artery project

8/2/2019 Artery Project

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Notes on Biofluid Dynamics

Annunziato Siviglia

Laboratory of Applied MathematicsUniversity of Trento, [email protected]

http://sites.google.com/site/nunziosivigliaita/home

Academic year 2010/2011

Annunziato Siviglia (University of Trento) Biofluid dynamics Academic year 2010/2011 1 / 61


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Table of contents I

1 Introduction

2 Mathematical Model

Review of the basic equationsTube lawviscous termsAssumptionsEquationsConservative formulationEigenstructure and characteristic fields

3 The Riemann ProblemMotivation for studying the Riemann ProblemPhysiological flow assumptionsWave patternsJump across shocks: Rankine-Hugoniot conditionsJump conditions across rarefactions

Solution of Riemann problemSample solutions

4 Conservative methodsIntegral form of hyperbolic systemsConservative numerical methodsStability conditionsSystem of governing equations

Boundary conditionsAnnunziato Siviglia (University of Trento) Biofluid dynamics Academic year 2010/2011 2 / 61


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Table of contents II

5 Notes on codingCode scheme

6 References



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Introduction

One dimensional blood flow model:analysis and exact solutions


I d i


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Introduction

Introduction

The theoretical study of blood flow phenomena in humans through mathematical models isclosely related to the study of flow of an incompressible liquid in thin-walled collapsible tubes. Infact the applicability of theoretical models for thin-walled collapsible tubes covers a wider variety

of physiological phenomena as well the design of clinical devises for practical medical applications.In this notes we are interested in theoretical models for blood flow in medium to large arteriesregarded as thin-walled collapsible tubes. We centre our attention on one-dimensional,time-dependent non-linear models. Classical works on this subject are, for example, Kamm andShapiro (1979) and Pedley (1980) and the many references therein. For more recent works seeAlastruey et al. (2007), Brook et al. (1999), Canic et al. (2006), Formaggia et al. (2003), Fullanaand Zaleski (2009), Quarteroni et al. (2000) and Sherwin et al. (2003) to name but a few.


M the ti l M del


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Mathematical Model

Mathematical Models

Consider the geometric situation described in Fig.(1), which depicts a model for a blood vessel.The mathematical model will assume one-dimensional flow in the axial direction x.

Figure: Assumed axially symmetric vessel configuration in three space dimensions at time t. Cross sectional

area A(x,

t) and wall thickness h0(x) are illustrated.


Mathematical Model


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Mathematical Model

Review of the basic equations

The basic equations for the flow of blood in medium-size to large arteries and veins are obtainedfrom the principles of conservation of mass

tA + x(uA) = 0 (1)

and momentum

t(uA) + x(Au2) + A

xp = Ru . (2)

is the density of blood (assumed constant;

R > 0 is the viscous resistance of the flow per unit length of the tube, assumed to be known

We assume = 1 in the momentum equation (2).


Mathematical Model


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Mathematical Model

Review of the basic equations

There are two governing partial differential equations, (1), (2), and four unknowns,

A(x, t) the cross-sectional area of the vessel or tube at position x and time t;

u(x, t) the averaged velocity of blood at a cross section;

p(x, t) pressure;

R viscous terms.

An extra relation is required to close the system. This is provided by the tube law, which relatesthe pressure p(x, t) to the wall displacement via the cross-sectional area A(x, t). The tube law

couples the elastic properties of the vessel to the fluid dynamics and is analogous to theequation

of state in gas dynamics Toro (2009).


Mathematical Model


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Mathematical Model

Tube law

We adopt a very simple tube law of the form

p = pe(x) + (A; K) , (3)

where(A; K) = p pe ptrans (4)

is the transmural pressure, the difference between the pressure in the vessel, the internal pressure,and the external pressure. Here we choose

(A; K) = K(x)

AA0

m 1 , (5)with

K(x) =

(1 2)E(x)h0(x)

A0(x). (6)

h0(x) is the vessel thickness;A0(x) is the cross-sectional area of the vessel at equilibrium (ptrans = 0);

E(x) is the Youngs modulus;

is the Poisson ratio;

m > 0 is the exponent of the tube law.

For arteries usually m is taken equal to12 .



