# An Iterative Method for Shape Optimal Design of Stokes–Brinkman Equations with Heat Transfer Model

Authors: Wenjing Yan; Feifei Jing; Jiangyong Hou; Zhiming Gao; Nannan Zheng
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This work is concerned with the shape optimal design of an obstacle immersed in the Stokes–Brinkman fluid, which is also coupled with a thermal model in the bounded domain. The shape optimal problem is formulated and analyzed based on the framework of the continuous adjoint method, with the advantage that the computing cost of the gradients and sensitivities is independent of the number of design variables. Then, the velocity method is utilized to describe the domain deformation, and the Eulerian derivative for the cost functional is established by applying the differentiability of a minimax problem based on the function space parametrization technique. Moreover, an iterative algorithm is proposed to optimize the boundary of the obstacle in order to reduce the total dissipation energy. Finally, numerical examples are presented to illustrate the feasibility and effectiveness of our method.


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## Body

## 1. Introduction

The optimal shape design for the fluid flows has wide applications in engineering design and computational fluid mechanics. The industrial applications include the design for wings profiles, impeller blades, and high-speed trains. In this paper, we focus on identifying the optimal shape of an obstacle located in the viscous and incompressible fluid, which is governed by Stokes–Brinkman equations strongly coupled with a thermal model. Our purpose is to effectively find the optimal shapes that minimize certain cost functional which may represent a given objective related to the specific characteristic features of the fluids, subject to mechanical and geometrical constraints.Different methods have been proposed to numerically solve the shape optimal problems, such as generic algorithm [1], complex Taylor series expansion approach [2], automatic differentiation method [3], one-shot method [4, 5], level set method [6, 7], domain derivative method [8], and adjoint method [9–11]. Among the popularly used approaches, the adjoint method has received plenty of attention. Especially for the shape optimal control in fluids, the cost of computing the gradients and sensitivities is independent of the number of design variables. Jameson first applied this method to solve the shape design of aircraft [12]. Srinath and Mittal presented a numerical method for shape optimization for unsteady viscous flows which is based on the continuous adjoint approach [13]. Yagi and Kawahara utilized the adjoint method to identify the optimal shape for a body located in incompressible flow [14].However, many authors considered the shape optimal problems in fluids without the heat transfer, steady state or not [15, 16]. In Reference [17], Chenais et al. solved the shape optimal problem in a potential flow coupled with a thermal model. Moreover, the number of publications on shape optimal problems for Stokes–Brinkman equations is relatively small when compared to Stokes equations [18–20].In shape optimization, the efficient computation requires a shape calculus which differs from its analog in vector spaces. The traditional approaches always involve the computation of the state derivative with respect to the domain, but the state parameters belong to the function spaces depending on the variable domain. Besides, the state differentiability is not necessary in many cases even if the state system is not differentiated. To avoid the differentiation of the state system, the adjoint method is employed to solve shape optimal problem, which just requires to solve only one extra adjoint system.This paper is organized as follows. In Section2, we briefly introduce the general approach of the shape optimal control problem in fluids. Section 3 briefly describes the shape optimal problem for Stokes–Brinkman equations and heat exchanges are considered. Section 4 is devoted to the velocity method which is used to perform the domain deformation. In addition, the definitions of Eulerian derivative and shape gradient are introduced. In Section 5, based on the minimax principle, the shape optimal problems can be expressed as the saddle point problems of some suitable Lagrangian functionals. Then, applying the function space parametrization technique and minimax differential theorem, we deduce the expression of shape gradient for the Lagrangian functional, which plays the key role of design variables in the optimal design framework. Finally, some numerical examples are presented to verify the effectiveness of the proposed method in Section 6.


