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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 152, pp. 1--8.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/152\hfil High-order topological asymptotic expansion]
{High-order topological asymptotic expansion
for Stokes equations}

\author[M. Abdelwahed, M. Barhoumi, N. Chorfi \hfil EJDE-2016/152\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, 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}

\thanks{Submitted February 27, 2016. Published June 20, 2016.}
\subjclass[2010]{35A15, 35B25, 35B40, 49K40}
\keywords{Stokes equations; topology optimization; sensitivity analysis;
\hfill\break\indent topological derivative; high order asymptotic expansion}

\begin{abstract}
 We use the topological sensitivity analysis method
 to solve various optimization problems.
 It consists of studying the asymptotic expansion of the objective
 function relative to a  perturbation  of the domain topology.
 This expansion becomes insufficient in some applications  when it is
 limited to the first order topological derivative.
 We present  a new topological sensitivity analysis for the Stokes equations
 based on a high order asymptotic expansion. The derived result is valid
 for different class of shape functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

The topological sensitivity technique is an optimization method used 
for different applications \cite{1,2,5,7,24}. The main idea consists on 
developing of an asymptotic expansion of the objective function in relation 
to the domain  topological perturbation. Many operators has been studied 
in the case of this method such as, the Laplace operator, 
the Stokes system, the Helmoltz equations, \ldots \cite{15,17,21,22}. 
The majority of the existing works using topological sensitivity method 
are limited to the first order expansion which is sufficient in the case 
where the size of the domain to be detected is of infinitesimal size and 
not close to the boundary. However,  In the case where this constraint 
is not ensured or if the first order term in the asymptotic expansion 
is equal to zero at some critical points, we need an extension of the 
expansion to the high order term. This concept was studied by Rocha et al
 \cite{10,11} in the case of two dimensional Laplace operator and for a second 
order topological asymptotic. Hassine et al. \cite{20} generalized this 
work to three dimensional case and for higher order development. 
We present in this work an extension of this concept to the Stokes equations
\begin{gather*}
 -\Delta u+\nabla p=F \quad\text{in } \Omega\\
 \nabla . u = 0   \quad\text{in } \Omega\\
 u= 0   \quad\text{on } \Gamma,  \\
 \end{gather*}
where $\Omega \subset\mathbb R^3$ with smooth boundary 
$\partial \Omega=\Gamma$, $u$ is the velocity, $p$ the pressure and
 $F$ is an external force.

We define $\omega_{z,\varepsilon}$ a small
geometry perturbation of $\Omega$  that is centered at $z\in \Omega$ 
and has the form $\omega_{z,\varepsilon}=z+\varepsilon\omega$,
where $\omega \in \mathbb R^3$ is a given fixed and bounded regular 
domain containing the origin.

Let the shape function $j$ be defined by
 \begin{equation}
 j(\Omega\backslash \overline{\omega_{z,\varepsilon}})
=J_{\varepsilon}(u_\varepsilon)
 \end{equation}
where $J_{\varepsilon} \in H^1(\Omega\backslash \overline{\omega_{z,\varepsilon}})^3$
and $u_\varepsilon$ the solution to the Stokes problem
in the perturbed domain $\Omega_{z,\varepsilon}
=\Omega\backslash \overline{\omega_{z,\varepsilon}}$
 with homogeneous Dirichlet condition on $\partial \omega_{z,\varepsilon}$
 \begin{equation}\label{2}
\begin{gathered}
 -\Delta u_\varepsilon+\nabla p_\varepsilon=F \quad\text{in } \Omega_{z,\varepsilon}\\
 \nabla . u_\varepsilon = 0  \quad\text{in } \Omega_{z,\varepsilon}\\
 u_\varepsilon = 0 \quad\text{on } \Gamma \\
 u_\varepsilon = 0  \quad\text{on } \partial  \omega_{z,\varepsilon}.
 \end{gathered}
 \end{equation}
The weak formulation of \eqref{2} consists in finding
$u_\varepsilon \in V_\varepsilon$ that satisfies
  \begin{equation}\label{3}
a_\varepsilon(u_\varepsilon,\omega)=l_\varepsilon(\omega),\quad \forall 
 \omega \in V_\varepsilon,
\end{equation}
 where
   \begin{gather}
 V_0=\{v \in (H_0^1(\Omega_{z,\varepsilon}))^3 : \nabla\cdot v=0\}, \\
a_\varepsilon(v,\omega)=\int_{\Omega_{z,\varepsilon}} \nabla u : \nabla v \,dx 
= \int_{\Omega_{z,\varepsilon}} \operatorname{tr}(\nabla u \cdot\nabla v) \,dx, \quad 
\forall  v,\omega \in V_0,
 \\
 l_\varepsilon(\omega)=\int_{\Omega_{z,\varepsilon}}F.\omega\, dx, \quad
\forall \omega \in V_0.
\end{gather}
Note that  Problem \eqref{3} has a unique solution \cite{13}.