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Mathematical Model


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Viscous terms

If we assume a Poiseuille flow distribution we end up with:

R = 22 ( = 0.0035Pa/s) (7)


Mathematical Model


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Assumptions

We consider a simple mathematical model consisting of the partial differential equations (1)-(2),along with the tube law (3)-(5) with m > 0. We assume:

h0, E, A0, pe constants;

Then the pressure gradient in (2) is

xp = AxA + KxK , (8)

with

A =

A =

mK

A A

A0m

, K = 0 . (9)


Mathematical Model


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Equations

The complete system reads

tA + x(uA) = 0 ,

t(uA) + x(Au2) +A

AxA = Ru .

(10)

The system (10) in quasi-linear form reads

tQ + A(Q)xQ = S(Q) , (11)

where

Q = q1q2

A

Au , S(Q) = s1

s2

0

Ru , A(Q) = 0 1

A

A u2

2u . (12)

Next we study some mathematical properties of the equations.


Mathematical Model


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Conservative formulation

It is possible to express the equations in conservation-law form. The equations inconservation-law form are

tQ + xF(Q) = S , (13)

in terms of the redefined vector of conserved variables

Q =

q1q2

A

Au (14)

flux vector

F(Q) =

f1f2

Au

Au2 + BAm+1

, (15)

and source terms

S = s1s2

0

Ru (16)

with

B =mK

(m + 1)Am0: constant . (17)


Mathematical Model


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Eigenstructure and characteristic fields

The eigenstructure of the first-order system (11), (12) is that of the principal part of the system

and is given by the eigenvalues and corresponding eigenvectors.Proposition 3.1. The eigenvalues of (11) are all real and given by

1 = u c , 2 = u + c , (18)

where

c =A

A (19)

is the wave speed, analogous to the sound speed in gas dynamics Toro (2009).Proof. By definition the eigenvalues of system (11), (12) are the eigenvalues of the matrix A,which in turn are the roots of the characteristic polynomial

P() = Det(A I) = 0 , (20)

where I is the identity matrix and is a parameter. Simple calculations give

P() =

2 2u + u2 c2 = 0 ,from which the result (18) follows.


Mathematical Model


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Proposition 3.2. The right eigenvectors of A corresponding to the eigenvalues (18) are

R1 = 1

1

u c

, R2 = 2

1

u + c

, (21)

where 1 and 2 are arbitrary scaling factors.Proof. For an arbitrary right eigenvector R = [r1, r2]T we have

AR = R , (22)

which gives the algebraic system

r2 = r1 ,(u2

c2)r1 + 2ur2 = r2 . (23)

By substituting in (23) by the appropriate eigenvalues in (18) in turn we arrive at the soughtresult.


Mathematical Model


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Proposition 3.3. The 1- and 2- characteristic fields are genuinely non-linear if the tube lawexponent m > 0.Proof. Some algebraic manipulations give

1R1 = m(m + 2)K A

A0m

2A

mK

AA0

m , 2R2 = m(m + 2)K A

A0m

2A

mK

AA0

m .

Therefore the 1(Q)- and 2(Q)- characteristic fields are genuinely non-linear provided m = 2,and the proof is complete.


Mathematical Model


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The Riemann Problem

Assume an artery that a certain time has two uniform areas, at rest (velocity is equal 0),separated by thin layer of material at position x = 0. If the thin layer separating the two uniform

areas is suddenly removed, two dominant features emerge from the process in the form of waves.

0.5 0 0.5

0.02

0.01

0

0.01

0.02

x [m]

Vesseldiameter[m]

0.5 0 0.51

0.5

0

0.5

1

x [m]

uvelocity[m/s]

Figure: Initial conditions for the Riemann Problem: time =0.0 s


Mathematical Model


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The Riemann Problem

A right facing wave travels into the smaller portion of the fluid rising the area abruptly. The leftfacing wave travels into the largest region and has the effect of reducing the area. The details ofthe physical processes occuring in the vicinity of the wall shortly after the collapse of the wall are

indeed extremely complex and are not correctly modelled (not even remotely) by the modelemployed, but if the wall collapses in a sufficiently short time the wave pattern emerging is almostthat of a centred wave system with a left rarefaction wave and a right shock wave; such wavesystem may be approximated by the mathematical model that we have described previously.

