Topological sensitivity analysis for a three-dimensional parabolic type problem

Abstract

This work focuses on the topological sensitivity analysis of a three-dimensional parabolic type problem. The considered application model is described by the heat equation. We derive a new topological asymptotic expansion valid for various shape functions and geometric perturbations of arbitrary form. The used approach is based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the perturbed temperature field.

Keywords

Geometric inverse problem topological sensitivity analysis asymptotic expansion parabolic operator

1. Introduction

The topological sensitivity analysis method [2,4,9,10,19,23] consists of establishing an asymptotic expansion describing the behavior of a shape function $G$ regarding a small geometric perturbation $H_{z, ε}$ of the domain Ω of the form $\begin{matrix} G (Ω ∖ {\overline{H}}_{z, ε}) = G (Ω) + f (ε) T (z) + o (f (ε)), \forall z \in Ω, \end{matrix}$ where $f (ε)$ is a scalar positive function going to zero with ε and $T$ is the topological sensitivity function. This concept allows to find the best location of $H_{z, ε}$ in Ω that would bring the best improvement of the shape function $G$ . It corresponds to where $T$ is most negative.

Using the topological sensitivity analysis, one can build a numerical algorithm that can solve various optimization problems. In this context, one can cite the developed algorithms for fluid flow optimal shape design [3], image restoration [8], cracks detection [5], flaws identification [9], structural shape optimization [4], … etc.

Theoretically, the topological sensitivity analysis method has been investigated by various researchers and for different operators in the stationary regime; one can consult [23] for the Helmholtz equation, [16] for the elasticity problem, [20] for the Stokes systems, [1,11,21] for the acoustic problem and [19] for the Laplace equation. The non stationary case has been discussed in the case where the domain Ω is perturbed by the presence of a small inhomogeneity $H_{z, ε}$ . In such a case, the perturbed solution of the studied partial differential equation is considered in the entire domain Ω with a continuity boundary condition on $H_{z, ε}$ (see, e.g., [6,15]). We also refer to [18,22] for related sensitivity results in the context of evolution problems.

We consider in this work the heat transfer equation as a parabolic model problem and we extend the topological sensitivity notion for the non-stationary case. Moreover, the solution of the perturbed problem is defined in $Ω_{z, ε} : = Ω ∖ {\overline{H}}_{z, ε}$ with a Dirichlet boundary condition on $\partial H_{z, ε}$ . Unlike the inhomogeneity case, we deal here with a more singular geometric perturbation, where the extended perturbed solution on Ω satisfies a discontinuous condition on the perturbation boundary.

The two dimensional case is treated in [17]. We examine in this work the three-dimensional case and we propose an accurate approach based on a rigorous mathematical framework describing and analyzing the asymptotic behavior of the temperature variation with respect to the presence of a small Dirichlet geometric perturbation inside the heat conduction domain. The leading term of the temperature variation admits an explicit expression involving the fundamental solution of the steady state part of the operator. The proposed theoretical method is general and can be adapted for various parabolic operators.

An outline of the paper is as follows:

In Section 2, we examine the influence of a small geometric perturbation on the heat transfer problem’s solution.

Section 3 is focused on the topological sensitivity analysis of the considered parabolic operator.

The application of the developed asymptotic expansion for some useful shape functions is presented in Section 4.

2. Geometric perturbation influence

Consider a heated domain $Ω \subset R^{3}$ with a smooth boundary $Σ : = \partial Ω$ . Let $H_{z, ε}$ be a small geometric perturbation created inside Ω; $H_{z, ε} = z + ε H$ , $ε \in] 0, 1 [$ and $H \subset R^{3}$ is a fixed bounded domain containing the origin. We denote by $Φ_{ε}$ the solution to the heat transfer problem in the perturbed domain $Ω ∖ {\overline{H}}_{z, ε}$ with a Dirichlet boundary condition on the boundary $\partial H_{z, ε}$ , given by $\begin{matrix} (2.1) & \{\begin{matrix} \frac{\partial Φ_{ε}}{\partial t} - div (γ (x) \nabla Φ_{ε}) = Q & in Ω ∖ {\overline{H}}_{z_{ε}} \times (0, T), \\ Φ_{ε} = φ_{D} & on Σ \times (0, T), \\ Φ_{ε} = 0 & on \partial H_{z, ε} \times (0, T), \\ Φ_{ε} (\cdot, 0) = 0 & in Ω ∖ {\overline{H}}_{z_{ε}}, \end{matrix} \end{matrix}$ where $γ \in C^{1} (\overline{Ω})$ is the thermal conductivity, $Q \in L^{2} (0, T, L^{2} (Ω))$ is the heat generated source, $φ_{D} \in L^{2} (0, T, H^{1 / 2} (Σ))$ is the boundary temperature and $T > 0$ is the computational time. We assume that there exist $γ_{\min}$ and $γ_{\max}$ such that $0 < γ_{\min} ⩽ γ (x) ⩽ γ_{\max}$ , $\forall x \in Ω$ .

For $ε = 0$ , the temperature field $Φ_{0}$ is defined in the non-perturbed domain Ω and satisfies $\begin{matrix} (2.2) & \{\begin{matrix} \frac{\partial Φ_{0}}{\partial t} - div (γ (x) \nabla Φ_{0}) = Q & in Ω \times (0, T), \\ Φ_{0} = φ_{D} & on Σ \times (0, T), \\ Φ_{0} (\cdot, 0) = 0 & in Ω . \end{matrix} \end{matrix}$ Problem (2.2) admits a unique solution $Φ_{0} \in H^{1} (0, T; H^{1} (Ω))$ (see [7]).

Next, we discuss the small geometric perturbation influence on the temperature distribution. Our aim is to derive the asymptotic behavior of the variation $ϕ_{ε} : = Φ_{ε} - Φ_{0}$ with respect to the perturbation size ε.

