A Shape Optimal Design Problem in FluidMechanics with Thermal Transfers

Jérôme Monnier

Abstract. The following is a study of a shape opti-mal design problem for a stationnary Navier-Stokes flowwith thermal transfers. The thermal transfers are con-vective, diffusive and radiative with multiple reflections(grey bodies). The state equation is a non-linear integro-differential system and the problem is to minimize a costfunction which depends on the solution, with respect tothe domain of the equations. We prove the differentiabil-ity of the solution with respect to the domain, the differ-entiability of the cost function, we introduce a classicaladjoint state equation and we give an expression of thedifferential of the cost function.

1 POSITION OF THE PROBLEMA shape optimal design problem in a potential flow cou-pled with the present thermal model was studied in [4].The present paper constitutes the sequel of the work donewith D. Chenais and J.P. Vila ([9], [4], [10]): the potentialflow model is replaced by the Navier-Stokes model.The following deals with the shape optimal design of aforced convection problem. The flow is viscous, station-nary, tridimensionnal, incompressible and laminar. Thephysical model is the Navier-Stokes equations weaklycoupled with a thermal model taking into account con-ductive, convective and radiative heat exchanges withmultiple reflections (the surfaces are greys, opaque andseparated by a radiatively non-participating media). Theproblem is to optimize the shape of the domain of theequations in order to minimize a smooth cost functionwhich depends on the solution.The initial motivation was an optimal design problemwhich arises from a cooling problem under a car bonnet,see [4], [9]. This industrial problem fits into the abstractframework presented here.

Laboratoire de Modelisation et Calcul (LMC-IMAG), TourIRMA, BP 53, F-38041 Grenoble Cedex 9, FranceProjet IDOPT (CNRS-INPG-INRIA-UJF)

This paper is organized as follows. We present the phys-ical model and the optimal design problem. In Section2, we recall results of existence and uniqueness of thesolutions of the state equation and the linearized stateequation. In Section 3, first we recall some results ofshape derivatives, then we state precisely the shape op-timal design problem and we prove the differentiabilityof the solution with respect to the domain. Finally, weintroduce a classical adjoint state equation and we givean expression of the differential of the cost function.Let us notice that we do not treat of the implementationof the equations. We refer to [4], [9] for the applicationto the cooling problem under a car bonnet.

The physical model The unknows of the full modelare the fluid velocity , the fluid pressure , the fluidtemperature and the radiosity (the radiosity is theradiant energy which flows away from a surface). Thedimensionless numbers are the Reynods number , thePeclet number , the Biot number and two radiativedimensionless numbers denoted and , [10].Let be a lipschitz bounded open set of ( =2 or3), the dimensionless model is the following. Findsatisfying (in the sense of distributions):

in (1)

in (2)on (3)on (4)

where and . The func-

tion belongs to , it satisfies andit vanishs in , where is a neighbourhood “smallenough” of .Given the fluid velocity , find satisfying (in the

c 1998 J. MonnierECCOMAS 98.Published in 1998 by John Wiley & Sons, Ltd.

sense of distributions):

in (5)

on (6)

on (7)

on (8)

on (9)

where

(10)

(11)

and is the mapping identity. We have, the measure of is strictly positive. The tempera-

tures and are positives and belong respectively toand . Moreover, is as-

sumed to vanish in , where is a neighbourhood“small enough” of .The kernel is the angle factor,it is positive, symmetric. The function is the em-missivity coefficient, it satisfies and it isassumed to be a lipschitz function, [10].Assumption 1 The parts of the boundary and aresuch that on i.e. the fluid flow is outgoingwhere the boundary temperature is not given.

We set and . Then, the problem(1)-(9) is equivalent to find satisfying (in the senseof distributions):

in (12)in (13)on (14)

with ; then tofind satisfying (in the sense of distributions):

in (15)

on (16)

on (17)

on (18)

on (19)

with and.

