aniterativemethodforshapeoptimaldesignof stokes...

Research ArticleAn Iterative Method for Shape Optimal Design ofStokesndashBrinkman Equations with Heat Transfer Model

Wenjing Yan 1 Feifei Jing2 Jiangyong Hou3 Zhiming Gao4 and Nannan Zheng1

1School of Mathematics and Statistics Xirsquoan Jiaotong University Xirsquoan 710049 China2School of Mathematics and Statistics Xirsquoan Key Laboratory of Scientific Computation and Applied StatisticsNorthwestern Polytechnical University Xirsquoan 710129 China3School of Mathematics Northwest University Xirsquoan 710127 China4Institute of Applied Physics and Computational Mathematics Beijing 100088 China

Correspondence should be addressed to Wenjing Yan wenjingyanxjtueducn

Received 19 April 2020 Accepted 7 May 2020 Published 29 June 2020

Academic Editor Yumin Cheng

Copyright copy 2020Wenjing Yan et al (is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

(is work is concerned with the shape optimal design of an obstacle immersed in the StokesndashBrinkman fluid which is alsocoupled with a thermal model in the bounded domain (e shape optimal problem is formulated and analyzed based on theframework of the continuous adjoint method with the advantage that the computing cost of the gradients and sensitivities isindependent of the number of design variables (en the velocity method is utilized to describe the domain deformation and theEulerian derivative for the cost functional is established by applying the differentiability of a minimax problem based on thefunction space parametrization technique Moreover an iterative algorithm is proposed to optimize the boundary of the obstaclein order to reduce the total dissipation energy Finally numerical examples are presented to illustrate the feasibility and ef-fectiveness of our method

1 Introduction

(e optimal shape design for the fluid flows has wide ap-plications in engineering design and computational fluidmechanics(e industrial applications include the design forwings profiles impeller blades and high-speed trains In thispaper we focus on identifying the optimal shape of anobstacle located in the viscous and incompressible fluidwhich is governed by StokesndashBrinkman equations stronglycoupled with a thermal model Our purpose is to effectivelyfind the optimal shapes that minimize certain cost functionalwhich may represent a given objective related to the specificcharacteristic features of the fluids subject to mechanicaland geometrical constraints

Different methods have been proposed to numericallysolve the shape optimal problems such as generic algorithm[1] complex Taylor series expansion approach [2] auto-matic differentiation method [3] one-shot method [4 5]level set method [6 7] domain derivative method [8] and

adjoint method [9ndash11] Among the popularly used ap-proaches the adjoint method has received plenty of atten-tion Especially for the shape optimal control in fluids thecost of computing the gradients and sensitivities is inde-pendent of the number of design variables Jameson firstapplied this method to solve the shape design of aircraft [12]Srinath and Mittal presented a numerical method for shapeoptimization for unsteady viscous flows which is based onthe continuous adjoint approach [13] Yagi and Kawaharautilized the adjoint method to identify the optimal shape fora body located in incompressible flow [14]

However many authors considered the shape optimalproblems in fluids without the heat transfer steady stateor not [15 16] In Reference [17] Chenais et al solved theshape optimal problem in a potential flow coupled with athermal model Moreover the number of publications onshape optimal problems for StokesndashBrinkman equationsis relatively small when compared to Stokes equations[18ndash20]

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 9405018 11 pageshttpsdoiorg10115520209405018

In shape optimization the efficient computation requiresa shape calculus which differs from its analog in vectorspaces (e traditional approaches always involve thecomputation of the state derivative with respect to thedomain but the state parameters belong to the functionspaces depending on the variable domain Besides the statedifferentiability is not necessary in many cases even if thestate system is not differentiated To avoid the differentiationof the state system the adjoint method is employed to solveshape optimal problem which just requires to solve only oneextra adjoint system

(is paper is organized as follows In Section 2 we brieflyintroduce the general approach of the shape optimal controlproblem in fluids Section 3 briefly describes the shapeoptimal problem for StokesndashBrinkman equations and heatexchanges are considered Section 4 is devoted to the velocitymethod which is used to perform the domain deformationIn addition the definitions of Eulerian derivative and shapegradient are introduced In Section 5 based on the minimaxprinciple the shape optimal problems can be expressed asthe saddle point problems of some suitable Lagrangianfunctionals (en applying the function space parametri-zation technique and minimax differential theorem wededuce the expression of shape gradient for the Lagrangianfunctional which plays the key role of design variables in theoptimal design framework Finally some numerical exam-ples are presented to verify the effectiveness of the proposedmethod in Section 6

2 Shape Optimal Control Problem in Fluids

In this section we present the general structure to solve theoptimal control problems which will be applied to theparticular case of shape optimal problem in StokesndashBrink-man flow with heat transfer in the following section

Our work is to minimize a cost functional J whichconsists of the solution of the state equations

minJ J(w φ)

Aw f + Bφ1113896 (1)

where w is the state variable A represents an elliptic dif-ferential operator f stands for the source term and B

denotes a differential operator acting on the control variableφ Now we introduce the Lagrangian functional L andLagrangian multiplier λ

L(w λφ) ≔ J(wφ) +langλ f + Bφ minus Awrang (2)

For the linear case problem (1) satisfiesnablaL(w λφ) 0Suppose that W and V are two suitable Hilbert spaces forw isinW and φ isin V we obtain the variational form for stateequation (1)

a(wψ) (fψ) + b(φψ) forallψ isin V (3)

where (middot middot) denotes the inner product a(middot middot) is a bilinearform with respect to a linear elliptic operator andb(φψ) langBφψrang (erefore

L(w λφ) J(wφ) + b(φ λ) +(f λ) minus a(wφ) (4)

We need to solve following problem to obtain the op-timal solution

seek(w λφ) isinW times W times V such thatnablaL(w λφ) 0

(5)

Usually we can apply an iterative method to solve thecontrol problem by choosing an initial value for the variableφ0 At each step we compute the state equations and thenevaluate the cost functional and solve the adjoint equationsWhen φj is available we give a suitable stopping criterionand derive the cost functional derivative Jprime[21 22]