0.5 0 0.5

0.02

0.01

0

0.01

0.02

x [m]

Vesseldiameter[m]

0.5 0 0.50

0.5

1

1.5

2

x [m]

uvelocity[m

/s]

0.5 0 0.50

0.1

0.2

0.3

0.4

0.5

x [m]

Fr[]

0.5 0 0.50

0.002

0.004

0.006

0.008

0.01

x [m]

time[s]



Mathematical Model


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The Riemann Problem



0.5 0 0.5

0.02

0.01

0

0.01

0.02

x [m]

Vesseldiameter[m]

0.5 0 0.50

0.5

1

1.5

2

x [m]

uvelocity[m

/s]

0.5 0 0.50

0.1

0.2

0.3

0.4

0.5

x [m]

Fr[]

0.5 0 0.50

0.005

0.01

0.015

0.02

0.025

0.03

x [m]

time[s]



Mathematical Model


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The Riemann Problem



0.5 0 0.5

0.02

0.01

0

0.01

0.02

x [m]

Vesseldiamet

er[m]

0.5 0 0.50

0.5

1

1.5

2

x [m]

uvelocity[m/s]

0.5 0 0.50

0.1

0.2

0.3

0.4

0.5

x [m]

Fr[]

0.5 0 0.50

0.01

0.02

0.03

0.04

0.05

x [m]

time[s]



The Riemann Problem


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The Riemann Problem

Here we pose and solve exactly the Riemann problem for system (11), that is the special Cauchyproblem with piece-wise constant initial condition, namely

tQ + A(Q)xQ = 0 , x R , t > 0 ,

Q(x, 0) =

QL if x < 0 ,

QR if x > 0 .

(24)

Figure: Structure of the solution of the Riemann problem (24) .


The Riemann Problem


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The Riemann Problem

The structure of the similarity solution of the problem is shown below in the entire x-t half plane.There are two wave families. The left family is associated with the eigenvalue 1, while the right

wave family is associated with 2. Waves associated with the genuinely non-linear characteristicfields 1 and 2 are either shocks (discontinuous solutions) or rarefactions (smooth solutions).The entire solution consists of three constant states, namely QL (data), Q, and QR (data),separated by two waves. The unknown states to be found are Q. If any of the 1 and 2 wavesis a rarefaction then there will be a smooth transition between two adjacent constant states. Inorder to solve exactly the entire initial-value problem we need to establish appropriate jumpconditions across each characteristic field to connect the unknown state Q to the initial

conditions QL (left) and QR (right) respectively. In what follows we establish such jumpconditions across each characteristic field.

Figure: Structure of the solution of the Riemann problem (24).


The Riemann Problem Motivation for studying the Riemann Problem

h Ri P bl


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The Riemann Problem

Why is relevant the study of the RiemannProblem

Because we can build exact solutions to compare with numerical ones;

Because it is usually used for building numerical methods.


The Riemann Problem Physiological flow assumptions

Th Ri P bl


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The Riemann Problem

Physiological flow assumptions

The cross-sectional area A remains bounded away from zero

0 < Amin A(x, t) Amax <

The blood velocity u is less than the wave propagation speed c, this implies:

1(A, uA) < 0 and 2(A, uA) > 0


The Riemann Problem Wave patterns

Th Ri P bl


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The Riemann Problem

Under the physiological flow assumptions, only the following four wave patterns are available:

Figure: Possible wave patterns under physiological conditions.



El t l ti f th Ri P bl


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Elementary wave solution of the Riemann Problem

In this section we study the much simpler case in which the initial data states for the Riemannproblem are connected by a single wave, that is, the solution of the Riemann problem consists ofa single non-trivial wave; all other waves are assumed to have zero strength. This assumption is

entirely justified as we can always solve the Riemann problem with general data and then selectthe constant states on either side of a particular wave as the initial data for a Riemann problem.In later slides we consider the general case in which all types of waves might simultaneously bepresent in the solution of the Riemann problem. First we recall some useful mathematical waverelations.



Generalized Riemann invariants


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The generalized Riemann invariants are relations that are valid across simple waves. These aremost conveniently expressed as a set of ordinary differential equations in phase space, see Jeffrey(1976) for details.Proposition 3.4. For a given hyperbolic system of M unknowns [w1, w2, . . . , wM]

T, for any

i-characteristic field with right eigenvector Ri = [r

1i,r

2i, . . . ,r

Mi]

T

the generalized Riemanninvariants are solutions of the following M 1 ordinary differential equations in phase spacedw1

r1i=

dw2

r2i= . . .

dwn

rni. (25)

Proof. (omitted). See Jeffrey (1976).





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Proposition 3.5. The generalized Riemann invariants for 1 = u c is

2m

c + u = constant . (26)

Proof. Application of (25) from Proposition 3.4 to = 1 = u c with R1 = 1[1, u c]T, with1 = 1 gives

dA

1=

d(Au)

u

c. (27)

Manipulating the equality we obtain

c(A)

AdA + du = 0 .