The aim of this work is to derive a high order topological asymptotic 
expansion for  $j$ relative to the presence
   of the geometry perturbation $\omega_{z,\varepsilon}$ in the domain $\Omega$. 
The idea is to develop $j(\Omega_{z,\varepsilon})-j(\Omega)$ with respect to 
$\varepsilon$ and establishing an asymptotic formula on the form
   \begin{equation}
       j(\Omega_{z,\varepsilon})-j(\Omega)=\sum_{k=1}^N f_k(\varepsilon)\delta_j^k(z)+o(f_N(\varepsilon))
   \end{equation}
where
   \begin{itemize}
   \item $f_k$, $1\le k\le N$ are positive scalar functions verifying 
$f_{k+1}(\varepsilon)=o(f_{k}(\varepsilon))$ and
   $$
   \lim_{\varepsilon\rightarrow}f_k(\varepsilon)=0,
   $$
   \item $\delta_j^k$ denotes the $k^{th}$ topological derivative of the
 shape function $j$.
   \end{itemize}
This work is a generalization of the topological sensitivity method. 
The presented result is of higher interest and is valid for different shape 
functions.

We begin by presenting the asymptotic formulation in section 2. 
Section 3 is devoted to the main result corresponding to the high order 
asymptotic expansion formula. Finally, an application of the developed 
result is presented for two different shape function examples.

\section{Asymptotic formula for the velocity variation}\label{asymptotic-formula}

In this section, we discuss the influence of the geometry perturbation 
$\omega_{z,\varepsilon}$ on the Stokes solution $(u_\varepsilon,p_\varepsilon)$. 
More precisely, we derive an asymptotic formula outlining the velocity 
field $u_\varepsilon$ (resp. the pressure field $p_\varepsilon$) variation with 
respect to the perturbation size $\varepsilon$. We begin our analysis by 
the next preliminary estimate

\begin{lemma}\label{Lemma0}
If the perturbation $\omega_{z,\varepsilon}$ is strictly embedded into $\Omega$, 
then the perturbed Stokes solution $(u_\varepsilon,p_\varepsilon)$ satisfies 
\begin{gather*}%\label{w0}
u_\varepsilon(x)-u_0(x) =  W_0((x-z)/\varepsilon)\, +\, O(\varepsilon)  \quad 
 \text{in } \Omega_{z,\varepsilon},\\
p_\varepsilon(x)-p_0(x) = \frac{1}{\varepsilon} Q_0((x-z)/\varepsilon) + O(\varepsilon) \quad 
 \text{in } \Omega_{z,\varepsilon}
\end{gather*}
where the leading term $(W_0,Q_0)$ is defined as the solution to the 
Stokes exterior problem
\begin{equation} \label{w0}
\begin{gathered}
-\Delta W_0 + \nabla Q_0=0 \quad\text{in } \mathbb{R}^3\setminus\overline{\omega},\\
\nabla \cdot  W_0  =0 \quad \text{in } \mathbb{R}^3\setminus\overline{\omega},\\
W_0  \to  0 \quad \text{at } \infty\\
W_0  =-u_0(z) \quad \text{on } \partial\omega.
\end{gathered}
\end{equation}
\end{lemma}

The proof of the above lemma is similar to that in 
\cite[Proposition 3.1]{1}; so we mit it.
Next, we will give a generalization of this estimate to the 
high-order case. The obtained asymptotic behavior is illustrated by 
the following result.


