D Paper Imagawa

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TOHOKU UNIVERSITY

Graduate School of Engineering

Study of Measurement-Integrated Simulation Design

A dissertation submitted for the degree of Doctor of Philosophy

(Engineering)

Department of Bioengineering and Robotics

by

Kentaro IMAGAWA

17 January 2010


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Table of Contents

1

Table of Contents

List of Figures 3

List of Tables 4

1 Introduction 5

2 Eigenvalue Analysis of Linearized Error Dynamics of Measurement Integrated

Simulation13

2.1 Introduction 14

2.2 Formulation 15

2.2.1. Basic Equations 15

2.2.2 Eigenvalue Problem 17

2.3 Numerical Experiment 20

2.3.1 Method 20

2.3.2 Results and Discussion 22

2.4 Conclusions 30

3. Eigenvalue Analysis in Arbitrary Flow Geometries 33

3.1 Introduction 34

3.2 Formulation 36

3.2.1 Eigenvalue Analysis for Arbitrary Flow Geometries 36

3.2.2 UMI Simulation for Two-Dimensional Blood Flow in Aneurysmal Aorta 37

3.3 Numerical Experiment 39

3.4 Conclusions 44

4. Critical Feedback Gain of Measurement Integrated Simulation 454.1 Introduction 46

4.2 Mechanism of Critical Feedback Gain 46

4.3. Numerical Scheme without Limitation of Critical Feedback Gain 49

4.4. Numerical Experiment 49

4.4.1 Method 50

4.4.2 Comparison with Result of MI Simulation 51

4.5 Conclusions 54


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Table of Contents

2

5. Conclusions 55

Acknowledgements 58

References 59

Appendix 64

Appendix. A 64

Appendix. B 70


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List of Figures

3

List of Figures

Fig. 1.1 Block diagram of measurement integrated simulation 9

Fig. 2.1 Schematic diagram for projection of vector field 19

Fig. 2.2 Domain and coordinate system 21

Fig. 2.3 Eigenvalue distribution of ordinary simulation (case A) 23

Fig. 2.4 Eigenvalue distribution of MI simulations (ku=8) 24

(a) Case (B) 24

(b) Case (C) 24

Fig. 2.5 Root locus of the most unstable mode for cases (B) and (C) 25

Fig. 2.6 Distribution of eigenvectors for Case (A) (a in Fig.2.5) 26

Fig. 2.7 Distribution of eigenvectors for Case (C) with ku=8 (b in Fig.2.5) 27

Fig. 2.8Comparison of variation of error norm between eigenvalue analysis and

numerical simulation28

Fig. 2.9 Steady error norm with feedback gain 29

Fig. 2.10 Time constant with feedback gain 30

Fig. 3.1 Block diagram of ultrasonic-measurement-integrated simulation 35

Fig. 3.2 Computational grid 38

Fig. 3.3 Schmatic diagram of ultrasonic measurement 38

Fig. 3.4 Distribution of standard solution 40

Fig. 3.5 Eigenvalue distribution of ordinary simulation 41

Fig. 3.6Comparison of variation of error norm between numerical simulation (solid

lines) and eigenvalue analysis (broken lines)42

Fig. 3.6 Time constant with feedback gain 43

Fig. 4.1 Domain and coordinate system 50

Fig. 4.2Steady error norm with feedback gain with first-order implicit scheme for time

dependent term52

Fig. 4.3Steady error norm with feedback gain with second-order implicit scheme for

time dependent term53


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List of Tables

4

List of Tables

Table 1.1 Comparison with integrated methods 10

Table 2.1 Computational condition 21


Table 3.2 Relationship between feedback gain and the least stable eigenvalue 40



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Chapter 1. Introduction

5

Chapter 1

Introduction


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In flow analysis, experimental measurement [1-4] and numerical simulation [5-10] are commonly

used. Experimental measurement is a direct way to obtain the state of real flow phenomena. However,

it is difficult to obtain complete information on the flow state extending widely in the spatial and time

domains. On the other hand, numerical simulation provides the full state of flow phenomena on grid

points of the computational domain. However, it is difficult to reproduce the instantaneous state of a

real flow because of uncertainly in the initial and/or boundary conditions, unknown disturbances, or

sensitivity in initial conditions.

Since both methods have advantages and disadvantages for correct reproduction of real flows, a

method to combine them has been investigated in many fields. The 4-dimensional variational

assimilation (4D-Var) is widely used in weather prediction [11-21]. 4D-Var is the method for the

assimilation to obtain unsteady flow field to correspond spatially and temporally to unsteady

measurement data by minimization of the cost-function defined as the difference between the

measurement and simulation. In this method, the computation of the numerical model and that of

adjoint equation [38] are repeatedly solved to obtain the optimal solution. The adjoint equation

computes the gradient of the cost-function which is necessary for the minimization of the function.

This method can make assimilation accurately using the measurement data without linear relationship

to analytical variables and/or the measurement data at different times, considering the time evolution

based on the physical relationship. 4D-Var has been put into practical use for the weather prediction at

ECMWF (European Center for Medium-Range Weather Forecasts) [13], Metro-France[14], UK Met

Office[15] and JMA (Japan Meteorological Agency) [16], and also applied to reproduction analysis of

the weather phenomenon that actually happened [17]This method has been applied to the analysis of

the ocean analysis [18-20] and the clear air turbulence in airports [21]. The validity of 4D-Var has

been confirmed in many applications; however, the 4D-Var has an disadvantage that the computational

cost of 4D-Var is relatively high because the computation of the numerical model and that of the

adjoint equation are alternately repeated, and that the update of the adjoint model is necessary at the

update of the numerical model.

The Kalman filter has been widely used in flow problems [22-31]. The Kalman filter [22]

assimilates flow fields with the minimum error covariance between the measurement and the


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simulation by the linear unbiased estimate at every time step, and estimates the state of the system

sequentially by time-developing of the minimum error covariance. The Kalman filter is constructed by

the four assumptions; (1) the linearity of the system, (2) the white noise for the system noise and the

measurement error, (3) the Gaussian noise for the system noise and the measurement error, and (4) the

least square error criterion. In spite of these limitations, the validity of the Kalman filter has

been confirmed in many applications. On the other hand, for the nonlinear system such as the flow

analysis, the Extended Kalman filter (EKF) [23] and the Unscented Kalman filter [24] are applied. The

EKF is applied for the flow analysis with using the linearized Navier-Stokes equation as the model

[25]. However, for the case the large difference between the assumptions and the considerations of the

real problem, the Kalman filter needs the expansion or the revision of the theory. For example, the

EKF applied for a nonlinear system sometimes becomes unstable [26]. Although the theory of the

Kalman filter is established, the application for the large-dimensional model such as the flow analysis

is rather difficult due to large computational cost for the error covariance evaluation. Therefore, in the

field for the large-dimensional problem such as the weather prediction, the Ensemble Kalman filter

(EnKF) which evaluates the covariance matrix not from the model but from the ensemble calculation

[27, 28], the Local Ensemble Transform Kalman filter (LETKF) with using the localizing technique to

reduce the computational cost [29], and the extended method of the LETKF [30, 31] are used. In

Canada the EnKF is practically used for the weather prediction.

The theories of these methods have already been established, and 4D-Var and EnKF are

practically used for the weather prediction around the world. The comparative studies between 4D-Var

and EnKF were also done [38-42]. It was found that the analysis with using EnKF is more accurate

than that with 4D-Var in the case of using the exact model, but EnKF is sensitive to the model error

and it may be less accurate for the cases using real measurement data. As these methods use the model

assuming the measurement error distribution as Gaussian, analytical precision decreases remarkably in

the case that the measurement error distribution is different from the Gaussian distribution.

