code. Therefore,
it is not possible to solve the equation stated above with FEniCS. It has to be modified in a way that we
solve a variational problem on the whole domain but do not affect the terms on the boundary by the
inside. This is done in the following way:
Z
0.0001 ∗
Z
∇w : ∇u dx + ζ
Ω
s
Z
∇ : σ (u) ds =
Γs
wP · n dx.
Γs
A noslip Dirichlet boundary condition is imposed on all boundary domains except the wall.
We get the following initialization algorithm:
1. Solve the steady rigid wall problem
2. Initialize v 0 as zero on the fluid-solid interface and solve Eq. (3.26) to obtain u0
3. Solve the steady deformable wall problem
(3.26)
Special Discretization Schemes for the Fluid and Solid Model and Implementation Aspects
19
Section 4.1.1 will discuss the applicability of the two different approaches, its convergence and stability.
Numerical Behaviour and Results
20
4. Numerical Behaviour and Results
This section presents the results obtained with the method discussed in the previous sections. First of
all, a simple tube flow is used as a model for an idealized artery and solved with the SUPG stabilized
scheme, (see Sec. 3.2). Within this context we discuss the impact of a deformable wall in contrast to
the rigid case, compare the two different initialization methods (Sec. 3.4.3), show the importance of the
membrane approach in contrast to Hooke’s law, and present results with pulsatile inflow conditions.
In the next step, we discuss stability issues when solving the same application with the IPCS scheme.
Finally, results of a more complex spinal cord like geometry are examined.
4.1. Tube Flow as an Example for an Idealized Artery
In the following we want to describe blood flow in an artery as a test problem. For this purpose we use
a simple cylindrical model which can be seen as an idealized artery (Fig. 4.1). The radius and vessel
length are 0.3 cm and 10 cm, respectively. The mesh used for the computations consits of 120019 elements and 10754 nodes and the timestep size is 2 ms. To begin with, we use the SUPG stabilized scheme
presented in Sec. 3.2.
It is necessary to fix the wall at the inlet and outlet ring [2]. This can be done by imposing a noslip
boundary condition which is identically to constraining the degrees-of-freedom of these nodes. However it was not possible to apply a Dirichlet boundary conditions to points or lines within FEniCS. In
order to circumvent this, a small part of the wall at the inlet and outlet is defined rigid which yields
the same effect. Additionally this approach ensures fully developed inflow and outflow velocity profils.
The parts of the boundary can be seen in Fig. 4.1 where 3 denotes the deformable parts of the wall and
4 is the rigid part.
Figure 4.1: Tube flow model with boundary domains
At the inlet, we prescribe a parabolic velocity profile in positive x-direction with a maximum flow
velocity of 45 cm/s. The outlet is described by a constant pressure boundary condition, where p =
1330 dyn/cm2 . Table 4.1 and 4.2 give physiologically realistic values of the material parameters [19].
Name
Young’s modulus
Poisson’s ratio
Wall density
Wall thickness
Identifier
E
νs
ρs
ζ
Value
4.07 × 106
0.5
1.0
0.03
Unit
dyn/cm2
g/cm3
cm
Table 4.1: Wall properties for the idealized artery model.
Numerical Behaviour and Results
Name
Fluid density
Dynamic viscosity
21
Identifier
ρ
µ
Value
1.06
0.04
Unit
g/cm3
dyn s/cm2
Table 4.2: Fluid properties for the idealized artery model.
4.1.1. Steady Run and Discussion about Initialization Steps
Sec. 3.4.3 points out the importance for the unsteady simulation to start from a stable, converged solution for the initial displacement field. This section presents and analyses results obtained with the two
different initialization methods.
It is important to keep in mind that the mesh is kept fixed. Therefore, the velocity and pressure profiles
for a fully converged deformable steady flow solution have to be identical to the rigid one. This can be
seen in Eq. (3.24) because no change in the displacement requires zero wall velocity and acceleration
fields. In this situation, the additional boundary terms of the wall act like a Nitsche noslip condition.
Before starting with the deformable computation a rigid wall problem is solved to get initial conditions
for velocity and pressure fields. We get an average pressure of 1750 dyn/cm2 in the tube. This hydrostatic condition yields the initial displacement given in Fig. 4.2.