The shape optimal design problem We seek to min-imize a smooth cost function with respect to ,where is the solution of the equa-tions (12)-(19) posed in .For example, in the cooling problem studied in [4], weconsider a part of the boundary, , and we seekto minimize with respectto the shape of .

Remark 1 We consider a shape optimal design problemwhere the cost function depends onand or dependson someof these variablesonly. How-ever, it is implicit that this cost function depends at leaston , otherwise the thermal model would be useless.In others respects, let us recall that many shape optimaldesign problems in a Navier-Stokes flow were treated bymany authors, see e.g. [12], [1], ...

2 THE STATE EQUATION

From now, we voluntarily omit the arrows above vecto-rials entities. We denote: and wedefine: . We write:

where. The state equation

is the following:

Find such that

(20)where is the sum of the variationnal formula-tions of the Navier-Stokes equations, the thermal partialdifferential equation and the integral equation. This stateequation (20) is equivalent to:

Find such that:

(21)

Shape optimal design in fluid mechanics 2 J. Monnier

where denotes the inner product in .

Given findsuch that:

(22)

Existence and uniqueness of the physical solutionOne knows that there exists ,solution of (21) and if the Reynolds number is smallenough, the solution is unique (see e.g. [7]). In others re-spects, there exists a unique physical solution to (22) -see[10]- i.e. a unique couplesuch that satisfies the weak maximumprinciple: a.e., with

and

.

Therefore there exists a unique physical solutionto (20). Let us notice that since satisfies the

weakmaximum principle and belongs to ,belongs to and the terms

and make sense with testfunctions in and , even in threedimensions of space.

The linearized problem In next Section (Lemma 1),we need to study under which conditions the linearizedstate equation is well posed. Concerning the fluid model,one knows that if the Reynolds number is small enough,then the linearizedNavier-Stokes equations arewell posed(see e.g. [5], Lemma IV.3.2), and concerning the thermalmodel, if the Peclet number is small enough, then thelinearized equations are well posed (see [10], Proposition8).

3 THE SHAPE OPTIMAL DESIGNPROBLEM

We have in view to minimize the cost function usingan algorithm of descent. To this end, we prove that the

solution of the state equation is differentiable with respectto the domain (Lemma 1), then the cost function is alsodifferentiable and finally, using the classical adjoint statemethod, we give an expression of the differential of thecost function (Theorem 1).

3.1 Some recalls in shape derivativesWe use the classical technic of transport which consiststo consider the admissible domains as perturbations of areference domain. The following results can be found in[11], [2], [3], [13], [8], [6], [4]...Let be an open subset of with a lipschitz boundary,we introduce the following functionnals spaces:

such that:

and

bijection of onto

Then, we define the domains space as follows:

In the remainder of the paper, we consider the followingparticular case: the boundary of is in two partsand , see Fig. 1. The part of the boundary denoted byis fixed and we seek to optimize the shape of . A

such situation arises in the cooling problem under a carbonnet considered in [4], it arises as well for examplein the shape optimization of a profile, see e.g. [12], [1],[6]. We denote by a neighborhood of , “largeenough” (Fig. 1), and we consider the affine subspace:

in where denotes theidentity of .

!

B

F=I

!

Figure 1. The reference domain

For , “small enough”, we define the domainby: (Fig. 2). Then, we define the Banach

Shape optimal design in fluid mechanics 3 J. Monnier

space: and the affinesubspace: . Let ,we define by ( in

).

" "

#

F=I+VF

F0

y

"F=I+V

#

yRI

Figure 2. Change of variables

If is close enough to in then is an open setof with a lipschitz boundary and . In theremainder of the paper, we study the differentiability ofthe cost function with respect to when belongs to asuch neighbourhood of . For a given cost function ,

, we define by:

And the derivative with respect to the domain satisfies:

3.2 The shape optimal design problemLet be the “reference” domain, ,“small enough”. We consider an open set

where . The function is continuous frominto , hence , the boundary of , is theimage of by . We denote by andthe image by of , respectively,

and . We make thefollowing assumption.

