Applied Mathematics and Computation 168 (2005) 1258–1272

www.elsevier.com/locate/amc

Slot nozzle design with specified pressure ina given subregion by using embedding method

H.H. Mehne a,b,*, M.H. Farahi b, J.A. Esfahani c

a Aerospace Research Institute, Thehran, 15875-3885, Iranb Department of Mathematics, Ferdowsi University of Mashhad, Mashhad 917751159, Iran

c Department of Mechanical Engineering, Ferdowsi University of Mashhad, Iran

Abstract

In this paper we study a shape optimization problem connected with controlled pres-

sure in a subregion of a 2D slot nozzle. Feasible shapes for this problem are character-

ized only with upper walls. This enables us to seek for the optimum in a class of

functions and their derivatives instead of a class of shapes. The shape optimization

problem can be written as an optimal control problem, then the resulting distributed

control problem is expressed in a measure theoretical form, in fact an infinite dimen-

sional linear programming problem. The optimal measure representing optimal shape

is approximated by the solution of a finite dimensional linear programming problem.

A brief sensitivity analysis on model parameters is given.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Linear programming; Optimal shape design; Measure theory

0096-3003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.10.018

* Corresponding author.

E-mail addresses: h_mehneh@yahoo.com, hashemi@math.um.ac.ir (H.H. Mehne), farahi@

math.um.ac.ir (M.H. Farahi), Abolfazl@ferdowsi.um.ac.ir (J.A. Esfahani).

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1259

1. Introduction

There are many industrial devices consisting of a 2D slot nozzle. In jet wip-

ing process applied in galvanization industry for example, the liquid film is

dragged on the surface of a moving strip and undergoes the effect of air knives

created by a 2D slot nozzle (see [9]).There have been a number of efforts dealing with the effect of the nozzle

shape on the devices� performance. For example Buchlin et al. [3] have stud-ied the effect of nozzle titling on splashing in jet wiping. The problem of find-

ing the best shape is considered in optimal shape design (OSD). In this area

we may refer to Butt [5] where a gradient method is developed for the opti-

mal shape design of a nozzle problem described by variational inequalities,

Pironneau [15] and Mohammadi and Pironneau [14] where adjoint methods

and finite element techniques are used to discuss the optimal designing ofnozzle. Laumen [12] proposed a Newton�s method in function space and

derived an algorithm to solve the discretized form of some OSD problems.

See also [1] for direct design of shapes and [10] for a review on diffuser

design.

In the above mentioned methods, the computation of the solution is time-

consuming task, because they need to solve many boundary value problems.

More over, these methods are iterative and they need to have an initial guess

for solution.In this article we translate the OSD problem to an optimal control problem

accompanied with partial differential equations. The converted problem can be

solved by extending measure theoretical method proposed by Rubio [17]. The

advantage of the proposed method lies in the fact that the method is not iter-

ative and it is self-starting and it does not need to solve corresponding bound-

ary value problem. Methods of this type were frequently and successfully

applied to different problems. For example the numerical estimation of the dis-

tributed control of a wave equation [8], optimal control of the heat equationwith a thermal source [11], and solving OSD problems governed by ordinary

differential equations [6], and optimal shape design for a nozzle with specified

velocity in a given region [7].

2. The optimal shape design problem

Consider the steady flow of a constant density fluid q in a converging 2Dslot nozzle (Fig. 1). According to the Bernoulli equation, the pressure decreases

from inlet to the outlet section. We are interested in designing the shape of the

nozzle that leads to a given pressure Pd in some specified region D inside of the

geometry. The resulting problem is to find a shape which minimizes the follow-

ing functional:

Fig. 1. Geometry of the problem.