Figure 4.2: Hydrostatic displacement
Based on this, the steady deformable flow problem is solved. The solution is sufficiently converged
when the wall velocity values become ||v|| < 10−4 .
For the hydrostatic displacement initialization the solution converges after ~120 time steps. Fig. 4.4
shows that pressure and velocity fields of the deformable and rigid solution are identical. This confirms
the assumption of convergence. The steady displacement field is symmetric and takes higher values
closer to the inlet, as one can expect, (see Fig. 4.3).
Figure 4.3: Displacement field after convergence
Numerical Behaviour and Results
(a) pressure field along the x-axis
22
(b) velocity profile of the cross section at x = -5
Figure 4.4: Comparison of pressure and velocity field of the rigid and deformable solution after 120 time
steps
Another possibility is to let the deformable run start with a zero displacement field. This steady deformable computation converges after ~300 time steps. That are twice as many iterations in contrast
to the non-zero initial displacement field. Thus, the non-zero initial displacement method seems to be
more advanced. Furthermore, it has to be taken into account that these computations have been solved
on a very simple geometry. In the rigid run the velocity field is always tangential to the wall. In more
complex geometries this is naturally not always the case. Without any initial displacement to begin with,
there is no initial force acting that counterbalances the fluid pushing outwards over the wall boundary.
This might cause large waves within the wall which can lead to instability. This phenomena can already
be observed in this simple example. Significantly bigger waves are going back and forth inside the wall
in contrast to a non-zero initial displacement. Therefore an initial displacement based on the hydrostatic
pressure promises more stable results and will be used in further computations.
4.1.2. Comparison of the Membrane Approach in Contrast to Hooke’s Law
In Sec. 3.4.1 two different constitutive equations for the wall stress tensor have been introduced, Hooke’s
law and a membrane approach. In the previous section the membrane approach has already been used.
For a steady deformable computation it has been shown that it leads to stable and physical reasonable
results. However it is still necessary to have a closer look at Hooke’s law. The reason is that no extra
vector computations have to be solved in order to add it to the variational formulation. In contrast to
that, for the membrane approach it is necessary to eliminate the normal component of the strain tensor
and map it to the global coordinate system, (see Sec. 3.4.1). It is not possible to compile these forms
with the current FEniCS version. Only the recent development of UFL Analyser and Compiler System
(UFLACS) allows the usage of the membrane approach. It translates tensor intensive symbolic expressions into a low level expression representation and C++ code. This component is only available in the
unstable developer version of FEniCS. Therefore Hooke’s law has been the only possibility to use the
coupled method in the FEniCS environment at the moment. Thus, steady deformable computations will
be shown in the following.
The same material parameters, mesh and time-stepping are used. Instead of Young’s modulus and the
Poisson ratio we need Lame’s first and second parameter as material constants. They can be computed
from Young’s modulus and the Poisson ratio as follows:
Numerical Behaviour and Results
23
λ=
Eν s
,
(1 + ν s ) (1 − 2ν s )
µs =
E
.
2 (1 + ν s )
Note that if ν s = 0.5 s.t. λ would go to infinity. Therefore, we slightly change the Poisson ratio to
ν s = 0.49. This yields λ = 66758389 dyn/cm2 and µs = 1353333 dyn/cm2 . The hydrostatic displacement
initialization does not yield stable results for Hooke’s law, s.t. the zero displacement initialization is
used. In the steady wall computation the wall velocity decreases quite fast and basically no changes in
the displacement field happen after ~100 timesteps. Fig. 4.5 gives the corresponding displacement field
after convergence. The displacement field is not symmetric. One can observe pattern of bigger radial
displacement going through the wall. The first idea is that this phenomenon is related to a noslip condition at the inlet and outlet. Looking at Fig. 4.1 shows that the interface between deformable wall and
the noslip boundary domain is zig-zag shaped. We assumed that this interface caused unstable waves
that result in the strange displacement field pattern. In order get reasonable results several changes to
the noslip-deformable interface have been done without any promising effects. Therefore this strange
behaviour must have a different origin. Recall that the displacement field is defined in the whole computational domain. Although, it only has a physical meaning at the fluid-solid interface. In contrast to
the membrane approach, no normal derivates are neglected with Hooke’s law. Thus normal derivatives
are built with displacement values that have no physical meaning, what seems to cause the observed
instabilities. This study shows that the CMM-FSI is only possible in case of a membrane structure.
