\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 233, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/233\hfil Determination of obstacles]
{Determination of obstacles in Stokes flow by boundary measurement}

\author[M. Abdelwahed,  M. Barhoumi, N. Chorfi \hfil EJDE-2017/233\hfilneg]
{Mohamed Abdelwahed,  Montassar Barhoumi, Nejmeddine Chorfi}

\address{Mohamed Abdelwahed \newline
Department of Mathematics,
College of Sciences,
King Saud University, Riyadh, Saudi Arabia}
\email{mabdelwahed@ksu.edu.sa}

\address{Montassar Barhoumi \newline
Department of Mathematics,
ESSTHS, Sousse University,
Sousse University, Tunisia}
\email{montassar.barhoumi2014@gmail.com}

\address{Nejmeddine Chorfi \newline
Department of Mathematics,
College of Sciences,
King Saud University, Riyadh, Saudi Arabia}
\email{nchorfi@ksu.edu.sa}


\dedicatory{Communicated by Vicen\c{t}iu D. R\u{a}dulescu}

\thanks{Submitted July 29, 2017. Published September 25, 2017.}
\subjclass[2010]{35N25, 49K40}
\keywords{Stokes flow; reciprocity gap function; calculus of variations;
\hfill\break\indent   topological sensitivity}


\begin{abstract}
 We study the determination of some obstacles in a Stokes flow domain
 with overdetermined boundary data. We use a method based on the topological
 sensitivity technique associated to the reciprocity gap function concept.
 We develop an asymptotic formula between the flow parameters and the
 boundary data. The obtained formula is interesting and serve as a useful
 tool to develop an accurate and robust numerical method in geometry
 inverse problems.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Let $\Omega$ be a regular domain in $\mathbb R^3$ occupied by a  homogeneous  
incompressible fluid flow. We assume that the fluid flow is in laminar
 regime in such way that the convection term can be neglected and the
 Navier-Stokes equations can be approximated by the Stokes system.

The velocity fluid $w$ and the pressure $q$ describing the fluid flow 
in $\Omega$ satisfy the following system
\begin{equation}\label{p0}
 \begin{gathered}
 -\nu \Delta w+\nabla q=G \quad\text{in } \Omega\\
 \nabla\cdot w = 0  \quad\text{in } \Omega\\
 w= w_d  \quad\text{on } \Gamma_d,  \\
 \sigma(w,q)n = g   \quad\text{on } \Gamma_n,
 \end{gathered}
\end{equation}
where $G$ is a source term (gravitational force), $\nu$ is the fluid 
viscosity,  $w_d$ is a given boundary velocity and $g$ is a given boundary 
force. Hence $\Gamma_d$ and $\Gamma_n$ are two parts of the boundary 
$\partial \Omega$ verifying 
$\overline{\partial\Omega}=\overline{\Gamma_d}\cup \overline{\Gamma_n}$ 
and ${\Gamma_d}\cap {\Gamma_n}=\emptyset$.

We suppose that the fluid flow domain $\Omega$ contains a finite number of 
unknowns obstacles $\mathcal O_i$, $i=1,\ldots,m$ that are well separated 
and not close to the boundary $\partial \Omega$.
In this work we assume that each obstacle $\mathcal O_i$ is characterized 
by its center $\xi_i\in \Omega$, its size $r_i$ and its shape $S_i$ 
with $r_i>0$ and $S_i\subset \mathbb R^3$ is a fixed bounded and 
smooth domain containing the origin. In other word, each obstacle $\mathcal O_i$ 
can be defined as $\mathcal O_i=\xi_i + r_i S_i$, $1\le i \le m$.

