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This paper is concerned with the numerical simulation for shape reconstruction of the unsteady advection-diffusion problems. The continuous dependence of the solution on variations of the boundary is established, and the explicit representation 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 examples indicate that the proposed algorithm is feasible and effective for the practical purpose.

The advection-diffusion problem is important to many branches of science and engineering. Many physical and chemical phenomena, such as the diffusion of polluted substances in water and air, the diffusion of heat and salinity in the ocean, and even economics and financial forecasting, can be described as advection-diffusion problems.

For the shape reconstruction problems by the domain derivative method, many people are contributed to it. Hettlich solved the inverse obstacle scattering problem for sound soft and sound hard obstacles [

This paper is organized into four parts. In Section

In this paper, we pay our attention on reconstructing the shape of a bounded and smooth domain from observed information. Let

The purpose of this paper is to investigate the feasibility of recovering the unknown boundary

First of all, we introduce the following functional spaces which will be used throughout this paper. Let

Multiplying the advection-diffusion equations (

Find

This section is devoted to deriving the domain derivative of the solution of the advection-diffusion equations.

A derivative of operator

Similarly, if the vector field

Let us consider

The Jacobian matrix of

Furthermore, we can prove the following important theorem which is the main theoretical result of the paper.

Assume that

Firstly, we will establish the continuous dependence of the solution

From the first-order approximations (

Secondly, in order to prove the differentiability of the solution

Finally, we have to show that

From Green formula and the above equation, we have

In this section, we will present a regularized Gauss-Newton algorithm and numerical examples in two dimensions to verify that our methods could be very useful and efficient for the shape reconstruction problem of unsteady advection-diffusion equations.

From the numerous methods which have been developed for the solution of inverse boundary value problems of this type, we note two groups of approaches, namely, regularized Gauss-Newton iterations and decomposition methods. We choose the regularized Gauss-Newton method in this paper.

Newton method is based on the observed information. We define an operator

However, since the linearized version of (

A numerical implementation requires a parametrization of the boundary. Here we apply the parametric representations

We define

For

The iterative algorithm can be summarized as follows.

Choose an initial boundary for

Solve the advection-diffusion equations (

For a given

Apply the regularized Gauss-Newton method to obtain the new approximation of boundary

We carry out the numerical examples to demonstrate the feasibility and validity of the proposed algorithm in Section

In the following, we consider the shape reconstruction of the advection-diffusion process for the transport of a contaminant in two dimensions. We choose

We will reconstruct the shapes of solid

An elliptic curve is defined by

A cone-shaped curve is parametrized by

The dimension of the space

Figures

Case

Case

Case

Case

Case

Case

In this paper, we discuss the shape reconstruction problem governed by unsteady advection-diffusion equations. The differentiability of solution of the initial boundary value problem with respect to the boundary in the sense of the domain derivative is established, which is the theoretical foundation for the Newton method. A regularized Gauss-Newton scheme is effectively applied to the shape determination problem. Numerical experiments indicate the feasibility of the proposed method.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (nos. 11371288, 11001216, and 11371289) and the Fundamental Research Funds for the Central Universities.