(a) z-normal pointing outwards
(b) z-normal pointing inwards
Figure 4.5: Steady displacement field with Hooke’s law for the wall stress tensor after 100 iterations
Numerical Behaviour and Results
24
4.1.3. Pulsatile Flow Model
In this section more realistic boundary conditions are used to describe the idealized artery model. In
the previous section we have only solved a steady flow problem, but blood flow in arteries is not steady
by nature. To change this, the inflow boundary conditions becomes time-depent. A sinusoidal inflow is
used to model a simplified pulsatile flow caused by the heart pulse. It has a period of T= 0.8 s and is once
again mapped to a parabolic profil. Fig. 4.6 displays the maximum velocity in the inlet cross-section.
Figure 4.6: Inflow pulse for the idealized artery model
The results are obtained with the same time stepping, mesh, material parameters as before for a total of
three cardiac cycles. The results from 4.1.1 are used as an initial condition for the time-dependend run.
In order to evaluate the flow and pressure behaviour an unsteady rigid model is computed as well.
Fig. 4.7 gives the maximum and minimum radial displacement occuring during a whole period. Note
that only a slice along the x-axis is shown since the solution is totally symmetric. These maximum and
minimum conditions correspond to zero radial velocity and maximum and minimum pressure drop, respectively. The maximum radial displacement is located close to the inlet and its value is ∆r = 0.002 cm.
The corresponding maximum circumferential strain is 00 = 0.67%. The time differences between maximum and minimum radial displacement is about half a period lenght, what can be expected for a
sinusoidal inflow. Similar effects can be observed for maximum and minimum radial velocity, (see Fig.
4.7). Furthermore, they occur at states of maximum pressure change. This shows that pressure and velocity are related by the same sinusoidal function without any phase shift, what perfectly matches with
the physics described by the wall equations.
Numerical Behaviour and Results
25
(a) maximum radial displacement at t = 0.47
(b) minimum radial displacement at t = 0.88
(c) maximum velocity in normal direction at t = 0.26
(d) minimum velocity in normal direction at t = 0.67
Figure 4.7: Maximum and minimum velocity and displacement fields in the unsteady idealized artery
model.
Numerical Behaviour and Results
26
Figure 4.8: Phase leg between rigid and deformable model at t = 0.3
Figure 4.9: Phase leg between rigid and deformable model at t = 0.7
Another effect that can be observed in deformable flow problems is a phase leg between the inflow
and outflow waves. It is hard to determine within this computation because of a relatively small radial
deformation (∆r = 0.002 cm). The mesh is quite coarse such that maximum velocities differ along
the x-axis due to interpolation 1 . However, the effect causing this phase leg can be seen. In moments
of an increasing pressure drop the normal wall velocity is positive according to Fig. 4.7 and shows
higher values closer to the inlet. This can be interpreted in a way that some extra mass is “stored”
and therefore, lower velocities occur downstream compared to a rigid wall simulation. Fig. 4.8 is a
plot along the center of the tube in x-direction. This plot clearly shows the difference in the velocity
magnitude between the rigid and deformable computation. The maximum difference is reached around
t = 0.3 s shortly after the state of maximum relative velocity. The same effect happens the other way
around while the pressure drop decreases. The “stored” flow goes back into the system such that higher
velocities in contrast to the rigid model occur. This effect becomes obvious in Fig. 4.9.
All this results show that the obtained is physical.
1
Computation have been performed on a laptop and thus, only coarse grids could be used.
Numerical Behaviour and Results
27
4.2. Analysis of IPCS Stability
So far, only the SUPG stabilized method has been examined. It has proven to work quite well for
the idealized artery model. In the next step we want to apply the IPCS method to the same model.