## 2. Shape Optimal Control Problem in Fluids

In this section, we present the general structure to solve the optimal control problems, which will be applied to the particular case of shape optimal problem in Stokes–Brinkman flow with heat transfer in the following section.Our work is to minimize a cost functionalJ which consists of the solution of the state equations:(1)minJ=Jw,φ,Aw=f+ℬφ,where w is the state variable, A represents an elliptic differential operator, f stands for the source term, and ℬ denotes a differential operator acting on the control variable φ. Now, we introduce the Lagrangian functional ℒ and Lagrangian multiplier λ:(2)ℒw,λ,φ≔Jw,φ+λ,f+ℬφ−Aw.For the linear case, problem (1) satisfies ∇ℒw,λ,φ=0. Suppose that W and V are two suitable Hilbert spaces; for w∈W and φ∈V, we obtain the variational form for state equation (1):(3)aw,ψ=f,ψ+bφ,ψ,∀ψ∈V,where ⋅,⋅ denotes the inner product, a⋅,⋅ is a bilinear form with respect to a linear elliptic operator, and bφ,ψ=ℬφ,ψ. Therefore,(4)ℒw,λ,φ=Jw,φ+bφ,λ+f,λ−aw,φ.We need to solve following problem to obtain the optimal solution:(5)seekw,λ,φ∈W×W×V,such that∇ℒw,λ,φ=0.Usually, we can apply an iterative method to solve the control problem by choosing an initial value for the variableφ0. At each step, we compute the state equations and then evaluate the cost functional and solve the adjoint equations. When φj is available, we give a suitable stopping criterion and derive the cost functional derivative J′[21, 22].


## 3. Shape Optimal Problem for Stokes–Brinkman Equations with Heat Transfer

In this section, we focus on the shape optimal problem of modeling flow through porous and partially porous media, which is described by Stokes–Brinkman equations with heat transfer. Suppose thatΩ⊂ℝNN=2or3 is a bounded Lipschitz domain which is filled with the incompressible viscous fluid of the kinematic viscosity ν. The boundary ∂Ω of the domain Ω is smooth and consists of four parts. Γn denotes the inflow boundary, Γw is the boundary corresponding to the fluid wall, Γo represents the outflow boundary, and Γs is the boundary of the obstacle S which is to be optimized.The fluid is described by the Stokes–Brinkman equations strongly coupled with a thermal model; the unknowns are the fluid velocityu=u1,…,uNT:Ω⟶ℝN, the pressure p:Ω⟶ℝ, and the temperature T:Ω⟶ℝ:(6)−divσu,p+Mu=λjTinΩ,(7)divu=0inΩ,(8)u=0onΓw∪Γs,(9)u=g,onΓn,(10)−αΔT+u⋅∇T=0inΩ,(11)∂T∂n=0onΓo∪Γs,(12)T=T1onΓw,(13)T=T2onΓn,where the stress tensor σu,p is defined by σu,p≔−pI+2ν∈u with the rate of deformation tensor ∈u≔Du+DTu/2, DTu denotes the transpose of the matrix Du, and I is the identity tensor. The matrix-valued function M:Ω⟶ℝN×N, α denotes the inverse of Peclet number, λ is the Grashof number, and j equals 0,1T.In this paper, our work is to identify the optimal shape of the boundaryΓs that minimizes the following cost functional J:(14)minΩ∈OJΩ=2ν∫Ω∈u⟶2dx+12∫Ω∇T2dx,where u⟶ and T denote the velocity and the temperature and ∈u is the rate of deformation tensor. The shape admissible set O is given by(15)O≔Ω⊂ℝN:Γn∪Γw∪Γois fixed,∫Ωdx=constant.This