From (2.1) and (2.2), $ϕ_{ε}$ is solution to $\begin{matrix} (2.3) & \{\begin{matrix} \frac{\partial ϕ_{ε}}{\partial t} - div (γ (x) \nabla ϕ_{ε}) = 0 & in Ω ∖ {\overline{H}}_{z_{ε}} \times (0, T), \\ ϕ_{ε} = 0 & on Σ \times (0, T), \\ ϕ_{ε} = - Φ_{0} & on \partial H_{z, ε} \times (0, T), \\ ϕ_{ε} (\cdot, 0) = 0 & in Ω ∖ {\overline{H}}_{z_{ε}} . \end{matrix} \end{matrix}$ We will prove that the leading term of the variation $Φ_{ε} - Φ_{0}$ can be approximated by a scalar function $L$ having the form $\begin{matrix} (2.4) & L (x, t) = Φ_{0} (z, t) ξ ((x - z) / ε), \forall (x, t) \in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \end{matrix}$ where $ξ \in W^{1} (R^{3} ∖ \overline{H})$ is the solution of the following exterior problem: $Find ξ \in W^{1} (R^{3} ∖ \overline{H}) such that$ $\begin{matrix} (2.5) & \{\begin{matrix} - Δ ξ = 0 & in R^{3} ∖ \overline{H}, \\ ξ ⟶ 0 & at \infty, \\ ξ = - 1 & on \partial H . \end{matrix} \end{matrix}$ To derive the expected asymptotic behavior, we start our analysis by establishing some technical results.

2.1. Preliminary results

In this paragraph, we present some preliminary estimates which will be used in the sequel. Firstly, we introduce some notations. Let $ϱ > 0$ such that ${\overline{H}}_{z, ε} \subset B (z, ϱ)$ and $\overline{B (z, ϱ)} \subset Ω$ . We denote by $Ω_{ϱ}$ the truncated domain $Ω_{ϱ}$ defined by $Ω_{ϱ} : = Ω ∖ \overline{B (z, ϱ)}$ . Also, we will make use of the following notations: $\begin{matrix} (2.6) & Ω_{z, ε} : = Ω ∖ {\overline{H}}_{z, ε}, {\tilde{Ω}}_{z, ε} : = \frac{Ω_{z, ε} - z}{ε}, {\tilde{Ω}}_{ϱ} : = \frac{Ω_{ϱ} - z}{ε} . \end{matrix}$ The first preliminary result is summarized in following lemma. It is devoted to some estimates for the exterior problem solution.

Lemma 2.1.
The function ξ solution of ( 2.5 ) admits the following estimates: there exist constants $c > 0$ and $ε_{0} > 0$ , independent of ε, such that for all $ε \in] 0, ε_{0} [$ $\begin{array}{l} (2.7) & ‖ ξ ‖_{L^{2} ({\tilde{Ω}}_{ϱ})} ⩽ c ε^{- 1 / 2}, \\ (2.8) & ‖ \nabla ξ ‖_{L^{2} ({\tilde{Ω}}_{ϱ})} ⩽ c ε^{1 / 2}, \\ (2.9) & ‖ ξ ‖_{L^{2} ({\tilde{Ω}}_{z, ε})} ⩽ c ε^{- 1 / 2}, \\ (2.10) & ‖ \nabla ξ ‖_{L^{2} ({\tilde{Ω}}_{z, ε})} ⩽ c, \\ (2.11) & ‖ \nabla ξ ‖_{L^{p} ({\tilde{Ω}}_{z, ε})} ⩽ c ε^{\frac{2 p - 3}{p}}, \forall 0 < p < \frac{3}{2} . \end{array}$
Proof.
To prove the desired estimates, we need to recall the integral representation of the function ξ, solution to the exterior problem (2.5). In fact, using the single layer potential (see [14] or [19]), ξ can be written as $\begin{matrix} (2.12) & ξ (y) = \int_{\partial H} Γ (y - x) q (x) d s (x), \forall y \in \partial H, \end{matrix}$ where $q \in H^{- 1 / 2} (\partial H)$ is the solution of the following boundary integral equation $\begin{matrix} (2.13) & \int_{\partial H} Γ (y - x) q (x) d s (x) = - 1, \forall y \in \partial H . \end{matrix}$ Here Γ is the fundamental solution of the Laplace operator in $R^{3}$ defined by $\begin{matrix} Γ (y) = \frac{1}{4 π | y |}, \forall y \in R^{3} . \end{matrix}$

As it is explained in [19], the solution ξ can be decomposed as $ξ = ξ_{1} + ξ_{2}$ such that for a large $r = ‖ y ‖$ , we have $\begin{array}{l} | ξ_{1} (y) | ⩽ \frac{c}{r}, | ξ_{2} (y) | ⩽ \frac{c}{r^{2}}, \\ | \nabla ξ_{1} (y) | ⩽ \frac{c}{r^{2}} and | \nabla ξ_{2} (y) | ⩽ \frac{c}{r^{3}} . \end{array}$ Using the above inequalities, the estimates (2.7)–(2.10) follow straightforwardly. For more details, one can consult [19, Lemma 7.1].

To examine the term $‖ \nabla ξ ‖_{L^{p} ({\tilde{Ω}}_{z, ε})}$ , we use the fact that the domain Ω is bounded in such a way that there exists $R > 0$ such that $\overline{Ω} \subset B (z, R)$ , $\forall z \in Ω$ . Then, one can easily check that $\begin{matrix} (2.14) & {\tilde{Ω}}_{z, ε} \subset C (0, ϱ, R / ε) : = {y \in R^{2}, ϱ < ‖ y ‖ < \frac{R}{ε}} . \end{matrix}$ Using the spherical coordinates, one can deduce that for all $0 < p < \frac{3}{2}$ , we have $\begin{matrix} (2.15) & ‖ \nabla ξ ‖_{L^{p} ({\tilde{Ω}}_{z, ε})} ⩽ ‖ \nabla ξ ‖_{L^{p} (C (0, ϱ, R / ε))} ⩽ c {| \int_{ϱ}^{R / ε} r^{2 - 2 p} d r |}^{1 / p} . \end{matrix}$ Consequently, for each $ε > 0$ sufficiently small, we have $\begin{matrix} ‖ \nabla ξ ‖_{L^{p} ({\tilde{Ω}}_{z, ε})} ⩽ c {| \frac{R^{3 - 2 p}}{3 - 2 p} |}^{1 / p} {| ε^{2 p - 3} - {(\frac{ϱ}{R})}^{3 - 2 p} |}^{1 / p} = O (ε^{\frac{2 p - 3}{p}}) . \end{matrix}$ □