Integration in phase space gives

c(A)

AdA + u = constant .

After expressing c(A) explicitly as a function of A and performing exact integration leads to thesecond result in (26).



Rankine Hugoniot jump conditions


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Rankine-Hugoniot jump conditions

An important concept that is applicable to discontinuous solutions of hyperbolic conservationlaws is that of Rankine-Hugoniot jump conditions or simply Rankine-Hugoniot conditions. Theyapply to a discontinuous wave travelling with speed S, which is related to jumps in conservedvariables Q and fluxes F(Q) across the wave as follows

Fahead Fbehind = S(Qahead Qbehind)

Here subscript ahead denotes the state immediately ahead of the discontinuity and behinddenotes the state immediately behind of the discontinuity.



Jump across shocks: Rankine-Hugoniot conditions


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In studying shock waves, one requires the conservative formulation of the equations to enforce theRankine-Hugoniot conditions. We can still provide the data in terms of the primitive variables.In the case of a left shock the idea is to find a relationship between left QL data and Q whichare the right data and for a Right shock the idea is to find a relationship between the right dataQR and the left data Q.

Figure: Single wave solution of the Riemann problem is an isolated left (left panel) and right (right panel) shock





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Jump across shocks: Rankine Hugoniot conditions

Proposition 4.1 (left shock). If the left 1-wave is a left-facing shock wave of speed SL then

u = uL

fL , fL =B(A AL)(Am+1 Am+1L )

ALA(28)

and the shock speed is given as

SL = uL ML

AL, ML =

BALA(A

m+1 Am+1L )

A AL. (29)

Proof. In Fig. 8 we illustrate the function fL that connects the velocity uL to the left data state

and the unknown AL. Let us assume that the left 1-wave is a left-facing shock wave of speedSL. We need to establish relations across the shock, for which one uses standard techniques, seeToro (2001) and Toro (2009). We first transform the equations to a stationary frame via

uL = uL SL , u = u SL . (30)Then the jump conditions become

Au = ALuL ,

Au2 + BA

m+1 = ALu

2L

+ BAm+1L

.

(31)

Now from the first equation in (31) define ML = ALuL = ALuL. In fact this is the mass fluxthrough the wave, which is constant. Use of ML into the second equation in (31) followed bysuitable manipulations leads to the sought relations (28). Details of the calculation of the shock

speed SL are omitted.Annunziato Siviglia (University of Trento) Biofluid dynamics Academic year 2010/2011 32 / 61




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Jump across shocks: Rankine Hugoniot conditions

Proposition 4.2 (right shock). If the right 3-wave is a right-facing shock wave of speed SR then

u = uR + fR , fR =

B(A AR)(Am+1 Am+1R )

ARA(32)

and the shock speed is given as

SR = uR +MR

AR, MR =

BARA(Am+1 Am+1R )A AR

. (33)

Proof. The proof follows the same methodology as for a left shock and details are thus omitted.

A right shock occur when the characteristic coming from the left travels faster than characteristic

travelling from the right side of the shock, i.e.:

c > cR A > AR



Jump conditions across rarefactions


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It is possible to establish jump relations across rarefactions waves by means of generalizedRiemann invariants.Proposition 4.3 (left rarefaction wave). Across a left rarefaction wave associated with thecharacteristic field 1 = u c the following relations hold

u = uL fL , fL =2

m(c cL) . (34)

Proof. From the left generalized Riemann invariants (26) we can write

2

mc + u =

2

mcL + uL ,

from which we have

u = uL (2

m c 2

m cL)

and the result follows.





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Proposition 4.4 (right rarefaction wave). Across a right rarefaction wave associated with thecharacteristic field 3 = u+ c the following relations hold

u = uR + fR , fR =2

m(c cR) . (35)

Proof. The proof uses the right generalized Riemann invariants (26) and is entirely analogous tothe previous case.

A right rarefaction occur when the characteristic coming from the left travels faster thancharacteristic travelling from the right side of the shock, i.e.:

c

cR

A

AR



Solution of Riemann problem


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p

In previous sections we have put in place all the necessary relations to obtain the solution of theRiemann problem in the Star Region, which is the region in physical space located between the

waves associated with the outer characteristic fields. The procedure to find the solution isembodied in the following proposition.



