Shape Identification for Stokes-Oseen Problem Based on Domain Derivative Method

In this paper, we consider the shape identification problem of a body immersed in the incompressible fluid governed by Stokes-Oseen equations. Based on the domain derivative method, we derive the explicit representation of the derivative of solution with respect to the boundary. Then, according to the boundary parametrization technique, we propose a regularized Gauss-Newton algorithm for the shape inverse problem. Finally, numerical examples indicate that the iterative algorithm is feasible and effective for the practical purpose.


Introduction
The purpose of this paper is to determine a shape of the body located in an incompressible viscous Stokes-Oseen flow by applying a formulation of the domain derivative to a numerical simulation.
Shape inverse problem usually consists in reconstructing or recovering the geometry shapes from the measured (observed) data.This kind of problems usually entails very large computational costs: besides numerical approximation of partial differential equations, it requires also a suitable approach for representing and deforming efficiently the shape of the underlying geometry.The control variable is the shape of the domain; the object is to recover the unknown boundary from the data which may be given by the designers.
For the domain derivative method, many people are contributed to it.Kress proposed a quasi-Newton method to solve inverse scattering problem in [1].Hettlich solved the inverse obstacle scattering problem for sound obstacles problem [2], and discussed a discontinuity in a conductivity from a single boundary measurement [3].
Chapko et al. dealt with the inverse boundary problem for the time-dependent heat equation only in the case of perfectly conducting and insulating inclusions [4] [5].Serranho presented a hybrid method for inverse scattering for shape and impedance [6].Harbrecht and Tausch considered the numerical solution of a shape identification problem for the heat equation [7].Yan et al. recovered the shape of a solid in the incompressible fluid driven by the Stokes flow [8], and considered the shape optimization problem of a body immersed in the incompressible fluid governed by Navier-Stokes equations coupling with a thermal model in [9].
The structure of the paper is as follows.In Section 2, we briefly introduce the shape reconstruction problem of the steady Stokes-Oseen equations.In Section 3, we describe the domain perturbation method which is used for the characterization of the deformation of the shapes, and derive the explicit representation of the derivative of solution with respect to the boundary.This will serve as the theoretical foundation of the Newton method for the approximation solution.Section 4 is devoted to the regularized Gauss-Newton scheme applied to the numerical shape identification problem.The performance of the numerical method is discussed and illustrated by numerical examples.

Shape Identification Problem
In this paper, we consider the shape identification of an immersed body in the incompressible viscous fluid which is driven by the steady-state Stokes-Oseen equations, Here u denotes the velocity field, w is the equilibrium solution of the Navier-Stokes equation, p is the pressure, and ν is the kinematic viscosity of the incompressible fluid.For a Newtonian fluid the stress tensor is given as ( ) ( ) where * Du denotes the transpose of the matrix.n is the unit normal vector on the smooth boundary ∂Ω which consists of four parts.in Γ is the inflow boundary, out Γ denotes the outflow boundary, w Γ represents the boundary corresponding to the fluid wall, and s Γ is the boundary to be recovered.For a given domain Ω , it is well known that this boundary value problem has a unique solution [10].
The purpose of this paper is to investigate the feasibility of recovering the unknown boundary s Γ from the measured (observed) data.We define the operator F on the admissible set X by ( ) , where M is the measured (observed) data and may represent a given objective related to specific characteristic features of the incompressible fluid.The inverse problem is both ill-posed and nonlinear.
If , f u and p are smooth functions satisfying (2.1), taking the scalar product of (2.1) with a function ( ) where , , D d .
Throughout the paper we will use the standard notation for Sobolev spaces.Specially ( ) r H Ω , where r is an integer greater than zero, will denote the Sobolev space of real-valued functions with square integrable derivatives of order up to r equipped with the usual norm which we denote r ⋅ .

( )
r Ω H will denote the space of vector-valued functions each of whose n components belong to

Domain Derivative Method
In this section, we will discuss how to derive the explicit representation of the derivative of solution with respect to the boundary.This will serve as the theoretical foundation of the numerical algorithm in next section.
A derivative of operator F at boundary Γ can be defined as follows [11]: For any real vector field ( ) δ is small enough.Now we define the domain derivative of F at boundary Γ in the direction δ by ( ) ( ) ( ) where the limit should exist uniformly.
Similarly, we denote a perturbation of the interior boundary s Γ by is sufficiently small.We choose an extension of ( ) which vanishes in the exterior of a neighbourhood of s Γ , and define the diffeomorphism ( ) ( ) If the inverse function of ϕ is denoted by ψ , J ϕ and J ψ are Jacobian matrices.