The second preliminary result concerns an energy estimate for the non-stationary heat problem. Lemma 2.2.
Consider $Q \in L^{2} (0, T; H^{1} (Ω))$ and $p \in] \frac{6}{5}, + \infty [$ . Let $Φ \in H^{1} (0, T; H^{1} (Ω))$ be the solution of the following parabolic problem $\begin{matrix} \{\begin{matrix} \frac{\partial Φ}{\partial t} - div (γ (x) \nabla Φ) = Q & in Ω \times (0, T), \\ Φ = 0 & on Σ \times (0, T), \\ Φ (\cdot, 0) = 0 & in Ω . \end{matrix} \end{matrix}$ There exists a constant $c > 0$ such that $\begin{matrix} ‖ Φ ‖_{L^{\infty} (0, T; L^{2} (Ω))} + ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} ⩽ c ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} . \end{matrix}$
Proof.
Using the weak formulation, one can obtain $\begin{matrix} \int_{Ω} \frac{\partial Φ}{\partial t} (x, t) Φ (x, t) d x + \int_{Ω} γ (x) {| \nabla Φ (x, t) |}^{2} d x = \int_{Ω} Q (x, t) Φ (x, t) d x . \end{matrix}$ Let $t_{0} \in [0, T]$ . Using the fact that $\frac{\partial Φ}{\partial t} (x, t) Φ (x, t) = \frac{1}{2} \frac{d}{d t} {(Φ (x, t))}^{2}$ , and $Φ (\cdot, 0) = 0$ in Ω, one can obtain $\begin{array}{l} \frac{1}{2} \int_{Ω} {| Φ (\cdot, t_{0}) |}^{2} d x + \int_{0}^{t_{0}} \int_{Ω} γ (x) {| \nabla Φ (x, t) |}^{2} d x d t & = \int_{0}^{t_{0}} \int_{Ω} Q (x, t) Φ (x, t) d x d t \\ (2.16) & ⩽ \int_{0}^{T} \int_{Ω} | Q (x, t) Φ (x, t) | d x d t . \end{array}$ Let $q \in N^{}$ such that $\frac{1}{p} + \frac{1}{q} = 1$ . Making use of Cauchy–Schwarz inequality, we get $\begin{matrix} \int_{0}^{T} \int_{Ω} | Q (x, t) Φ (x, t) | d x d t ⩽ ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} ‖ Φ ‖_{L^{2} (0, T; L^{q} (Ω))} . \end{matrix}$ From the fact that $p > \frac{6}{5}$ , it follows that $q \in] 1, 6 [$ and we have $H^{1} (Ω) ↪ L^{q} (Ω)$ . Then, $\begin{matrix} (2.17) & \frac{1}{2} \int_{Ω} {| Φ (\cdot, t_{0}) |}^{2} d x + \int_{0}^{t_{0}} \int_{Ω} γ (x) {| \nabla Φ (x, t) |}^{2} d x d t ⩽ c_{1} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} . \end{matrix}$ Next, we will examine the quantities $‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))}$ and $‖ Φ ‖_{L^{\infty} (0, T; L^{2} (Ω))}$ separately.

∙ Estimate of* $‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))}$ : Choosing $t_{0} = T$ in (2.17), one can deduce $\begin{matrix} (2.18) & γ_{\min} ‖ \nabla Φ ‖_{L^{2} (0, T; L^{2} (Ω))}^{2} ⩽ c_{1} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} . \end{matrix}$ Similarly, for each $t_{0} \in [0, T]$ , we have $\begin{matrix} (2.19) & \int_{Ω} {| Φ (\cdot, t_{0}) |}^{2} d x ⩽ 2 c_{1} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} . \end{matrix}$ Integrating between 0 and T, it follows $\begin{matrix} (2.20) & ‖ Φ ‖_{L^{2} (0, T; L^{2} (Ω))}^{2} ⩽ 2 T c_{1} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} . \end{matrix}$ Then, from (2.20) and (2.18), one can prove that there exists $c_{2} > 0$ such that $\begin{matrix} (2.21) & ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} ⩽ c_{2} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} . \end{matrix}$ Here, one can choose $c_{2} = (\frac{1}{γ_{\min}} + 2 T) c_{1}$ .

∙ Estimate of $‖ Φ ‖_{L^{\infty} (0, T; L^{2} (Ω))}$ : Based on the estimates (2.19) and (2.21), we get $\begin{matrix} \int_{Ω} {| Φ (\cdot, t_{0}) |}^{2} d x ⩽ 2 c_{1} c_{2} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))}^{2}, \forall t_{0} \in [0, T] . \end{matrix}$ Then, we have $\begin{matrix} {‖ Φ (\cdot, t_{0}) ‖}_{L^{2} (Ω)} ⩽ 2 c_{1} \sqrt{T + \frac{1}{2 γ_{\min}}} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))}, \forall t_{0} \in [0, T] . \end{matrix}$ Consequently, there exists a constant $c_{3} > 0$ such that $\begin{matrix} (2.22) & ‖ Φ ‖_{L^{\infty} (0, T; L^{2} (Ω))} ⩽ c_{3} ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} . \end{matrix}$

∙ The desired estimate: Combining (2.21) and (2.22), one can deduce that there exists a constant $c > 0$ , such that (one can choose $c = max (c_{2}, c_{3})$ ) $\begin{matrix} (2.23) & ‖ Φ ‖_{L^{\infty} (0, T; L^{2} (Ω))} + ‖ Φ ‖_{L^{2} (0, T; H^{1} (Ω))} ⩽ c ‖ Q ‖_{L^{2} (0, T; L^{p} (Ω))} . \end{matrix}$

The second estimate has been proven in [13] for the case $γ (x) = 1$ . Using the fact that $γ \in C^{1} (\overline{Ω})$ and uniformly bounded, one can adapt the same techniques. □

2.2. Asymptotic behavior of the temperature field variation

We are now ready to derive the asymptotic behavior of the variation $Φ_{ε} - Φ_{0}$ with respect to the perturbation size ε. As we have previously mentioned, the leading term of this variation will be described by function $L$ which is defined as $\begin{matrix} (2.24) & L (x, t) = Φ_{0} (z, t) ξ ((x - z) / ε), \forall (x, t) \in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \end{matrix}$ where $ξ \in W^{1} (R^{3} ∖ \overline{H})$ is the solution to (2.5). The established estimate is presented in the following theorem.