Once again we use a steady inflow to begin with. The initial displacement field is obtained with the
same hydrostatic pressure as in Sec. 4.1.1. After the rigid wall computation the solution after the first
deformable time step is shown in Fig. 4.10,
(a) y-velocity after tentative velocity step
(b) pressure field after pressure correction step
(c) y-velocity after velocity correction step
Figure 4.10: First deformable time step with hydrostatic initial displacement
It can be observed that no reasonable results are computed after the velocity correction step. The wall
velocity after computing the tentative velocity step is still symmetric and has comprehensible values.
Due to the fact that the pressure field is constant in this computation, whereas it increases in the first
deformable run for the SUPG stabilized scheme, the wall velocities in radial direction are negative. In
the pressure correction step a huge increase of the pressure drop can be observed. The difference of the
pressure drop in contrast to the SUPG stabilized method is ∆pdrop = 4040 dyn/cm2 . This is not physical
and could be a reason for unstable results in the velocity update, s.t the computation blows up after
some other time steps.
In order to avoid the negative radial velocity, a zero initial displacement field is used. The results can
be seen in Fig. 4.10. The tentative velocity field looks like what is physically expected. It is symmetric
Numerical Behaviour and Results
28
and has a positive radial velocity. The pressure drop increases in comparison to the rigid wall solution
after the pressure correction. This can also be considered as physically reasonable. Only the velocity
field after the velocity correction obtains unstable and unphysical results. It seems that this equation
causes trouble. In a rigid wall IPCS scheme noslip boundary conditions would be imposed here but in
this method no additional terms are added, (see Sec. 3.3).
(a) y-velocity after tentative velocity step
(b) pressure field after pressure correction step
(c) y-velocity after velocity correction step
Figure 4.11: First deforamble timestep with zero initial displacement
Another possibility is that the immediate change from noslip to deformable wall causes trouble. To
circumvent this effect we use the converged displacement field from Sec. 4.1.1 as initial displacement
field. This does not have any effect and the simulation crashes as well.
Therefore, one has to conclude that CMM-FSI applied to IPCS or operator splitting schemes in general
does not yield physical and stable results. This might be caused by the strong dependency of pressure
and displacement and the fact that the displacement field is only introduced in the tentative velocity
step where the pressure is kept constant.
Numerical Behaviour and Results
29
4.3. Application in CSF flow
Sec.1 pointed out that CMM-FSI is an interesting an promising approach for CSF flow in order to investigate the impacts of Chiari I malformation. Based on [11] we use an idealized geometry to model
obstructions in the cervical spinal subarachnoid space. The geometry consits of the domain between two
cylinders with diameters 1.5 cm and 1.0 cm each and a length of 10 cm. The inner cylinder is enlarged
at the midpoint with a diameter of 1.2 cm mapped to an elliptic profil. The enlargement has a length
of 2 cm. The corresponding mesh consits of 60324 cells and 12038 nodes. We assume that the cervical
spinal subarachnoid space is surrounded by bones and other material that is constant. Therefore only
the tissue in the middle can be deformed. Fig. 4.12 shows the mesh and the boundary parts, where 3
denotes again the deformable tissue wall and a noslip condition is applied at 4.
Figure 4.12: Idealized subarachnoid space model with bounday domains
The material parameters for brain tissue found in literature vary a lot. Young’s modulus and Poisson
ratio for a nearly incompressible brain presented by Dutta-Roy [16] are used. The tissue density is
assumed to be equivalent to the one of CSF that is water density at body temperature [11]. The wall
thickness takes the radius of the inner cylinder. Table 4.3 summarizes all the chosen material parameters.
Name
Young’s modulus
Poisson’s ratio
Wall density
Wall thickness
CSF density
CSF dynamic viscosity
Identifier
E
νs
ρs
ζ
ρ
µ
Value
4.67 × 103
0.5
1.0
0.1
1.0
0.007
Unit
dyn/cm2
g/cm3
cm
g/cm3
dyn s/cm2
Table 4.3: CSF and brain tissue properties
We use a sinusoidal inflow that is mapped to a parabolic profile. Fig. 4.13 shows the time-dependend
inflow velocity. Its period length is 0.8 s. Note that the CSF flow changes direction in reality what is
not used in this case because an outward-pointing inflow condition could cause instabilities. A constant
pressure is applied at the outflow boundary with pout = 100 dyn/cm2 . We run the simulation for three
period lengths.