type of optimal problem often occurs in the design and control of many industrial equipment. The weak formulation associated with (6)–(13) can be written as(16)Findu,p,T∈VgΩ×QΩ×∈H1Ω,such that,∫Ω2ν∈u:∈v+vTMu−pdivvdx=∫ΩλTj⋅vdx,∀v∈V0Ω,∫Ωdivuqdx=0,∀q∈QΩ,∫Ωα∇T⋅∇S+u⋅∇TSdx=0,∀S∈H1Ω,where the functional spaces are defined as follows:(17)V0Ω≔w∈H1Ω2:w=0onΓw∪Γs∪Γn,QΩ≔p∈L2Ω:∫Ωpdx=0,VgΩ≔w∈H1Ω2:w=0onΓw∪Γs,w=gonΓn.Shape optimal problems usually involve very large computational costs, besides the numerical approximation of partial differential equations and optimization. To avoid the differentiation of the state system and save the computational cost, the adjoint method applied to solve shape optimal problem can be summarized as follows: first, we establish the saddle point problem and the Lagrangian functional associated with the cost functional and weak form of the state system. Then, we are able to perform the shape sensitivity analysis of the Lagrangian functional by the minimax principle concerning the differentiability problem. Last but not least, applying the function space technique, we obtain the Euler derivative of cost functional by the first variation of the cost functional with respect to the domain.First, we give the following Lagrangian functional which is associated with (14) and (16):(18)LΩ,u,p,T,v,q,S=JΩ−WΩ,u,p,T,v,q,S,where(19)WΩ,u,p,T,v,q,S=∫Ω2ν∈u:∈v+vTMu−pdivv−divuq−λTj⋅vdx+∫Ωα∇T⋅∇S+u⋅∇TSdx.Now, problem (18) can be transformed into the saddle point form:(20)minΩ∈Ominu,p,T∈VgΩ×QΩ×H1Ωmaxv,q,S∈V0Ω×QΩ×H1ΩLΩ,u,p,T,v,q,S.Then, we use the minimax framework to avoid the analysis of the state derivative with respect to the variable domains and establish the first optimality condition of the shape optimal problem to deduce the adjoint equations:(21)minu,p,T∈VgΩ×QΩ×H1Ωmaxv,q,S∈V0Ω×QΩ×H1ΩLΩ,u,p,T,v,q,S.Since the adjoint equations are defined from the Euler–Lagrange equations of the corresponding Lagrange functionalL, the variation of L with respect to v,q,S can recover the state system and its weak formulation. Furthermore, we can differentiate L with respect to the state variables (u,p,T) to deduce the adjoint state system.Differentiating Lagrangian functionalL with respect to p in the direction δp, we have(22)∂L∂pΩ,u,p,T,v,q,S⋅δp=∫Ωδpdivvdx=0.Owing toδp with compact support in Ω, it leads to divv=0. Moreover, we differentiate L with respect to u in the direction δu and apply Green formula:(23)∂L∂uΩ,u,p,T,v,q,S⋅δu=∫Ω−2νΔu+νΔv−∇q−Mv⋅δudx−∫ΩS∇T⋅δudx−2ν∫∂Ω∈v⋅n⋅δuds+4ν∫∂Ω∈u⋅n⋅δuds+∫∇Ωqδu⋅nds=∫Ω−2νΔu+νΔv−∇q−Mv−S∇T⋅δudx−∫∂Ωσv,q⋅n⋅δuds+4ν∫∂Ω∈u⋅n⋅δuds.Sinceδu has compact support in Ω, we obtain(24)−νΔv+Mv+∇q+S∇T=−2νΔu.To varyδu on the boundary Γo, we deduce(25)σv,q⋅n−4ν∈u⋅n=0.Similarly, we obtain the adjoint equation with respect toT:(26)∂L∂TΩ,u,p,T,v,q,S⋅δT=∫ΩαΔS+u⋅∇S+λTj⋅v⋅δTdx−∫Γo∪Γsα∂S∂n+u⋅S⋅nδTds−∫ΩΔT⋅δTdx+∫∂Ω∇T⋅δT⋅nds.Finally, we derive the following adjoint state system associated with (6)–(13):(27)−νΔv+Mv+∇q+S∇T=−2νΔuinΩ,divv=0inΩ,σv,q⋅n−4ν∈u⋅n=0onΓo,v=0onΓn∪Γw∪Γs,(28)αΔS−u⋅∇S−λj⋅v=−ΔTinΩ,α∂S∂n+u⋅S⋅n=∇T⋅nonΓo∪Γs,S=0onΓn∪Γw.