Solution of Riemann problem


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p

Proposition 4.6 (solution of Riemann problem). The solution of the Riemann problem in theStar Region is given by the solution of the following non-linear system

f1(x1, x2) = x2

uL + fL(x1) = 0 ,

f2(x1, x2) = x2 uR fR(x1) = 0 , (36)where the unknowns of the problem are

X = [x1, x2] [A, u] , (37)with

fL(x1) =B(x1AL)(xm+11 Am+1L )

ALx1if A > AL ,

2m

Dx

m/21 cL

if A AL ,(38)

and

fR(x1) =

B(x1AR)(x

m+11 A

m+1R

)

ARx1if A > AR ,

2m

Dx

m/21 cR

if A AR ,(39)

D =

mK

Am0. (40)

The wave speeds cL and cR are evaluated on the data according to (19) and B is given by (61).

Proof: The proof involves putting together the results stated previously. Details are omitted.Annunziato Siviglia (University of Trento) Biofluid dynamics Academic year 2010/2011 37 / 61


Sample solutions


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p

We consider three test problems for a long, straight tube of length 1 m, of constant equilibriumcross sectional area A0 = 2.1124 104 m2, of wall thickness h0 = 8.2 104 m, Youngsmodulus E = 3.0 105 N/m2, exponent in tube law m = 1

2and Poisson ratio = 1

2. The

coefficient K is taken asK0 =

43

E(x)h0A0

,

while blood density is = 1050kg/m3 .



Shock-Shock case


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The initial values are:

AL = 3.42E 5 m2 uL = 1.2 m/s AR = 2.8E 5 m2 uR = 0.2 m/s;

0.5 0 0.5

6

4

2

0

2

4

6

x 103

x [m]

Vesseldiameter[m]

0.5 0 0.50.2

0.4

0.6

0.8

1

1.2

x [m]

uvelocity[m/s]

0.5 0 0.5

0.1

0.2

0.3

0.4

x [m]

u/c[]

0.5 0 0.50

0.02

0.04

0.06

0.08

x [m]

time[s]

Figure: Exact solution for the Shock-Shock case at the output time 0.08 s



Rarefaction-Shock case


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0.5 0 0.50.02

0.01

0

0.01

0.02

x [m]

Vesseldiameter[m]

0.5 0 0.5

0.6

0.8

1

1.2

x [m]

uvelocity[m/s]

0.5 0 0.50.1

0.15

0.2

0.25

x [m]

u/c[]

0.5 0 0.50

0.02

0.04

0.06

0.08

x [m]

time[s]

Figure: Exact solution for the Rarefaction-Shock case at the output time 0.08 s



Rarefaction-Rarefaction case


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0.5 0 0.50.02

0.01

0

0.01

0.02

x [m]

Vesseldiameter[m]

0.5 0 0.55

0

5

x [m]

uvelocity[m/s]

0.5 0 0.5

1

0.5

0

0.5

1

x [m]

u/c[]

0.5 0 0.50

0.005

0.01

0.015

0.02

0.025

0.03

x [m]

time[s]

Figure: Exact solution for the Rarefaction-Rarefaction case at the output time 0.03 s


Conservative methods Integral form of hyperbolic systems

Integral form


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Consider the following one-dimensional system of conservation laws:

Qt + F(Q)x = 0

This is called the differential form of the conservation laws and is valid only for the case in whichthe solution is smooth throughout. In the presence of discontinuities, one must use the integralform.

[Qdx F(Q)dt] = 0 (41)

where the line integration is performed along the boundary of the domain in counterclokwisemanner.


Conservative methods Integral form of hyperbolic systems

Integral form of hyperbolic systems


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We choose a rectangular control volume Vi in

the x t plane (see the figure), of dimension[xi1/2, xi+1/2] [tn, tn+1]

By integrating along the boundary of the control

volume:Figure: Control volume in the x t space.

xi+1/2

xi1/2

[Q(x, tn)dx +

tn+1

tn

[

F(Q(xi+1/2, t))]dt +

xi1/2

xi+1/2

[Q(x, tn+1)dx +

tn

tn+1

[

F(Q(xi1/2, t))]dt = 0

xi+1/2xi1/2

[Q(x, tn+1)dx =

xi+1/2xi1/2

[Q(x, tn )dx

tn+1tn

F(Q(xi+1/2, t))dttn+1tn

F(Q(xi1/2, t))dt

(42)