Numerical Behaviour and Results
30
Figure 4.13: Inflow pulse for the idealized subarachnoid space model
Fig 4.14, 4.15 and 4.16 show the results obtained within the third periodic cycle. For t = 0 it can be
observed that the velocity magnitude in front of the obstruction and behind it is quite similar. The wall
pushes outwards due to an increasing pressure gradient. At t = 0.2 the effect of storing flow becomes
very clear. The velocity has further increased in front of the obstruction whereas it decreases behind
it. This effect can be explained by looking at the radial displacement field that increases towards the
inlet. Comparing the pressure distribution at t = 0.4 and t = 0.6, one can observe that the pressure
drop decreases. Thus the “stored” mass is put back into the system, what can be seen by the fact that
the radial wall velocities are negative. The inflow velocity reaches its minimum at t = 6. In contrast
to t = 0, at this point velocity magnitudes are significantly higher behind the obstruction compared to
those in front.
Thus an interplay of increasing velocities in the front of the obstruction and decreasing behind alongside with growing pressure gradients and vis versa, can be clearly shown and is physically in general.
Looking at CSF flow in the subarachnoid space this does not perfectly fit anymore. Deformability only
plays a role when looking at the pressure drop and does not have these huge effects to the velocity as
observed in this model. There might be several reasons for the strong effects on the velocity field. One
might be that the simplified model is too simple in order to translate effects to real live. Furthermore
the choice of the material parameters of the tissue might not be correct. As already mentioned before
there is a huge variation and discussion about that in literature. For a more detailed analysis further
simulations have to be done.
Numerical Behaviour and Results
31
(a) t = 0
(b) t = 0.2
(c) t = 0.4
(d) t = 0.6
Figure 4.14: Velocity field in idealized subarachnoid space model with glyphs prescibing the wall velocity scaled by a factor of 3 in the third periodic cycle.
Numerical Behaviour and Results
32
(a) t = 0
(b) t = 0.2
(c) t = 0.4
(d) t = 0.6
Figure 4.15: Radial displacement field in idealized subarachnoid space model in the third periodic cycle.
Numerical Behaviour and Results
33
(a) t = 0
(b) t = 0.2
(c) t = 0.4
(d) t = 0.6
Figure 4.16: Pressure field in idealized subarachnoid space model in the third periodic cycle.
Summary and Discussion
34
5. Summary and Discussion
The goal of this thesis was a numerical investigation of the stability and feasibility of the CMM-FSI
method for different fluid solver schemes with applications to physiological problems. CMM-FSI was
applied to a SUPG stabilized FEM and to IPCS. Both schemes were implemented with FEniCS. Furthermore, two different constitutive equations for the wall stress tensor were discussed as well as two
different initial conditions.
First, stability and feasibility tests were performed for an idealized artery model with steady inflow for
the SUPG stabilized FEM. Zero initialization for the displacement field as well as a hydrostatic pressure displacement initialization converged. However, the hydrostatic approach seems more advanced,
since it has a faster convergence rate and promises to be more stable for complex problems. In the next
step, the two constitutive equations were compared for the same model. In contrast to the membrane
approach, instabilities emerged for Hooke’s law. This could be explained by the presence of unphysical
normal derivatives that are eliminated in the membrane model. In order to examine a more realistic problem, we changed the steady inflow velocity to a sinusoidal one. The results showed physical
relations between pressure, velocity and displacement. Furthermore, the well-known phenomena of
storing mass during increasing pressure phases and thus maintaining higher velocities during pressure
decrease could be observed. Therefore, it can be concluded that CMM-FSI applied to SUPG stabilized
FEM yields stable results.
Subsequently, the IPCS approach was tested for the same steady model. The solution was not physical. Several changes to the initial conditions were made without any positive outcome. It is not clear
why this method does not work. A possible reason could be that the constant pressure in the tentative
velocity step causes unphysical results, because of the strong interaction of pressure, velocity and displacement. Another possible explanation is that no additional terms are added at the boundary for the
velocity correction whereas no-slip conditions are applied here for rigid computations.