3 Shape Optimal Problem forStokesndashBrinkman Equations withHeat Transfer

In this section we focus on the shape optimal problem ofmodeling flow through porous and partially porous mediawhich is described by StokesndashBrinkman equations with heattransfer Suppose that Ω sub RN(N 2 or 3) is a boundedLipschitz domain which is filled with the incompressibleviscous fluid of the kinematic viscosity ] (e boundary zΩof the domain Ω is smooth and consists of four parts Γndenotes the inflow boundary Γw is the boundary corre-sponding to the fluid wall Γo represents the outflowboundary and Γs is the boundary of the obstacle S which isto be optimized

(e fluid is described by the StokesndashBrinkman equationsstrongly coupled with a thermal model the unknowns arethe fluid velocity u (u1 uN)T Ω⟶ RN the pres-sure p Ω⟶ R and the temperature T Ω⟶ R

minus div σ(u p) + Mu λjT inΩ (6)

div u 0 inΩ (7)

u 0 on Γw cupΓs (8)

u g on Γn (9)

minus αΔT + u middot nablaT 0 inΩ (10)

zT

zn 0 on Γo cup Γs (11)

T T1 on Γw (12)

T T2 on Γn (13)

where the stress tensor σ(u p) is defined by σ(u p) ≔ minus

pI + 2] isin (u) with the rate of deformation tensorisin (u) ≔ (Du + DTu)2 DTu denotes the transpose of thematrix Du and I is the identity tensor (e matrix-valuedfunction M Ω⟶ RNtimesN α denotes the inverse of Pecletnumber λ is the Grashof number and j equals (0 1)T

In this paper our work is to identify the optimal shape ofthe boundary Γs that minimizes the following cost functional J

2 Mathematical Problems in Engineering

minΩisinO

J(Ω) 2]1113946Ω

|isin ( urarr

)|2dx +

12

1113946Ω

|nablaT|2dx (14)

where urarr and T denote the velocity and the temperature and

isin (u) is the rate of deformation tensor (e shape admissibleset O is given by

O ≔ Ω sub RN Γn cupΓw cupΓo is fixed 1113946

Ωdx constant1113882 1113883

(15)

(is type of optimal problem often occurs in the designand control of many industrial equipment (e weak for-mulation associated with (6)ndash(13) can be written as

Find(u p T) isin Vg(Ω) times Q(Ω) times isin H1(Ω) such that

1113938Ω 2] isin (u) isin (v) + vTMu minus p div v1113858 1113859dx 1113938ΩλTj middot v dx forallv isin V0(Ω)

1113938Ωdiv u q dx 0 forallq isin Q(Ω)

1113938Ω(αnablaT middot nablaS + u middot nablaTS)dx 0 forallS isin H1(Ω)

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)

where the functional spaces are defined as follows

V0(Ω) ≔ w isin H1(Ω)1113872 1113873

2 w 0 on Γw cup Γs cup Γn1113882 1113883

Q(Ω) ≔ p isin L2(Ω) 1113946

Ωpdx 01113882 1113883

Vg(Ω) ≔ w isin H1(Ω)1113872 1113873

2 w 0 on Γw cup Γs w g on Γn1113882 1113883

(17)

Shape optimal problems usually involve very largecomputational costs besides the numerical approximationof partial differential equations and optimization To avoidthe differentiation of the state system and save the com-putational cost the adjoint method applied to solve shape

optimal problem can be summarized as follows first weestablish the saddle point problem and the Lagrangianfunctional associated with the cost functional and weak formof the state system (en we are able to perform the shapesensitivity analysis of the Lagrangian functional by theminimax principle concerning the differentiability problemLast but not least applying the function space technique weobtain the Euler derivative of cost functional by the firstvariation of the cost functional with respect to the domain

First we give the following Lagrangian functional whichis associated with (14) and (16)

L(Ω u p T v q S) J(Ω) minus W(Ω u p T v q S) (18)

where

W(Ω u p T v q S) 1113946Ω

2] isin (u) isin (v) + vTMu minus p div v minus div u q minus λTj middot v1113960 1113961dx + 1113946

Ω(αnablaT middot nablaS + u middot nablaTS)dx (19)

Now problem (18) can be transformed into the saddlepoint form

minΩisinO

min(upT)isinVg(Ω)timesQ(Ω)timesH1(Ω)

max(vqS)isinV0(Ω)timesQ(Ω)timesH1(Ω)

L(Ω u p T v q S) (20)

(en we use the minimax framework to avoid theanalysis of the state derivative with respect to the variable

domains and establish the first optimality condition of theshape optimal problem to deduce the adjoint equations



L(Ω u p T v q S) (21)

Since the adjoint equations are defined from theEulerndashLagrange equations of the corresponding Lagrange

functional L the variation of L with respect to (v q S) canrecover the state system and its weak formulation

Mathematical Problems in Engineering 3

Furthermore we can differentiate L with respect to the statevariables (u p T) to deduce the adjoint state system

Differentiating Lagrangian functional L with respect to p

in the direction δp we havezL

zp(Ω u p T v q S) middot δp 1113946

Ωδp div v dx 0 (22)

Owing to δp with compact support in Ω it leads todiv v 0 Moreover we differentiate L with respect to u inthe direction δu and apply Green formula

zL

zu(Ω u p T v q S) middot δu

1113946Ω

(minus 2]Δu + ]Δv minus nablaq minus Mv) middot δu dx minus 1113946Ω

SnablaT middot δu dx minus 2]1113946zΩisin (v) middot n middot δu ds + 4]1113946

zΩisin (u) middot n middot δu ds + 1113946

nablaΩqδu middot n ds

1113946Ω

(minus 2]Δu + ]Δv minus nablaq minus Mv minus SnablaT) middot δu dx

minus 1113946zΩσ(v q) middot n middot δu ds + 4]1113946

zΩisin (u) middot n middot δu ds

(23)

Since δu has compact support in Ω we obtain

minus ]Δv + Mv + nablaq + SnablaT minus 2]Δu (24)

