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

(is 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. (e 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. (en, 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.


Introduction
e optimal shape design for the fluid flows has wide applications in engineering design and computational fluid mechanics. e 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][10][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][19][20].
In shape optimization, the efficient computation requires a shape calculus which differs from its analog in vector spaces. e 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. is paper is organized as follows. In Section 2, 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. en, 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.

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 functional J which consists of the solution of the state equations: min J � J(w, φ), 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: For the linear case, problem (1) satisfies ∇L(w, 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): where (·, ·) denotes the inner product, a(·, ·) is a bilinear form with respect to a linear elliptic operator, and b(φ, ψ) � 〈Bφ, ψ〉. erefore, We need to solve following problem to obtain the optimal solution: seek(w, λ, φ) ∈ W × W × V, such that ∇L(w, λ, φ) � 0.

(5)
Usually, we can apply an iterative method to solve the control problem by choosing an initial value for the variable φ 0 . At each step, we compute the state equations and then evaluate the cost functional and solve the adjoint equations. When φ j is available, we give a suitable stopping criterion and derive the cost functional derivative J ′ [21,22].

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 that Ω ⊂ R N (N � 2 or 3) is a bounded Lipschitz domain which is filled with the incompressible viscous fluid of the kinematic viscosity ]. e boundary zΩ of the domain Ω is smooth and consists of four parts. Γ n denotes the inflow boundary, Γ w is the boundary corresponding to the fluid wall, Γ o represents the outflow boundary, and Γ s is the boundary of the obstacle S which is to be optimized. e fluid is described by the Stokes-Brinkman equations strongly coupled with a thermal model; the unknowns are the fluid velocity u � (u 1 , . . . , u N ) T : Ω ⟶ R N , the pressure p: Ω ⟶ R, and the temperature T: Ω ⟶ R: where the stress tensor σ(u, p) is defined by σ(u, p) ≔ − pI + 2] ∈ (u) with the rate of deformation tensor ∈ (u) ≔ (Du + D T u)/2, D T u denotes the transpose of the matrix Du, and I is the identity tensor. e matrix-valued function M: Ω ⟶ R N×N , α denotes the inverse of Peclet number, λ is the Grashof number, and j equals (0, 1) T . In this paper, our work is to identify the optimal shape of the boundary Γ s that minimizes the following cost functional J: 2 Mathematical Problems in Engineering where u → and T denote the velocity and the temperature and ∈ (u) is the rate of deformation tensor. e shape admissible set O is given by is type of optimal problem often occurs in the design and control of many industrial equipment. e weak formulation associated with (6)- (13) can be written as where the functional spaces are defined as follows: 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. en, 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): Now, problem (18) can be transformed into the saddle point form: en, 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: Since the adjoint equations are defined from the Euler-Lagrange equations of the corresponding Lagrange functional L, the variation of L with respect to (v, q, S) can recover the state system and its weak formulation.

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Furthermore, we can differentiate L with respect to the state variables (u, p, T) to deduce the adjoint state system.
Differentiating Lagrangian functional L with respect to p in the direction δp, we have zL zp Owing to δp with compact support in Ω, it leads to div v � 0. Moreover, we differentiate L with respect to u in the direction δu and apply Green formula: Since δu has compact support in Ω, we obtain To vary δu on the boundary Γ o , we deduce Similarly, we obtain the adjoint equation with respect to T: Finally, we derive the following adjoint state system associated with (6)- (13):

The Velocity Method
In this section, we will apply the velocity method to describe the domain deformation. For shape optimal problem, the set of domain Ω 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 realvalued 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 C k and the velocity field belongs to [D k (Ω)] N for each ϵ. It can generate transformations F ϵ (V)X � x(ϵ, X) through the following dynamical system: with the initial value X. e flow with respect to V can be defined as the mapping F ∈ : is the solution of (30). e transformed domain F ∈ (V)(Ω) can be denoted by Ω ∈ (V) at ∈ ≥ 0, and its boundary Γ ∈ ≔ F ∈ (∇Ω). Next, we introduce two definitions for shape sensitivity analysis. e Eulerian de rivative of the cost functional J(Ω) at Ω for the velocity field V → is defined as [23] lim Moreover, if the map V↦dJ(Ω; V): E k ⟶ R is linear and continuous, J is shape differentiable at Ω. In the distributional sense, it leads to When J has a Eulerian derivative, ∇J is called the shape gra di ent of J at Ω.

