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This paper is concerned with the optimal design of an obstacle located in the viscous and incompressible fluid which is driven by the steady-state Oseen equations with thermal effects. The structure of shape gradient of the cost functional is derived by applying the differentiability of a minimax formulation involving a Lagrange functional with a space parametrization technique. A gradient type algorithm is employed to the shape optimization problem. Numerical examples indicate that our theory is useful for practical purpose and the proposed algorithm is feasible.

In this paper, we consider the shape optimization of an immersed body in the viscous and incompressible fluid which is driven by the Oseen equations coupling with heat transfer. Shape optimization problem is to find the geometry shapes that minimize certain objective functional, for instance, the energy dissipation, subject to mechanical and geometrical constraints. The research of shape optimization is a branch of optimal control governed by PDEs and has a very wide range of applications in engineering such as in the design of impeller blades, aircraft wings, high-speed train heads, and bridges in medically bypassing surgeries. The optimal shape design in fluids has been a challenging task for a long time, and has been investigated by many mathematicians and engineers.

Shape optimization problem usually entails very large computational costs: besides numerical approximation of partial differential equations and optimization, it requires also a suitable approach for representing and deforming efficiently the shape of the underlying geometry, as well as for computing the shape gradient of the cost functional to be minimized. The control variable is the shape of the domain; the object is to minimize a cost functional that may be given by the designer, and finally the optimal shapes can be obtained.

In the last few decades, the shape optimization problems have attracted the interests of many specialists. Pironneau [

In this paper, we will consider the energy minimization problem for Oseen flow with convective heat transfer despite of its lack of rigorous mathematical justification in case where the Lagrange formulation is not convex. We shall show how this theorem allows at least formally bypassing the study of material derivative and obtaining the expression of shape gradient for the dissipated energy functional. For the numerical solution of the viscous energy minimization problem, we introduce a gradient type algorithm with mesh adaptation technique, while the partial differential systems are discretized by means of the finite element method. Finally, we give some numerical examples concerning with the optimization of a two-dimensional solid body in the viscous flow.

This paper is organized as follows. In Section 2, we briefly give the description of the shape optimization problem of the Oseen flow taking account of conductive heat transfer, and we employ a velocity method to describe a variational domain in the optimization process. In addition, we introduce the definitions of Eulerian derivative and shape gradient. Then we draw the divergence-free condition directly into the Lagrangian functional which leads to a saddle point formulation of the shape optimization problem for the state equations. In Section 3, we obtain the continuous gradient of the cost functional with respect to the boundary shape with the adjoint equations and a function space parametrization technique, which plays the role of design variables in the optimal design framework. In Section 4, we present a gradient-type algorithm for the shape optimization problem, and numerical examples demonstrate that our method is efficient and useful in the numerical implementations.

Before closing this section, we state some notations to be used in this paper.

where

The boundary Γ consists of four parts: Γ_{in} is the inflow boundary; Γ_{out} denotes the outflow boundary; Γ_{w} represents the boundary corresponding to the fluid wall; and Γ_{s} is the boundary which is to be optimized.

We consider a typical problem to design an obstacle S with the boundary

The fluid is modeled by the Oseen flow taking account of thermal effects, and the unknowns are the fluid velocity

and

where

In this paper, our purpose is to optimize the shape of the boundary

The boundary

Let Ω be of piecewise

Now, we choose an open set Ω in

belongs to

with the initial value X given. We denote the transformed domain by

Moreover, we suppose

Furthermore, if the map

When J has a Eulerian derivative, we say that

Generally, there is a few approaches to compute the exact differential or the shape gradient. In the direct differentiation, it requires to derive the state equations with respect to the shape variables. In practice, it implies to solve as many PDEs systems as discrete shape variables. To avoid this extra computational cost, we use the classical adjoint state method which requires to solve only one extra PDE system. There are two ways for it. The first one is to discretize the equations, using a finite element method for example, and to derive the discrete equations and obtain the discrete shape gradient. The second one is to calculate the expression of the exact differential of the cost functional and to discretize it. In this paper, we follow the latter approach. We will derive the structure of the shape gradient for the cost functional