The problem that we consider can be formulated as follows:
\\
$\bullet$  Given two boundaries data on the accessible part $\Gamma_a$ 
of the boundary
 $\Gamma_n$  a measured velocity $w_m$ and an imposed force $g$.
\\
$\bullet$ Find the unknown obstacle $\mathcal O=\cup_{i=1}^{m} \mathcal O_i$ 
such that the velocity field $w_{\mathcal O}$ and the pressure $q_{\mathcal O}$ 
in the perturbed domain $\Omega\backslash\overline{\mathcal O}$ satisfy the 
 boundary value problem
 \begin{gather*}
 -\nu \Delta w_{\mathcal O}+\nabla q_{\mathcal O}=G \quad\text{in }
 \Omega\backslash\overline{\mathcal O}\\
 \nabla\cdot w_{\mathcal O} = 0   \quad\text{in } 
 \Omega\backslash\overline{\mathcal O}\\
 w_{\mathcal O}= w_m   \quad\text{on } \Gamma_a  \quad \text{[accessible boundary]}\\
 \sigma(w_{\mathcal O},q_{\mathcal O})n = g   \quad\text{on } \Gamma_a\quad 
\text{[accessible boundary]}\\
  \sigma(w_{\mathcal O},q_{\mathcal O})n = 0   \quad\text{on } 
\Gamma_i^1\quad\text{(in and out) [inaccessible boundary]}\\
   w_{\mathcal O}= 0  \quad\text{on } \Gamma_i^2 \quad\text{(the wall) 
[inaccessible boundary]}\\
    w_{\mathcal O}= 0  \quad\text{on } \partial \mathcal O.
 \end{gather*}


In this formulation, the fluid flow domain $\Omega\backslash\overline{\mathcal O}$ 
is unknown since the obstacle geometry is unknown. It is well known that this
 kind of problem is ill-posed in the sense of Hadamard. The majority of 
investigation focusing on this type of problems fall into the category of shape 
optimization and utilize the shape derivation technics.

In this work, we suggest a new formulation of the above inverse problem  
based on the reciprocity gap concept \cite{3,1,2} and the topological 
sensitivity analysis method \cite{4,11,12,5,6,13,14,15}. 
More precisely, we will derive an asymptotic formula connecting the known boundary 
data and the unknown obstacle properties (its location $\xi_i$, its size $r_i$ 
and its shape $\delta_i$).

This article is organized as follows. 
In section 2, we introduce the reciprocity gap functional. 
A preliminary estimate describing the variation of the reciprocity gap 
functional with respect to the presence of an obstacle $\mathcal O=\xi+rs$ 
inside the fluid flow domain $\Omega$ is presented in Proposition 1.
 To derive the expected formula, we start our analysis by studying the 
influence of the presence of the obstacle on the velocity field. 
We derive a high order asymptotic expansion of the perturbed velocity 
with respect to the obstacle size $r$ in section 3. 
Finally, section 4 is devoted to the  derivation of  a high order 
topological sensitivity analysis for the reciprocity gap function.

\section{Reciprocity gap functional and Stokeslet sub-space}

The reciprocity gap function is a function defined on the boundary 
$\partial\Omega$. it describes the fluid response to an imposed force on 
the boundary. This function associated to the presence of an obstacle 
$\mathcal O_{\xi,r}$ in the flow domain $\Omega$ is defined by
$\mathcal F_{\xi,r}: H^{1}(\Omega)\times L^2(\Omega) \to \mathbb{R}$:
\[
 \mathcal F_{\xi,r}(u,p)=\int_{\partial \Omega}
\sigma(u,p)n w_r \, ds -\int_{\partial \Omega} \sigma(w_r,q_r)n u \, ds,
\]
where $w_r,q_r$ is the solution of the Stokes problem in the presence of an
 obstacle $\mathcal O_{\xi,r}$,
 \begin{gather*}
 -\nu \Delta w_r+\nabla q_r=G \quad\text{in } 
\Omega\setminus\overline{\mathcal O_{\xi,r}}\\
 \nabla\cdot w_r = 0   \quad\text{in } \Omega\setminus\overline{\mathcal O_{\xi,r}}\\
 w_r= w_d   \quad\text{on } \Gamma_d,  \\
 \sigma(w_r,q_r)n = g   \quad\text{on } \Gamma_n.
 \end{gather*}
In the absence of the obstacle $\mathcal O_{\xi,r}$, the reciprocity gap 
function is denoted by $\mathcal F_0$ and is defined on 
$H^{1}(\Omega)\times L^2(\Omega)$ by
$$
\mathcal F_0(u,p)=\int_{\partial \Omega} \sigma(u,p)n w_0 \, ds 
-\int_{\partial \Omega} \sigma(w_0,q_0)n u \, ds,
$$
where $(w_0,q_0)$ is the solution of \eqref{p0}.