Conservative methods Conservative numerical methods

Derivation of conservative numerical formulation


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Let us now define the mesh size xi and the time step t:

xi = xi+1/2 xi1/2, t = tn+1 tn

We divide through xi and multiply the whole of the second term within the square brackets onthe right-hand side by t/t:

1

xi

xi+1/2xi1/2

Q(x, tn+1)dx =1

xi

xi+1/2xi1/2

[Q(x, tn)dx txit

tn+1tn

F(Q(xi+1/2, t))dttn+1tn

F(Q(xi1/2

(43)We define integral averages of Q(x, t) at times t = tn+1 and t = tn respectively over the lengthxi (it is a length in the one-dimensional case, in general it is a volume):

Qn+1i

=1

xi

xi+1/2

xi1/2

Q(x, tn+1)dx, Qni =1

xi

xi+1/2

xi1/2

Q(x, tn)dx (44)

We also define time integral averages of the flux F(Q) at position x = xi+1/2 and x = xi1/2:

Fi+1/2 =1

t

tn+1tn

F(Q(xi+1/2, t))dt, Fi1/2 =1

t

tn+1tn

F(Q(xi1/2, t))dt (45)



Conservative numerical methods


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Substituting the definitions (44) and (45) in the equation (43), we obtain the formula thatconstitute the basis of conservative numerical methods:

Qn+1i = Q

ni

t

xi

Fi+1/2 Fi1/2 (46)So far the derived expressions are exact expressions and do not involve numerical approximations.However the formula (44) can be interpreted in a numerical sense, if the complete spatial domainis discretized into a set of control volumes Ii = [xi1/2, xi+1/2] called cells, i = 1, 2, ..., m. Theterm Fi+1/2 is called the inter-cell numerical flux corresponding to the inter-cell boundary atx = xi+1/2 between cells i and i + 1 . In general the numerical flux is of the form

Fi+1/2 = Fi+1/2

QniKL

, ..., QniKR

, where the non-negative integers KL and KR depend on the

particular choice of the numerical flux.

The conservative numerical methods for solving the general initial-boundary value problem for thehyperbolic system are of two distinct classes:

the upwind methods, or Godunov-type methods, that use the wave propagation information,essential property of the hyperbolic partial differential equations, to construct the numericalschemes;

the centred methods, that do not explicitly require the provision of wave propagationinformation to construct the numerical schemes, i.e. they require the explicit solution of theRiemann Problem..



Centred methods: Lax-Friedrichs


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Here we present schemes that do not require the (explicit) solution of the Riemann problem.These schemes are not biased by the wave propagation direction, which distinguishes upwindmethods, and are called centred or symmetric schemes.

Numerical schemes may be obtained from the conservative formula(46) by giving appropriatedefinitions for the inter-cell flux F

i+ 12. The Lax-Friedrichs method results if we choose:

FLFi+ 1

2

= 12

F(Qni+1) + F(Q

ni ) 1

2xt

Qni+1 Qni

(47)

This first-order accurate scheme is exceedingly simple to implement but, as is well known, is toodiffusive to be used in practical computations. The scheme is monotone and has linearisedstability condition:

t

CFL x

maxwith CFL = 0.9 in practical computations.



Centred methods: Lax-Wendroff


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The Lax-Wendroff method results if we choose:

FLWi+ 1

2

= F

Q

n+ 12i+ 12

, (48)

with

Qn+ 1

2

i+

1

2

=1

2

(Qni + Qni+1)

1

2

t

xF(Qni+1) F(Q

ni ) , (49)

The two-step Lax-Wendroff scheme is not monotone This scheme is second-order accurate and isnot monotone and thus produces spurious oscillations in the vicinity of high gradients, such as inshock waves. It has linearised stability condition:

t CFL xmax

with CFL =0.9 in practical computations.



Centred methods: FORCE


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Another possible choice of flux is that of the force scheme (First ORder CEntred) proposed by

Toro (2001); this may be written as:

FFORCEi+ 1

2

=1

2

FLF

i+ 12

+ FLWi+ 1

2

, (50)

The force scheme is first-order accurate and has half the numerical viscosity of the Lax-Friedrichsmethod. The scheme has been proved to be monotone and stable, with linearised stability

conditiont CFL x

max

with CFL =0.9 in practical computations. The FORCE scheme can be written in a two-stepstaggered grid version as

Qn+ 12

i+12

=1

2(Qn

i

+ Qn

i+1

)

1

2

t

xF(Qn

i+1

)