In the field of the flow visualization, there is a method in which the pressure field of a flow is

obtained by substituting velocity data of Particle Image Velocimetry (PIV) into the pressure equation

[32]. A similar method to use Particle Tracking Velocimetry (PTV) data has also been studied [33]. In


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the case of PTV, in which defined points of velocity are discrete, a correction method has been

proposed for velocity data to satisfy the continuity equation [34].

The Tikhonov regularization method, which is common in inverse problems, has been applied to

aerospace problems [35]. The problem to determine the flow field from the measurement data is an

inverse problem in contrast to the direct problem in which measurement data is obtained from the flow

field. The solution has uniqueness when the problem is well-posed; however, most inverse problem is

ill-posed and the solution is not unique. Ill-posed inverse problems are solved with using the

Tikhonovs regularization method. The method with the Tikhonovs regularization estimates the flow

field from the well-posed problem which approximates the original ill-posed problem using the

regularization parameter [35]

POD (Proper Orthogonal Decomposition) used in flow control [36] is one of the statistical

method, in which the low dimensional model is constructed from the orthogonal eigenvectors

preserving primary kinetic energy of the flow field. The observer based on the low order model

constructed from POD has been investigated [36]

Hayase et al. proposed measurement-integrated simulation (hereafter abbreviated as MI

simulation), which is integration of measurement and simulation by application of a flow observer

(Figure 1.1) [43]. MI simulation is a SIMPLER-based flow simulation scheme modified by adding the

feedback signal proportional to the difference between the real flow (OUTPUT1) and the simulation

(OUTPUT2). The main feature of MI simulation, which distinguishes it from other existing observers,

is usage of the CFD scheme as a mathematical model of the physical flow. The validity of MI

simulation has been shown in many applications. Hayase et al. numerically showed that the MI

simulation reproduces the fully developed turbulent flow in a square duct by adding feedback signals as

pressure difference between the upstream and downstream boundaries which are proportional to the

difference between calculated axial velocity and the real one on one cross section of the duct [44]. Nakao et

al. developed MI simulation using a turbulence model for steady and unsteady oscillatory airflows

passing an orifice plate in a pipeline. As a result, velocity and pressure feedbacks were conducted and

both feedback methods showed good agreement with the experimental results, and the calculation time

was demonstrated to be significantly reduced compared with ordinary simulation [45]. Nisugi et al.


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developed a hybrid wind tunnel by integrating an experimental wind tunnel and numerical simulation

in the framework of the MI simulation [46]. In the hybrid wind tunnel, a Karman vortex street behind a

square cylinder was treated, and the reproduction of the velocity field was demonstrated by utilizing

the pressure information on the sidewall of the cylinder. As a result, the hybrid wind tunnel showed the

ability to reconstruct the real flow and an advantage over ordinary numerical simulation in its

computational efficiency [47]. Yamagata et al. reproduced the flow field with using PIV measurement

in a hybrid wind tunnel and considered the influence of the feedback data rate for the reproducibility

of the flow field [48]. MI simulation was also applied to the medical engineering. Funamoto et al.

developed Ultrasonic Measurement Integrated (UMI) simulation employing medical ultrasonography [49].

The medical ultrasonography visualizes the cross section of a blood vessel and velocity information

noninvasively, but the obtainable velocity information is the Doppler velocity that is a part of blood flow

velocity projected in the direction of an ultrasonic beam. Funamoto et al. showed from the numerical

experiment that the UMI simulation reproduces the two/three dimensional steady/unsteady complex blood

flows in an aneurismal aorta from the feedback of the Doppler velocity [49-53]. As mentioned above, MI

simulation has been applied widely and the validity has been shown in these applications.

However, the general theory of observer cannot be applied to MI simulation, due to large

dimension and its nonlinearity, and the feedback law is designed by trial and error based on physical

OUTPUT(2)

Feedback law

Model flow

(CFD)

I.C.(2)

Physical flow

I.C.(1)

OUTPUT(1)

Boundarycondition

Nmu

Nu

OUTPUT(2)

Feedback law

Model flow

(CFD)

I.C.(2)

Physical flow

I.C.(1)

OUTPUT(1)

Boundarycondition

Nmu

Nu

+

OUTPUT(2)

Feedback law

Model flow

(CFD)

I.C.(2)

Physical flow

I.C.(1)

OUTPUT(1)

Boundarycondition

Nmu

Nu

OUTPUT(2)

Feedback law

Model flow

(CFD)

I.C.(2)

Physical flow

I.C.(1)

OUTPUT(1)

Boundarycondition

Nmu

Nu

+

Figure 1.1. Block diagram of measurement integrated simulation.


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considerations. Furthermore, in every applications of MI simulation, the error increases for the

feedback gain larger than some critical value. Although the numerical experiment of the former study

[52] suggested that the critical feedback gain is proportional to the reciprocal of the time increment

the mechanism has not been clarified. It is preferred that MI simulation has small time constant which

means that the MI simulation can follow the high frequency component of unsteady flow, and small

steady error which means the reconstruction of the real flow field with high accuracy. However, small

time constant and steady error of MI simulation means a large feedback gain and small time increment

with large computation cost.

Among the above integrated methods, the characters of MI simulation, Kalman filter and 4D-Var are

compared in Table 1.1. Computational cost of 4D-Var is very high due to repeating computation of

numerical model and adjoint equation. The cost of Kalman filter is lowest, because generally low-order

model dynamics and the error covariance matrix is calculated only once in time series. The cost of MI

simulation is nearly the same as that of an ordinary simulation, being lower than that of 4D-Var and

generally higher than that of Kalman filter. From the view point of design method, the method to design

4D-Var and Kalman filter are established, while the method of MI simulation is not well established but the

feedback law is designed by using the trial and error method. The accuracy of the Kalman filter may be less

than the other methods due to usage of a simple linear model. From above mentioned, the establishment of

a general theory to design the feedback law is indispensable to use MI simulation in various areas. In

Table 1.1. Comparison with integrated methods.

MI simulation Kalman filter 4D-Var

Computational cost

Design method

Accuracy

: Good : Average : Poor


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this study, as a fundamental consideration to construct a general theory of MI simulation, we consider

a design method of feedback law from the eigenvalue of the system matrix for the linearized error

dynamics of MI simulation to express time development of the error between the simulation and the

real flow. We also clarify the mechanism of the critical feedback gain above which the MI simulation

diverges, and propose a numerical method in which limitation due to the critical feedback gain is

removed.

This dissertation is composed of 5 chapters. In chapter 2, as a fundamental consideration to

construct a general theory of MI simulation, we formulate a linearized error dynamics equation to

express time development of the error between the simulation and the real flow, and derive the

governing equation for eigenvalue analysis. The validity of the method is investigated by comparison

of the eigenvalue analysis and the result of the MI simulation for a low-order model problem of the

turbulent flow in a square duct with various feedback gains in the cases of feedback with all three

velocity components and two velocity components. Chapter 3 describes the eigenvalue analysis of the

linearized error dynamics for MI simulation applicable to arbitrary flow geometries. On the analysis,

the system matrix is constructed using variables in complex flow geometry defined with solid and

fluid flags in Cartesian coordinate. The validity of the method is investigated by comparison between

the result of the eigenvalue analysis and that of the MI simulation for a low-order model problem of a

two-dimensional blood flow in an aneurismal aorta. In chapter 4, we considered the phenomenon that

the error increases with excessive feedback gain above the critical feedback gain in the MI simulation.