In order to analyse the feasibility of the SUPG stabilized FEM applied to CMM-FSI, this method was
tested on an idealized model representing obstructions in the cervical spinal subarachnoid space. The
obtained results showed physical behaviour but a tremendous effect due to deformability could be observed that does not occur like this in real life. It is not clear whether this is caused by inappropriate
material parameters of the wall or whether the model might be too idealized to be compared with reality.
All in all, the results of this thesis show that CMM-FSI implemented with SUPG stabilized FEM yields
stable solutions for simplified test cases. However, it is not applicable to IPCS.
5.1. Outlook
Further work should first of all concentrate on improving the implemented SUPG stabilized FEM method
to make it more efficient. This is necessary to run patient-specific simulations. A huge aspect is that up to
now this method does not run in parallel. To achieve this the computation of the element level stabilization parameters has to be parallelized. Furthermore only direct solvers have been used so far. Therefore
extra work has to be done for preconditioning of the method to be able to apply iterative solvers. A
potentially more efficient method could then be used for more complex models. In the context of CSF
flow in the cervical spinal subarachnoid space, the material parameters must be analysed in more detail
before using this approach in patient-specific models.
Furthermore, a detailed analysis of the equations for IPCS should be done in order to get a better understanding of what causes the unphysical behaviour of the system. This might also help to find measures
to ensure the stability of this method.
Acknowledgments
35
6. Acknowledgments
First of all I have to thank Kent-Andre Mardal from Simula Research Laboratory for letting me work
in his group. He was my supervisor at Oslo as well and one could not possbibly imagine a better one.
He had always an open door for discussions and guidance when needed but still allowing me to work
independent. Within this context I also have to thank the whole BioComp group at Simula that accepted
and treated me like an equal member from the beginning. The daily discussion gave me new insights
that go way beyond this thesis.
I wish to thank Prof. Marek Behr and Lutz Pauli from the Chair for Computational Analysis of Technical
Systems for giving me the opportunity to write my thesis abroad by agreeing to be my supervisors at
RWTH Aachen University, as well as final guidence at the end of the thesis.
Last but not least, what is Norway in the winter without the perfect skiing technique? In this context
I would like to thank all friends and collegues for all the nice moments during my stay, that made it
unforgettable.
Acknowledgments
36
References
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Method. Springer, 2012.
[2] C. Figueroa, I.E. Vignon-Clementel, K.E. Jansen, T.J.R. Hughes, C.A. Taylor. A coupled momentum
method for modeling blood flow in three-dimensional deformable arteries. Computer Methods in
Applied Mechanics and Engineering, 2006.
[3] C.A. Brebbia. Modelling in Medicine and Biology VIII. WIT, 2009.
[4] D.M. Sforza, C.M. Putman and J.R. Cebral. Computational fluid dynamics in brain aneurysms.
International Journal for Numerical Methods in Biomedical Engineering, 28:801–808, 2012.
[5] J. Donea and A. Huerta. Finite Element Methods for Flow Problems. John Wiley & Sons, 2003.
[6] E. Magosso, A. Feliciani, and M. Ursino. A mathematical model of cardiovascular response to
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[7] Ø. Evju. Sensitiviy analysis of simulated blood flow in cerebral aneurysms. Master Thesis, University of
Oslo, 2011.
[8] T.J.R. Hughes F. Shakib and Z. Johan. A new finite element formulation for computational fluid
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[9] K. Valen-Sendstad, A. Logg, K.-A. Mardal, H. Narayanan, and M. Mortensen. A comparison of
some common finite element schemes for the incompressible Navier-Stokes equations. In: Automated Solution of Differential Equations by the Finite Element Method, ed. by Logg, Anders and Mardal,
Kent-Andre and Wells, Garth N.. Springer, vol. 84, chap. 21, pp. 395-418. Lecture Notes in Computational
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[10] K. Valen-Sendstad, K.-A. Mardal, M. Mortensen, B. Anders P. Reif and H.P. Langtangen. Direct numerical simulation of transitional flow in a patient-specific mca aneurysm. Journal of Biomechanics,
44, 2011.