## 4. The Velocity Method

In this section, we will apply the velocity method to describe the domain deformation. For shape optimal problem, the set of domainΩ is not a vectorial space, but we need an expression of the differential of the cost functional. In order to overcome this difficulty, we define the derivative of a real-valued function with respect to the domain so that we can present the differential expression for the cost functional to establish a gradient-type algorithm.Let boundary∇Ω be piecewise Ck and the velocity field V∈Ek≔C0,τ;DkΩ¯N, where τ is a small positive real number and DkΩ¯N denotes the space of all k−times continuous differentiable functions with compact support contained in Ω. The velocity field(29)V∈x=V∈,x,x∈Ω,∈≥0belongs to DkΩ¯N for each ϵ. It can generate transformations FϵVX=xϵ,X through the following dynamical system:(30)dxd∈∈,X=V∈,x∈,x0,X=X,with the initial value X. The flow with respect to V can be defined as the mapping F∈:ℝN⟶ℝN with F∈X=x∈,X, where x∈,X is the solution of (30). The transformed domain F∈VΩ can be denoted by Ω∈V at ∈≥0, and its boundary Γ∈≔F∈∇Ω.Next, we introduce two definitions for shape sensitivity analysis. TheEulerianderivative of the cost functional JΩ at Ω for the velocity field V⟶ is defined as [23](31)lim∈↘01∈JΩ∈−JΩ≔dJΩ;V.Moreover, if the mapV↦dJΩ;V:Ek⟶ℝ is linear and continuous, J is shape differentiable at Ω. In the distributional sense, it leads to(32)dJΩ;V=∇J,VDkD¯N×DkD¯N.WhenJ has a Eulerian derivative, ∇J is called the shapegradient of J at Ω.