1260 H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272

ZD

1

2qkrUðx,yÞk2 þ Pd �

1

2qV in � P in

��������dxdy, ð1Þ

where Vin and Pin are inlet velocity and pressure, respectively, that we set them

equal to 1, D is a given bounded set of R2, and U satisfies the incompressible

potential flow equation

DU ¼ 0, ð2Þin region R surrounded by predefined walls C1,C2,C3, and unspecified wall C.Moreover U must satisfy in the following Neumann conditions:

oUon

¼�1, on C1,

0, on C3 [ C,jC1jjC2j, on C2,

8><>: ð3Þ

where n is the normal vector to the boundary of R and jCij denotes the lengthof Ci. We assume that C1 = {0} · [0,a], C2 = {L} · [0,b] and C3 = [0,L] · {0},where a > b > 0. If we consider the upper wall as a function C : ½0,L� ! R of

the variable x, then the problem of finding the optimal shape is reduced to a

problem in the function space C1([0,L]); the space of real-valued differentiable

functions on [0,L] with continuous derivative. This enables us to convert the

problem of minimizing (1) subject to (2,3), to an optimal control problem. This

procedure has been used in some OSD�s literature, see for example nozzle andwing design in [15].

3. Measure theoretical formulation

Consider the generalized form of (2) and (3) (see [13]):Z L

0

Z CðxÞ

0

rUrmdy dx ¼ am 8m 2 C1ðRÞ, ð4Þ

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1261

where C1ðRÞ is the space of all real-valued differentiable functions with contin-uous first partial derivatives on R, and

am ¼ab

Z b

0

mðL,yÞdy �Z a

0

mð0,yÞdy: ð5Þ

Choosing the derivative of the boundary function C(x) as a control variable, wechange the OSD problem to an optimal control problem where the task is to

minimize (1) subject to the variational constraints (4) and ordinary differential

control system:

d

dxCðxÞ ¼ #ðxÞ, x 2 ð0,LÞ, ð6Þ

Cð0Þ ¼ a, CðLÞ ¼ b, ð7Þ

where C(Æ) is the trajectory and #(Æ) is the control.

Definition 3.1. We shall say that the triple p = (#,C,U) is admissible if thefollowing conditions hold:

(i) The control function # takes its values in a setU and C(Æ) is the response ofthe system (6,7) to #.

(ii) U is the solution of (4) corresponding to C(x).

We assume that the set of all admissible triples is nonempty and denote it by

P. Let X ¼ ½0,L� � ½0,a� � ½0,a� �U� K, where U is a known compact set in Rand K is a subset of R2, such that the vector DU(x,y) gets its value for eachðx,yÞ 2 R in this set.

By the Definition 3.1, each shape R can be replaced by exactly one admis-sible triple p ¼ ð#,C,UÞ 2 P. Of course this is an injection corresponding. Thusthe minimization (1) over the class of shapes is equivalent to minimization of

(1) over P. Now we transfer the problem of minimization of (1) over P, intoanother, nonclassical problem which appears to have some better properties

in some aspects. Define the following mapping:

Kp : F !Z L

0

Z CðxÞ

0

F ðx,y,CðxÞ,#ðxÞ,rUÞdy dx, F 2 CðXÞ, ð8Þ

where C(X) is the space of all continuous real-valued functions F. Being an

injection, the transformation p # Kp provides us to describe the set of all

admissible triples P as a subset of the set of all linear continuous mappings

on C(X). Moreover by Riesz representation theorem [19], corresponding toKp there is a positive Radon measure l on X, so that

1262 H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272

KpðF Þ ¼Z

XF dl ¼ lðF Þ, 8F 2 CðXÞ: ð9Þ

Also we need to convert (6) and (7) to its integral form. For this purpose, let B

be an open ball in R2 containing [0,L] · [0,a], and C1(B) be the space of all

real-valued continuous differentiable functions with continuous first partial

derivatives on B. Let u 2 C1(B) and define functions u# as follows:

u#ðx,y,CðxÞ,#ðxÞ,rUÞ ¼ uyðx,CðxÞÞ#ðxÞ þ uxðx,CðxÞÞ, ð10Þ

for each (x,y,C(x),#(x),$U) 2 X. The function u# is in the space C(X), for eachadmissible triple (#,C,U) we haveZ L