\begin{theorem}\label{thm-asym} 
If the geometry perturbation $\omega_{z,\varepsilon}=z+\varepsilon\omega$ 
is strictly embedded in the fluid flow domain $\Omega$, then the velocity 
and pressure fields satisfy the following asymptotic behavior
\begin{gather}
u_\varepsilon (x)=  \sum_{k=0}^{N} \varepsilon^k [U_k (x)+ W_k ((x-z)/\varepsilon)) ]+ O(\varepsilon^{N+1}) 
\quad\text{in } \Omega_{z,\varepsilon}, \\
p_\varepsilon (x)=  \sum_{k=0}^{N} \varepsilon^k [P_k (x)+ \frac{1}{\varepsilon} Q_k ((x-z)/\varepsilon)) ]
+ O(\varepsilon^{N+1}) \quad\text{in } \Omega_{z,\varepsilon},
\end{gather}
where
 $(U_k,P_k)_{0\leq k\leq N}$ are smooth functions, solutions to a 
sequence of Stokes problems in $\Omega$,
and $(W_k,Q_k)_{0\leq k\leq N}$ are smooth functions, solutions to a 
sequence of exterior problems in $\mathbb{R}^3\setminus\overline{\omega}$.
\end{theorem}

\begin{proof}
The sequences $(U_k,P_k)_{0\leq k\leq N}$ and 
$(W_k,Q_k)_{0\leq k\leq N}$ are constructed using an iterative process 
with $(U_0,P_0)=(u_0,p_0)$ and $(W_0,Q_0)$ is the solution to \eqref{w0}.
The proof is made in three steps.
\smallskip

\noindent\textbf{Step 1:} 
We derive the asymptotic behavior  of the functions $W_k$, $0\leq k\leq N$ 
relative to $\varepsilon$. Due to a single layer potential \cite{13}, 
$W_k$, $0\leq k\leq N$ can be written as
\[
W_k(y)=\int_{\partial \omega}E(y-t)\,\eta_k(t)ds(t), \quad
\forall y\in \mathbb{R}^3\setminus\overline{\omega},
\]
where $E$ is the fundamental solution of Stokes system in $\mathbb{R}^3$ 
and $\eta_k$ is the solution to a boundary integral equation defined on 
$\partial \omega$.
It is easy to see that for each 
$x\in \mathbb{R}^3\setminus\overline{\omega_{z,\varepsilon}}$ we have
\[
 W_k((x-z)/\varepsilon)=\int_{\partial \omega}E((x-z)/\varepsilon-t)
\eta_k(t)ds(t) =\varepsilon \int_{\partial \omega}E((x-z)-\varepsilon t)\eta_k(t)ds(t).
\]
From the fact that $\omega_{z,\varepsilon}$ is not close to the boundary 
$\partial\Omega$, one can remark that for all $t\in \partial\omega$ 
and for all $x$ in a neighborhood of $\Gamma$ the function 
$\Pi_{x-z,t}:\varepsilon \mapsto \Pi_{x-z,t}(\varepsilon)=\varepsilon\,E((x-z)-\varepsilon\,t)$
is smooth with respect to $\varepsilon$ and admits the asymptotic expansion
\[
  \Pi_{x-z,t}(\varepsilon)=\sum_{p=1}^{N}\frac{\varepsilon^p}{p!}\Pi^{(p)}_{x-z,t}(0)
+ O(\varepsilon^{N+1}),
\]
where $\Pi^{(p)}_{x-z,t}(0)$ is the $p$-th derivative 
of $\Pi_{x-z,t}$ at $\varepsilon=0$. It depends on the $p$-th derivative of the function 
$E$ at the point $x-z$. Consequently, the function 
$x\mapsto W_k((x-z)/\varepsilon)$ satisfies the following asymptotic behavior
\begin{equation} \label{fun-wkp0}
 W_{k}((x-z)/\varepsilon)=\sum_{p=1}^{N}\varepsilon^p\,W_k^{(p)}(x-z)
+O(\varepsilon^{N+1}),
\end{equation}
with $W_k^{(p)}$ is the smooth function defined in
 $\mathbb{R}^3\setminus\overline{\omega}$ by
\begin{equation}\label{fun-wkp}
W_k^{(p)}(x-z)=\frac{1}{ p!}\int_{\partial\omega}\Pi^{(p)}_{x-z,t}(0)
\eta_k(t)ds(t),\quad \forall x\in \mathbb{R}^3\setminus\overline{\omega}.
\end{equation}
\smallskip