Our goal is to establish a relation between the boundary data and the obstacles 
$\mathcal O_{\xi,r}$ properties $\xi$, $r$ and $S$. We begin this study 
by the following estimation.

\subsection{Preliminary estimations}
We consider the  subspace
$$
\mathcal V =\{(u,p)\in H^{1}(\Omega)\times L^2(\Omega); 
 -\nu \Delta u+\nabla p=0 \text{ in }  \Omega  \text{ and }  \nabla\cdot u = 0 \;
\text{  in} \, \Omega \}.
$$
The restriction of the reciprocity gap function $\mathcal F_{\xi,r}$ 
to the subspace $\mathcal V$ gives the following estimation.

\begin{proposition}\label{prop1}
For all $(u,p)\in \mathcal V$, we have
\begin{equation}\label{0}
\mathcal F_{\xi,r}(u,p)-\mathcal F_0 (u,p)
=-\int_{\mathcal O_{\xi,r}} \nu \nabla u : \nabla  w_0 \, dx
 +\int_{\partial \mathcal O_{\xi,r}} \sigma(w_r-w_0,q_r-q_0)n u \, ds.
\end{equation}
\end{proposition}

\begin{proof}
Using Green's formula and the fact that $w_r=0$ on $\partial \mathcal O_{\xi,r}$, 
one can obtain
\begin{gather*}
\int_{\partial \Omega} \sigma(u,p)n w_r \, ds
=\int_{\Omega_{\xi,r}}  \nabla u : \nabla  w_r \, dx,\\
\int_{\partial \Omega} \sigma(w_r,q_r)n u \, ds
=\int_{\Omega_{\xi,r}}  \nabla u : \nabla  w_r \, dx
-\int_{\partial \mathcal O_{\xi,r}} \sigma(w_r,q_r)n u \, ds
\end{gather*}
which implies
\begin{equation}\label{1}
\mathcal F_{\xi,r}(u,p)
=-\int_{\partial \mathcal O_{\xi,r}} \sigma(w_r,q_r)n u \, ds \quad 
\forall(u,p)\in \mathcal V.
\end{equation}
In the same way we obtain
\begin{gather*}
\int_{\partial \Omega} \sigma(u,p)n w_0 \, ds
=\int_{\Omega_{\xi,r}}  \nabla u : \nabla  w_0 \, dx
 -\int_{\partial \mathcal O_{\xi,r}} \sigma(u,p)n w_0 \, ds, \\
\int_{\partial \Omega} \sigma(w_0,q_0)n u \, ds
=\int_{\Omega_{\xi,r}}  \nabla w_0 : \nabla  u \, dx
-\int_{\partial \mathcal O_{\xi,r}} \sigma(w_0,q_0)n u \, ds.
\end{gather*}
It follows that
\begin{equation}\label{2}
\mathcal F_0(u,p)=-\int_{\partial \mathcal O_{\xi,r}} \sigma(u,p)n w_0 \, ds 
+ \int_{\partial \mathcal O_{\xi,r}} \sigma(w_0,q_0)n u \, ds.
\end{equation}
Using \eqref{1}, \eqref{2} and the fact that 
$-\nu \Delta u + \nabla p=\; in\; \mathcal O_{\xi,r}$ and 
$\nabla.u=0 \; in\; \mathcal O_{\xi,r}$ we deduce
$$
\mathcal F_{\xi,r}(u,p)-\mathcal F_0(u,p)
=-\int_{\Omega_{\xi,r}}  \nabla w_0 : \nabla  u \, dx
+\int_{\partial \mathcal O_{\xi,r}} \sigma(w_r-w_0,q_r-q_0)n u \, ds.
$$
\end{proof}

\subsection{Stokeslet sub-space}

To make relation \eqref{0} more explicit we introduce the so called Stokeslet sub-space  \cite{7,8}.