F(Qn

i

) , (51)Qn+1

i=

1

2

Q

n+ 12

i 12

+ Qn+ 1

2

i+ 12

1

2

t

x

F(Q

n+ 12

i+ 12

) F(Qn+12

i 12

)

, (52)


Conservative methods Stability conditions

Stability condition


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To enforce the stability condition, we need explain the meaning of the Courant-Friedrichs-Lewynumber or CFL number, or Courant number. It is defined as:

CFL =

t

x

(53)

To comprise the physical and numerical meaning of this number, we consider the update formula

(46), for a generic conservative numerical method, applied to the linear advection equation:

qn+1i = qni

t

xi

fi+1/2 fi1/2

= qni t

xi

qi+1/2 qi1/2

= qni t

xi

qi+1/2 qi1/2

Note that CFL is a dimensionless quantity, it is the ratio of two velocities:

CFL =

t

x =

x/t =

speed in PDE

speed of mesh (54)



Stability condition


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If we impose a value of the CFL coefficient, we enforce the extension of the numerical domain ofdependency, related to the true domain of dependency.The stability of a method depends on the relation between the two domains. The stabilitycondition of all the centred scheme presented in the previous slides is:

CFL 1, (55)

which means that the speed of the mesh must be grater than the speed of the PDE. Accountingfor the numerical approximations, we take CFL 0.9 Therefore in the numerical solution we mustdetermine the maximum value of the time step t according to the stability condition as:

t =CFLx

max, (56)

where max is the maximum value of in the cells at the time step that we consider.



System of governing equations


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The equations in conservation-law form that we want to solve are

tQ + xF(Q) = S , (57)

in terms of the redefined vector of conserved variables

Q =

q1q2

A

Au

(58)

flux vector

F(Q) =

f1f2

AuAu2 + BAm+1

, (59)

and source terms

S =

s1s2

0Ru

(60)

with

B = mK(m + 1)Am0

( = 1050 kg/m3) , (61)

andR = 22 ( = 0.0035 Pa/s) , (62)



Domain discretization


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The domain is discretised in M volumes as in the figure below.

Figure: Discretisation of the domain.


Conservative methods Boundary conditions

Boundary conditions


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At each time step we update the value of the unknown in all cells from the time tn to the time

tn+1. The update formula in each cell requires the knowledge of the state of the nearby cells.Therefore the cells 1 and M require also the data in the cells 0 and M + 1 respectively.

Downstream Boundary conditions

Transmissive boundary conditions

A(M + 1) = A(M)

u(M + 1) = u(M)

Reflective boundary conditions

A(M + 1) = A(M)

u(M + 1) = u(M)



Boundary conditions

U B d di i


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Upstream Boundary conditions

Transmissive boundary conditions

A(0) = A(1)

u(0) = u(1)

Reflective boundary conditions

A(0) = A(1)u(0) = u(1)

Imposed Pressure Since the relationship between the pressure and area given by the tubelaw, imposition of pressure is equivalent to assign the value of the cross section in 0.

A(0, t) =(0, t)

K + 11/m

(63)

while the value u(0, t) is obtained solving the following compatibility (characteristic)equations:

( 2u) dAdt

+duA

dt+ Ru = 0 = u c (64)



Boundary conditions: solution strategy for the Imposed pressure case


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The system of governing equations (57) can be cast in the so called characteristic form (CE) as

( 2u)

dA

dt +duA

dt + Ru = 0 . (65)

These equations are valid along the characteristics:

dx

dt= = u c (66)

We recall that under physiological conditions we have that + = u + c > 0 and

= u c < 0,thus in the x t plane we have (see Figure):

Figure: Characteristic lines in the x t plane




I th f i d i th t ti k f th


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In the case of imposed pressure in the upstream cross section, we can make use of thecharacteristic equations (65) in order to obtain the value u(0):

A(0, t) is given by the pressure law;

u(0, t) is obtained from equations (65)In particular the u(0, t) is obtained using the characteristic (blu line in Figure below):

Figure: Upstream boundary condition in the x t plane

In the following we neglect viscous forces R = 0. Along the (blue line in the figure) we haveto solve the following equation:

( 2u) dAdt

+duA

dt= (u + c) dA

dt+

duA

dt= 0 (67)