Mechanism of the critical feedback gain is analytically investigated. Then, based on the consideration,

we propose a numerical scheme in which limitation due to the critical feedback gain is removed.

Finally, the conclusions in this study are summarized in chapter 5.


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Chapter2. Eigenvalue Analysis of Linearized Error Dynamics of Measurement-Integrated Simulation

13

Chapter 2

Eigenvalue Analysis of Linearized Error

Dynamics of Measurement Integrated

Simulation


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2.1 Introduction

In flow analysis, experimental measurement and numerical simulation are commonly used.

Howeverboth methods have advantages and disadvantages for correct reproduction of real flows, a

method to combine them has been investigated in many fields: the 4-dimensional variational

assimilation is widely used in weather prediction [11-21], the Kalman filter has been widely used in

flow problems [22-31], the Tikhonov regularization method which is common in inverse problems is

applied to aerospace problems [35], the Particle Tracking Velocimetry (PTV) is combined with flow

simulation in flow visualization measurement [32-34], and POD (Proper Orthogonal Decomposition)

is used in flow control [36].

MI simulation is proposed by Hayase et al., which is integration of measurement and simulation

by application of a flow observer [43]. MI simulation is a SIMPLER-based flow simulation scheme

modified by adding the feedback signal proportional to the difference between the real flow and the

simulation. The main feature of MI simulation, which distinguishes it from other existing observers, is

usage of the CFD scheme as a mathematical model of the physical flow. The validity of MI simulation

has been proved in many applications, such as reproduction of the fully developed turbulent flow in a

square duct [44], steady and unsteady oscillatory airflows passing an orifice plate in a pipeline with

using a turbulence model [45], hybrid wind tunnel to reproduce Karman vortex street [46-48] or an

ultrasonic measurement integrated simulation of blood flow [49-53]. However, the design method of

MI simulation is not established and the feedback law is designed by using the trial and error method. The

establishment of a general theory to design the feedback law is indispensable to use MI simulation in

various areas.

In this chapter, as a fundamental consideration to construct a general theory of MI simulation, we

formulate a linearized error dynamics equation to express time development of the error between the

simulation and the real flow, and derive the governing equation for eigenvalue analysis. The validity of

the method is investigated by comparison of the eigenvalue analysis and the result of the MI

simulation for a low-order model problem of the turbulent flow in a square duct. Section 2.2 describes

formulation of the linearized error dynamics equation and derivation of the governing equations of its

eigenvalue analysis. In Section 2.3, the validity of the method is investigated by comparison of the


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result of the eigenvalue analysis and that of the MI simulation for a low-order model problem of

turbulent flow in a square duct with various feedback gains in the cases of feedback with all three

velocity components and two velocity components. Section 2.4 presents the conclusion of this chapter.

2.2 Formulation

2.2.1 Basic Equations

This paper deals with incompressible and viscous fluid flow. The dynamic behavior of the flow

field is governed by the Navier-Stokes equation:

( ) pt

= + +

uu u u f (2.1)

and the equation of continuity:

0 =u (2.2)

as well as by initial and boundary conditions of the flow field. In the Navier-Stokes equation (2.1), f

denotes the external force term as the feedback signal in the MI simulation, fdenotes the body force,

and p denotes pressure divided by density of fluid. The pressure equation is derived from Eqs. (2.1)

and (2.2) as

( ){ }p div = + u u f (2.3)

We use Eqs. (2.1) and (2.3) as the fundamental equations. In the following, Eqs. (2.1) and (2.3) are

simplified as Eqs. (2.4).

( )

( )

gt

p q

= + = +

uu p f

u f

(2.4)

where

( ) ( )

( ) ( ){ }

g

q div

= +

=

u u u u

u u u(2.5)

The basic equations of the numerical simulation are represented as a spatially discretized form of

the governing equations (2.4):


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( )

( )

NN N N N N

TN N N N N N

dg

dt

q

= +

= +

uu p f

p u f

(2.6)

where uN and pN are computational results for the 3N-dimensional velocity vector and the

N-dimensional pressure vector, respectively,Ndenotes the number of grid points, and N and N are

matrices which express the discrete form of operators and .

We define the operator ( )N D to generate theN-dimensional vector consisting of the values of

a scalar field sampled at Ngrid points. Definition ofDN is naturally extended to the case when the

variable is a velocity vector field as 1 2 3( ) ( ) ( ) ( )T

T T TN N N N u u u = D u D D D . Applying the

operator to the Navier Stokes equation and the pressure equation, we obtain the sampling of these

equations atNgrid points as,

( ) ( )( ) ( )

( ) ( )( )

N N N

N N

dp

dt

p q

= =

D u D u D

D D u

. (2.7)

We assume that there is no external force (DN (f) =0) in the real flow. On the other hand, we apply

external force denoted by a function of real flow and numerical simulation in the MI simulation. In

this study, we consider the case in which external force fN is denoted by a linear function of the

difference of velocity and pressure between real flow and numerical simulation:

( )( ){ } ( )( ){ }N N N N N p= + +u u u u p p p pf K C u C D u K C p C D (2.8)

where Ku denotes the 3N-by-3Nfeedback gain matrix of velocity, Kp denotes the 3N-by-Nfeedback

gain matrix of pressure, Cuand Cpdenote the 3N-by-3NandN-by-Ndiagonal matrices consisting of

diagonal elements of 1 for measurable points or 0 for immeasurable points, and 3N-dimensional vector

u andN-dimensional vectorp mean measurement error. By substituting Eq. (2.8) into Eq. (2.6), we

derive the general formulation of MI simulation:


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( ) ( )( ){ }

( )( ){ }

( ) ( )( ){ }( )( ){ }

NN N N N N N

N N

TN N N N N N N

TN N N

d

dt

p

p

= +

+

= + +

u u u

p p p

u u u

p p p

ug u p K C u D u

K C p D

p q u K C u D u

K C p D

(2.9)

Here, we derive the linearized error dynamics of MI simulation from the difference of the

governing equation between real flow and numerical simulation. Disregarding the second order and

higher order terms in the Taylor expansion for the difference between real flow in Eq. (2.7) and the

basic equation of the MI simulation in Eq. (2.9) with respect to uN-DN(u) and pN-DN(p), we can derive

the linearized error dynamics:

( )( ) ( )( )

( ) ( )( )

( )( ) ( )( ){ } ( ) ( )( )

N

NN N N N

N

N N N

N N N N N N

dd

dt d

p

p p

=

+

+ + +

+

u u

u

p p

u u

gu D u K C u D u

u

K C p D

g D u D g u D D

K

(2.10)

and complementary static equation for pressure error:

( )( ) ( ) ( )( )

( ) ( )( ) ( ){ }

( ) ( )

1

1

1

N

T TNN N N N N N N

N

TN N N N N N

T TN N N

dp

d

p

= +

+ +

+ + +

p p u u

u

p p

p p u u p p

qp D K C K C u D u

u

K C g D u D

K C K K

(2.11)

where the underlined terms are caused by model error and the double-underlined terms are caused by

measurement error.