\begin{definition} \label{def2.2} \rm
We call Stokeslet of size $b\in \mathbb{R}^3$ and location $\eta \in \mathbb{R}^3$
the vectorial function $\mathcal S_{\eta,b}$ defined on $\mathbb{R}^3$ by
$$
\mathcal S_{\eta,b}(x)=\Big(U(x-\eta)b,P(x-\eta).b\Big)\quad \forall x \in \mathbb{R}^3,
$$
where $x \mapsto (U(x-\eta),P(x-\eta))$ is the fundamental solution of the 
Stokes operator with regards to a Dirac mass at point $\eta$.
\end{definition}

\begin{remark} \label{rmk2.3} \rm
For all $1\le i \le 3$, the function $x \mapsto (U^i(x-\eta)b,P^i(x-\eta))$ 
is the solution of
\begin{gather*}
 -\nu \Delta_x U^i(x-\eta)+\nabla_x P^i(x-\eta)
=\delta_{\eta} e_i  \quad \text{in } \mathbb{R}^3 \\
 \nabla_x .U^i(x-\eta)  = 0    \quad \text{in } \mathbb{R}^3
 \end{gather*}
with $U^i$ is the $i^{th}$ column of $U$ and $e_i$ is the $i^{th}$  
vector of canonical basis of $\mathbb{R}^3$.
\end{remark}

We introduce know the so-called  Stokeslet sub-space 
$\mathcal V_{\mathcal S,\Omega}$. It is a sub-space of $\mathcal V$ defined 
by the functions of Stokeslet type localized outside $\Omega$
$$
\mathcal V_{\mathcal S,\Omega}=\{x \mapsto {{{\mathcal S_{\eta,b}}{|}}_{\Omega}}, 
\, \eta \in \mathbb{R}^3 \backslash\overline{\Omega}, b \in \mathbb{R}^3 \}.
$$
We remark that the sub-space $\mathcal V_{\mathcal S,\Omega}$ 
is generated by the functions $(U^i(x-\eta)b,P^i(x-\eta));\ 1\le i \le 3$.

For all $1\le i \le 3$, we denote by $\mathcal F^i_{\xi,r}$ the reciprocity 
gap function associated to the Stokeslet $\mathcal S_{\eta,e_i}$ defined by
$$
\mathcal F^i_{\xi,r}=\int_{\partial\Omega}\sigma(U^i(x,\eta),P^i(x,\eta))n w_r \, ds 
- \int_{\partial \Omega} \sigma(w_r,p_r)n U^i(x,\eta) \, ds \quad 
\forall \eta\in \mathbb{R}^3 \backslash\overline{\Omega}.
$$
From Proposition \ref{prop1}, we deduce the following result.

\begin{corollary}\label{corolary1}
For all $1\le i \le 3$, the function $\mathcal F^i_{\xi,r}$ satisfies:
 for all $\eta \in \mathbb{R}^3 \backslash \overline{\Omega}$,
\begin{equation}\label{corol1}
\begin{aligned}
&\mathcal F^i_{\xi,r}(\eta)-\mathcal F^i_{0}(\eta) \\
&=-\int_{\mathcal O_{\xi,r}}\nu \nabla w_0:
\nabla u^i(x,\eta)\, dx +\int_{\partial \mathcal O_{\xi,r}} 
\sigma(w_r-w_0,q_r-q_0)n U^i(x-\eta) \, ds.
\end{aligned}
\end{equation}
\end{corollary}

\section{Main results}

 We derive an asymptotic formula linking the unknown properties of the obstacle 
(its position $z$, size $r$ and form $\mathcal O$) and the boundary data. 
We begin by studying the influence  of the obstacle on the flow state.

\subsection{Estimate of the perturbed fluid flow}
We give an estimate of the solution $(w_r,q_r)$ describing the flow in 
 presence of the obstacle $\mathcal O_{\xi,r}$.