0

u#ðx,y,CðxÞ,#ðxÞ,rUÞdx ¼Z L

0

uyðx,CðxÞÞ#ðxÞ þ uxðx,CðxÞÞdx

¼ uðL,CðLÞÞ � uð0,Cð0ÞÞ :¼ Du, ð11Þ

for all u 2 C1(B), where u(L,C(L)) and u(0,C(0)) are known. Define

Guðx,y,CðxÞ,#ðxÞ,rUÞ ¼ u#ðx,y,CðxÞ,#ðxÞ,rUÞCðxÞ , 8u 2 C1ðBÞ,

thus from (10) and above definition, we haveZ L

0

Z CðxÞ

0

Guðx,y,CðxÞ,#ðxÞ,rUÞdy dx ¼ bu,

where bu is the integral of Gu over R.Now the minimizing of the functional (1) over P is equivalent to the mini-

mizing of

I ¼ l ! lðf0Þ ð12Þover the set of measures l corresponding to admissible triples, which satisfy

lðF mÞ ¼ am, 8m 2 C1ðXÞ, ð13Þ

lðGuÞ ¼ bu, 8u 2 C1ðBÞ, ð14Þ

where f0ðx,y,CðxÞ,#,rUÞ ¼ vDj 12qkrUðx,yÞk2 þ Pd � 12qV in � P inj,vD is charac-

teristic function on D and (13,14) are integral form of (4) and (6) and (7),

respectively.

We define the set of all positive Radon measures on X satisfying (13,14) as

Q. Also we assume thatMþðXÞ be the set of all positive Radon measures on X.Now if we topologize the space MþðXÞ by the weak*-topology, it can be seen

from [17], that Q is compact. In the sense of this topology, the functional

I : Q ! R defined by (12) is a linear continuous functional on the compact

set Q, thus it attains its minimum on Q (see Theorem III.1 in [17]), and sothe measure-theoretical problem, which consists of finding the minimum of

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1263

the functional (12) over the subset Q of MþðXÞ, possesses a minimizing solu-tion, say l*, in Q. The minimizing problem (12)–(14) is an infinite dimensional

LP problem and we shall mainly interested in approximating it. It is possible to

approximate the solution of the problem (12)–(14) by the solution of a finite

dimensional linear program of sufficiently large dimension.

First we consider the minimization of (12) not only over the set Q but over asubset of it defined by requiring that only a finite number of constraints (13)

and (14) be satisfied.

Consider the equalities (13) and (14), let the set of functions fmi,i 2 Ng andfui,j 2 Ng are total, respectively in C1(X) and C1(B). Now we can prove:

Proposition 3.1. Let Q(M1,M2) be a subset MþðXÞ consisting of all measures

satisfying

lðF miÞ ¼ ami , i ¼ 1,2, . . . ,M1, ð15Þ

lðGujÞ ¼ buj

, j ¼ 1,2, . . . ,M2: ð16Þ

If gðM1,M2Þ ¼ infQðM1,M2Þlðf0Þ and g = infQl(f0), then g(M1,M2) ! g as

M1,M2 ! 1.

Proof. See [7]. h

The first stage of the approximation is completed successfully. As the secondstage, it is possible to develop a finite-dimensional linear program whose solu-

tion can be used to construct the solution of the infinite dimensional linear pro-

gram consisting of minimizing (12) subject to (13) and (14). From the Theorem

A.5 of [17], we can characterize a measure, say l*, in the set Q(M1,M2) at

which the function l ! l(f0) attains its minimum, it follows from a result of

[16] that:

Proposition 3.2. The measure l* in the set Q(M1,M2) at which the functionl ! l(f0) attains its minimum has the form

l� ¼XM1þM2

k¼1.�kdðz�kÞ ð17Þ

with z*k 2 X, and .�k P 0,k ¼ 1,2, . . . ,M1 þM2.

Here d(z) is a unitary atomic measure [18], characterized by d(z)(H) = H(z),

where H 2 C(X) and z 2 X. Now the measure theoretical optimization problemis equivalent to a nonlinear optimization problem, in which the unknownsare the coefficients .*k and supports fz�kg,k ¼ 1,2, . . . ,M1 þM2. It would be

convenient if we could minimize the function l ! l(f0) only with respect to

1264 H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272

the coefficients .�k ,k ¼ 1,2, . . . ,M1 þM2 in (17), this would be a linear program-

ming problem. However, we do not know the support of the optimal measure.