2.2.2 Eigenvalue Problem

Here, we derive the basic equation of eigenvalue analysis for the linearized error dynamics which

are formulated as Eqs. (2.10) and (2.11) in previous section. In this section, we consider the case of no


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model error, no measurement error, and feedback with only velocity components (Kp = 0). In this case,

Eq. (2.10) is written as

N

d

dt

= u u pe

Ae e (2.12)

where eu and ep are the difference in velocity and pressure, respectively, between the MI simulation

and the real flow defined, and A is the 3N-by-3Nmatrix defined as

( )

( )

N

N N

N N

N

N

p

dg

d

=

=

=

u

p

u

u

e u D u

e p D

A K C

u

(2.13)

In the following, we assume that the normal component of eu is 0 on the boundary NV . This

assumption is used to reduce the dimension of the velocity error vectoreu to 2Ndimension based on

the Weyl decomposition [54]. In Weyl decomposition, any vector field w can be uniquely decomposed

into the orthogonal vector fields as,

grad

div 0 and 0, V

= +

= =

v

v v n x (2.14)

The analysis in this study assumes the special case in which normal the component of eeeeuuuu

on the boundary is null, or eeeeuuuu is identical to vvvv and grad = 0000. In general cases where the

normal component of eeeeuuuu on the boundary is nonzero, the eeeeuuuu is divided into two components

of vvvv and grad . In that case, the result of the present work applies to non-autonomous

system for vvvv component of the error vector eeeeu, and the behavior of the system is affected

by not only the result of the present work but the forcing term of grad .

The discretized equation of mass continuity for the velocity error evaluated at each grid point is

written as,

( ) 0T T Ti u i N i N = =b e b u b D u . 1,2, ,i N= (2.15)

The 3N-by-Nmatrix N in Eq. (2.12) is constructed with bis as

[ ]1 2 N = b b b . (2.16)


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Referring to Fig.2.1, we define B as the range of N ,

( )Range N= B

and 3N-by-2Nmatrix B% consisting of 1 2 2, , , N b b b% % % the orthonomal basis of B , the orthogonal

complementary space of B.

1 2 2N = B b b b

% % % % (2.17)

Let P denote the projection operator onto B by projection theorem:

= TP BB% % . (2.18)

The projection of Eq. (2.12) onto B is given as

( ) ( )Nd

dt

=

u

p

eP P Ae P e . (2.19)

Considering ( )P 0N =pe as BN pe , and = uPe e as Bue , the equation is rewritten as

Td

dt=u

eBB Ae% % . (2.20)

A ueN e

( )B Range N=

P

ued

dtue( )B Range B = %

A ueN e

( )B Range N=

P

ued

dtue( )B Range B = %

Figure 2.1. Schematic Diagram for projection of vector field.


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Here, we define the 2Ndimensional vector e consisting of coefficients of ib% s in euas

=u ue Be% . (2.21)

By inserting Eq. (2.21) into Eq. (2.20), we derive the basic equation of eigenvalue analysis of the

errordynamics as:

A'd

dt =

ue e , (2.22)

where

T' =A B AB% % . (2.23)

We can analyze the linearized error dynamics from the eigenvalues of the 2N-by-2Nsystem matrix A.

2.3 Numerical Experiment

In this section, a numerical experiment is performed to examine the validity of the eigenvalue

analysis presented in the previous section. Eigenvalue analysis and MI simulation are performed for

the case of simple model turbulent flow through a square duct with feedback using all three velocity

components (Case (A)), or using the mainstream and one transverse velocity component (Case (B)).

2.3.1 Method

In the numerical experiment, we deal with a numerical solution of a fully developed turbulent

flow in a square pipe as the standard solution that is a model of real flow (Fig.2.2). In the following,

all the values are expressed in dimensionless form using the side length of the square cross section b~

,

the density of fluid % , and the mean axial velocity 0mu% given by 0 2mu p L= % % where the

coefficient of resistance 410316.02== eRLp is evaluated by means of the Blasius formula

[55]. As to the boundary condition, periodical velocity condition and the constant pressure difference

p corresponding to a specified Reynolds number 0 0 /e mR u b =% %% is assumed between the upstream

and downstream boundaries of the duct with the periodical length of 4. A non-slip condition is applied

on the walls [46].

Computational scheme used in this study is the same as that in the former study [44]. The

discretized representations of the governing equations (2.4) are obtained through the finite volume


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method on an orthogonal equidistant staggered grid system. Convection terms are discretized by a

reformulated QUICK scheme [56]. A two-time level implicit scheme is used for time dependent terms

[57]. The resultant set of finite difference equations is solved using the iterative procedure based on

the SIMPLER method [58].

The computational conditions in dimensionless value are shown in Table 2.1. Although the grid

resolution is not fine enough to reproduce the detailed structure of the turbulent flow, the numerical

solution has the fundamental characteristics of the relevant turbulent flow [44]. This simplification is

x3

x1

x2

0 =4p

1

1

x3

x1

x2

0 =4p

1

1

Figure 2.2. Domain and coordinate system.

Table 2.1. Computational conditions.

Pipe length 4

Pressure difference 0.0649

Reynolds number 9000

Grid points 20 10 10

Time increment 0.025

Residual at convergence 0.01


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justified because the purpose of this numerical experiment is not to investigate the turbulent flow butto

examine whether the eigenvalue analysis can be applied in designing the feedback law of the MI

simulation.

The standard solution or the model of the real flow was obtained using the final result of the

statistically steady flow solution for a fully developed turbulent flow in the former study [43] as the

initial condition. As to the MI simulation considered here, we use a computational scheme identical to

that for the standard solution. The feedback gain matrix Ku in Eq. (2.8) is assumed to be a diagonal

matrix whose diagonal components are all identical value ku:

uk=uK I . (2.24)

Hereafter, the orthogonal component ku is called the feedback gain. The resultant feedback signal

accelerates or decelerates the fluid in a control volume to reduce the error in velocity. As forCu in Eq.

(2.8), we consider two cases: all three velocity components, or the mainstream and one transverse

velocity component are available at all the grid points.

As to the eigenvalue analysis, we assume no measurement error. Terms due to model errors are

also ignored since we use the same computational scheme for both the standard solution and MI

simulation canceling out the model error terms in Eqs. (2.10) and (2.11). For calculation of system

matrix A in Eq. (2.22), % in Eq. (2.17) is numerically obtained from singular value decomposition

by using MATLAB R2006b (ver7.3, The MathWorks). The expression of matrix A is similar to the

expression of the basic equation of the SIMPLER method (omitted due to space limitation). The

eigenvalues of matrix A are calculated by the QR method by using SCSL library. Computation using

MATLAB was performed with SX-9 in Cyberscience Center, Tohoku University, and other

computation was performed with Altix 3700 Bx2 using one CPU in the Advanced Fluid Information

Research Center, Institute of Fluid Science, Tohoku University.

2.3.2 Results and Discussion

In the following we consider three cases: case (A), the ordinary simulation; case (B), MI

simulation with feedback using all three velocity components; and case (C), MI simulation with

feedback using the mainstream and one transverse velocity components.

Eigenvalues, i (i =1, 2,, 2N,whereN=2000) of the system matrix A of the error dynamics


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23

for the ordinary simulation are shown in Fig. 2.3. The right figure of Fig. 2.3 is the figure whose real

axis is enlarged to show unstable eigenvalues clearly. For ordinary simulation, a number of

eigenvalues are unstable, the most unstable eigenvalue pair being m=0.983.2j. This means that the

numerical simulation starting from an initial condition near the standard solution deviates from it

exponentially, representing a sensitive dependence on the initial condition, which is typical for

turbulent flows.