\begin{proposition}\label{prop2}
In  presence of the obstacle $\mathcal O_{\xi,r}$ inside the fluid flow domain $\Omega$, the Stokes solution $(w_r,q_r)$ admits the following estimate: $\forall x \in \Omega \backslash \overline{\mathcal O_{\xi,r}}$
\begin{gather*}
w_r(x)=\sum_{k=0}^{N}r^k\big[W_k(x)+Z_k\big(\frac{x-z}{r}\big)  \big]+O(r^{N+1}),\\
q_r(x)=\sum_{k=0}^{N}r^k\big[Q_k(x)+S_k\big(\frac{x-z}{r}\big)  \big]+O(r^{N+1}),
\end{gather*}
where
 $(W_k,Q_k)_{0\le k \le N}$ are regular functions defined in $\Omega$ and 
solutions of a sequence of Stokes problems;
 $(Z_k,S_k)_{0\le k \le N}$ are regular functions, solutions of a sequence 
of Stokes problems in the exterior domain 
$\mathbb{R}^3 \backslash\overline{\Omega}$.
\end{proposition}

\subsection{Preliminary calculus}
We give an estimate of each term in  variation \eqref{corol1}.

\begin{lemma}\label{lem1}
The first integral term in \eqref{corol1}, admits the  estimate
$$
\int_{\mathcal O} \nu \nabla w_o(x):U^i(x-\eta)\, dx 
=\sum_{j=0}^{N-3}r^{j+3} \mathcal I_{\eta,\mathcal O}^{i,j}(z)+o(r^N),
\quad \eta \in \mathbb{R}^3 \backslash\overline{\Omega},
$$
where the functions $z \mapsto \mathcal I_{\eta,\mathcal O}^{i,j}(z)$,
$0\le j \le N-3$ are defined by
$$
\mathcal I_{\eta,\mathcal O}^{i,j}(z)=-\sum_{q=0}^{j} \frac{1}{j!(j-q)!}
\int_{\mathcal O} \nu (\nabla^{(q+1)}w_0(z)y^q)\cdot
(\nabla^{(j-q+1)}U^i(z-\eta)y^{(j-q)})\, dy
$$
with $y^q=(y,\ldots,y)\in (\mathbb{R}^3)^q$ and $\nabla^{(p)}\varphi(z)$ 
is the $p^{th}$ derivative of $\varphi$ at the point $z$.
\end{lemma}

\begin{lemma}\label{lem2}
The second integral term in \eqref{corol1} satisfies the estimate
$$
\sum_{j=1}^{N}r^j \int_{\partial \mathcal O_{\xi,r}}\sigma(W_j,Q_j)n U^i(x-\eta)\, ds
=\sum_{j=0}^{N-3} r^{j+3} \mathcal K_{\eta,\mathcal O}^{i,j}(z)+o(r^N)\quad
 \forall \eta \in \mathbb{R}^3\backslash\overline{\Omega},
$$
where the functions $z \mapsto \mathcal K_{\eta,\mathcal O}^{i,j}(z)$ are 
defined by
$$
\mathcal K_{\eta,\mathcal O}^{i,j}(z)
=\sum_{k=0}^{j}\sum_{l=0}^{k}\frac{1}{l!(k-l)!}
\int_{\partial \mathcal O}[\mathcal A_{j-k+1}^{(l)}(z)(y)n(y)]
[ \nabla^{(k-l)}U^i(z-\eta)(y)^{(k-l)}]\,ds(y)
$$
with $\mathcal A_{j-k+1}^{(l)}(z)(y)$ is the matrix defined by
$$
\Big(\mathcal A_{j-k+1}^{(l)}(z)(y)\Big)_{p,q}
=\nabla^{(l)}(\sigma(W_j,Q_j))_{p,q}(z)(y^l)\quad \forall 1 \le p,q \le 3.
$$
\end{lemma}

\begin{lemma}\label{lem3}
The third integral term in \eqref{corol1} admits the estimate
\[
\sum_{j=0}^{N} r^j \int_{\partial \mathcal O_{\xi,r}}\sigma (Z_j,S_j)
\big(\frac{x-z}{r}\big) n(x) U^i(x-\eta)\, ds(x)
=\sum_{j=0}^{N-1} r^{j+1} \mathcal L_{\eta,\mathcal O}^{i,j}(z)+o(r^N)
\]
for all $\eta \in \mathbb{R}^3\backslash\overline{\Omega}$,
where the functions $z \mapsto \mathcal L_{\eta,\mathcal O}^{i,j}(z)$ 
are defined by
$$
\mathcal L_{\eta,\mathcal O}^{i,j}(z)=\sum_{q=0}^{j}
\frac{1}{q!}\int_{\partial \mathcal O}
[\sigma(Z_{j-q},S_{j-q})(y) n(y)]\cdot
[\nabla^{(q)}U^i(z-\eta)(y^{(q)}) ]\, ds(y)+o(r^N)
$$
for all $\eta \in \mathbb{R}^3\backslash\overline{\Omega}$.
\end{lemma}