The answer lies in approximation of this support, by introducing a dense set in

X.

Proposition 3.3. Let x be a countable dense subset X. Given � > 0, a measurek 2 MþðXÞ can be found such that

jðl� � kÞðf0Þj < �,

jðl� � kÞðF miÞj < �, i ¼ 1,2, . . . ,M1,

jðl� � kÞðGujÞj < �, j ¼ 1,2, . . . ,M2

the measure k has the form

k ¼XM1þM2

k¼1.�kdðzkÞ,

where the coefficients .�k are the same as in the optimal measure (17) and zk 2 x.

Proof. See the proof of Proposition III.3 of [17]. h

Thus the infinite dimensional linear programming (12) with restrictions de-fined by (13) and (14) can be approximated by the following linear program-

ming, where zi, i = 1,2, . . . ,N, belong to a dense subset of X. These resultssuggest the following linear program:

MinimizeXNk¼1

.kf0ðzkÞ ð18Þ

over the set .k P 0, k = 1,2, . . . ,N, subject to

XNk¼1

.kF miðzkÞ ¼ ami , i ¼ 1,2, . . . ,M1, ð19Þ

XNk¼1

.kGujðzkÞ ¼ buj

, j ¼ 1,2, . . . ,M2: ð20Þ

The procedure to extract a piecewise constant control function from the solu-

tion {.k P 0,k = 1,2, . . . ,N} of linear programming problem (18)–(20) which

approximates the action of the optimal measure, is based on the analysis in

[17]. The trajectory is then simply found by solving the differential equation

(6) with initial condition C(0) = a. The resulting solution is a piecewise linearfunction.

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1265

4. Numerical treatments

Consider the problem (1)–(3) where assumed a = 0.5, b = 0.25, L = 1,

D = [0.13,0.30] · [0, 0.25], q = 2, Pd = 0.5, Vin = 1, and Pin = 1. We solved the

problem by procedure described in Section 3, where assumed M1 = 56,

M2 = 6 and N = 13,500, we find the piecewise control and trajectory, respec-tively as Figs. 2 and 3.

Approximating the infinite dimensional nonlinear problem by a finite

dimensional linear one, may contains unwished errors that lead to undesirable

results. In this section we explain some of these errors and propose a correction

technique in each case.

(a) Miss distance: When we solve the corresponding LP problem we find a

piecewise constant control (derivative of the trajectory). Fig. 2 shows such a

control function. The wall of the nozzle is then a trajectory made by using con-trol values and initial condition C(0) = a as well. Fig. 3 indicates the corre-

sponding trajectory for control of Fig. 2. Because of some approximations,

the resulting piecewise linear trajectory may not satisfy the final condition

C(L) = b, as it is clear in Fig. 3. Let define jC(L) � bj as the miss distance. Thismiss distance can be reduced by increasing the number of total functions in (19)

Fig. 2. The piecewise constant control function.

Fig. 3. The trajectory corresponding to the control of Fig. 2.

1266 H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272

and (20), but increasing the number of total functions may cause difficulties in

solving the linear programming problem (18)–(20). To succeed in dealing with

this difficulty, we add the following weighted functional to the objective

function:Z L

d2

xjCðxÞ � bjdx

Adding this functional to the cost function we have better upper walls, see Fig.

4, where the miss distance in Fig. 3 is improved by adding the above functional

to the objective.

(b) Minimum length: It can be found from Fig. 4 that the piecewise lineartrajectory constructed by using control values may have suboptimal length.

Reduction in the length of the wall may cause reduction in material used to

construct it. So the designer may be interested in design a nozzle with minimum

length as it is important in applications, for example in gas dynamic laser (see

[2]). For improving the method to handle this case we propose to add the fol-

lowing functional to the objective:Z L

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ #ðxÞ2

qdx:

Implementing this technique, we find a shorter wall as in Fig. 5.

(c) Smoothing methods: If the aim of design is a smooth wall without sharp

corners, we can benefit from numerical approximation methods. We have

examined fitting methods based on least square and interpolation methods. Fit-

ting methods provide smooth walls but they may cause miss distance at the endpoints.