## 5. Function Space Parametrization

In this section, we derive the expression of the shape gradient for the cost functionalJΩ by the function space parametrization techniques.The velocity method is applied to describe the domain deformations. We only perturb the boundaryΓs and consider the mapping F∈V and the flow of the velocity field:(33)V∈Vad≔V∈C00,τ;C2ℝNN:V=0in the neighorhood ofΓn∪Γw∪Γo.The perturbed domain is denoted byΩ∈=F∈VΩ.We aim to evaluate the derivative ofj∈ with respect to ∈, where(34)j∈≔minu∈,p∈,T∈∈VgΩ∈×QΩ∈×H1Ω∈maxv∈,q∈,S∈∈V0Ω∈×QΩ∈×H1Ω∈LΩ∈,u∈,p∈,T∈,v∈,q∈,S∈,and u∈,p∈,T∈ and v∈,q∈,S∈ satisfy corresponding state and adjoint systems on the perturbed domain Ω∈, respectively. However, the Sobolev spaces VgΩ∈, V0Ω∈, QΩ∈, and WΩ∈ depend on the perturbation parameter ϵ. Consequently, we need to apply the function space parametrization technique to get rid of it. The advantage of this technique is being able to transport different quantities defined on the variable domain Ω∈ back into the reference domain Ω which is entirely unrelated to ∈. Then, we can employ the differential calculus since the functionals involved are defined in a fixed domain Ω with respect to the parameter ∈.Now, we define the following parametrization functions:(35)VgΩ∈=u∘F∈−1:u∈VgΩ,V0Ω∈=v∘∈−1:v∈V0Ω,QΩ∈=p∘F∈−1:p∈QΩ,H1Ω∈=T∘F∈−1:T∈H1Ω,where “°” denotes the composition of the two maps.Note thatF∈ and F∈−1 are diffeomorphisms, so the parametrization will not change the value of the saddle point. We can rewrite (34) as(36)j∈=minu,p,Tmaxv,q,SLΩ∈,u∘F∈−1,p∘F∈−1,T∘F∈−1,v∘F∈−1,q∘F∈−1,S∘F∈−1,where the Lagrangian functional(37)LΩ∈,u∘F∈−1,p∘F∈−1,T∘F∈−1,v∘F∈−1,q∘F∈−1,S∘F∈−1=I1∈+I2∈+I3∈,with(38)I1∈≔2ν∫Ωϵ∈u∘F∈−12dx+12∫Ωϵ∇T∘F∈−12dx,I2∈≔−∫Ωt2ν∈v∘F∈−1:∈u∘F∈−1+v∘F∈−1TMu∘F∈−1−p∘F∈−1divv∘F∈−1−divu∘F∈−1q∘F∈−1−λjT∘F∈−1v∘F∈−1dx,I3∈≔−∫Ωϵα∇T∘F∈−1⋅∇S∘F∈−1+u∘F∈−1⋅∇T∘F∈−1S∘F∈−1dx.Next work is to differentiate the perturbed Lagrangian functionalLΩ∈,u∘F∈−1,p∘F∈−1,T∘F∈−1,v∘F∈−1,q∘F∈−1,S∘F∈−1, so we introduce the following Hadamard formula to perform the differentiation:(39)dd∈∫Ω∈T∈,xdx=∫Ω∈∇T∇ϵ∈,xdx+∫∇Ω∈T∈,xV⋅n∈ds,for a sufficiently smooth functional T:0,τ×ℝN⟶ℝ.Applying (39), we have(40)∂ϵLΩϵ,u∘Fϵ−1,p∘Fϵ−1,T∘Fϵ−1,v∘Fϵ−1,q∘ϵ−1,S∘Fϵ−1ϵ=0=I1′0+I2′0+I3′0,where(41)I1′0=4ν∫Ω∈u:ϵ−Du⋅Vdx+2ν∫Γs∈u⟶2Vnds+∫Ω∇T⋅∇−DT⋅Vdx+12∫Γs∇T2Vnds,(42)I2′0=−∫Ω2ν∈−Du⋅V⋅ϵv+2ν∈u⋅ϵ−Dv⋅V+vTM⋅−Du⋅V+Mu⋅−Dv⋅V−−∇p⋅Vdivv−pdiv−Dv⋅V−−∇q⋅Vdivu−λjT⋅−Dv⋅V−λj−DT⋅V⋅v−qdiv−Du⋅Vdx+∫Γs−2ν∈u:∈v−vTMu+pdivv+divuqVnds,(43)I3′0=−∫Ωα∇−DS⋅V⋅∇Tdx−∫Ωα∇S⋅∇−DT⋅Vdx−∫Ω−Du⋅V⋅∇TSdx−∫Ωu⋅∇−DT⋅VSdx+∫Ωu⋅∇TDS⋅VSdx−∫Γsu⋅−DT⋅VS⋅nds−∫Γsα∇T⋅∇S+u⋅∇TSVnds−∫Γsα∇T⋅−DS⋅V⋅ndx−∫Γsα∇S⋅−DT⋅V⋅ndx.In order to simplify the above identities, we introduce the following lemma.Lemma 1 (see [23]).
If vector functionsu and v vanish on the boundary Γs, the following identities hold on the boundary Γs:(44)Du⋅V⋅n=divuVn,∈u:∈v=∈u⋅n⋅∈v⋅n,∈u⋅n⋅Dv⋅V=∈u⋅n⋅∈v⋅nVn.
We apply Lemma1 and obtain(45)I1′0=−2ν∫ΩΔu⋅−Du⋅Vdx−2ν∫Γs∈u2Vnds−∫ΩΔT⋅−DT⋅Vdx+∫Γs∇T⋅n⋅−∇T⋅Vds+12∫Γs∇T2Vnds.
Recallingu,p,T and v,q,S satisfies the state and adjoint system, respectively, and (42) can be reduced to(46)I2′0=∫ΩνΔu−Mu−∇p−λjT⋅−Dv⋅Vdx−∫Γs2ν∈u:∈vVnds+∫ΩνΔv−vTM−∇q⋅−Du⋅Vdx+∫Ωλj−DT⋅V⋅vdx−∫Γsσu,p⋅n⋅−Dv⋅V+σv,q⋅n⋅−Du⋅Vds=∫Ω2νΔu+∇TS−Du⋅Vdx+∫Ωλj−DT⋅V⋅v⟶dx+∫Γs2ν∈u:∈vVnds.Similarly, (43) can be rewritten as(47)I3′0=∫ΩαΔT−u⋅∇T−DS⋅Vdx+∫ΩαΔS+u⋅∇S−DT⋅Vdx−∫Ω−Du⋅V⋅∇TSdx−∫Γsα∇T⋅∇S+u⋅∇TSVnds−∫Γsu⋅−DT⋅VS⋅nds−∫Γsα∇T⋅−DS⋅V⋅nds−∫Γsα∇S⋅−DT⋅V⋅nds=∫ΩΔT−λj⋅v−DT⋅Vdx−∫Ω∇TS⋅−Du⋅Vdx+∫Γsα∇T⋅∇S+u⋅∇TSVnds−∫Γsu⋅−DT⋅VS⋅nds−∫Γsα∇T⋅−DS⋅V⋅nds−∫Γsα∇S⋅−DT⋅V⋅nds.Finally, we have the boundary expression for the Eulerian derivative ofJΩ:(48)dJΩ;V=2ν∫Γs∈u:∈v−∈u2Vnds+12∫Γs∇T2Vnds+∫Γsα∇T⋅∇SVnds.According to (32), we derive the expression of the shape gradient for the cost functional:(49)∇J=2ν∈u:∈v−∈u2+12∇T2+α∇T⋅∇Sn.