[11] K.H. Støverud, H.P. Langtangen, V. Haughton, K.A. Mardal. CSF pressure and velocity in obstructions of the subarachnoid spaces. NRJ Digital - The Neuroradiology Journal, 3:163–171, 2013.
[12] M.S. Alnæs and A. Logg. Ufl specification and user manuel. URL http://fenicsproject.org, 2009.
[13] N. Shaffer, B. Martin, F. Loth. Cerebrospinal fluid hydrodynamics in type i chari malformation.
Neurological Research, 2011.
[14] G. Rutkowska. Computational Fluid Dynamics in Patient-Specific Models of Normal and Chiari I Geometries. Master Thesis, University of Oslo, 2011.
[15] W. Schröder. Fluidmechanik. Aachener Beiträge zur Strömungsmechanik, third edition, 2010.
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Appendix
A. Appendix
A.1. Computation of modified stress tensor for a membrane approach
# Compute f a c e t n o r m a l v e c t o r
n = FacetNormal ( mesh )
norm_n = ( n [ 0 ] ∗ n [ 0 ] + n [ 1 ] ∗ n [ 1 ] + n [ 2 ] ∗ n [ 2 ] ) ∗ ∗ ( 1 / 2 . 0 )
n = 1/norm_n∗n
# Compute f i r s t t a n g e n t v e c t o r
t 1 = a s _ v e c t o r ( [ n[1] −n [ 2 ] , n[2] −n [ 0 ] , n[0] −n [ 1 ] ] )
norm_t1 = ( t 1 [ 0 ] ∗ t 1 [ 0 ] + t 1 [ 1 ] ∗ t 1 [ 1 ] + t 1 [ 2 ] ∗ t 1 [ 2 ] ) ∗ ∗ ( 1 / 2 . 0 )
t 1 = 1/norm_t1 ∗ t 1
# Compute s e c o n d t a n g e n t v e c t o r
t2 = as_vector ( [ n[1]∗ t1 [2 ] − n[2]∗ t1 [ 1 ] , n[2]∗ t1 [0 ] − n[0]∗ t1 [ 2 ] , n[0]∗ t1
[1] − n[1]∗ t1 [ 0 ] ] )
norm_t2 = ( t 2 [ 0 ] ∗ t 2 [ 0 ] + t 2 [ 1 ] ∗ t 2 [ 1 ] + t 2 [ 2 ] ∗ t 2 [ 2 ] ) ∗ ∗ ( 1 / 2 . 0 )
t 2 = 1/norm_t2 ∗ t 2
# Computation o f m o d i f i e d e p s i l o n
def e p s i l o n ( u ) :
print " blabla "
e = grad ( u )
ev = a s _ v e c t o r ( [ dot ( dot ( t1 , e ) , t 1 ) , \
dot ( dot ( t2 , e ) , t 2 ) , \
0∗ dot ( dot ( n , e ) , n ) , \
dot ( dot ( t1 , e ) , t 2 ) + dot ( dot ( t2 , e ) , t 1 ) , \
0∗ dot ( dot ( t1 , e ) , n ) + dot ( dot ( n , e ) , t 1 ) , \
0∗ dot ( dot ( t2 , e ) , n ) + dot ( dot ( n , e ) , t 2 ) ] )
r e t u r n ev
k = 5.0/6.0
# C o m p u t a t i o n o f membrane s t r e s s t e n s o r
def sigma_s ( u ) :
D _ c o e f f i c i e n t = E/(1 − nu_s ∗ nu_s )
D = as_matrix ( [
[ 1 , nu_s , 0 , 0 , 0 , 0 ] ,
[ nu_s , 1 , 0 , 0 , 0 , 0 ] ,
[0 ,0 ,0 ,0 ,0 ,0] ,
[ 0 , 0 , 0 , 0 . 5 ∗ ( 1 − nu_s ) , 0 , 0 ] ,
[ 0 , 0 , 0 , 0 , 0 . 5 ∗ k∗(1− nu_s ) , 0 ] ,
[ 0 , 0 , 0 , 0 , 0 , 0 . 5 ∗ k∗(1− nu_s ) ] ] )
D = D _ c o e f f i c i e n t ∗D
eps = e p s i l o n ( u )
s = dot (D, eps )
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Appendix
return s
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