Figures 2.4 (a) and (b) show the eigenvalues of the MI simulation on cases (B) and (C) with the

feedback gain ku = 8, respectively. In each case, all eigenvalues have a negative real part, implying that

the error dynamics is stable due to the effect of feedback, and the error of the MI simulation

decreasesexponentially. The result of case (B) in Fig. 2.4 (a) is a translation of the result of Fig. 2.3 in

the negative real direction with an amount of the feedback gain ku. This is obvious from the definition

of A in Eq. (2.13). The least stable eigenvalue pair are m=-7.023.2j (see right figure of Fig. 2.4 (a)).

In the result for case (C) in Fig. 2.4 (b), the eigenvalues also shift to the left, but the amount of the

-25 -20 -15 -10 -5 0 5

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-1 0 1 2

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-25 -20 -15 -10 -5 0 5

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-1 0 1 2

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

Figure 2.3. Eigenvalue distribution of ordinary simulation (case A).


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24

-25 -20 -15 -10 -5 0 5

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-9 -8 -7 -6

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-25 -20 -15 -10 -5 0 5

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-9 -8 -7 -6

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

(a) Case (B)

-25 -20 -15 -10 -5 0 5

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-1 0 1 2

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-25 -20 -15 -10 -5 0 5

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

-1 0 1 2

-15

-10

-5

0

5

10

15

Imaginaryaxis

Real axis

(b) Case (C)

Figure 2.4. Eigenvalue distribution of MI simulations (ku=8).


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25

shift is less than in case (B) for some eigenvalues. The least stable eigenvalue pair are

m=-0.480.029j (see right figure of Fig. 2.4 (b)).

Figure 2.5 shows the root loci of the most unstable or least stable eigenvalue with the feedback

gain in cases (B) and (C). The broken line with circle shows the result of case (B), and the solid line

with box shows the result of case (C). The real part of the eigenvalue monotonically decreases with the

feedback gain. The eigenvalue crosses imaginary axis at ku=0.98 for case (B) orku=1.67 for case (C).

With feedback gain larger than these critical values, all the eigenvalues are stable and the error of the

MI simulation decreases to 0 exponentially from any initial condition.

-4 -3 -2 -1 0 1 2-1

0

1

2

3

4

5

Case (C)

Case (B)

ku=16

ku=3

ku=2

b

a

u1,u

2

ku=64 k

u=8

ku=1.67

ku=0.98k

u=4 k

u=0

Imaginaryaxis

Real axis

Figure 2.5. Root loci of the most unstable mode for cases (B) and (C).


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26

Figures 2.6 and 2.7 show the distribution of the 3N-dimensional eigenvector of the most

unstable eigenvalue for the ordinary simulation (case (A)) shown as point a in Fig. 2.5 and that for

the MI simulation in case (C) with the feedback gain ku = 8 shown as point b in Fig. 2.5, respectively.

The result of the ordinary simulation in Fig. 2.6 has a complex structure. For case (C) in Fig. 2.7, on

the other hand, the component in the x3 direction is larger than the other components. On the x2-x3

plane, six slender vortices appear. This probably implies that the eigenvalues corresponding to the

eigenvectors with large x1 orx2 component shift to the left due to the feedback and that eigenvalue

corresponding to the eigenvectors with large x3 components are little affected by the feedback.

In the following, the results of the eigenvalue analysis and the MI simulation are shown and

compared. Here, the norm of velocity error is defined as

0 1 2 3 4

0.0

0.5

1.0

x3

x1

x2=0.5

0.0 0.5 1.0

0.0

0.5

1.0

x3

x2

0.0 0.5 1.0

0.0

0.5

1.0

x3

x2

0.0 0.5 1.0

0.0

0.5

1.0

x3

x2

x1=1 x1=2 x1=3

Figure 2.6. Distribution of eigenvectors with ku=0 (a in Fig.2.5).


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27

121

3

T

uEN

u ue e . (2.24)

Time-variation of the error normEu for the MI simulation with the feedback gain ku = 8 in the cases of

(B) and (C) are indicated in Fig. 2.8 by the solid lines. In case (B), the error norm first decreases

exponentially and then remains in a certain range. On the other hand, in case (C), the error norm first

decreases exponentially in the same way as in case (B), but the reducing rate changes around t= 0.6

and the error decreases more slowly afterwards. Broken lines in the figure represent the variation of

the error norm for the least stable mode obtained from the eigenvalue analysis for cases (B) and (C).

These are calculated using the real part of the eigenvalue and the initial magnitude identical to that of

the MI simulation.

0 1 2 3 4

0.0

0.5

1.0

x3

x1

x2=0.5

0.0 0.5 1.0

0.0

0.5

1.0

x3

x2

0.0 0.5 1.0

0.0

0.5

1.0

x3

x2

0.0 0.5 1.0

0.0

0.5

1.0

x3

x2

x1=1 x1=2 x1=3

Figure 2.7. Distribution of eigenvectors Case (C) with ku=8 (b in Fig.2.5).


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In MI simulation, the error norm reaches some steady value as time passes. Figure 2.9 shows

steady error with the feedback gain. As shown in this figure, the steady error norm Eus decreases to

order of 10-4

in the range of 0.5


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29

of 1/e. In MI simulation, as shown in Fig. 2.8, the rate of the error norm reduction is almost constant

for case (B), while it changes around t= 0.6 for case (C). We evaluated the time constant at t= 0 for

case (B), or the value at t = 3 for case (C). For eigenvalue analysis,the time constant ofEu is

estimated as

1

k

= , (2.25)

where k is the real part of the eigenvalue for the least stable mode. Generally, as time passes, the least

stable mode becomes the dominant mode.

10-2

10-1

100

101

102

10-5

10-4

10-3

10-2

10-1

100

101

Simulation

Case (C)Case (B)

Eigenvalue

Case (C)

Case (B)

Eus

ku

Figure 2.9. Steady error norm with feedback gain.


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The variation of the time constant with the feedback gain is compared between the MI

simulation and the eigenvalue analysis for cases (B) and (C) in Fig. 2.10. The results of eigenvalue

analyses agree well with those of the MI simulations except for case (C) with 4uk < .

10-2

10-1

100

101

102

10-2

10-1

100

101

102

Simulation

Case (C)

Eigenvalue

Case (B)

Simulation

Case (B)

Eigenvalue

Case (C)

ku

Figure 2.10. Time constant with feedback gain.

2.4 Conclusions

In this chapter, as a fundamental consideration to construct a general theory of MI simulation, we

formulated a linearized error dynamics equation to express time development of the error between the


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31

simulation and the real flow, and derived the governing equation for eigenvalue analysis. The validity

of the method was investigated by the comparison of the eigenvalue analysis and the result of the

numerical experiment for a low-order model problem of the turbulent flow in a square duct with

various feedback gains in the case of feedback with all velocity components and u1, u2 velocity

components.

From the eigenvalue analysis in the case without feedback (case (A)), the error dynamics is

unstable and the error increased exponentially. When ku > 0.98 with feedback of all velocity

components (case (B)) or ku > 1.67 with feedback of u1, u2 velocity components (case (C)), all

eigenvalues were stable. As to the eigenvector of the least stable eigenvalue in case (C), the x3

component without feedback was larger than the other components and slender vortices appeared inx3

direction on thex2-x3 plane. In the numerical experiment, the critical feedback gains correspond to the

lower limit of the feedback gain to reduce the steady error. In the comparison of the time constant for

the reduction of the error norm, the time constant obtained from the eigenvalue analysis agreed with

that from the numerical experiment.

The above-mentioned results indicate that the eigenvalue analysis of the linearized error

dynamics formulated in this chapter is effective for evaluating the effect of the feedback gain of the

MI simulation. In this chapter, we performed the eigenvalue analysis for the simple flow geometry.