To vary δu on the boundary Γo we deduce

σ(v q) middot n minus 4] isin (u) middot n 0 (25)

Similarly we obtain the adjoint equation with respect toT

zL

zT(Ω u p T v q S) middot δT

1113946Ω

(αΔS + u middot nablaS + λTj middot v) middot δT dx minus 1113946ΓocupΓs

αzS

zn+ u middot S middot n1113888 1113889δT ds

minus 1113946ΩΔT middot δT dx + 1113946

zΩnablaT middot δT middot n ds

(26)

Finally we derive the following adjoint state systemassociated with (6)ndash(13)

minus ]Δv + Mv + nablaq + SnablaT minus 2]Δu inΩ

div v 0 inΩ

σ(v q) middot n minus 4] isin (u) middot n 0 on Γo

v 0 on Γn cup Γw cupΓs

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)

αΔS minus u middot nablaS minus λj middot v minus ΔT inΩ

αzS

zn+ u middot S middot n nablaT middot n on Γo cupΓs

S 0 on Γn cupΓw

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)

4 The Velocity Method

In this section we will apply the velocity method to describethe domain deformation For shape optimal problem the setof domain Ω is not a vectorial space but we need an ex-pression of the differential of the cost functional In order toovercome this difficulty we define the derivative of a real-valued function with respect to the domain so that we canpresent the differential expression for the cost functional toestablish a gradient-type algorithm

Let boundary nablaΩ be piecewise Ck and the velocity fieldV isin Ek ≔ C([0 τ] [Dk(Ω)]N) where τ is a small positivereal number and [Dk(Ω)]N denotes the space of all kminus timescontinuous differentiable functions with compact supportcontained in Ω (e velocity field

V(isin)(x) V(isin x) x isin Ω isin ge 0 (29)


belongs to [Dk(Ω)]N for each ϵ It can generate transfor-mations Fϵ(V)X x(ϵ X) through the following dynamicalsystem

dx

d isin(isin X) V(isin x(isin)) x(0 X) X (30)

with the initial value X (e flow with respect to V can bedefined as the mapping Fisin R

N⟶ RN withFisin(X) x(isin X) where x(isin X) is the solution of (30) (etransformed domain Fisin(V)(Ω) can be denoted byΩisin(V) atisinge 0 and its boundary Γisin ≔ Fisin(nablaΩ)

Next we introduce two definitions for shape sensitivityanalysis(e Eulerian de rivative of the cost functional J(Ω)

at Ω for the velocity field Vrarr

is defined as [23]

limisin0

1isin

J Ωisin( 1113857 minus J(Ω)1113858 1113859 ≔ dJ(Ω V) (31)

Moreover if the map V↦dJ(Ω V) Ek⟶ R is linearand continuous J is shape differentiable at Ω In the dis-tributional sense it leads to

dJ(Ω V) langnablaJ Vrang Dk(D)N( )timesDk(D)N (32)

When J has a Eulerian derivative nablaJ is called theshape gra di ent of J at Ω

5 Function Space Parametrization

In this section we derive the expression of the shape gra-dient for the cost functional J(Ω) by the function spaceparametrization techniques

(e velocity method is applied to describe the domaindeformations We only perturb the boundary Γs and con-sider the mapping Fisin(V) and the flow of the velocity field

V isin Vad ≔ V isin C0

[0 τ] C2R

N1113872 11138731113960 1113961

N1113874 1113875 V 0 in the neighorhood of Γn cupΓw cupΓo1113882 1113883 (33)

(e perturbed domain is denoted by Ωisin Fisin(V)(Ω) We aim to evaluate the derivative of j(isin) with respect toisin where

j(isin) ≔ minuisin pisin Tisin( )isinVg Ωisin( )timesQ Ωisin( )timesH1 Ωisin( )

maxvisin qisin Sisin( )isinV0 Ωisin( )timesQ Ωisin( )timesH1 Ωisin( )

L Ωisin uisin pisin Tisin visin qisin Sisin( 1113857 (34)

and (uisin pisin Tisin) and (visin qisin Sisin) satisfy corresponding stateand adjoint systems on the perturbed domain Ωisin respec-tively However the Sobolev spaces Vg(Ωisin) V0(Ωisin)Q(Ωisin) and W(Ωisin) depend on the perturbation parameterϵ Consequently we need to apply the function space pa-rametrization technique to get rid of it(e advantage of thistechnique is being able to transport different quantitiesdefined on the variable domain Ωisin back into the referencedomain Ω which is entirely unrelated to isin (en we canemploy the differential calculus since the functionals in-volved are defined in a fixed domain Ω with respect to theparameter isin

Now we define the following parametrization functions

Vg Ωisin( 1113857 u ∘Fminus 1isin u isin Vg(Ω)1113966 1113967

V0 Ωisin( 1113857 v ∘ minus 1isin v isin V0(Ω)1113966 1113967

Q Ωisin( 1113857 p ∘Fminus 1isin p isin Q(Ω)1113966 1113967

H1 Ωisin( 1113857 T ∘Fminus 1

isin T isin H1(Ω)1113966 1113967

(35)

where ldquodegrdquo denotes the composition of the two mapsNote that Fisin and Fminus 1

isin are diffeomorphisms so the pa-rametrization will not change the value of the saddle pointWe can rewrite (34) as

j(isin) min(upT)

max(vqS)