\noindent\textbf{Step 2:} 
We are now ready to present the leading terms of the expected formulas. 
Let us suppose that we have already derived the terms $(U_i,P_i)$ and
 $(W_i,Q_i)$ for all $0\leq i \leq k-1$. The $k$-th order term is  described
 by the function $x\mapsto (U_k(x),P_k(x)) + (W_k((x-z)/\varepsilon), 
\frac{1}{\varepsilon} Q_k((x-z)/\varepsilon))$ which is constructed as follows:
\begin{itemize}
\item $(U_k,P_k)$ depends on $W_{j}$, $0\leq j \leq k-1$ and solves 
the  interior problem
\begin{equation}\label{Uk}
\begin{gathered}
-\Delta U_k + \nabla P_k=0 \quad \text{in } \Omega,\\
\nabla \cdot U_k =0 \quad \text{in } \Omega,\\
U_k = -\sum_{p=1}^{k}W_{k-p}^{(p)}(x-z)\quad \text{on } \Gamma,
\end{gathered}
\end{equation}
with $W_j^{(p)}$ is defined by \eqref{fun-wkp}.

\item $(W_k,Q_k)$ depends on $U_j$, $0\leq j \leq k$ and solves the 
 exterior problem
\begin{equation}\label{Wk}
\begin{gathered}
-\Delta W_{k} + \nabla Q_k=0 \quad\text{in } 
 \mathbb{R}^3\setminus\overline{\omega},\\
\nabla\cdot W_k=0 \quad \text{in } \mathbb{R}^3\setminus\overline{\omega},\\
W_k  \to  0 \quad \text{at }  \infty\\
W_k =-U_k(z)-
\sum_{p=1}^{k}\frac{1}{p!}D^p{U}_{k-p}(z)(y^p) \quad \text{on }
\partial\omega,
\end{gathered}
\end{equation}
where $D^p{U}_{k-p}(z)$ is the $p$-th derivative of the function 
${U}_{k-p}$ and $y^p=(y,\dots,y)\in (\mathbb{R}^3)^p$.
\end{itemize}
\smallskip

\noindent\textbf{Step 3:} 
We check that the used iterative process leads to the expected asymptotic 
formulas. Posing
$R_{N,\varepsilon}(x)=\sum_{k=0}^{N} \varepsilon^k [U_k (x)+ W_k ((x-z)/\varepsilon)) ]-u_\varepsilon$
and $S_{N,\varepsilon}(x)=\sum_{k=0}^{N} \varepsilon^k [P_k (x)
+  \frac{1}{\varepsilon} Q_k ((x-z)/\varepsilon)) ]-p_\varepsilon$. 
One can easily verify that $(R_{N,\varepsilon},S_{N,\varepsilon})$ solves the Stokes 
system in $\Omega_{z,\varepsilon}$
\begin{equation}
\begin{aligned}
-\Delta R_{N,\varepsilon} + \nabla S_{N,\varepsilon}
= 0 \quad \text{in } \Omega_{z,\varepsilon},\\
\nabla \cdot R_{N,\varepsilon} = 0 \quad \text{in } \Omega_{z,\varepsilon},
\end{aligned}
\end{equation}
and satisfies the  boundary conditions:
\begin{itemize}
\item on $\partial\omega_{z,\varepsilon}$:  Using the systems \eqref{Uk}-\eqref{Wk},
the multi-linearity of $D^p{U}_{k-p}(z)$, Taylor's Theorem and the fact that
$\|x-z\| = O(\varepsilon)$ on $\partial \omega_{z,\varepsilon}$, one can  derive
\[
 R_{N,\varepsilon}(x)=\sum_{k=0}^{N}\varepsilon^k \Big[U_k(x)
-\sum_{p=0}^{N-k}\frac{1}{p!}D^p{U}_{k}(z)((x-z)^p)\Big]=O(\varepsilon^{N+1}) .
\]

\item on $\Gamma$: From \eqref{Uk}, \eqref{Wk} and  the asymptotic
expansion \eqref{fun-wkp0}, one can obtain
\begin{align*}
R_{N,\varepsilon}(x)&=\varepsilon^N W_N((x-z)/\varepsilon)
 + \sum_{k=0}^{N-1}\varepsilon^k \big[W_k((x-z)/\varepsilon) \\
&- \sum_{p=1}^{N-k} \varepsilon^p W_{k}^{(p)}(x-z)\big]= O(\varepsilon^{N+1}).
\end{align*}
\end{itemize}
\end{proof}