The eigenvalue analysis for arbitrary flow geometries in practical flow problems of MI simulation is

discussed in the next chapter.


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Chapter 3. Eigenvalue Analysis in Arbitrary Flow Geometries

33

Chapter 3

Eigenvalue Analysis in Arbitrary Flow

Geometries


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34

3.1 Introduction

In flow analysis, experimental measurement and numerical simulation are commonly used.

Howeverboth methods have advantages and disadvantages for correct reproduction of real flows, a

method to combine them has been investigated in many fields.

MI simulation is proposed by Hayase et al., which is integration of measurement and simulation by

application of a flow observer [43]. The validity of MI simulation has been proved in many

applications [44-53]. The design method of MI simulation is not established but the feedback law is

designed by using the trial and error method. Then, the establishment of a general theory to design the

feedback law is indispensable to use MI simulation in various areas.

In this thesis, we focus on the error dynamics to express time development of the error between

the MI simulation and the real flow, and consider a design method of feedback law from the

eigenvalue of the system matrix of the linearized error dynamics. In chapter 2, we formulated the

linearized error dynamics of the MI simulation and derived the basic equation of the eigenvalue

analysis. The validity of the method was investigated by the comparison of the eigenvalue analysis and

the result of the MI simulation for a low-order model problem in a simple flow geometry of a turbulent

flow in a square duct. However, the eigenvalue analysis for arbitrary flow geometries is indispensable

for applying MI simulation to practical flow problems.

In this chapter, we deal with the eigenvalue analysis of the linearized error dynamics of the MI

simulation applied to the arbitrary flow geometries on the orthogonal grid system. Validity of the

method is investigated by the comparison of the eigenvalue analysis with the result of the MI

simulation for a low-order model problem of the Ultrasonic-Measurement-Integrated simulation (UMI

simulation) [49-53] for two-dimensional blood flow in the aneurismal aorta.

Circulatory diseases occupy a main causes of death in many countries [59]Many experimental

and numerical studies has been performed on the aneurismal aorta, pointing out that hemodynamics

closely relate to progress of the aneurismal aorta. For the advanced diagnosis, accurate and detailed

information of blood flow field is essential. Funamoto et al. proposed the UMI simulation using

measured data by ultrasonic measurement equipment [49]. Although the measurable flow information

from the ultrasonic measurement is limited to the Doppler velocity, a component of the velocity vector


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35

along the ultrasonic beam originating from the ultrasound probe, it has been commonly used to

measure the blood flow with the advantages of noninvasive and real-time measurement. The schematic

diagram of the UMI simulation is shown in Fig.3.1. This method has been successfully applied to the

analysis of the blood flow in the aneurismal aorta [49-53]. However, the general theory of observer

cannot be applied to the design of the feedback law for the UMI simulation and to the analysis of its

convergence, and the feedback law of these studies was designed by trial and error based on physical

considerations. The establishment of a general theory for designing the feedback law is indispensable

to apply the UMI simulation for various medical applications.

In this chapter, we construct the eigenvalue analysis of the MI simulation for arbitrary flow

geometries and confirm its validity for a simple model problem of UMI simulation of 2D steady blood

flow in aneurismal aorta. In section 3.2, we formulate the eigenvalue analysis of the linearized error

dynamics for the MI simulation applied to arbitrary flow geometries. On the analysis, the system

matrix is constructed using variables in a flow domain in complex geometry specified with the flags to

distinguish fluid and solid in Cartesian coordinates. The UMI simulation for two-dimensional

blood flow in an aneurismal aorta is described. In Section 3.3, the validity of the method is

Figure 3.1. Block diagram of ultrasonic-measurement-integrated simulation.


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36

investigated by comparison of the eigenvalue analysis and the result of the UMI simulation for a

low-order model problem of two-dimensional blood flow in the aneurismal aorta. Section 3.4 presents

the conclusion of this chapter.

3.2 Formulation

3.2.1 Eigenvalue Analysis for Arbitrary Flow Geometries

Equations (2.21) and (2.22) formulated in chapter 2 are used as the basic equation of the eigenvalue

analysis of the linearized error dynamics.

d

dt

=u

e A'e , (3.1)

T=A' B AB% % . (3.2)

As described in the former chapter, we assume that the normal component of velocity erroreu is 0 on

the boundary NV . In the above equations, A is the system matrix derived from the linearized error

dynamics and given as Eqs. (3.3) and (3.4).

N

N

N

dg

d=

uu

A K Cu

. (3.3)

( ) ( )N N N N Ng = + u u u u (3.4)

The system matrix A is the function of the velocity vectoruN and is computed by putting the velocity

field into Eq. (3.4) (see Appendix A).

In the eigenvalue analysis for arbitrary flow geometries, we first define the flag to distinguish

solid and fluid region on the orthogonal grid system. The system matrix A and the matrix B%

representing the projection onto the subspace satisfying the continuity equation are constructed

automatically by using the flag to choose the velocity component in the flow domain. The eigenvalue

analysis for arbitrary flow geometries is made for the system matrix A in Eq. (3.2). We assume no

measurement error and no model error in the analysis. The matrix B% is numerically obtained from

singular value decomposition by using MATLAB R2006b (ver7.3, The MathWorks). The expression

of matrix A in Eq. (3.3) is similar to the expression of the basic equation of the SIMPLER method (see


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37

Appendix A). The eigenvalues of matrix A are calculated by the QR method by using IMSL library.

3.2.2 UMI Simulation for Two-Dimensional Blood Flow in Aneurysmal Aorta

In this chapter, the aneurismal aorta treated by Funamoto et al. [49] shown in Fig. 3.2 is

considered. As a fundamental consideration, a numerical simulation of the two-dimensional steady

flow corresponding to the mean velocity disregarding the effect of heart pulsation was used for the

standard solution as a model of a real blood flow. For the standard solution, parabolic velocity profile

was used as the upstream boundary condition, and a free-flow condition was used as the downstream

boundary condition. For the UMI simulation, velocity profile same as the standard solution was used

as the upstream and downstream boundary conditions, from these boundary conditions, the assumption

in the formulation of the eigenvalue analysis that the normal component ofeu is 0 on the boundary is

satisfied. The computational condition is shows in Table 3.1. In this chapter, all the values are

expressed in dimensionless form using the equivalent diameter of the artery D at the upstream

boundary, the density of fluid , and the mean axial velocity u. Computational scheme used in this

study is the same as that in the former study. The discretized representations of the governing

equations are obtained through the finite volume method on an orthogonal equidistant staggered grid

system. Convection terms are discretized by reformulated QUICK scheme [56]. A two-time level

implicit scheme is used for time dependent terms [57]. The resultant set of finite difference equations

is solved using the iterative procedure based on the SIMPLER method [58].

Table 3.1. Computational condition.

Reynolds number 1000

Grid spacing in each direction 0.0526 (1.487103

m)

GridNxNy 6540

Time increment 0.05014 (0.01s)

Residual at convergence 1.010-5


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38

x

y

x

y

x

y

Figure 3.2. Computational grid.

Ultrasonic beam

cV

eV

eu

Feedback point

Ultrasonic measurement probe

EV

EU

cu

Ultrasonic beam

cV

eV

eu

Feedback point

Ultrasonic measurement probe

EV

EU

cu

Figure 3.3. Schematic diagram of ultrasonic measurement.