L Ωisin u ∘Fminus 1isin p ∘Fminus 1

isin T ∘Fminus 1isin v ∘Fminus 1

isin q ∘Fminus 1isin S ∘Fminus 1

isin1113872 1113873 (36)

where the Lagrangian functional




isin1113872 1113873 I1(isin) + I2(isin) + I3(isin) (37)

with


I1(isin) ≔ 2]1113946Ωϵisin u ∘Fminus 1

isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx +

12

1113946Ωϵnabla T ∘Fminus 1

isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx

I2(isin) ≔ minus 1113946Ωt

2] isin v ∘Fminus 1isin1113872 1113873 isin u ∘Fminus 1

isin1113872 1113873 + v∘Fminus 1isin1113872 1113873

TM u ∘Fminus 1

isin1113872 1113873 minus p ∘Fminus 1isin1113872 1113873div v ∘Fminus 1

isin1113872 1113873 minus div u ∘Fminus 1isin1113872 1113873 q ∘Fminus 1

isin1113872 1113873 minus λj T ∘Fminus 1isin1113872 1113873 v ∘Fminus 1

isin1113872 11138731113876 1113877dx

I3(isin) ≔ minus 1113946Ωϵ

αnabla T ∘Fminus 1isin1113872 1113873 middot nabla S ∘Fminus 1

isin1113872 1113873 + u ∘Fminus 1isin1113872 1113873 middot nabla T ∘Fminus 1

isin1113872 1113873 S ∘Fminus 1isin1113872 11138731113960 1113961dx

(38)

Next work is to differentiate the perturbed Lagrangianfunctional L(Ωisin u ∘Fminus 1

isin p ∘Fminus 1isin T ∘Fminus 1

isin v ∘Fminus 1isin q ∘Fminus 1

isin

S ∘Fminus 1isin ) so we introduce the following Hadamard formula to

perform the differentiationdd isin

1113946ΩisinT(isin x)dx 1113946

Ωisin

nablaTnablaϵ

(isin x)dx + 1113946nablaΩisin

T(isin x)V middot nisinds

(39)

for a sufficiently smooth functional T [0 τ] times RN⟶ RApplying (39) we have

zϵL Ωϵ u ∘Fminus 1ϵ p∘Fminus 1

ϵ T ∘Fminus 1ϵ v ∘Fminus 1

ϵ q ∘ minus 1ϵ S∘Fminus 1ϵ1113872 1113873

11138681113868111386811138681113868ϵ0 I1prime (0) + I2prime (0) + I3prime(0) (40)

where

I1prime (0) 4]1113938Ωisin (u) ϵ(minus Du middot V)dx + 2]1113938Γs|isin ( u

rarr)|2Vnds + 1113938ΩnablaT middot nabla(minus DT middot V)dx +

12

1113946Γs

|nablaT|2Vnds (41)

I2prime (0) minus 1113946Ω

2] isin (minus Du middot V) middot ϵ(v) + 2] isin (u) middot ϵ(minus Dv middot V) + vTM middot (minus Du middot V) + Mu middot (minus Dv middot V) minus (minus nablap middot V)div v1113960

minus p div(minus Dv middot V) minus (minus nablaq middot V)div u minus λjT middot (minus Dv middot V) minus λj(minus DT middot V) middot v minus q div(minus Du middot V)1113859dx

+ 1113946Γs

minus 2] isin (u) isin (v) minus vT

Mu + p div v + div uq1113872 1113873Vnds

(42)

I3prime (0) minus 1113946Ωαnabla(minus DS middot V) middot nablaTdx minus 1113946

ΩαnablaS middot nabla(minus DT middot V)dx minus 1113946

Ω(minus Du middot V) middot nablaTS dx minus 1113946

Ωu middot nabla(minus DT middot V)Sdx + 1113946

Ωu

middot nablaT(DS middot V)Sdx minus 1113946Γs

u middot (minus DT middot V)S middot nds minus 1113946Γs

(αnablaT middot nablaS + u middot nablaTS)Vnds minus 1113946ΓsαnablaT middot (minus DS middot V) middot ndx minus 1113946

ΓsαnablaS

middot (minus DT middot V) middot ndx

(43)

In order to simplify the above identities we introducethe following lemma

Lemma 1 (see [23]) If vector functions u and v vanish on theboundary Γs the following identities hold on the boundary Γs

Du middot V middot n div uVn

isin (u) isin (v) (isin (u) middot n) middot (isin (v) middot n)

(isin (u) middot n) middot (Dv middot V) (isin (u) middot n) middot (isin (v) middot n)Vn

(44)

We apply Lemma 1 and obtain

I1prime (0) minus 2]1113946ΩΔu middot (minus Du middot V)dx minus 2]1113946

Γs|isin (u)|2Vnds minus 1113946

ΩΔT middot (minus DT middot V)dx

+ 1113946Γs

(nablaT middot n) middot (minus nablaT middot V)ds +12

1113946Γs

|nablaT|2Vnds

(45)


Recalling (u p T) and (v q S) satisfies the state andadjoint system respectively and (42) can be reduced to

I2prime(0) 1113946Ω

[(]Δu minus Mu minus nablap minus λjT) middot (minus Dv middot V)]dx minus 1113946Γs

(2] isin (u) isin (v))Vnds

+ 1113946Ω

]Δv minus vTM minus nablaq1113872 11138731113960 1113961 middot (minus Du middot V)dx + 1113946

Ωλj(minus DT middot V) middot vdx minus 1113946

Γs[σ(u p) middot n middot (minus Dv middot V) + σ(v q) middot n middot (minus Du middot V)ds]

1113946Ω

(2]Δu + nablaTS)(minus Du middot V)dx + 1113946Ωλj(minus DT middot V) middot v

rarrdx

+ 1113946Γs


(46)

Similarly (43) can be rewritten as

I3prime(0) 1113946Ω

(αΔT minus u middot nablaT)(minus DS middot V)dx + 1113946Ω

(αΔS + u middot nablaS)(minus DT middot V)dx

minus 1113946Ω

(minus Du middot V) middot nablaTSdx minus 1113946Γs

(αnablaT middot nablaS + u middot nablaTS)Vnds minus 1113946Γs

u middot (minus DT middot V)S middot nds minus 1113946ΓsαnablaT middot (minus DS middot V) middot nds

minus 1113946ΓsαnablaS middot (minus DT middot V) middot nds

1113946Ω

(ΔT minus λj middot v)(minus DT middot V)dx minus 1113946ΩnablaTS middot (minus Du middot V)dx

+ 1113946Γs


u middot (minus DT middot V)S middot nds

minus 1113946ΓsαnablaT middot (minus DS middot V) middot nds minus 1113946

ΓsαnablaS middot (minus DT middot V) middot nds

(47)