## 6. Numerical Examples

This section is devoted to present the numerical algorithm and examples for the shape optimization problem in two dimensions.We consider the optimal design of a body immersed in a Stokes–Brinkman flow and aim at reducing the dissipation energy acting on its surface. Namely, we solve the minimization problem(50)minΩ∈OJΩ=2ν∫Ω∈u2dx+12∫Ω∇T2dx,subject to (6)–(13).For the minimization problem (50), we rather work with the following minimization problem:(51)minΩ∈ℝ2GΩ=JΩ+lAΩ,where AΩ≔∫Ωdxdy, l is a positive Lagrangian multiplier, and GΩ satisfies the following equation:(52)dG=∫Γs∇G⋅Vds,where dG is the shape gradient with(53)dG=2νϵu:∈v−∈u2+12∇T2+α∇T⋅∇S+ln.Taking no account of regularization, a descent direction is sought by(54)V=−hk∇G,and then the shape of domain Ω can be updated as(55)Ωk=I+hkVΩ,where hk is a small descent step at k-th iteration. We obtain the iterative scheme:(56)Jk+1=Jk−hk∇Jk,∇Jk0,Jk≔JΩk.To avoid shape oscillations, we have to project or smooth the variation intoH1Ω. Therefore, we choose the descent direction d∈H1Ω2 which is the unique solution of the problem(57)∫ΩDd:DVdx=−dJΩ;V,∀V∈H1Ω2.It is obvious thatd is a descent direction which guarantees the decrease of the cost functional JΩ. The computation of d is seemed as a regularization of the shape gradient.Then, we consider how to choose the Lagrangian multiplierl in the optimization problem. In order to satisfy the fixed constraint, the value of l is updated at each iteration. As a result of the high cost in moving the mesh, we do not impose exactly the volume constraint before convergence. If the present area is smaller than the target area, we decrease the multiplier l; otherwise, we increase it. We suppose(58)dGΩ;V=dJΩ;V+ldVΩ;V=0,at least in the average sense on the boundary Γs,(59)l=−∫ΓsdJds∫Γsds.Therefore, we update the Lagrange multiplier by(60)lk+1=lk+l+l2+mAΩk−AsΩAsΩ,where m is a small positive parameter and AsΩ denotes the target area.We propose the numerical algorithm for solving the shape optimal problem in a Stokes–Brinkman flow with convective transfer. The algorithm is terminated when the relative decrease (denoted by∈re) of two consecutive objective is less than a given tolerance ∈ (Algorithm 1).Algorithm 1: Iterative algorithm for shape optimal control.
Choose an initial shapeΩ0 and initial step h0 and a Lagrangian multiplier L0whileϵre≤ϵ, doStep 1: solve state system (6)–(13).Step 2: compute adjoint system (27) and (28).Step 3: evaluate the cost functional.Step 4: compute the descent directiondk by (56).Step 5: setΩk+1=I+hkdkΩk and a suitable Lagrange multiplier lk+1, where hk is a small positive real number.endwhileWe restrict the shape optimal problem posed on a bounded rectangular domain by introducing an artificial boundary∂D, and Ω=D\S¯ is the effective domain with its boundary ∂Ω=Γn∪Γw∪Γo. The fluid enters horizontally from the left boundary Γn and exits from the right boundary Γo. We choose the initial shapes of the obstacle S to be different curves:Case 1: a circle whose center is at origin with radius 0.5.Case 2: an elliptic curve:x=0.45∗ost+0.4,y=0.08∗sint,t∈0,2π.For the two examples, the inflow velocities are assumed to be parabolic with the profilesg0,y=1−y4,0T and g0,y=0.25−y2,0T, respectively. Also, the no-slip boundary conditions are imposed at all the other boundaries. The admissible set is defined by(61)O=Ω⊂ℝ2:Γn∪Γw∪Γois fixed,the areaAtargetΩ=constant.Figures1 and 2 show the comparison between the initial shape and optimal shape for the computing meshes, the contours of the velocity u⟶=u1,u2T, and the pressure p with different Reynolds numbers. Figures 3 and 4 demonstrate that the proposed method is effective, stable, and rapidly convergent. We also observe that when the Reynolds number increases, the cost of the optimization procedure rises due to the increase of computation of the state and adjoint system.Figure 1
Case 1: comparison of the initial shape and optimal shape (Reynolds number = 1000). (a) Mesh for initial shape. (b)u1 for initial shape. (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 for optimal shape.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 2
Case 2: comparison of the initial shape and optimal shape (Reynolds number = 500). (a) Mesh for initial shape. (b)u1 for initial shape. (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 for optimal shape.
(a)(b)(c)(d)(e)(f)(g)(h)Figure 3
Case 1: convergence history of the cost functional (Reynolds number = 1000).Figure 4
Case 2: convergence history of the cost functional (Reynolds number = 500).


## 7. Conclusion

This work focuses on the optimal shape determination in an incompressible viscous Stokes–Brinkman flow, with the consideration of heat transfer. Based on the adjoint method and the function space parametrization technique, we derive the shape gradient of the cost functional by involving a Lagrangian functional, which plays the key role of design variables in the optimal design framework. Moreover, we propose a gradient-type algorithm for the minimization dissipation energy problem. Finally, we present numerical examples to demonstrate the proposed algorithm is feasible and effective for the quite high Reynolds number problems.



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