Theorem 2.3.
Let $H_{z, ε}$ be a small geometric perturbation created inside the background domain Ω with a Dirichlet boundary condition on $\partial H_{z, ε}$ . Then, there exist constants $c > 0$ and $ε_{1} > 0$ , independent of ε, such that for all $ε \in] 0, ε_{1} [$ we have $\begin{matrix} ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$
Proof.
Firstly, to simplify the presentation, we introduce the following function $\begin{matrix} D_{ε} (x, t) = Φ_{ε} (x, t) - Φ_{0} (x, t) - L (x, t), \forall (x, t) \in Ω ∖ {\overline{H}}_{z, ε} \times (0, T) . \end{matrix}$ Using the system (2.3) and (2.24), one can easily check that $D_{ε}$ is the solution of the following system $\begin{matrix} (2.25) & \{\begin{matrix} \frac{\partial D_{ε}}{\partial t} - div (γ (x) \nabla D_{ε}) = - \frac{\partial Φ_{0} (z, t)}{\partial t} ξ ((x - z) / ε) \\ - Φ_{0} (z, t) \nabla γ (x) . \nabla ξ ((x - z) / ε) & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ D_{ε} = - Φ_{0} (z, t) ξ ((x - z) / ε) & on Σ \times (0, T), \\ D_{ε} = Φ_{0} (z, t) - Φ_{0} (x, t) & on \partial H_{z, ε} \times (0, T), \\ D_{ε} (\cdot, 0) = 0 & in Ω ∖ {\overline{H}}_{z, ε} . \end{matrix} \end{matrix}$

Next, we aim to prove that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} (2.26) & ‖ D_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + ‖ D_{ε} ‖_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$ To this end, we consider the following decomposition $\begin{matrix} D_{ε} : = D_{ε}^{1} + D_{ε}^{2} + D_{ε}^{3} + D_{ε}^{4}, \end{matrix}$ where the functions $D_{ε}^{1}$ , $D_{ε}^{2}$ , $D_{ε}^{3}$ and $D_{ε}^{4}$ will be defined and estimated successively.

∙ The term $D_{ε}^{1}$ : It is chosen as the solution to the following system $\begin{matrix} (2.27) & \{\begin{matrix} \frac{\partial D_{ε}^{1}}{\partial t} - div (γ (x) \nabla D_{ε}^{1}) = 0 & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ D_{ε}^{1} = 0 & on Σ \times (0, T), \\ D_{ε}^{1} = - Φ_{0} (x, t) + Φ_{0} (z, t) & on \partial H_{z, ε} \times (0, T), \\ D_{ε}^{1} (\cdot, 0) = 0 & in Ω ∖ {\overline{H}}_{z, ε} . \end{matrix} \end{matrix}$ Let $\tilde{D_{ε}^{1}}$ be an extension of $D_{ε}^{1}$ in Ω (with continuity condition on $\partial H_{z, ε} \times (0, T)$ ), defined by $\begin{matrix} (2.28) & \tilde{D_{ε}^{1}} (x, t) = \{\begin{matrix} D_{ε}^{1} (x, t) & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ - Φ_{0} (x, t) + Φ_{0} (z, t) & in H_{z, ε} \times (0, T) . \end{matrix} \end{matrix}$ The smoothness of $D_{ε}^{1}$ and $Φ_{0}$ implies that $\tilde{D_{ε}^{1}} \in H^{1} (0, T; H^{1} (Ω))$ and satisfies the following system $\begin{matrix} (2.29) & \{\begin{matrix} \frac{\partial \tilde{D_{ε}^{1}}}{\partial t} - div (γ (x) \nabla \tilde{D_{ε}^{1}}) = Q_{ε}^{1} & in Ω \times (0, T), \\ \tilde{D_{ε}^{1}} = 0 & on Σ \times (0, T), \\ \tilde{D_{ε}^{1}} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ with $Q_{ε}^{1} (x, t) = [Q (x, t) - \frac{\partial Φ_{0} (z, t)}{\partial t}] χ_{H_{z, ε}} (x)$ .

By a standard energy estimate for the parabolic heat problem, we derive that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} {‖ \tilde{D_{ε}^{1}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{1}} ‖}_{L^{2} (0, T; H^{1} (Ω))} ⩽ c {‖ Q_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω))} . \end{matrix}$ Using the smoothness of $Φ_{0}$ and the Cauchy–Schwarz inequality, the term $Q_{ε}^{1}$ admits the following estimate $\begin{matrix} {‖ Q_{ε}^{1} ‖}_{L^{2} (0, T; L^{2} (Ω))} ⩽ | H_{z, ε} |^{1 / 2} {‖ Q (x, t) ‖}_{L^{2} (0, T; L^{\infty} (Ω))} + | H_{z, ε} |^{1 / 2} {‖ \frac{\partial Φ_{0} (z, t)}{\partial t} ‖}_{L^{2} (0, T)}, \end{matrix}$ where $| H_{z, ε} | = | H | ε^{3}$ denotes the Lebesgue measure of $H_{z, ε}$ in $R^{3}$ . Then, one has $\begin{matrix} {‖ \tilde{D_{ε}^{1}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{1}} ‖}_{L^{2} (0, T; H^{1} (Ω))} = o (ε) . \end{matrix}$ Since $\tilde{D_{ε}^{1}} = D_{ε}^{1}$ in $Ω ∖ {\overline{H}}_{z, ε}$ , one can conclude that $\begin{matrix} (2.30) & {‖ D_{ε}^{1} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + {‖ D_{ε}^{1} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} = o (ε) . \end{matrix}$