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Equation (67) can be solved following these steps:

i A first value of xR [a, b], the position of the foot of the characteristic, can be obtained from

xR = xPt[(u c)]tn

0 (68)

where []tn0 means that the value in the square brackets is computed in the cross section 0at time tn. In case xR / [0, 1] we can use the following:

xR = xP

t[(u

c)]t

n

1 (69)

ii Values of A and u are computed in xR as a simple averages:

u(xR, tn ) = [u]n1 ([u]n1 [u]n0)

x1 xRx

A(xR, tn ) = [A]n1

([A]n1

[A]n0)

x1 xRx

iii Equation (67) is used in differential form in order to find the sought value at 0 , tn+1

[uA]tn+1

0 [uA]tn

xR [(u + c)]tnxR([A]t

n+1

0 [A]tn

xR) = 0


Notes on coding Code scheme

Code scheme


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The aim of this project is to write a code that determines the numerical solution of the 1D

equations governing the flow in a compliant vessel neglecting the dissipations (i.e. R = 0). Theprogram can be coded using Matlab or Fortran, or any other high-level language. The structureof the code can be the following:

Reading data related to the geometry of the problem, the initial conditions, and all theoptions concerning the numerical calculations (CFL, number of cells, time for output,Youngs modulus E, h0, A0, ...)

Definition of the initial conditions: we must define the variables involved in the problem asvectors: the elements are respectively the values of the variables at the initial time t = 0, foreach cell of the domain. In our problem we can define a vectorA = (A[0], A[1],..., A[i],..., A[M], A[M + 1]) and a vectoru = (u[0], u[1],..., u[i],..., u[M], u[M + 1]), of which the elements are the values of A and u

in the center of each volume 0, 1, 2, ..., i,..., M, M + 1.


Notes on coding Code scheme

Code scheme


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Time cycle: Here start the cycle operated on the time steps, from the initial value (at timet = 0) to the output time at which we desire the results.For each time step we need:

Boundary conditions Definition of the values of the variables in cells 0 and M + 1, as indicated inthe previous slides.

Stability conditions Evaluation of the value of the time step t, according to the stability condition

of the numerical method chosen;

Fluxes computations We define a flux vector F = (F[0], F[1], ..., F[i], ..., F[M], F[M]) in which F[i]represents the value of the flux at the interface + 12 . It is a function only of the old values (time t

n)of the unknowns. We can use any of the fluxes (47), (48), (50);

Update For each cell, the unknowns are updated from time t

n

, to the new value at t

n+1

using theUpdate formula (46).

Output: We can print the useful results obtained by the calculations and plotting them.


References

References I


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J. Alastruey, J. Parker, K. Peiro, S. Byrd, and S. Sherwin. Modelling the circle of Willis to assessthe effects of anatomical variations and occlusions on cerebral flows. Journal of Biomechanics,40:17941805, 2007.

B. Brook, S. Falle, and T. Pedley. Numerical solutions for unsteady gravity-driven flows incollapsible tubes: evolution and roll-wave instability of a steady state. J. Fluid Mech., 396:223256, 1999.

S. Canic, C. Hartley, D. Rosenstrauch, J. Tamba, G. Guidoboni, and A. Mikelic. Blood flow incompliant arteries: an effective viscoelastic reduced model, numerics, and experimentalvalidation. Annals of Biomedical Engineering, 34(4):575592, 2006.

L. Formaggia, D. Lamponi, and A. Quarteroni. One-dimensional models for blood flow in arteries.Journal of Engineering Mathematics,, 47:251276, 2003.

M. Fullana and S. Zaleski. NA branched one-dimensional model of vessel networks. J. FluidMech., 621:183204, 2009.

A. Jeffrey. Quasilinear Hyperbolic Systems and Waves. Pitman, 1976.

R. Kamm and A. Shapiro. Unsteady flow in a collapsible tube subjected to external pressure orbody forces. J. Fluid Mech., 95:178, 1979.

T. J. Pedley. The fluid dynamics of large blood vessels, Third Edition. Cambridge UniversityPress, 1980.

A. Quarteroni, M. Tuveri, and A. Veneziani. Computational vascular fluid dynamics: problems,models and methods. Computing and Visualization in Science, 2(4):16397, 2000.


References

References II


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S. Sherwin, L. Formaggia, J. Peiro, and V. Franke. Computational modelling of 1D blood flowwith variable mechanical properties and its application to the simulation of wave propagation inthe human arterial system. International Journal for Numerical Methods in Fluids, 43:673700,,2003.

E. F. Toro. Shock capturing methods for free surface shallow water flows, Third Edition. Wileyand Sons, 2001.

E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics, Third Edition.SpringerVerlag, 2009.


artery project

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