\section{High-order topological asymptotic expansion}\label{asymp-expansion}

We derive in this section a high-order terms in the topological asymptotic 
expansion for the Stokes operator. The obtained results are an extension 
of the the topological derivative notion for the high-order case and  
are valid for all shape function $j$ defined by
$$
j(\Omega_{z,\varepsilon})= J_\varepsilon(u_\varepsilon), 
$$
with $J_\varepsilon$ is a scalar function in $H^1(\Omega_{z,\varepsilon})^3$,  
satisfying the following hypothesis:
\begin{itemize}
\item[(H1)]
 The function  $J_0$ is differentiable with respect  to $u$.

 There exist real numbers  $\delta^1 J(z),\dots,\delta^N J(z) $, 
such that 
$$ 
J(u_\varepsilon)-J_0(u_0)=DJ_0(u_0)(u_\varepsilon-u_0)
+\sum_{k=1}^{N}\varepsilon^k\delta^k J(z) 
+ o(\varepsilon^N)\, ,\quad \forall\varepsilon>0.
$$
\end{itemize}
In the term $DJ_0(u_0)(u_\varepsilon-u_0)$, the velocity field
 $u_\varepsilon$ is extended by zero inside the domain $\omega_{z,\varepsilon}$. 
Its extension will be denoted by $u_\varepsilon$ throughout the rest of the paper.

Under  hypothesis (H1), the variation of the shape function $j$ reads
$$ 
j(\Omega_{z,\varepsilon})-j(\Omega)=\int_{\Omega_{z,\varepsilon}}
\nabla (u_0-u_\varepsilon):\nabla v_0dx+\sum_{k=1}^{N} \varepsilon^k \,
\delta^k J(z) +o(\varepsilon^N),
$$
where $u_0$ and $v_0$ are respectively solutions to the Stokes and its
 associated adjoint problems.
Using Green formula and  Theorem \ref{thm-asym}, the integral term can
 be decomposed as
\begin{equation} \label{e77}
\begin{aligned}
&\int_{\Omega_{z,\varepsilon}}\nabla (u_0-u_\varepsilon):\nabla v_0dx\\
&=\int_{\omega_{z,\varepsilon}}\nabla u_0:\nabla v_0dx
 - \sum_{k=0}^{N} \varepsilon^k \int_{\partial\omega_{z,\varepsilon}}
 \nabla_x W_k ((x-z)/\varepsilon)) n \cdot v_0ds\\
&\quad - \sum_{k=1}^{N} \varepsilon^k
\int_{\partial\omega_{z,\varepsilon}}
\nabla U_k(x)n(x)\cdot v_0(x)ds +  O(\varepsilon^{N+1}).
\end{aligned}
\end{equation}
To derive the high-order topological asymptotic expansion for $j$,
we establish an estimate for all terms on the right side of  equality \eqref{e77}.

\subsection{Preliminary estimates}\label{preliminary-estimates}

\begin{lemma}\label{Lemma1}
The first integral term in \eqref{e77} satisfies 
$$
\int_{\omega_{z,\varepsilon}}\nabla u_0 : \nabla v_0dx
=\sum_{k=3}^{N}\varepsilon^k \,\mathcal G_{u_0,v_0}^{1,k-3}(z)+
 O(\varepsilon^{N+1}),
$$
where the functions $z\mapsto \mathcal G_{u_0,v_0}^{1,k}(z)$,
$0\leq k\leq N$ are defined in $\Omega$ by
\begin{equation}
\mathcal G_{u_0,v_0}^{1,k}(z)=
\sum_{p=0}^{k}\frac{1}{p!(k-p)!} \int_{\omega}  \nabla^{(p+1)} u_0
(z)(y^p):  \nabla^{(k-p+1)} v_0 (z)(y^{k-p})dy,
\end{equation}
with $y^k=(y,\dots,y)\in (\mathbb{R}^3)^k$ and $\nabla^{(k)}w(z)$ 
denotes the $k$-th derivative of the function $w$ at the point $z$.
\end{lemma}