We can use interpolation methods to give importance to the inlet and outlet

sections. We may use for example nearest neighbor interpolation, cubic spline

interpolation, piecewise cubic Hermite interpolation (see [4]). We have used the

last method to have a more smooth wall as in Fig. 6.

Fig. 4. The optimal shape without miss distance.

Fig. 5. The wall with minimum length.

Fig. 6. The smoothed wall of Fig. 5 with interpolation.

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1267

The above mentioned numerical treatments are used in all the problems of

the next section.

5. Sensitivity analysis

The effect of model parameters on the optimal shape is studied in this sec-

tion. We consider the effect of changing in the position of D, the amount ofPd, the size of D, and the ratio between a and b.

(a) Position of D: We choose Pd = 0.5 which is between the inlet and outlet

pressure and change the place of D along the nozzle length. The anticipated

curve would be a curve with negative slope followed by a straight line to exit

when the position of the D is near inlet. Similarly when the position of D is near

exit, we should expect a straight line from inlet section up to D followed by a

curve with negative slope. Resulting shapes are depicted in Figs. 7–9. For this

shapes all of the other parameters have been taken unchanged.(b) The amount of Pd: Now we are interested to consider the sensitivity of the

optimal shape with respect to changes in the value of Pd. Let consider a trivial

case Pd = 1, that is the desired pressure in D equals to the inlet pressure. The

Fig. 7. D = [0.13,0.30] · [0, .25].

Fig. 8. D = [0.33,0.50] · [0, .25].

Fig. 9. D = [0.53,0.70] · [0, .25].

1268 H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272

optimal solution must be a straight line to the upstream of D followed by a

curve to the outlet section. We have solved the problem by the method in Sec-

tion 3, and found interesting result confirmed this physical fact (Fig. 10). When

the value of Pd is larger than the inlet pressure we expect for a diverging–con-

verging wall, that is the wall must be diverge on D and converge when it pasts

D to the outlet section. On the other hand, when Pd is less than the outlet pres-

sure, we anticipate a converging-diverging wall. Generally the larger value for

Fig. 10. Pd = 1.

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1269

Pd leads to a wall with larger height on D. These theoretical justifications are

evident in the resulting shapes for Pd = 0.2, 0.4, 0.8, 1.6 (Fig. 11).(c) The size of D: Since the pressure changes along the nozzle length, the

changes in the size of D along the stream direction will change the optimal

shape, but the transversal changes in the size of D do not have effect on the

shape, because the pressure remains constant perpendicular to the stream.

Solving problem with different sizes of subregion D, we find that small

changes in the size of D do not give rise to considerable variations in the shape

Fig. 11. Changing in the value of Pd.

Fig. 12. D = [0.13,0.30] · [0,0.25].

1270 H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272

but when the longitudinal altering is large, the shape shows different behavior

(Figs. 12 and 13), while the transversal changing does not alter the optimal

shape (Fig. 14). In Fig. 14 four values for the height of D is examined; 0.05,

0.15, 0.25, 0.35. Pd is 1 in Figs. 12 and 13 and is 0.5 in Fig. 14.

(d) Changing in inlet-outlet section: Small perturbation in inlet velocity gives

rise to small changing in the corresponding shape. Since we normalize the

Fig. 14. Transversal changing in the size of D and corresponding shapes.

Fig. 13. D = [0.13,0.50] · [0,0.25].

Fig. 15. Small perturbation in inlet section.

H.H. Mehne et al. / Appl. Math. Comput. 168 (2005) 1258–1272 1271

velocities in terms of the inlet velocity, we conclude that small changes in inlet

section may lead to small changes in the corresponding shape. The correspond-

ing shapes with respect to (a,b) = (0.48,0.25) and (a,b) = (0.49,0.25) have been

compared in Fig. 15. The value of Pd is 0.8 in each case.

6. Concluding remarks

In this paper, we have shown that the embedding method (embedding the

admissible set into a subset of measures), which has been successfully used

for solving optimal control problems, is also applicable to solve OSD prob-

lems. The method is not iterative. It does not need to any initial guess of the

solution. Other objectives for example the length of the wall can be added to

the objective simply without change in structure of the problem and itscomplexity.

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