∙ The term $D_{ε}^{2}$ : It is defined as $\begin{matrix} (2.31) & D_{ε}^{2} (x, t) = T_{ε} (x) Φ_{0} (z, t), \forall (x, t) \in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \end{matrix}$ where the function $T_{ε}$ is chosen as the solution of the following Laplace problem $\begin{matrix} (2.32) & \{\begin{matrix} - div (γ (x) \nabla T_{ε}) = 0 & in Ω ∖ {\overline{H}}_{z, ε}, \\ T_{ε} = ξ ((x - z) / ε) & on Σ, \\ T_{ε} = 0 & on \partial H_{z, ε} . \end{matrix} \end{matrix}$ Using the elliptic energy estimate, one can deduce that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} (2.33) & ‖ T_{ε} ‖_{H^{1} (Ω ∖ {\overline{H}}_{z, ε})} ⩽ c {‖ ξ ((x - z) / ε) ‖}_{H^{1 / 2} (Σ)} . \end{matrix}$ Using (2.31), one can derive $\begin{matrix} (2.34) & {‖ D_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + {‖ D_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c {‖ Φ_{0} (z, t) ‖}_{L^{2} (0, T)} {‖ ξ ((x - z) / ε) ‖}_{H^{1 / 2} (Σ)} . \end{matrix}$ The trace theorem implies $\begin{matrix} (2.35) & {‖ ξ ((x - z) / ε) ‖}_{L^{2} (0, T; H^{1 / 2} (Σ))} ⩽ c {‖ ξ ((x - z) / ε) ‖}_{H^{1} (Ω_{ϱ})} . \end{matrix}$ By the change of variables $x = z + ε y$ and Lemma 2.1, one can derive $\begin{array}{l} (2.36) & {‖ ξ ((x - z) / ε) ‖}_{L^{2} (Ω_{z, ε})} ⩽ ε^{3 / 2} ‖ ξ ‖_{L^{2} ({\tilde{Ω}}_{z, ε})} ⩽ c ε, \\ (2.37) & {‖ ξ ((x - z) / ε) ‖}_{H^{1} (Ω_{ϱ})} ⩽ ε^{3 / 2} ‖ ξ ‖_{L^{2} ({\tilde{Ω}}_{ϱ})} + ε^{1 / 2} ‖ \nabla_{y} ξ ‖_{L^{2} ({\tilde{Ω}}_{ϱ})} ⩽ c ε . \end{array}$ Then, one can deduce that there exists $c > 0$ , independent of ε, such that $\begin{matrix} (2.38) & {‖ D_{ε}^{2} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + {‖ D_{ε}^{2} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$

∙ The term $D_{ε}^{3}$ : The function $D_{ε}^{3}$ is defined as the solution of the following system $\begin{matrix} (2.39) & \{\begin{matrix} \frac{\partial D_{ε}^{3}}{\partial t} - div (γ (x) \nabla D_{ε}^{3}) = Φ_{0} (z, t) \nabla γ (x) . \nabla Γ (x - z) & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ D_{ε}^{3} = 0 & on Σ \times (0, T), \\ D_{ε}^{3} = 0 & on \partial H_{z, ε} \times (0, T), \\ D_{ε}^{3} (\cdot, 0) = 0 & in Ω ∖ {\overline{H}}_{z, ε} . \end{matrix} \end{matrix}$ Let $\tilde{D_{ε}^{3}}$ be an extension by zero of $D_{ε}^{3}$ in $H_{z, ε}$ i.e. $\begin{matrix} (2.40) & \tilde{D_{ε}^{3}} = \{\begin{matrix} D_{ε}^{3} & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ 0 & in H_{z, ε} \times (0, T) . \end{matrix} \end{matrix}$ Using the continuity condition $D_{ε}^{3} = 0$ on $\partial H_{z, ε} \times (0, T)$ , the function $\tilde{D_{ε}^{3}}$ belongs to $H^{1} (0, T; H^{1} (Ω))$ and satisfies the system $\begin{matrix} (2.41) & \{\begin{matrix} \frac{\partial \tilde{D_{ε}^{3}}}{\partial t} - div (γ (x) \nabla \tilde{D_{ε}^{3}}) = Q_{ε}^{3} (x, t) & in Ω \times (0, T), \\ \tilde{D_{ε}^{3}} = 0 & on Σ \times (0, T), \\ \tilde{D_{ε}^{3}} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ with $Q_{ε}^{3} (x, t) = Φ_{0} (z, t) \nabla γ (x) . \nabla ξ ((x - z) / ε) χ_{Ω ∖ {\overline{H}}_{z, ε}} (x)$ .

Applying Lemma 2.2, one can derive that there exists a constant $c > 0$ , independent of ε, such that $\begin{array}{l} {‖ \tilde{D_{ε}^{3}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{3}} ‖}_{L^{2} (0, T; H^{1} (Ω))} \\ ⩽ c {‖ Φ_{0} (z, t) \nabla γ (x) . \nabla ξ ((x - z) / ε) ‖}_{L^{2} (0, T; L^{p} (Ω))}, \forall p \in] 6 / 5, 3 / 2 [. \end{array}$ Since γ and $Φ_{0}$ are sufficiently smooth, it follows $\begin{matrix} (2.42) & {‖ \tilde{D_{ε}^{3}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{3}} ‖}_{L^{2} (0, T; H^{1} (Ω))} ⩽ c {‖ \nabla ξ ((x - z) / ε) ‖}_{L^{p} (Ω ∖ {\overline{H}}_{z, ε})} . \end{matrix}$ By a change of variable, one has $\begin{matrix} (2.43) & {‖ \nabla ξ ((x - z) / ε) ‖}_{L^{p} (Ω ∖ {\overline{H}}_{z, ε})} ⩽ c ε^{\frac{3}{p} - 1} ‖ \nabla ξ ‖_{L^{p} ({\tilde{Ω}}_{z, ε})} . \end{matrix}$ Using Lemma 2.1, we obtain $\begin{matrix} {‖ \tilde{D_{ε}^{3}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{3}} ‖}_{L^{2} (0, T; H^{1} (Ω))} ⩽ c ε . \end{matrix}$ Since $D_{ε}^{3} = \tilde{D_{ε}^{3}}$ in $Ω ∖ {\overline{H}}_{z, ε}$ , we deduce $\begin{matrix} (2.44) & {‖ D_{ε}^{3} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + {‖ D_{ε}^{3} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$