Finally we have the boundary expression for theEulerian derivative of J(Ω)

dJ(Ω V) 2]1113946Γsisin (u) isin (v) minus |isin (u)|

21113960 1113961Vnds +

12

1113946Γs

|nablaT|2Vnds + 1113946

ΓsαnablaT middot nablaSVnds (48)

According to (32) we derive the expression of the shapegradient for the cost functional

nablaJ 2] isin (u) isin (v) minus |isin (u)|2

1113872 1113873 +12|nablaT|

2+ αnablaT middot nablaS1113876 1113877n

(49)

6 Numerical Examples

(is section is devoted to present the numerical algorithmand examples for the shape optimization problem in twodimensions

We consider the optimal design of a body immersed in aStokesndashBrinkman flow and aim at reducing the dissipationenergy acting on its surface Namely we solve the mini-mization problem

minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)

subject to (6)ndash(13)For the minimization problem (50) we rather work with

the following minimization problem

minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)


Choose an initial shape Ω0 and initial step h0 and a Lagrangian multiplier L0while ϵre le ϵ doStep 1 solve state system (6)ndash(13)Step 2 compute adjoint system (27) and (28)Step 3 evaluate the cost functionalStep 4 compute the descent direction dk by (56)Step 5 set Ωk+1 (I + hkdk)Ωk and a suitable Lagrange multiplier lk+1 where hk is a small positive real numberen dwhile

ALGORITHM 1 Iterative algorithm for shape optimal control

(a) (b)

(c) (d)

(e) (f )

(g) (h)

Figure 1 Case 1 comparison of the initial shape and optimal shape (Reynolds number 1000) (a) Mesh for initial shape (b) u1 for initialshape (c) Mesh for optimal shape (d) u1 for optimal shape (e) u2 for initial shape (f ) p for initial shape (g) u2 for optimal shape (h) p foroptimal shape


where A(Ω) ≔ 1113938Ωdx dy l is a positive Lagrangian multi-plier and G(Ω) satisfies the following equation

dG 1113946ΓsnablaG middot V ds (52)

where dG is the shape gradient with

dG 2] ϵ(u) isin (v) minus |isin (u)|2

1113872 1113873 +12|nablaT|

2+ αnablaT middot nablaS + l1113876 1113877n

(53)

Taking no account of regularization a descent directionis sought by

(a) (b)

(c) (d)

(e) (f )

(g) (h)



V minus hknablaG (54)

and then the shape of domain Ω can be updated as

Ωk I + hkV( 1113857Ω (55)

where hk is a small descent step at k-th iteration We obtainthe iterative scheme

Jk+1 Jk minus hk nablaJknablaJk( 11138570 Jk ≔ J Ωk( 1113857 (56)

To avoid shape oscillations we have to project or smooththe variation into H1(Ω) (erefore we choose the descentdirection d isin H1(Ω)2 which is the unique solution of theproblem

1113946Ω

D d DVdx minus dJ(ΩV) forallV isin H1(Ω)

2 (57)

It is obvious that d is a descent direction which guar-antees the decrease of the cost functional J(Ω) (e com-putation of d is seemed as a regularization of the shapegradient

(en we consider how to choose the Lagrangian mul-tiplier l in the optimization problem In order to satisfy thefixed constraint the value of l is updated at each iteration Asa result of the high cost in moving the mesh we do notimpose exactly the volume constraint before convergence Ifthe present area is smaller than the target area we decreasethe multiplier l otherwise we increase it We suppose

dG(ΩV) dJ(ΩV) + ldV(ΩV) 0 (58)

at least in the average sense on the boundary Γs

l minus1113938Γs

dJds

1113938Γsds

(59)

(erefore we update the Lagrange multiplier by

lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)

where m is a small positive parameter and As(Ω) denotes thetarget area

We propose the numerical algorithm for solving theshape optimal problem in a StokesndashBrinkman flow withconvective transfer (e algorithm is terminated when therelative decrease (denoted by isinre) of two consecutive ob-jective is less than a given tolerance isin (Algorithm 1)

We restrict the shape optimal problem posed on abounded rectangular domain by introducing an artificialboundary zD and Ω DS is the effective domain with itsboundary zΩ Γn cup Γw cup Γo (e fluid enters horizontallyfrom the left boundary Γn and exits from the right boundaryΓo We choose the initial shapes of the obstacle S to bedifferent curves

Case 1 a circle whose center is at origin with radius 05Case 2 an elliptic curve x 045lowast ost + 04 y 008lowastsin t t isin [0 2π]

For the two examples the inflow velocities are assumedto be parabolic with the profiles g(0 y) (1 minus y4 0)T andg(0 y) (025 minus y2 0)T respectively Also the no-slipboundary conditions are imposed at all the other bound-aries (e admissible set is defined by

O Ω sub R2 Γn cupΓw cupΓo is fixed the areaAtarget(Ω) constant1113966 1113967 (61)

Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration

Figure 3 Case 1 convergence history of the cost functional(Reynolds number 1000)

Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration



Figures 1 and 2 show the comparison between the initialshape and optimal shape for the computing meshes thecontours of the velocity u

rarr (u1 u2)

T and the pressure p

with different Reynolds numbers Figures 3 and 4 demon-strate that the proposed method is effective stable andrapidly convergent We also observe that when the Reynoldsnumber increases the cost of the optimization procedurerises due to the increase of computation of the state andadjoint system

7 Conclusion

(is work focuses on the optimal shape determination in anincompressible viscous StokesndashBrinkman flow with theconsideration of heat transfer Based on the adjoint methodand the function space parametrization technique we derivethe shape gradient of the cost functional by involving aLagrangian functional which plays the key role of designvariables in the optimal design framework Moreover wepropose a gradient-type algorithm for the minimizationdissipation energy problem Finally we present numericalexamples to demonstrate the proposed algorithm is feasibleand effective for the quite high Reynolds number problems

Data Availability

(e data and code used in this study cannot be shared at thistime as the data also form a part of an ongoing study