\begin{lemma}\label{Lemma2}
 The second integral term in \eqref{e77} satisfies 
\[
 \sum_{k=0}^{N} \varepsilon^k
\int_{\partial\omega_{z,\varepsilon}}\nabla_x W_k
((x-z)/\varepsilon)) n \cdot v_0ds  
= -\sum_{k=1}^{N} \varepsilon^{k}  \mathcal G_{W,v_0}^{2,k-1}(z) +  O(\varepsilon^{N+1}),
\]
where the functions $z\mapsto \mathcal G_{W,v_0}^{2,k}(z)$,
$0\leq k\leq N$ are defined in $\Omega$ by
\begin{equation}
\mathcal G_{W,v_0}^{2,k}(z)
= -\sum_{p=0}^{k}\frac{1}{p!}
\int_{\partial\omega}\nabla_y W_{k-p}(y)
n(y)\cdot[\nabla^{(p)} v_0 (z)(y^{p})]ds(y).
\end{equation}
\end{lemma}

\begin{lemma}\label{Lemma3}
 The third integral term in \eqref{e77} satisfies
\[
\sum_{k=1}^{N} \varepsilon^k
\int_{\partial\omega_{z,\varepsilon}} \nabla
U_k(x) n(x)\cdot v_0(x)ds 
=-\sum_{k=3}^{N} \varepsilon^{k} \mathcal
G_{U,v_0}^{3,k-3}(z) + O(\varepsilon^{N+1}).
\]
where the functions $z\mapsto \mathcal G_{U,v_0}^{3,k}(z)$,
$0\leq k\leq N$ are defined in $\Omega$ by
\begin{align*}
&\mathcal G_{U,v_0}^{3,k}(z)\\
&=- \sum_{p=0}^{k}
\sum_{q=0}^{p}\frac{1}{q!(p-q)!}
\int_{\partial\omega} [ \nabla^{(q+1)} U_{k-p+1}
(z)(y^q)]  n(y) \cdot [\nabla^{(p-q)} v_0 (z)(y^{p-q})]ds(y).
\end{align*}
\end{lemma}

\subsection{Asymptotic expansion}\label{asymptotic}

Based on the previous estimates, we derive in theorem \ref{thm-top} 
a high-order topological asymptotic expansion valid for all shape function 
that meets  hypothesis (H1). Propositions \ref{prop1} and \ref{prop2} 
are devoted to two particular examples of shape functions.


\begin{theorem}\label{thm-top}  
Let $\omega_{z,\varepsilon}=z+\varepsilon\omega$ be a geometry perturbation 
strictly embedded in $\Omega$. If $J_\varepsilon$ satisfies {\rm (H1)},
then the associated shape function $j$  satisfies
$$
j(\Omega_{z,\varepsilon})-j(\Omega)=\sum_{k=1}^{N}
\varepsilon^k\delta^k j(z)+o(\varepsilon^N),
$$
where $\delta^k j$ is the $k-$th topological derivative order, defined 
in $\Omega$ by
\[
\delta^k j(z)= \begin{cases}
\mathcal G_{W,v_0}^{2,k-1}(z)+\delta^k J(z) & \text{if } k=1,2\\[4pt]
\mathcal G_{u_0,v_0}^{1,k-3}(z)+\mathcal G_{W,v_0}^{2,k-1}(z)
 +\mathcal G_{U,v_0}^{3,k-3}(z)+\delta^k J(z) &\text{if } 3\leq k\leq N.
\end{cases}
\]
\end{theorem}

To discuss  hypothesis (H1). We consider two examples of shape 
functions satisfying (H1) and we derive their 
variations $\delta^1J$, $\delta^2J$, \dots, and $\delta^N J$.

\begin{proposition}\label{prop1}
 Let $g\in L^2(\Omega)^3$ be a given function. 
The function $u \mapsto  J_{\varepsilon}(u)=\int_{\Omega_{z,\varepsilon}}
g\cdot udx$, for $u\in H^1(\Omega_{z,\varepsilon})$ 
satisfies (H1) with
\[
  D J_0(w)=\int_{\Omega}g\cdot w\,dx,\quad \forall w\in H^1(\Omega),
\]
and $\delta^k J(z)=0$  in  $\Omega$ $k=1,\dots,N$.
\end{proposition}