\subsection{Asymptotic formula for the reciprocity gap function}

In this section, we derive an asymptotic formula describing the variation 
of the reciprocity gap function with respect to the presence of the obstacle
$\mathcal O_{\xi,r}$ in the flow domain $\Omega$.

From corollary \ref{corolary1} and proposition \ref{prop2}, we have
\begin{align*}
&\mathcal F^i_{\xi,r}(\eta)-\mathcal F^i_{0}(\eta)\\
&=-\int_{\mathcal O_{\xi,r}}\nu \nabla w_0:\nabla u^i(x-\eta)\, dx 
 +\sum_{j=1}^{N}r^j\int_{\partial \mathcal O_{\xi,r}} \sigma(W_j,Q_j)n U^i(x-\eta)
 \, ds \\
&\quad +\sum_{j=0}^{N}r^j\int_{\partial \mathcal O_{\xi,r}} 
\sigma(Z_j,S_j)(\frac{x-z}{r})n(x) U^i(x-z) \, ds+o(r^N).
\end{align*}
Using Lemmas \ref{lem1}, \ref{lem2} and \ref{lem3}, 
we obtain the following theorem.

\begin{theorem}\label{th1}
Let $\mathcal O_{\xi,r}=z+r\mathcal O$ an  obstacle immersed in the fluid 
flow domain $\Omega$. For each $1\le i \le 3$ the reciprocity gap function
$\mathcal F_{\eta,\mathcal O}^{i}$ satisfies the following asymptotic formula
\begin{equation}
\mathcal F_{\eta,\mathcal O}^{i}(\eta)-\mathcal F_{0}^{i}(\eta)
=\sum_{j=1}^{N} r^j  \Psi_{\eta,\mathcal O}^{i,j}(z)+o(r^N)\quad 
\forall \eta \in \mathbb{R}^3 \backslash\overline{\Omega}.
\end{equation}
where $ \Psi_{\eta,\mathcal O}^{i,j}(z)$, $1\le i \le 3$, $1\le j \le N$
 are defined by
$$
\Psi_{\eta,\mathcal O}^{i,j}(z)
=\begin{cases}
\mathcal L_{\eta,\mathcal O}^{i,j-1}(z)  & \text{if } 1\le j \le 2, \\
\mathcal L_{\eta,\mathcal O}^{i,j-1}(z)+\mathcal K_{\eta,\mathcal O}^{i,j-3}(z)
+\mathcal I_{\eta,\mathcal O}^{i,j-3}(z)  & \text{if } 3\le j \le N.
\end{cases}
$$
\end{theorem}

\section{Conclusion}

The asymptotic formula derived in Theorem \ref{th1} can be used as the basis 
of a numerical algorithm serving to reconstruct an unknown obstacle
$\mathcal O_{\xi,r}$ from boundary measured data. In fact
\begin{itemize}
\item The force $\sigma(w_r,q_r)n$ is imposed on $\Gamma_n$ and measured on 
$\Gamma_d$.
\item The velocity field $w_r$ is imposed on $\Gamma_d$ and measured on $\Gamma_n$.
\end{itemize}
Then the variation
\begin{align*}
L^i(\eta)
&=\mathcal F_{\eta,\mathcal O}^{i}(\eta)-\mathcal F_{0}^{i}(\eta)\\
&= \int_{\partial \Omega} \sigma(U^i,P^i)n (w_r-w_s) \, ds
-\int_{\partial \Omega} \sigma(w_r-w_0,q_r-q_0)n U^i(x-\eta) \, ds
\end{align*}
can be used as a measured datum on $\partial \Omega$ for all 
$\eta \in \mathbb{R}^3 \backslash\overline{\Omega}$.