The weak formulation of (2.1)-(2.8) can be expressed as follows: find

and seek

We will utilize the differentiability of a minimax formulation involving a Lagrangian functional with the function space parametrization technique. First of all, we introduce the following Lagrangian functional associated with (3.1) and (3.2):

where

Thus, the minimization problem (2.9) reads as the following form

The minimax framework can be applied to avoid the study of the state derivative with respect to the shape of the domain. The Karusch-Kuhn-Tucker conditions will furnish the shape gradient of the cost functional

Conversely, the adjoint equations are defined from the Euler-Lagrange equations of the Lagrange functional G. Obviously, the variation of G with respect to

Considering

Similarly, we differentiate G with respect to u in the direction

Taking

Then varying

Finally, we obtain the following adjoint state system of (2.1)-(2.4),

By the same technique, we differentiate G with respect to T in the direction

The adjoint state system of (2.5)-(2.8) can be read as

Now we introduce the so-called function space parametrization technique, which consists in transporting the different quantities defined in the variable domain

We only perturb the boundary

The perturbed domain can be defined by

where

where “

Since

Correspondingly, the Lagrangian functional is given by

where

We introduce the following Hadamard formula [

for a sufficiently smooth functional

where

We introduce the following lemma to simplify (3.15)-(3.17).

Lemma 4.1. [

hold on the boundary

According to Lemma 4.1, it follows that

Since

Similarly, (3.17) can be written as

Summing the three integrals together, we finally derive the boundary expression for the Eulerian derivative of

Since the mapping

This section is devoted to present the numerical algorithm and examples for the shape optimization problem in two dimensions.

In all computations, the finite element discretization is effected using the P_{1} bubble-P_{1} pair of finite element spaces on a triangular mesh. The mesh is performed by a Delaunay-Voronoi mesh generator (see [_{h} can be approximated by using a recovery method, such as the Zienkiewicz-Zhu recovery procedure [^{2} projection

where ^{2} projection on the P_{1} Lagrange finite element space (see [

Taking no account of regularization, a descent direction is found by defining

and then we can update the shape Ω as

where

which guarantees the decrease of the cost functional

In the numerical implementation, we choose the descent direction

It is clear that d is a descent direction which guarantees the decrease of J. The computation of d can also be interpreted as a regularization of the shape gradient, and the choice of

The numerical algorithm can be summarized as follows:

・ Step 1: Give an original shape

・ Step 2: Solve the state system and adjoint state system, and evaluate the descent direction

・ Step 3: Set

Let us now characterize the framework of Section 3 to a problem of interest in fluid dynamics, namely the optimal design of a body immersed in a fluid flow, aiming at reducing the dissipation energy acting on its surface. We solve the minimization problem

subject to

and

The outer boundary is a rectangle which is fixed, and the inner boundary

The flow is around an obstacle S in a fixed rectangular domain

We choose the initial shape of the body S to be different shapes:

Case 1: A circle with center

Case 2: An elliptic curve:

The state system and the adjoint system are discretized by a mixed finite element method. Spatial discretization is effected using the Taylor-Hood pair [

Figures 1-5 and Figures 7-11 demonstrate the comparison between the initial shape and optimal shape for the computing mesh, the contours of the velocity

We run many iterations in order to show the good convergence and stability properties of our algorithm, however it is clear that it has converged in a small number of iterations (see

In this paper, we consider the shape optimization problem of a body immersed in the incompressible fluid governed by Oseen equations coupling with a thermal model. Based on the continuous adjoint method, we formulate and analyze the shape optimization problem. Then we derive the structure of shape gradient for the cost functional by employing the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. Moreover, we propose a gradient-type algorithm for the shape optimization problem, and the numerical examples indicate that the proposed algorithm is feasible and effective for the low Reynolds numbers.

This work is supported by the National Natural Science Foundation of China (No. 11371288), and the Research Foundation of Department of Education of Shaanxi (No. 11JK0494).

WenjingYan,AxiaWang,GuoxingGuan, (2015) A Numerical Method for Shape Optimal Design in the Oseen Flow with Heat Transfer. Journal of Applied Mathematics and Physics,03,1295-1307. doi: 10.4236/jamp.2015.310158