\begin{proposition}\label{prop2} 
Let $U_d$ be a given desired state, smooth in $\omega_{z,\varepsilon}$. 
The function $u\mapsto J_{\varepsilon}(u)
= \int_{\Omega_{z,\varepsilon}}\left|\nabla u-\nabla U_d\right|^2dx$, 
$u\in H^1(\Omega_{z,\varepsilon})$ satisfies (H1) with
$$ 
D J_0(w)= 2\int_{\Omega}\nabla (u_0-U_d) :\nabla
wdx,\quad \forall w\in H^1(\Omega),
$$
and
\[
\delta^k J(z)= \begin{cases}
\mathcal G_{W,u_0}^{2,k-1}(z) &\text{if } k=1,2\\[4pt]
\mathcal G_{W,u_0}^{2,k-1}(z)+
\mathcal G_{u_0,u_0}^{1,k-3}(z)+\mathcal G_{U_d,U_d}^{1,k-3}(z)
+\mathcal G_{U,u_0}^{3,k-3}(z)
&\text{if } 3\leq k\leq N.
\end{cases}
\]
\end{proposition}

\subsection*{Conclusion}
The present work generalizes the topological derivative notion for 
the high-order case. The obtained results are based on the asymptotic 
formulas describing the variations of the velocity and pressure fields 
relative to the presence of a geometry perturbation 
$\omega_{z,\varepsilon}=z+\varepsilon\omega$ in the fluid flow domain 
$\Omega$. The presented mathematical analysis is general. 
It can be extended to different partial differential equations.

\subsection*{Acknowledgments}
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).

\begin{thebibliography}{00}

\bibitem{1} M. Abdelwahed, M. Hassine;
\emph{Topological optimization method for a geometric
control problem in Stokes flow}. Appl. Numer. Math. 59(8), 1823-1838, 2009.

\bibitem{2} M. Abdelwahed, M. Hassine, M. Masmoudi M;
\emph{Optimal shape design for fluid flow using topological perturbation technique}, 
J. Math. Anal. and Applic., 356, 548-563, 2009.

\bibitem{5} S. Amstutz, I. Horchani, M. Masmoudi;
\emph{Crack detection by the topological
gradient method. Control and Cybernetics}, 34(1), 81-101, 2005.

\bibitem{7} A. Ben Abda, M. Hassine, M. Jaoua, M. Masmoudi;
\emph{Topological sensitivity analysis for the location of small 
cavities in Stokes flow}, SIAM J. Contr. Optim, 48(5), 2871-2900, 2009.

\bibitem{10} J. Rocha de Faria, A. A. Novotny;
\emph{On the second order topologial asymptotic expansion}. Structural 
and Multidis-ciplinary Optimization, 39(6), 547-555, 2009.

\bibitem{11} J. Rocha de Faria, A. A. Novotny, R. A. Feijoo, E. Taroco;
\emph{First and second-order topological sensitivity analysis for inclusions}
 J. Inverse Problems in Science and Engineering, 17(5), 665-679, 2009.

\bibitem{13} R. Dautray, J. Lions;
\emph{Analyse math\'emathique et calcul num´erique pour
les sciences et les techniques}. Collection CEA, Masson, 1987.


\bibitem{15} S. Garreau, Ph. Guillaume, M. Masmoudi;
\emph{The topological asymptotic for PDE systems: The elastics case}, 
SIAM J. contr. Optim. 39(6), 1756-1778, 2001.

\bibitem{17} Ph. Guillaume, K. Sid Idris;
\emph{Topological sensitivity and shape optimization
for the Stokes equations}, SIAM J. Control Optim. 43(1) 1-31, 2004.


\bibitem{20} M. Hassine, Kh. Khelifi;
\emph{On the high-order asymptotic expansion for shape functions}, 
Electronic journal of differential equations, vol. 2016 (2016), no. 110, 1-16, 

\bibitem{21} M. Hassine, M. Masmoudi;
\emph{The topological asymptotic expansion for the
quasi-Stokes problem}, ESAIM COCV J. 10(4) 478-504, 2004.

\bibitem{22} J. Pommier, B. Samet;
\emph{The topological asymptotic for the Helmholtz equation
with Dirichlet condition on the boundary of an arbitrarily shaped hole},
SIAM J. Control Optim. 43(3), 899-921, 2004.


\bibitem{24} J. Sokolowski, A. Zochowski;
\emph{On the topological derivative in shape optimization},
SIAM J. Control Optim. 37(4) 1251-1272, 1999.

\end{thebibliography}

\end{document}