Conflicts of Interest

(e authors declare that they have no conflicts of interest

Acknowledgments

(is study was supported by the National Natural ScienceFoundation of China (grant nos 11971377 91730306 and11701451) Natural Science Foundation of Shaanxi Province(grant nos 2019JM-367 2018JQ1077 and 2019JQ-173) andScientific Research Program funded by Shaanxi ProvincialEducation Department (grant no 17JK0787)

References

[1] S Milewski ldquoDetermination of the truss static state by meansof the combined FEGA approach on the basis of strain anddisplacement measurementsrdquo Inverse Problems in Science andEngineering vol 27 no 11 pp 1537ndash1558 2019

[2] C O E Burg and J C Newma III ldquoComputationally efficientnumerically exact design space derivatives via the complexTaylorrsquos series expansion methodrdquo Computers amp Fluidsvol 32 no 3 pp 373ndash383 2003

[3] L Sherman A Taylor L Green et al ldquoFirst and second orderaerodynamic sensitivity derivatives via automatic differenti-ation with incremental iterative methodsrdquo Journal of Com-putational Physics vol 129 no 2 pp 307ndash331 1994

[4] S B Hazra and A Jameson ldquoOne-shot pseudo-time methodfor aerodynamic shape optimization using the Navier-Stokesequationsrdquo International Journal for Numerical Methods inFluids vol 68 no 5 pp 564ndash581 2012

[5] L Kusch T Albring A Walther and N R Gauger ldquoA one-shot optimization framework with additional equality con-straints applied to multi-objective aerodynamic shape opti-mizationrdquo Optimization Methods and Software vol 33no 4ndash6 pp 694ndash707 2018

[6] P Fulmanski A Laurain J Scheid et al ldquoLevel set methodwith topological derivatives in shape optimizationrdquo Inter-national Journal of Computer Mathematics vol 85 no 10pp 1491ndash1514 2018

[7] S Zhu Q Wu and C Liu ldquoShape and topology optimizationfor elliptic boundary value problems using a piecewise con-stant level set methodrdquo Applied Numerical Mathematicsvol 61 no 6 pp 752ndash767 2011

[8] W Yan Y He and YMa ldquoA numerical method for the viscousincompressible Oseen flow in shape reconstructionrdquo AppliedMathematical Modelling vol 36 no 1 pp 301ndash309 2012

[9] B Mohammadi and O Pironneau ldquoShape optimization influid mechanicsrdquo Annual Review of Fluid Mechanics vol 36no 1 pp 255ndash279 2004

[10] M Giles and N Pierce ldquoAn introduction to the adjoint ap-proach to designrdquo Flow Turbulence and Combustion vol 65no 3-4 pp 393ndash415 2000

[11] E Katamine H Azegami T Tsubata and S Itoh ldquoSolution toshape optimization problems of viscous flow fieldsrdquo Inter-national Journal of Computational Fluid Dynamics vol 19no 1 pp 45ndash51 2005

[12] A Jameson ldquoComputational algorithms for aerodynamicanalysis and designrdquo Applied Numerical Mathematics vol 13no 5 pp 383ndash422 1993

[13] D N Srinath and S Mittal ldquoAn adjoint method for shapeoptimization in unsteady viscous flowsrdquo Journal of Compu-tational Physics vol 229 no 6 pp 1994ndash2008 2010

[14] H Yagi and M Kawahara ldquoOptimal shape determination of abody located in incompressible viscous fluid flowrdquo ComputerMethods in Applied Mechanics and Engineering vol 196no 49ndash52 pp 5084ndash5091 2007

[15] Y Ogawa and M Kawahara ldquoShape optimization of bodylocated in incompressible viscous flow based on optimalcontrol theoryrdquo International Journal of Computational FluidDynamics vol 17 no 4 pp 243ndash251 2003

[16] H Heck G Uhlmann G Uhlmann and J-N Wang ldquoRe-construction of obstacles immersed in an incompressible fluidrdquoInverse Problems amp Imaging vol 1 no 1 pp 63ndash76 2007

[17] D Chenais J Monnier and J P Vila ldquoShape optimal designproblem with convective and radiative heat transfer analysisand implementationrdquo Journal of Optimization Beory andApplications vol 110 no 1 pp 75ndash117 2001

[18] W Yan Y He and Y Ma ldquoShape inverse problem for the two-dimensional unsteady stokes flowrdquo Numerical Methods forPartial Differential Equations vol 26 no 3 pp 690ndash701 2010

[19] F Caubet andM Dambrine ldquoLocalization of small obstacles inStokes flowrdquo Inverse Problems vol 28 no 10 p 105007 2012

[20] A Abda M Hassine M Jaoua et al ldquoTopological sensitivityanalysis for the location of small cavities in stokes flowrdquo SIAMJournal on Control and Optimization vol 48 no 5pp 2871ndash2900 2009

[21] J Lions Optimal Control of Systems Governed by Partial Dif-ferential Equations Springer-Verlag New-York NY USA 1971

[22] O Pironneau Optimal Shape Design by Local BoundaryVariations Springer-Verlag Berlin Germany 1988

[23] M C Delfour and J-P Zolesio Shapes and GeometriesAnalysis Differential Calculus and Optimization Advance inDesign and Control Society for Industrial and AppliedMathematics Philadelphia PA USA 2001


In shape optimization the efficient computation requiresa shape calculus which differs from its analog in vectorspaces (e traditional approaches always involve thecomputation of the state derivative with respect to thedomain but the state parameters belong to the functionspaces depending on the variable domain Besides the statedifferentiability is not necessary in many cases even if thestate system is not differentiated To avoid the differentiationof the state system the adjoint method is employed to solveshape optimal problem which just requires to solve only oneextra adjoint system