Function Space Parametrization
In this section, we derive the expression of the shape gradient for the cost functional J(Ω) by the function space parametrization techniques. e velocity method is applied to describe the domain deformations. We only perturb the boundary Γ s and consider the mapping F ∈ (V) and the flow of the velocity field: e perturbed domain is denoted by Ω ∈ � F ∈ (V)(Ω). We aim to evaluate the derivative of j(∈) with respect to ∈, where and (u ∈ , p ∈ , T ∈ ) and (v ∈ , q ∈ , S ∈ ) satisfy corresponding state and adjoint systems on the perturbed domain Ω ∈ , respectively. However, the Sobolev spaces V g (Ω ∈ ), V 0 (Ω ∈ ), Q(Ω ∈ ), and W(Ω ∈ ) depend on the perturbation parameter ϵ. Consequently, we need to apply the function space parametrization technique to get rid of it. e 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 ∈. en, 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: where "°" denotes the composition of the two maps. Note that F ∈ and F − 1 ∈ are diffeomorphisms, so the parametrization will not change the value of the saddle point. We can rewrite (34) as where the Lagrangian functional with Mathematical Problems in Engineering (38) Next work is to differentiate the perturbed Lagrangian functional , so we introduce the following Hadamard formula to perform the differentiation: for a sufficiently smooth functional T: Applying (39), we have where In order to simplify the above identities, we introduce the following lemma.
Lemma 1 (see [23]). If vector functions u and v vanish on the boundary Γ s , the following identities hold on the boundary Γ s : We apply Lemma 1 and obtain Recalling (u, p, T) and (v, q, S) satisfies the state and adjoint system, respectively, and (42) can be reduced to Similarly, (43) can be rewritten as Finally, we have the boundary expression for the Eulerian derivative of J(Ω): According to (32), we derive the expression of the shape gradient for the cost functional: (49)

Numerical Examples
is 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 subject to (6)- (13).
For the minimization problem (50), we rather work with the following minimization problem:

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Choose an initial shape Ω 0 and initial step h 0 and a Lagrangian multiplier L 0 while ϵ re ≤ ϵ, do Step 1: solve state system (6)-(13).
Step 3: evaluate the cost functional.
Step 4: compute the descent direction d k by (56).
Step 5: set Ω k+1 � (I + h k d k )Ω k and a suitable Lagrange multiplier l k+1 , where h k is a small positive real number. en d while ALGORITHM 1: Iterative algorithm for shape optimal control. Figure 1: Case 1: comparison of the initial shape and optimal shape (Reynolds number � 1000). (a) Mesh for initial shape. (b) u 1 for initial shape. (c) Mesh for optimal shape. (d) u 1 for optimal shape. (e) u 2 for initial shape. (f ) p for initial shape. (g) u 2 for optimal shape. (h) p for optimal shape.

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where A(Ω) ≔ Ω dx dy, l is a positive Lagrangian multiplier, and G(Ω) satisfies the following equation: where dG is the shape gradient with Taking no account of regularization, a descent direction is sought by Figure 2: Case 2: comparison of the initial shape and optimal shape (Reynolds number � 500). (a) Mesh for initial shape. (b) u 1 for initial shape. (c) Mesh for optimal shape. (d) u 1 for optimal shape. (e) u 2 for initial shape. (f ) p for initial shape. (g) u 2 for optimal shape. (h) p for optimal shape.
and then the shape of domain Ω can be updated as where h k is a small descent step at k-th iteration. We obtain the iterative scheme: To avoid shape oscillations, we have to project or smooth the variation into H 1 (Ω). erefore, we choose the descent direction d ∈ H 1 (Ω) 2 which is the unique solution of the problem It is obvious that d is a descent direction which guarantees the decrease of the cost functional J(Ω). e computation of d is seemed as a regularization of the shape gradient.
en, we consider how to choose the Lagrangian multiplier l 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 at least in the average sense on the boundary Γ s , erefore, we update the Lagrange multiplier by where m is a small positive parameter and A s (Ω) denotes the target area. We propose the numerical algorithm for solving the shape optimal problem in a Stokes-Brinkman flow with convective transfer. e algorithm is terminated when the relative decrease (denoted by ∈ re ) of two consecutive objective is less than a given tolerance ∈ (Algorithm 1).
We restrict the shape optimal problem posed on a bounded rectangular domain by introducing an artificial boundary zD, and Ω � D\S is the effective domain with its boundary zΩ � Γ n ∪ Γ w ∪ Γ o . e fluid enters horizontally from the left boundary Γ n and exits from the right boundary Γ o . 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: For the two examples, the inflow velocities are assumed to be parabolic with the profiles g(0, y) � (1 − y 4 , 0) T and g(0, y) � (0.25 − y 2 , 0) T , respectively. Also, the no-slip boundary conditions are imposed at all the other boundaries. e admissible set is defined by O � Ω ⊂ R 2 : Γ n ∪ Γ w ∪ Γ o is fixed, the area A target (Ω) � constant .  Figures 1 and 2 show the comparison between the initial shape and optimal shape for the computing meshes, the contours of the velocity u → � (u 1 , u 2 ) T , 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.

Conclusion
is 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.
Data Availability e data and code used in this study cannot be shared at this time as the data also form a part of an ongoing study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.