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39

As the feedback law of the UMI simulation, the feedback points were defined at the all grid

points in the red box in Fig. 3.2. In the UMI simulation, fluid is accelerated or decelerated at each

feedback point in the direction to reduce the velocity error [49-53]. Referring to Fig. 3.3 the feedback

force is derived as,

( )u V u c sk k= = f E v v (3.5)

where ku is the feedback gain, vsand vc are the Doppler velocity of the standard solution and the UMI

simulation, respectively. In this equation, EV is the error vector derived as

( )V U c s= = E E u u (3.6)

where us

and ucare the velocity vector of the standard solution and the UMI simulation at the feedback

point, respectively, and is the projection matrix of the velocity vector at the feedback point onto the

ultrasonic beam direction. From these equations, the feedback vectorfNis derived as

( ) ( )( )N u N N N k D= f u u (3.7)

From the comparison between Eqs. (2.8) and (3.7), the feedback matrix Ku in the formulation of the

basic equation for the eigenvalue analysis at section 2.2 is derived for the UMI simulation as

u Nk= uK . (3.8)

3.3 Numerical Experiment

In this section, the validity of the proposed method is investigated by the comparison between

the result of the eigenvalue analysis and that of UMI simulation for the two-dimensional blood flow in

the aneurismal aorta. Figure 3.4 shows the velocity vectors of the standard solution used in the

numerical experiment.

Eigenvalues of the system matrix A obtained using the standard solution, are shown in Fig. 3.5.

In this figure, (a), (b), (c) and (d) are the eigenvalues with the feedback gain ku= 0, 1, 8 and 64,

respectively. The number of eigenvalue is 1216, and the time required for each calculation is about 30

minutes by SGI Altix 3700 Bx2 using one CPU. In the case of ku= 0 shown in Fig. 3.5 (a), which

means the ordinary simulation without feedback, all the eigenvalues are stable, with the least stable

eigenvalue of m=-0.42531. This means that the ordinary numerical simulation from any initial


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40

Figure 3.4. Distribution of standard solution.

Table 3.2. Relationship between feedback gain ku and the least stable eigenvalue.

Feedback gain ku Least stable eigenvalue

0 -0.42531

0.1 -0.51441

0.2 -0.57018

0.5 -0.69787

1 -0.84525

2 -0.94428

4 -1.087

8 -1.2376

16 -1.3431

32 -1.4117

64 -1.4553

128 -1.4824


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41

condition converges to the standard solution. It is a reasonable result since we use the steady flow

solution as the standard solution. In the case ofku= 1 and 8 shown in Figs. 3.5 (b) and (c), respectively,

eigenvalues move to the left direction, or into more stable area without changing the overall

eigenvalue distribution. It is noted, however, some eigenvalues near the origin were little affected by

the feedback. In the case of ku= 64 shown in Fig. 3.5. (d), eigenvalues move leftward with changing

the distribution pattern with some eigenvalues near the origin little affected by the feedback. The

-30 -20 -10 0-80

-60

-40

-20

0

20

40

60

80

ImaginaryAxis

Real Axis

(a) ku=0 (ordinary simulation)

-30 -20 -10 0-80

-60

-40

-20

0

20

40

60

80

ImaginaryAxis

Real Axis

(c) ku=8

-30 -20 -10 0-80

-60

-40

-20

0

20

40

60

80

ImaginaryAxis

Real Axis

(b) ku=1

-60 -50 -40 -30 -20 -10 0-80

-60

-40

-20

0

20

40

60

80

ImaginaryAxis

Real Axis

(d) ku=64

Figure 3.5. Eigenvalue distribution of ordinary simulation.


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42

relationship between the least stable eigenvalue and the feedback gain are shown in Table 3.2. In this

table, the least stable eigenvalue is the real value and it moves to more stable direction with increasing

the feedback gain.

In the following, the results of the eigenvalue analysis and UMI simulation are compared. Here,

the norm of velocity error is defined as

121

2

T

uEN

u u

e e (3.9)

0 10 20 3010

-5

10-4

10-3

10-2

10-1

100

Eu

t

ku = 0

ku = 8

0 10 20 3010

-5

10-4

10-3

10-2

10-1

100

Eu

t

ku = 0

ku = 8

Figure 3.6. Comparison of variation of error norm between numerical simulation (solid lines) and

eigenvalue analysis (broken lines).


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whereNis the number of all grid points in the flow field. Time-variation of the error norm Eu for the

UMI simulation with the feedback gain ku = 0 and 8 are plotted in Fig. 3.6 by the solid lines. In each

case, the error norm first decreases exponentially and then approaches in a certain value. Broken lines

in the figure represent the variation of the error norm of the least stable mode obtained from the

eigenvalue analysis, which agree well with those of MI simulation.

Next, we consider the time constant as the time in which the error norm decreases by a factor

of 1/e. For the eigenvalue analysis, the time constant ofEu is estimated as

0.01 0.1 1 10 1000

1

2

3

ku=0 (ordinary simulation)

From Numerical Experiment

ku

From Eigenvalue Analysis

Figure 3.7. Time constant with feedback gain.


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44

1

k

= (3.10)

where k is the real part of the eigenvalue of the least stable mode. The variation of the timeconstant

with the feedback gain is compared between the UMI simulation and the eigenvalue analysis in Fig.

3.6. The result obtained from eigenvalue analysis agree well with that from the UMI simulation.

3.4 Conclusions

In this chapter, the eigenvalue analysis of the linearized error dynamics for MI simulation with

arbitrary flow geometries was formulated. In the analysis, the system matrix is constructed using the

flag to distinguish the solid and fluid regions in the orthogonal grid system. The validity of the

proposed method was investigated by comparison of the eigenvalue analysis and the MI simulation for

a low-order model problem of two-dimensional UMI simulation of the blood flow in the aneurysmal

aorta. In the comparison of the time constant for the reduction of the error norm, the time constant

obtained from the eigenvalue analysis agreed well with that from the MI simulation. The results

indicate that the proposed method of the eigenvalue analysis of the linearized error dynamics is

effective for evaluating the effect of the feedback of the MI simulation in arbitrary flow geometries.


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Chapter 4. Critical Feedback Gain of Measurement Integrated Simulation

45

Chapter 4

Critical Feedback Gain of Measurement

Integrated Simulation


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46

4.1 Introduction

In flow analysis, experimental measurement and numerical simulation are commonly used. Since

both methods have advantages and disadvantages for correct reproduction of real flows, a method to

combine them has been investigated in many fields.

MI simulation is integration of measurement and simulation by application of a flow observer [43].

The validity of MI simulation has been proved in many applications [44-53]. In these applications, the

time constant and the steady error both reduces with increasing the feedback gain. However, the steady

error increases with further increasing the gain above some critical value in all the cases. The former

study [52] pointed out that the critical feedback gain is inversely proportional to the time increment,

but the mechanism has not been clarified yet.

It is preferred MI simulation that has small time constant which means that the MI simulation

can follow the high frequency component of unsteady flow, and small steady error which means the

reconstruction of the real flow field with high accuracy. However, small time constant and steady error

of the MI simulation means a large feedback gain and small time increment. To solve this problem, it

is needed the clarification of the mechanism of the critical feedback gain, and the construction of the

numerical scheme to remove the limitation of the critical feedback gain.

In this chapter, we aim to clarify the mechanism of critical feedback gain and propose a new

numerical scheme without the limitation of the critical feedback gain. In section 4.2, we consider the

mechanism of critical feedback gain from the general equation of MI simulation. Section 4.3 proposes

a new computational scheme without limitation of the critical feedback gain based on the result of

section 4.2. In section 4.4, the validity of the proposed scheme is investigated by the numerical

experiment for a low-order model problem of a turbulent flow in a square duct. Section 4.5 presents

the conclusions of this chapter.