(is paper is organized as follows In Section 2 we brieflyintroduce the general approach of the shape optimal controlproblem in fluids Section 3 briefly describes the shapeoptimal problem for StokesndashBrinkman equations and heatexchanges are considered Section 4 is devoted to the velocitymethod which is used to perform the domain deformationIn addition the definitions of Eulerian derivative and shapegradient are introduced In Section 5 based on the minimaxprinciple the shape optimal problems can be expressed asthe saddle point problems of some suitable Lagrangianfunctionals (en applying the function space parametri-zation technique and minimax differential theorem wededuce the expression of shape gradient for the Lagrangianfunctional which plays the key role of design variables in theoptimal design framework Finally some numerical exam-ples are presented to verify the effectiveness of the proposedmethod in Section 6

2 Shape Optimal Control Problem in Fluids

In this section we present the general structure to solve theoptimal control problems which will be applied to theparticular case of shape optimal problem in StokesndashBrink-man flow with heat transfer in the following section

Our work is to minimize a cost functional J whichconsists of the solution of the state equations

minJ J(w φ)

Aw f + Bφ1113896 (1)

where w is the state variable A represents an elliptic dif-ferential operator f stands for the source term and B

denotes a differential operator acting on the control variableφ Now we introduce the Lagrangian functional L andLagrangian multiplier λ

L(w λφ) ≔ J(wφ) +langλ f + Bφ minus Awrang (2)

For the linear case problem (1) satisfiesnablaL(w λφ) 0Suppose that W and V are two suitable Hilbert spaces forw isinW and φ isin V we obtain the variational form for stateequation (1)

a(wψ) (fψ) + b(φψ) forallψ isin V (3)

where (middot middot) denotes the inner product a(middot middot) is a bilinearform with respect to a linear elliptic operator andb(φψ) langBφψrang (erefore

L(w λφ) J(wφ) + b(φ λ) +(f λ) minus a(wφ) (4)

We need to solve following problem to obtain the op-timal solution

seek(w λφ) isinW times W times V such thatnablaL(w λφ) 0

(5)

Usually we can apply an iterative method to solve thecontrol problem by choosing an initial value for the variableφ0 At each step we compute the state equations and thenevaluate the cost functional and solve the adjoint equationsWhen φj is available we give a suitable stopping criterionand derive the cost functional derivative Jprime[21 22]

3 Shape Optimal Problem forStokesndashBrinkman Equations withHeat Transfer

In this section we focus on the shape optimal problem ofmodeling flow through porous and partially porous mediawhich is described by StokesndashBrinkman equations with heattransfer Suppose that Ω sub RN(N 2 or 3) is a boundedLipschitz domain which is filled with the incompressibleviscous fluid of the kinematic viscosity ] (e boundary zΩof the domain Ω is smooth and consists of four parts Γndenotes the inflow boundary Γw is the boundary corre-sponding to the fluid wall Γo represents the outflowboundary and Γs is the boundary of the obstacle S which isto be optimized

(e fluid is described by the StokesndashBrinkman equationsstrongly coupled with a thermal model the unknowns arethe fluid velocity u (u1 uN)T Ω⟶ RN the pres-sure p Ω⟶ R and the temperature T Ω⟶ R

minus div σ(u p) + Mu λjT inΩ (6)

div u 0 inΩ (7)

u 0 on Γw cupΓs (8)

u g on Γn (9)

minus αΔT + u middot nablaT 0 inΩ (10)

zT

zn 0 on Γo cup Γs (11)

T T1 on Γw (12)

T T2 on Γn (13)

where the stress tensor σ(u p) is defined by σ(u p) ≔ minus

pI + 2] isin (u) with the rate of deformation tensorisin (u) ≔ (Du + DTu)2 DTu denotes the transpose of thematrix Du and I is the identity tensor (e matrix-valuedfunction M Ω⟶ RNtimesN α denotes the inverse of Pecletnumber λ is the Grashof number and j equals (0 1)T

In this paper our work is to identify the optimal shape ofthe boundary Γs that minimizes the following cost functional J


minΩisinO

J(Ω) 2]1113946Ω

|isin ( urarr

)|2dx +

12

1113946Ω

|nablaT|2dx (14)





(15)






⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)


V0(Ω) ≔ w isin H1(Ω)1113872 1113873


Q(Ω) ≔ p isin L2(Ω) 1113946

Ωpdx 01113882 1113883

Vg(Ω) ≔ w isin H1(Ω)1113872 1113873


(17)





where

W(Ω u p T v q S) 1113946Ω




minΩisinO



L(Ω u p T v q S) (20)





L(Ω u p T v q S) (21)










zL


1113946Ω





1113946Ω




(23)






zL


1113946Ω


αzS




(26)



div v 0 inΩ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)


αzS


S 0 on Γn cupΓw

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)







dx






is defined as [23]

limisin0

1isin









[0 τ] C2R

N1113872 11138731113960 1113961












isin T isin H1(Ω)1113966 1113967

(35)



j(isin) min(upT)

max(vqS)




isin1113872 1113873 (36)






with



isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx +

12


isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx




TM u ∘Fminus 1




isin1113872 11138731113876 1113877dx





(38)




isin




Ωisin

nablaTnablaϵ



(39)






where



12

1113946Γs

|nablaT|2Vnds (41)




+ 1113946Γs



(42)





Ωu




ΓsαnablaS


(43)






(44)





+ 1113946Γs


1113946Γs

|nablaT|2Vnds

(45)



I2prime(0) 1113946Ω



+ 1113946Ω




1113946Ω


rarrdx

+ 1113946Γs


(46)


I3prime(0) 1113946Ω



minus 1113946Ω





1113946Ω


+ 1113946Γs





(47)



21113960 1113961Vnds +

12

1113946Γs





1113872 1113873 +12|nablaT|


(49)




minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)



minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)




(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References

























minΩisinO

J(Ω) 2]1113946Ω

|isin ( urarr

)|2dx +

12

1113946Ω

|nablaT|2dx (14)





(15)






⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

(16)


V0(Ω) ≔ w isin H1(Ω)1113872 1113873


Q(Ω) ≔ p isin L2(Ω) 1113946

Ωpdx 01113882 1113883

Vg(Ω) ≔ w isin H1(Ω)1113872 1113873


(17)





where

W(Ω u p T v q S) 1113946Ω




minΩisinO



L(Ω u p T v q S) (20)