Let
( ) be the solution of corresponding boundary value problem, i.e. satisfy the variational equation for all ( ) . Transporting the variables to the reference domain Ω leads to for all ( ) , where the notations ( ) 1 det div .
In order to prove the main theoretical result of the paper, we introduce some useful identities (see [2] [12]) , then the following identity holds: where ∈ Γ be a scalar function, and a vector field ( ) . The following decompositions hold: denote the solution of (2.1), and u  is defined in (3.2).Then u is differentiable at s Γ in the sense that there exists * u depending on h , such that Furthermore, ( ) u, where the domain derivative ′ u satisfies the following equations ( ) ( ) where n = ⋅ h h n is the normal component of the vector field h .Proof: Step 1: We establish the continuous dependence of the solution u on variations of the boundary s Γ .Considering the difference − u u  , the variational equation holds

u u v u u w v w u u v u v u v u v u v u v u v u v u v u w v w u v u w v u w v u w v w u v w u v w u v
From Equations (3.1) and (3.2), we have ( C Ω

− → → u u h 
Step 2: In order to show the differentiability, let ( ) for all ( ) From the properties of forms ( )

u u u v u u u w v w u u u v u u v u v u u w v u w v w u u v w u v
u is the solution of (3.11), we rewrite the above identity as )

u u v h h u v h v u u v h n h u v n
Notice that u satisfies the Stokes-Oseen Equation (2.1) and applies the geometrical decompositions formulae, and we can get

T h u v h v u h v f u u v h f u u
From Lemma 3.2, we have the identity, ( Step (4): Give the conditions on boundaries.It is known that | 0

u u u u h u h u h n h n n
Thus, ′ u satisfies the boundary value problem (3.10).The proof is completed.

Numerical Algorithm and Examples
In this section, we will propose a regularized Gauss-Newton algorithm and numerical examples in two dimensions, and the numerical results verify that our methods could be very feasible and effective for the shape inverse problem of the Stokes-Oseen equations.
To our knowledge, there are two groups of approaches for the solution of shape inverse problems of this type, namely regularized Gauss-Newton iterations and decomposition methods.In this paper, we choose the regularized Gauss-Newton method.Generally, Newton method is based on the observed information.We define an operator F on set X of admissible boundaries by ( ) where M is the measured (observation) data [12], , and ϕ is the parametrized form of boundary s Γ .However, since the linearized version of (4.1) inherits the ill-posedness, the Newton iterations need to be regularized.
First of all, we apply the following boundary parametrization technique in numerical implementations.Here the parametric representations are denoted by , ,0 2π , , . qr r r In the following, we fix N and Q, and obtain the following theorem as an application of Theorem 3.1.
u p ′ ′ are the solutions to the thermodynamic equations ( ) ( ) The numerical algorithm can be organized as follows: 1): Given an initial curve, parametrize it to 0 η by the boundary parametrization technique; 2): Solve the direct problem (2.1) by the finite element method; 3): For a given n η , calculate the discrete domain derivative Equation (4.3) and the Jacobian matrix; 4): Apply the regularized Gauss-Newton method, , The dimension of the admissible space N U is 2 1 49 N + = , and the number of observation points is 96.We use the finite element method to solve both the direct and inverse problems.Spatial discretization is effected using the Taylor-Hood pair of finite element spaces on a triangular mesh [13] [14], that is, the finite element spaces are chosen to be continuous piecewise quadratic polynomials for the velocity and continuous piecewise linear polynomials for the pressure.
For case 1, Figure 1 and Figure 2 give the comparison between the exact curve with the approximate curve for the viscosity coefficient ν = 0.01 and 0.0025, respectively.For case 2, Figure 3 and Figure 4 display the comparison between the target shape with the reconstructed shape for the viscosity coefficient ν = 0.01, 0.005.
The numerical examples indicate the feasibility of the proposed algorithm and further research is necessary on efficient implementations.

Conclusion
This paper is concerned with the numerical simulation for shape identification of the steady Stokes-Oseen problems.The continuous dependence of the solution on variations of the boundary is established, and the repre-sentation of domain derivative of corresponding equations is derived.This allows the investigation of iterative method for the ill-posed problem.By the parametric method, a regularized Gauss-Newton scheme is employed to the shape inverse problem.Numerical experiments indicate the feasibility of the proposed method.
where µ is a regularization parameter; otherwise go back to step(2).We carry out the numerical examples to demonstrate feasibility and validity of the proposed algorithm.In the following, we set D to be a rectangle [ ] [ ] of solid S is to be recovered in our simulations.We choose s Γ to be different curves: Case 1: A circle whose center is at the origin with radius 0.6, s Γ