Assumption 2 The parts of the boundary and areincluded in (they are fixed). Moreover, the datas ofDirichlet denoted and vanish in ; more precisely,

and are stricly included in , Fig. 1 (they are“small enough”).

Let us suppose given a positive function in ,and . (We denote

with the variables defined in , Fig. 2). We denote:. Hence

for , we have:.

We assume that the parts of the boundary and

are such that on , hence for ,Assumption 1 holds.Wewant theHilbert space preserved by the changeof variable. To this end and following [1] and [6], weintroduce the space of pressures:

This space is isomorph to the spaceand from now, we consider:

It follows that for , themappingis an isomorphism. (Let us notice that we

cannot consider divergence free velocity fields becausedoes not imply ).

Let be an observationfunction.We assumethat this observation function is con-tinuously differentiable and we define the cost function

, where is theunique physical solution of the state equation (20). Theproblem we seek to solve is the following:

Find such that

Transport of the equations We transport the equationson the “reference” domain . For any , welet:

with , Fig. 2. Themapping is supposed to be of class .The transported state equation is:

Find such that:

(23)where , being the unique physical solutionof (20). Since the mapping

is an isomorphism for , the transport statedequation has a unique physical solution .We write . The minimizationproblem we solve is now:

Find such that

Shape optimal design in fluid mechanics 4 J. Monnier

Differentiability of the solution

Lemma 1 Let be the the unique physicalsolution of (23). If the Reynolds and Peclet numbers aresmall enough then there exists a neighborhood of theidentity mapping in such that the mapping

is of class .

Proof. The mapping is with respect toand the linearized problem is well posed when

the Reynolds and Peclet numbers are small enough. Itfollows from the implicit function theorem that underthese conditions, the equation (23) defines a -mapping

in a neighborhood of .

Remark 2 Under Assumption 2, we haveand for all . Therefore, the physi-cal solution of (1)-(9) is continuously differentiable withrespect to the domain i.e. the mapping

, with, is of class

in a neighbourhood of . On the other hand,if Assumption 2 does not hold and if , thenin general and . And, fora function , or , the map-ping is of class from into([11], Lemma IV.4). Hence, in order to have the mapping

of class in , one needs to consider transformationsmore regular (for example, transformations of class

).

Differential of the cost function

Theorem 1 If the Reynolds andPeclet numbersare smallenoughthen there exists a neighborhood of the identitymapping in such that the cost function

is of class for all. Furthermore, for all , we have:

(24)

where is the solution of the state equation (20) posedin and is the unique solution of the adjoint stateequation:

Find such that

(25)

Proof. The observation function is of classand the solution is of class in (Lemma1), hencethe cost function is also of class for . Thecomputation method of the derivatives with respect tohas beendescribedpreviously:we transport the mappingsto the “reference” domain ; then, by definition, the par-tial derivative with respect to is the partial derivativeof the transported mappings with respect to (see e.g.[9], [4] for detailed computations). In others respects, theexpression (24) of the differential of the cost function fol-lows from the classical adjoint state method.

The term in (24) can be expressed by:

With the notations and , it gives:

where trace , and be-

ing two real matrix .

Shape optimal design in fluid mechanics 5 J. Monnier

where is the unit tangent vector to .

The adjoint state equation (25) is equivalent to a sys-tem of linear partial differential equations and an integralequation. These equations are solved in the reverse waycompared to the state equation and each of them is the ad-joint equation of the corresponding equation to the directproblem. They are:

Find such that:

Given findsuch that:

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[4] D. Chenais, J. Monnier, and J.P. Vila, ‘Analysis of anoptimal design problem in fluid mechanics with radiativethermal transfers’, submitted.

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Shape optimal design in fluid mechanics 6 J. Monnier