∙ The term $D_{ε}^{4}$ : It is chosen such that $D_{ε}^{4} = D_{ε} - D_{ε}^{1} - D_{ε}^{2} - D_{ε}^{3}$ . From (2.25), (2.27), (2.31) and (2.39), it follows that the term $D_{ε}^{4} \in H^{1} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))$ and satisfies the following system $\begin{matrix} (2.45) & \{\begin{matrix} \frac{\partial D_{ε}^{4}}{\partial t} - div (γ (x) \nabla D_{ε}^{4}) = \frac{\partial Φ_{0}}{\partial t} (z, t) [T_{ε} (x) - ξ ((x - z) / ε)] & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ D_{ε}^{4} = 0 & on Σ \times (0, T), \\ D_{ε}^{4} = 0 & on \partial H_{z, ε} \times (0, T), \\ D_{ε}^{4} (\cdot, 0) = 0 & in Ω ∖ {\overline{H}}_{z, ε} . \end{matrix} \end{matrix}$ Similarly to the previous term, we define the extension of $D_{ε}^{4}$ by zero in $H_{z, ε}$ $\begin{matrix} (2.46) & \tilde{D_{ε}^{4}} = \{\begin{matrix} D_{ε}^{4} & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ 0 & in H_{z, ε} \times (0, T) . \end{matrix} \end{matrix}$ Using the continuity condition $D_{ε}^{4} = 0$ on $\partial H_{z, ε} \times (0, T)$ , the function $\tilde{D_{ε}^{4}}$ belongs to $H^{1} (0, T; H^{1} (Ω))$ and satisfies $\begin{matrix} (2.47) & \{\begin{matrix} \frac{\partial \tilde{D_{ε}^{4}}}{\partial t} - div (γ (x) \nabla \tilde{D_{ε}^{4}}) = Q_{ε}^{4} & in Ω \times (0, T), \\ \tilde{D_{ε}^{4}} = 0 & on Σ \times (0, T), \\ \tilde{D_{ε}^{4}} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ where the source term is defined by $\begin{matrix} Q_{ε}^{4} (x, t) = \frac{\partial Φ_{0}}{\partial t} (z, t) [T_{ε} (x) - ξ ((x - z) / ε)] χ_{Ω ∖ {\overline{H}}_{z, ε}} (x), \forall (x, t) \in Ω \times (0, T) . \end{matrix}$ Besides, one has $\begin{matrix} {‖ \tilde{D_{ε}^{4}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{4}} ‖}_{L^{2} (0, T; H^{1} (Ω))} ⩽ {‖ Q_{ε}^{4} ‖}_{L^{2} (0, T; L^{2} (Ω))} . \end{matrix}$ Using the fact that $\frac{\partial Φ_{0}}{\partial t} (z, t)$ is sufficiently smooth along with the triangle inequality, one obtain $\begin{matrix} (2.48) & {‖ Q_{ε}^{4} ‖}_{L^{2} (0, T; L^{2} (Ω))} ⩽ c ({‖ T_{ε} (x) ‖}_{L^{2} (Ω ∖ {\overline{H}}_{z, ε})} + {‖ ξ ((x - z) / ε) ‖}_{L^{2} (Ω ∖ {\overline{H}}_{z, ε})}) . \end{matrix}$ Using (2.33) and a change of variable, we have $\begin{matrix} (2.49) & {‖ \tilde{D_{ε}^{4}} ‖}_{L^{\infty} (0, T; L^{2} (Ω))} + {‖ \tilde{D_{ε}^{4}} ‖}_{L^{2} (0, T; H^{1} (Ω))} ⩽ c ε . \end{matrix}$ Since $D_{ε}^{4}$ coincides with $\tilde{D_{ε}^{4}}$ in $Ω ∖ {\overline{H}}_{z, ε}$ , we deduce $\begin{matrix} (2.50) & {‖ D_{ε}^{4} ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} + {‖ D_{ε}^{4} ‖}_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$

∙ The desired estimate: Combining the estimates (2.30), (2.38), (2.44) and (2.50), one can deduce that there exists $c > 0$ , independent of ε, such that the function $D_{ε} = Φ_{ε} - Φ_{0} - L$ satisfies the following estimate $\begin{matrix} ‖ D_{ε} ‖_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{H_{z, ε}}))} + ‖ D_{ε} ‖_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$ □

In the case of a spherical geometric perturbation ( $H = B (0, 1)$ is the unit ball), the solution ξ of the exterior problem (2.5) admits an explicit expression, $\begin{matrix} ξ (y) = - 4 π Γ (y), \forall y \in R^{3} ∖ \overline{B (0, 1)} . \end{matrix}$ In this particular case, the perturbed temperature distribution admits the estimate $\begin{array}{l} {‖ Φ_{ε} - Φ_{0} + 4 π ε Φ_{0} (z, t) Γ (x - z) ‖}_{L^{\infty} (0, T; L^{2} (Ω ∖ \overline{H_{z, ε}}))} \\ + {‖ Φ_{ε} - Φ_{0} + 4 π ε Φ_{0} (z, t) Γ (x - z) ‖}_{L^{2} (0, T, H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{array}$
3. Topological sensitivity analysis

This section is devoted to the topological sensitivity analysis for the considered parabolic problem. We will establish an asymptotic expansion of the form $\begin{matrix} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) = f (ε) T (z) + o (f (ε)), \end{matrix}$ verified by all shape function $G$ defined by $\begin{matrix} G (Ω ∖ {\overline{H}}_{z, ε}) = \int_{0}^{T} g_{ε} (Φ_{ε} (\cdot, t)) d t, \end{matrix}$ with $Φ_{ε}$ is the solution to (2.1) and $g_{ε}$ is a cost function defined on $H^{1} (Ω ∖ {\overline{H}}_{z, ε})$ and satisfying the following assumptions:

Assumption ( $A$ ).

$\forall ε ⩾ 0$ such that $\overline{H_{z, ε}} \subset Ω$ , $g_{ε} (Φ_{ε} (\cdot, t)) \in L^{1} (0, T)$ .

$g_{0}$ is differentiable at $Φ_{0} (\cdot, t)$ , $\forall t \in (0, T)$ and we denote $D g_{0} (Φ_{0} (\cdot, t))$ its derivative.

There exists $δ G : Ω \to R$ and $f : R_{+} \to R_{+}$ in such away $\begin{array}{l} \int_{0}^{T} [g_{ε} (Φ_{ε} (\cdot, t)) - g_{0} (Φ_{0} (\cdot, t))] d t \\ (3.1) & = \int_{0}^{T} D g_{0} (Φ_{0} (\cdot, t)) (Φ_{ε} - Φ_{0}) (\cdot, t) d t + f (ε) δ G (z) + o (f (ε)), \end{array}$ with ${lim}_{ε \to 0} f (ε) = 0$ .