By neglecting the terms $o(r^N)$, Theorem \ref{th1} gives us a non linear
 system verified by the unknown parameters: the location $z$, the size $r$ 
and the form $\mathcal O$:
$$
\sum_{j=1}^{N}r^j \Psi_{\eta,\mathcal O}^{i,j}(z)=L^{i}(\eta)\quad 
\forall 1\le i \le 3,\; \forall \eta \in  \mathbb{R}^3 \backslash\overline{\Omega}.
$$
This system is difficult to solve but firstly, we can establish a 
numerical algorithm to identify the location $z$ and the size $r$, 
then we can use this system to have an approximation of the form $\mathcal O$. 
This numerical work will be subject of a forthcoming paper.

\section{Proofs of main results}

\begin{proof}[Proof of Lemma \ref{lem1}]
By the change of variable $x=z+ry$,
$$
\int_{\mathcal O_{\xi,r}}\nu \nabla w_0(x): \nabla U^i(x-\eta)\, dx
=r^3 \int_{\mathcal O}\nu \nabla_x w_0(z+ry).\nabla_x U^i(z-\eta+ry)\, dy.
$$
The functions $w_0$ and $U^i$ are sufficiently regular in $\mathcal O_{\xi,r}$. 
Using the Taylor-Young formula, we obtain
\begin{gather*}
\nabla w_0(z+ry)=\nabla w_0(z)+\sum_{j=1}^{N-1}\frac{r^j}{j!}\nabla^{(j+1)}
 w_0(z)(y^j)+O(r^N)\\
\nabla U^i(z-\eta+ry)=\nabla U^i(z-\eta)+\sum_{j=1}^{N-1}
\frac{r^j}{j!}\nabla^{(j+1)}U^i(z-\eta)(y^j)+O(r^N).
\end{gather*}
Using the Cauchy formula for the product of two polynomials, we deduce
\begin{align*}
&\int_{\mathcal O_{\xi,r}}\nu \nabla w_0(x): \nabla U^i(x-\eta)\, dx\\
&=r^3 \int_{\mathcal O}\Big[\nu \sum_{j=0}^{N-1}\frac{r^j}{j!}\nabla^{(j+1)}
w_0(z)(y^j)\Big]\Big[ \sum_{j=0}^{N-1}\frac{r^j}{j!}\nabla^{(j+1)}U^i(z)(y^j) \Big]
 +O(r^{N+1})\\
&=\sum_{j=0}^{N-3}r^{j+3}\Big[\sum_{q=0}^{j}\frac{1}{q!(j-q)!}
\int_{\mathcal O}\nu \Big( \nabla^{(q+1)}w_0(z)(y^{(q)})\Big) \\
&\quad \cdot\Big( \nabla^{(j-q+1)}U^i(z-\eta)(y^{(j-q)})\Big)\, dy\Big]
 + O(r^{N+1}).
\end{align*}
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem2}]
We have
\begin{align*}
&\int_{\partial \mathcal O_{\xi,r}}\sigma_x(W_j,Q_j)n U^i(x-\eta)\,ds(x) \\
&=r^2\int_{\partial \mathcal O}\sigma_x(W_j,Q_j)(z+ry)n(z+ry).U^i(z-\eta+ry)\, ds(y)
\end{align*}
Since the solution $(W_j,Q_j)$ is regular, for all $1\le p,q \le 3$ the function 
\[
y \mapsto [\sigma_x(W_j,Q_j)]_{p,q}(z+ry)
=\frac{1}{2}\Big(\frac{\partial (W_j)_p}{\partial x_q}
 +\frac{\partial (W_j)_q}{\partial x_p}\Big)+Q_j \delta_{p,q}
\]
 is regular in the neighborhood of $z$, and 
$$
[\sigma(W_j,Q_j)]_{p,q}(z+ry)
=\sum_{k=0}^{N}\frac{r^k}{k!}\nabla^{(k)} [\sigma(W_j,Q_j)]_{p,q}(z)(y^k)+O(r^N).
$$
In the same way,
$$
U^i(z-\eta+ry)=\sum_{k=0}^{N}\frac{r^k}{k!}\nabla^{(k)}U^i(z-\eta)(y^k)+O(r^N).
$$
We deduce
\begin{align*}
&\int_{\partial \mathcal O_{\xi,r}}\sigma(W_j,Q_j)n U^i(x-\eta)\, ds(x)\\
&=\sum_{k=0}^{N-2}r^{k+2}\sum_{l=0}^{k}\frac{1}{l!(k-l)!}
\int_{\partial \mathcal O} \mathcal A_j^{(l)}(z)(y) n(y)\cdot
\nabla^{(k-l)}U^i(z-\eta)y^{(k-l)}\, ds(y),
\end{align*}
where $\mathcal A_j^{(l)}(z)(y)$ is the matrix 
$[\mathcal A_j^{(l)}(z)(y)]_{p,q}=\nabla^{(l)}[\sigma(W_j,Q_j)]_{p,q}(z)(y^l)$
for $ 1\le p,q \le 3$.
Then we obtain
\begin{align*}
&\sum_{j=1}^{N}r^j\int_{\partial \mathcal O_{\xi,r}}
 \sigma(W_j,Q_j)(x)n(x).U^i(x-\eta)\, ds(x)\\
&=\sum_{j=1}^{N}r^j\sum_{k=0}^{N-2}r^{k+2}\sum_{l=0}^{k}
 \frac{1}{l!(k-l)!}\int_{\partial \mathcal O} \mathcal A_j^{(l)}(z)(y)n(y) \cdot
 \nabla^{(k-l)}U^i(z-\eta)(y^{k-l})\, ds(y) \\
&\quad +O(r^{N+1})\\
&=\sum_{j=3}^{N}r^j\sum_{k=0}^{j-3}\sum_{l=0}^{k}\frac{1}{l!(k-l)!}
 \int_{\partial \mathcal O} \mathcal A_{j-k+2}^{(l)}(z)(y) n(y) \cdot
 \nabla^{(k-l)}U^i(z-\eta)(y^{k-l})\, ds(y)\\
&\quad +O(r^{N+1}).
\end{align*}
\end{proof}