4.2 Mechanism of Critical Feedback Gain

Governing equations of MI simulation are Navier-Stokes equation and pressure equation,


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47

( )

( )

g pt

p q

= + = +

uu f

u f

(4.1)

wherep is the pressure divided by density of fluid and fis external force used as the feedback signal.

Discretized form of the above equations are

( )

( )

NN N N N

T

N N N N N N

dg

dt

q

= + = +

uu p f

p u f

. (4.2)

where uN is 3N-dimensional velocity vector (N: number of grid points), pN isN-dimensional vector of

the pressure divided by density. fNis the feedback signal expressed as

( )*N N= f K u u . (4.3)

where K is the feedback gain matrix and uN* is 3N-dimensional measurement velocity vector in which

element without measurement data is set to 0.

In former studies [43-44, 46-53], the discretized representations of the governing equations, Eqs.

(4.2) and (4.3) were obtained through the finite volume method on an orthogonal equidistant staggered

grid system. Convection term were discretized by a reformulated QUICK scheme [56]. A first or

second order time implicit scheme was used for time dependent terms [57]. The resultant set of finite

difference equations is solved using the iterative procedure based on the SIMPLER method [58].

In the following, the fundamental case with using first order time implicit scheme is described.

(Formulation with the second order scheme is given in Appendix B) In consideration of Eq. (4.3), Eq.

(4.2) the governing equation with one-time level implicit scheme is written as

, 1 *( ) ( )N N

N N N N N N t

=

u ug u p K u u , (4.4)

where the second subscript -1 at the left-hand side of Eq. (4.4) means the value of precious time step.

The first term in the right-hand side of Eq. (4.4) is nonlinear with respect to uN, and the term is

solved using the iterative procedure in the SIMPLER method. The iterative procedure is written as

1( ) ( )n n nN N N = +u F u G u . (4.5)

where n is the index of iteration. In this equation, F and G is given as


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48

1 1 1 * 1

, 1

( ) ( ) ( ))

( ) ( ) ( )

n n n

N N N N N

n n n n

N N N N N N N N

t t

t t t

=

= +

F u g u s u

G u u p K u u s u, (4.6)

where sN means the term moved to the source term in the process of descretization of convective term.

It is noted that the feedback term of former MI simulation uses the value of the last iteration. The norm

of the difference between velocities with index n and those ofn-1 is derived from triangle inequality

and mean-value theorem as

1 1 1 2

1 1 2

1 1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

sup ( ) sup ( )

n n n n n nN N N N N N

n n n nN N N N

n n n n

N N N N

= +

+

+

u u F u G u F u G u

F u F u G u G u

F u u G u u

, (4.7)

where F'()))), G'() are given as

'( )

'( )

N N

N N

N NN

N N

d dt t

d d

d dt t t

d d

=

= +

g sF

u u

p sG K

u u

, (4.8)

and and are the internally dividing points of nu and 1nN

u , and 1nu and 2nN

u , respectively.

From Eq. (4.7) we obtain the inequality as

1 1 2

sup ( )

1 sup ( )

n n n n

N N N N

G

u u u uF

. (4.9)

We assume the main term of the denominator of right-hand side of Eq. (4.9) is 1 and that of the

numerator is tK. In the case of time increment tbeing very small compared with 1 and the feedback

gain being relatively large, the condition in which the solution converges with iteration is given as

sup ( )

11 sup ( ) 1

t

1.67 with feedback of u1, u2 velocity components (case (C)), all

eigenvalues were stable. As to the eigenvector of the least stable eigenvalue in case (C), the x3

component without feedback was larger than the other components and slender vortices appeared inx3

direction on thex2-x3 plane. In the numerical experiment, the critical feedback gains correspond to the

lower limit of the feedback gain to reduce the steady error. In the comparison of the time constant for

the reduction of the error norm, the time constant obtained from the eigenvalue analysis agreed with


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Chapter 5. Conclusions

57

that from the numerical experiment. The above-mentioned results indicate that the eigenvalue analysis

of the linearized error dynamics formulated in this chapter is effective for evaluating the effect of the

feedback gain of the MI simulation.

In chapter 3, the eigenvalue analysis of the linearized error dynamics for MI simulation with

arbitrary flow geometries was formulated. In the analysis, the system matrix is constructed using the

flag to distinguish the solid and fluid regions in the orthogonal grid system. The validity of the

proposed method was investigated by comparison of the eigenvalue analysis and the MI simulation for

a low-order model problem of two-dimensional Ultrasonic measurement integrated (UMI) simulation

of the blood flow in the aneurismal aorta. In the comparison of the time constant for the reduction of

the error norm, the time constant obtained from the eigenvalue analysis agreed well with that from the

MI simulation. The results indicate that the proposed method of the eigenvalue analysis of the

linearized error dynamics is effective for evaluating the effect of the feedback gain of MI simulation in

arbitrary flow geometries.

In chapter 4, we considered the phenomenon that the error of the MI simulation rapidly increases

with increasing the feedback gain above a critical value in the former studies. We formulated the

condition of the critical feedback gain for the former computational scheme and proposed the

computational scheme by which rapid increase of the error above the critical feedback gain does not

occur. From the numerical experiment for a model problem of turbulent flow in a square duct, the

critical feedback gain for the former scheme agreed with that estimated by the formula. Rapid increase

of the error above the critical feedback gain does not occur with the proposed computational scheme.

This enables us to use a larger feedback gain of the MI simulation resulting in faster convergence and

better accuracy.

The author hopes that the obtained knowledge of this fundamental study on design of the

feedback law which is indispensable to use MI simulation in various areas helps the construction of

the general theory of MI simulation


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Acknowledgements

58

AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements

I would like to express my gratitude to Professor Toshiyuki HAYASE, my supervisor, for his

invaluable inspiration, kind guidance and continuous encouragement through this research. I really

enjoyed working with Professor HAYASE, who introduced me to the fascinating areas of fluid

engineering, and specifically to this challenging problem of overcoming the limitation both in

measurement and computation.

I wish to express my gratitude and appreciation to Professor Kazuhiro KOSUGE and Professor

Shigeru OBAYASHI for serving on the graduate committee and for their helpful comments on this

dissertation.

I am greatly indebted to Associate Professor Atsushi SHIRAI. His thoughtful suggestions and

helpful discussions throughout of my research made this dissertation possible.

The authors acknowledge the support from the Tohoku University Global COE Program Global

Nano-Biomedical Engineering Education and Research Network Centre and the program of

Development of System and Technology for Advanced Measurement and Analysis (SENTAN).

The computations were performed using the supercomputer systems ORIGIN 2000 and Altix

3700 B2 in the Institute of Fluid Science, Tohoku University. I would like to express my appreciation

to staff in the Advanced Fluid Information Research Center, Institute of Fluid Science, Tohoku

University for their support.

I would like to express sincere thanks to Associate Professor Kenichi FUNAMOTO, Associate

Professor Takayuki YAMAGATA at the Niigata University, and Dr. Lei LIU at the GE Healthcare,

from them I gained much knowledge for the MI simulation.

Many thanks are also to Secretary Kayo Saito and all fellows in the Super-Real-Time Medical

Engineering Laboratory, Transdisciplinary Fluid Integration Research Center, Institute of Fluid

Science for comfortable academic research environment

Lastly and finally, I would like to thank my parents, for providing the opportunity to study and

all they have given me. Without their understanding, I would not be where I am right now.


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