L(Ω u p T v q S) (21)










zL


1113946Ω





1113946Ω




(23)






zL


1113946Ω


αzS




(26)



div v 0 inΩ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)


αzS


S 0 on Γn cupΓw

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)







dx






is defined as [23]

limisin0

1isin









[0 τ] C2R

N1113872 11138731113960 1113961












isin T isin H1(Ω)1113966 1113967

(35)



j(isin) min(upT)

max(vqS)




isin1113872 1113873 (36)






with



isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx +

12


isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx




TM u ∘Fminus 1




isin1113872 11138731113876 1113877dx





(38)




isin




Ωisin

nablaTnablaϵ



(39)






where



12

1113946Γs

|nablaT|2Vnds (41)




+ 1113946Γs



(42)





Ωu




ΓsαnablaS


(43)






(44)





+ 1113946Γs


1113946Γs

|nablaT|2Vnds

(45)



I2prime(0) 1113946Ω



+ 1113946Ω




1113946Ω


rarrdx

+ 1113946Γs


(46)


I3prime(0) 1113946Ω



minus 1113946Ω





1113946Ω


+ 1113946Γs





(47)



21113960 1113961Vnds +

12

1113946Γs





1113872 1113873 +12|nablaT|


(49)




minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)



minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)




(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References































zL


1113946Ω





1113946Ω




(23)






zL


1113946Ω


αzS




(26)



div v 0 inΩ



⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(27)


αzS


S 0 on Γn cupΓw

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(28)







dx






is defined as [23]

limisin0

1isin









[0 τ] C2R

N1113872 11138731113960 1113961












isin T isin H1(Ω)1113966 1113967

(35)



j(isin) min(upT)

max(vqS)




isin1113872 1113873 (36)






with



isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx +

12


isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx




TM u ∘Fminus 1




isin1113872 11138731113876 1113877dx





(38)




isin




Ωisin

nablaTnablaϵ



(39)






where



12

1113946Γs

|nablaT|2Vnds (41)




+ 1113946Γs



(42)





Ωu




ΓsαnablaS


(43)






(44)





+ 1113946Γs


1113946Γs

|nablaT|2Vnds

(45)



I2prime(0) 1113946Ω



+ 1113946Ω




1113946Ω


rarrdx

+ 1113946Γs


(46)


I3prime(0) 1113946Ω



minus 1113946Ω





1113946Ω


+ 1113946Γs





(47)



21113960 1113961Vnds +

12

1113946Γs





1113872 1113873 +12|nablaT|


(49)




minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)



minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)




(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References


























dx






is defined as [23]

limisin0

1isin









[0 τ] C2R

N1113872 11138731113960 1113961












isin T isin H1(Ω)1113966 1113967

(35)



j(isin) min(upT)

max(vqS)




isin1113872 1113873 (36)






with



isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx +

12


isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx




TM u ∘Fminus 1




isin1113872 11138731113876 1113877dx





(38)




isin




Ωisin

nablaTnablaϵ



(39)






where



12

1113946Γs

|nablaT|2Vnds (41)




+ 1113946Γs



(42)





Ωu




ΓsαnablaS


(43)






(44)





+ 1113946Γs


1113946Γs

|nablaT|2Vnds

(45)



I2prime(0) 1113946Ω



+ 1113946Ω




1113946Ω


rarrdx

+ 1113946Γs


(46)


I3prime(0) 1113946Ω



minus 1113946Ω





1113946Ω


+ 1113946Γs





(47)



21113960 1113961Vnds +

12

1113946Γs





1113872 1113873 +12|nablaT|


(49)




minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)



minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)




(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References


























isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx +

12


isin1113872 111387311138681113868111386811138681113868

111386811138681113868111386811138682dx




TM u ∘Fminus 1




isin1113872 11138731113876 1113877dx





(38)




isin




Ωisin

nablaTnablaϵ



(39)






where



12

1113946Γs

|nablaT|2Vnds (41)




+ 1113946Γs



(42)





Ωu




ΓsαnablaS


(43)






(44)





+ 1113946Γs


1113946Γs

|nablaT|2Vnds

(45)



I2prime(0) 1113946Ω



+ 1113946Ω




1113946Ω


rarrdx

+ 1113946Γs


(46)


I3prime(0) 1113946Ω



minus 1113946Ω





1113946Ω


+ 1113946Γs





(47)



21113960 1113961Vnds +

12

1113946Γs





1113872 1113873 +12|nablaT|


(49)




minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)



minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)




(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References


























I2prime(0) 1113946Ω



+ 1113946Ω




1113946Ω


rarrdx

+ 1113946Γs


(46)


I3prime(0) 1113946Ω



minus 1113946Ω





1113946Ω


+ 1113946Γs





(47)



21113960 1113961Vnds +

12

1113946Γs





1113872 1113873 +12|nablaT|


(49)




minΩisinO

J(Ω) 2]1113946Ω

|isin (u)|2dx +

12

1113946Ω

|nablaT|2dx (50)



minΩisinR2

G(Ω) J(Ω) + lA(Ω) (51)




(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References



























(a) (b)

(c) (d)

(e) (f )

(g) (h)







1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References





























1113872 1113873 +12|nablaT|


(53)


(a) (b)

(c) (d)

(e) (f )

(g) (h)





Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References



























Ωk I + hkV( 1113857Ω (55)




1113946Ω


2 (57)





l minus1113938Γs

dJds

1113938Γsds

(59)


lk+1 lk+l + l( 1113857

2+ m

A Ωk( 1113857 minus As(Ω)1113868111386811138681113868

1113868111386811138681113868

As(Ω) (60)







Cost functional

011

012

013

014

015

016

017

018

019

Cos

t fun

ctio

nal

40 6020 80 1000Iteration


Cost functional

02

022

024

026

028

03

032

034

Cos

t fun

ctio

nal

20 40 60 80 1000Iteration




rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References


























rarr (u1 u2)



7 Conclusion


Data Availability




Acknowledgments


References

























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