\begin{proof}[Proof of Lemma \ref{lem3}]
We have
$$
\int_{\partial \mathcal O_{\xi,r}}\sigma_x(Z_k,S_k)(\frac{x-z}{r})n
\cdot U^i(x-\eta)\, ds(x)
=r\int_{\partial \mathcal O}\sigma_y(Z_k,S_k)(y)n U^i(z-\eta+ry)\, ds(y).
$$
As $\eta\in \mathbb{R}^3 \backslash\overline{\Omega}$, the function 
$x\mapsto U^i(x-\eta)$ is $C^{\infty}$ in the neighborhood of $z$. 
We can derive the expansion
$$
U^i(x-\eta)=U^i(z-\eta)+\sum_{j=1}^{N-1}\frac{r^j}{j!}\nabla^{(j)}U^i 
(z-\eta)(y^j)+O(r^N).
$$
Then we deduce
\begin{align*}
&\int_{\partial \mathcal O_{\xi,r}}\sigma_x(Z_k,S_k)
 (\frac{x-z}{r})n.U^i(x-\eta)\, ds(x)\\
&=\sum_{j=0}^{N-1}\frac{r^{j+1}}{j!}\int_{\partial \mathcal O}
 [\sigma_y(Z_k,S_k)n(y)].[\nabla^{(j)}U^i (z-\eta)(y^j)]+O(r^N).
\end{align*}
Therefore,
\begin{align*}
&\sum_{j=0}^{N}r^j\int_{\partial \mathcal O_{\xi,r}}\sigma(Z_j,S_j)
(\frac{x-z}{r})n.U^i(x-z)\, ds(x)\\
&=\sum_{j=1}^{N}r^j\sum_{q=0}^{j-1}\frac{1}{q!}
\int_{\partial \mathcal O}[\sigma(Z_{j-q-1},S_{j-q+1})(y)n(y)]\\
&\quad \cdot [\nabla^{(q)}U^i (z-\eta)(y^{(q)})]\, ds(y)+O(r^N).
\end{align*}
\end{proof}

\subsection*{Acknowledgements}
The authors would like to extend their sincere appreciation to the Deanship
 of Scientific Research at King Saud University for funding this Research
group No (RG-1435-026).

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\end{document}