In the last expansion, $Φ_{ε}$ is extended by zero inside the domain $H_{z, ε}$ . Inspired by the asymptotic behavior of the temperature variation (see Theorem 2.3), the positive function f is given by $f (ε) = ε$ .
3.1. Preliminaries

Using the assumption ( A ) , the variation of the shape function $G$ reads $\begin{matrix} (3.2) & G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) = \int_{0}^{T} D g_{0} (Φ_{0}) (Φ_{ε} - Φ_{0}) d t + f (ε) δ G (z) + o (f (ε)) . \end{matrix}$ To examine the integral term, we introduce the adjoint state $v_{0}$ solution of the following boundary value problem $\begin{matrix} (3.3) & \{\begin{matrix} - \frac{\partial v_{0}}{\partial t} - div (γ (x) \nabla v_{0}) = - D g_{0} (Φ_{0}) & in Ω \times (0, T), \\ v_{0} = 0 & on Σ \times (0, T), \\ v_{0} (\cdot, T) = 0 & in Ω . \end{matrix} \end{matrix}$ The weak formulation of the problem (3.3) implies that the shape function variation can be rewritten as follows $\begin{array}{l} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = - \int_{0}^{T} \int_{Ω} \frac{\partial (Φ_{ε} - Φ_{0})}{\partial t} v_{0} d x d t - \int_{0}^{T} \int_{Ω} γ (x) \nabla (Φ_{ε} - Φ_{0}) . \nabla v_{0} d x d t \\ + f (ε) δ G (z) + o (f (ε)) . \end{array}$ Applying Green formula to the system (2.3), we get $\begin{array}{l} (3.4) & G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = \int_{0}^{T} \int_{H_{z, ε}} \frac{\partial Φ_{0}}{\partial t} v_{0} d x d t + \int_{0}^{T} \int_{H_{z, ε}} γ (x) \nabla Φ_{0} . \nabla v_{0} d x d t \\ - \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n v_{0} d s (x) d t \\ (3.5) & + f (ε) δ G (z) + o (f (ε)) . \end{array}$ In order to derive the expected asymptotic expansion, we need to study the behavior of the integral terms with respect to ε.

Lemma 3.1.
Let $H_{z, ε}$ be a small geometric perturbation inserted inside the domain Ω. The integral terms in ( 3.4 ) satisfy the estimate $\begin{matrix} \int_{0}^{T} \int_{H_{z, ε}} \frac{\partial Φ_{0}}{\partial t} v_{0} d x d t + \int_{0}^{T} \int_{H_{z, ε}} γ (x) \nabla Φ_{0} \nabla v_{0} d x d t = O (ε^{3}) . \end{matrix}$
Proof.
Using the change of variables $x = z + ε y$ , the first integral term in (3.4) can be written as $\begin{array}{l} \int_{0}^{T} \int_{H_{z, ε}} \frac{\partial Φ_{0}}{\partial t} v_{0} d x d t & = ε^{3} \int_{0}^{T} \int_{H} (\frac{\partial Φ_{0}}{\partial t} (z + ε y, t) v_{0} (z + ε y, t) - \frac{\partial Φ_{0}}{\partial t} (z, t) v_{0} (z, t)) d y d t \\ + ε^{3} | H | \int_{0}^{T} \frac{\partial Φ_{0}}{\partial t} (z, t) v_{0} (z, t) d t, \end{array}$ where $| H |$ denotes the Lebesgue measure of the domain $H$ .

From the fact that $Φ_{0}$ and $v_{0}$ are smooth near z, one can derive $\begin{matrix} \int_{0}^{T} \int_{H_{z, ε}} \frac{\partial Φ_{0}}{\partial t} v_{0} d x d t = ε^{3} | H | \int_{0}^{T} \frac{\partial Φ_{0}}{\partial t} (z, t) v_{0} (z, t) d t + o (ε^{3}) = O (ε^{3}) . \end{matrix}$ The second integral term in (3.4) can be estimated using the same arguments. Therefore, the smoothness of $Φ_{0}$ and $v_{0}$ leads to $\begin{matrix} \int_{0}^{T} \int_{H_{z, ε}} γ (x) \nabla Φ_{0} \nabla v_{0} d x d t = O (ε^{3}), \end{matrix}$ which achieves the proof. □

Furthermore, to examine the integral term in (3.5), we will use the established estimate for the perturbed solution. The following lemma gives a preliminary approximation for the considered term. Lemma 3.2.
The integral term in ( 3.5 ) admits the estimate $\begin{array}{l} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n v_{0} d s (x) d t \\ (3.6) & = \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L (x, t) . n v_{0} (x, t) d s (x) d t + o (ε) . \end{array}$
Proof.
The integral term reads $\begin{array}{l} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n v_{0} d s (x) d t \\ (3.7) & = \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0} - L) . n v_{0} d s (x) d t + \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L . n v_{0} d s (x) d t . \end{array}$ The divergence formula gives $\begin{array}{l} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) | \nabla (Φ_{ε} - Φ_{0} - L) . n v_{0} | d s (x) d t \\ ⩽ c ‖ v_{0} ‖_{L^{2} (0, T; H^{1} (H_{z, ε}))} [‖ Φ_{ε} - Φ_{0} - L ‖_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} \\ + {‖ div (γ (x) \nabla (Φ_{ε} - Φ_{0} - L)) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}] . \end{array}$ The smoothness of the function $v_{0}$ near z implies $\begin{matrix} (3.8) & ‖ v_{0} ‖_{L^{2} (0, T; H^{1} (H_{z, ε}))} = O (ε^{3 / 2}) . \end{matrix}$ From Proposition 2.3, one has $\begin{matrix} (3.9) & ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε . \end{matrix}$ Using the triangle inequality, it follows $\begin{array}{l} {‖ div (γ (x) \nabla (Φ_{ε} - Φ_{0} - L)) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} & ⩽ {‖ div (γ (x) \nabla (Φ_{ε} - Φ_{0})) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} \\ + {‖ div (γ (x) \nabla L) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} . \end{array}$ Using (2.24), the smoothness of γ and the fact that $Δ L = 0$ in $Ω ∖ {\overline{H}}_{z, ε}$ , one gets $\begin{matrix} {‖ div (γ (x) \nabla L) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ‖ \nabla L ‖_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} . \end{matrix}$ Lemma 2.1 and a change of variables imply that there exists a constant $c > 0$ , independent of ε, such that $\begin{matrix} {‖ div (γ (x) \nabla L) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε^{1 / 2} ‖ \nabla ξ ‖_{L^{2} ({\tilde{Ω}}_{z, ε})} . \end{matrix}$ Besides, since $ϕ_{ε} = Φ_{ε} - Φ_{0}$ satisfies (2.3), one can define $\tilde{ϕ_{ε}}$ the extension of $ϕ_{ε}$ in $H_{z, ε}$ , i.e. $\begin{matrix} (3.10) & \tilde{ϕ_{ε}} = \{\begin{matrix} ϕ_{ε} & in Ω ∖ {\overline{H}}_{z, ε} \times (0, T), \\ - Φ_{0} & in H_{z, ε} \times (0, T) . \end{matrix} \end{matrix}$ Using a continuity boundary condition on $\partial H_{z, ε} \times (0, T)$ , the function $\tilde{ϕ_{ε}}$ belongs $H^{1} (0, T; H^{1} (Ω))$ and satisfies $\begin{matrix} \{\begin{matrix} \frac{\partial \tilde{ϕ_{ε}}}{\partial t} - div (γ (x) \nabla \tilde{ϕ_{ε}}) = Q_{ε} & in Ω \times (0, T), \\ \tilde{ϕ_{ε}} = 0 & on Σ \times (0, T), \\ \tilde{ϕ_{ε}} (\cdot, 0) = 0 & in Ω, \end{matrix} \end{matrix}$ where the source term is defined by $Q_{ε} (x, t) = Q (x, t) χ_{H_{z, ε}} (x)$ . Using Lemma 2.2, one has $\begin{matrix} {‖ div (γ (x) \nabla \tilde{ϕ_{ε}}) ‖}_{L^{2} (0, T; L^{2} (Ω))} ⩽ c {‖ Q_{ε} (x, t) ‖}_{L^{2} (0, T; L^{2} (Ω))} . \end{matrix}$ Since $\tilde{ϕ_{ε}}$ and $ϕ_{ε}$ coincides in $Ω ∖ {\overline{H}}_{z, ε}$ , we get $\begin{matrix} {‖ div (γ (x) \nabla ϕ_{ε}) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c | H_{z, ε} |^{1 / 2} {‖ Q (x, t) ‖}_{L^{2} (0, T; L^{\infty} (Ω))} . \end{matrix}$ Then, one can deduce that $\begin{matrix} (3.11) & {‖ div (γ (x) \nabla (Φ_{ε} - Φ_{0} - L)) ‖}_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))} ⩽ c ε^{3 / 2} . \end{matrix}$ Combining (3.8), (3.9) and (3.11), one can conclude that $\begin{matrix} (3.12) & \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0} - L) . n v_{0} d s (x) d t = o (ε), \end{matrix}$ which achieves the proof. □

3.2. Topological asymptotic expansion

In order to compute the topological asymptotic expansion, we introduce the integral representation of the function ξ. In fact, using a single potential layer, the solution ξ of the exterior problem (2.5) can be written as (see [14]), $\begin{matrix} ξ (y) = \int_{\partial H} Γ (y - x) q (x) d s (x), \forall y \in \partial H, \end{matrix}$ where $q \in H^{- 1 / 2} (\partial H)$ is the solution of the following boundary integral equation $\begin{matrix} (3.13) & \int_{\partial H} Γ (y - x) q (x) d s (x) = - 1, \forall y \in \partial H . \end{matrix}$ We are now ready to present the main result of this section. We will derive a topological asymptotic expansion for parabolic operator. From the established preliminary estimate in Lemma 3.2, one can deduce that the term $\int_{0}^{T} D g_{0} (Φ_{0}) (Φ_{ε} - Φ_{0}) d t$ admits the following asymptotic behavior $\begin{matrix} (3.14) & \int_{0}^{T} D g_{0} (Φ_{0}) (Φ_{ε} - Φ_{0}) d t = ε \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L (x, t) . n v_{0} (x, t) d s (x) d t + o (ε) . \end{matrix}$

In the following theorem, we derive a topological asymptotic expansion valid for all objective function $g_{ε}$ satisfying the assumption ( A ) with $f (ε) = ε$ and arbitrary shaped perturbations.

Theorem 3.3.
Let $H_{z, ε} = z + ε H$ be a small geometric perturbation inserted in the background domain Ω. If the function $g_{ε}$ satisfies the assumption ( A ) with $f (ε) = ε$ , then the shape function $G$ admits the following asymptotic expansion $\begin{matrix} (3.15) & G (Ω ∖ {\overline{H}}_{z, ε}) = G (Ω) + ε T (z) + o (ε), \end{matrix}$ where $T$ is the topological sensitivity function, defined in Ω by $\begin{matrix} T (x) = - γ (x) \int_{0}^{T} Φ_{0} (x, t) v_{0} (x, t) d t [\int_{\partial H} q (y) d s (y)] + δ G (x), \forall x \in Ω . \end{matrix}$
Proof.
Using (3.2) with $f (ε) = ε$ and Lemma 3.2, the shape function variation reads $\begin{matrix} (3.16) & G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) = - \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L (x, t) . n v_{0} (x, t) d s (x) d t + ε δ G (z) + o (ε) . \end{matrix}$ To derive the expected asymptotic expansion, we examine the integral term $\begin{array}{l} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L . n v_{0} d s (x) d t & = \int_{0}^{T} \int_{\partial H_{z, ε}} γ (z) \nabla L . n v_{0} (z, t) d s (x) d t \\ + \int_{0}^{T} \int_{\partial H_{z, ε}} \nabla L . n [γ (x) v_{0} (x, t) - γ (z) v_{0} (z, t)] d s (x) d t . \end{array}$ Using Cauchy–Schwarz’s inequality, trace theorem and the smoothness of γ, we get $\begin{array}{l} | \int_{0}^{T} \int_{\partial H_{z, ε}} \nabla L . n [γ (x) v_{0} (x, t) - γ (z) v_{0} (z, t)] d s (x) d t | \\ ⩽ c ‖ \nabla L . n ‖_{L^{2} (0, T; H^{- 1 / 2} (Ω ∖ {\overline{H}}_{z, ε}))} {‖ γ (x) v_{0} (x, t) - γ (z) v_{0} (z, t) ‖}_{L^{2} (0, T; H^{1} (H_{z, ε}))} . \end{array}$ Using the change of variable $x = z + ε y$ and the fact that $Δ L = 0$ in $Ω ∖ {\overline{H}}_{z, ε}$ , one has $\begin{matrix} ‖ \nabla L . n ‖_{L^{2} (0, T; H^{- 1 / 2} (\partial H_{z, ε}))} ⩽ c (ε^{3 / 2} ‖ ξ ‖_{L^{2} ({\tilde{Ω}}_{z, ε})} + ε^{1 / 2} ‖ \nabla_{y} ξ ‖_{L^{2} ({\tilde{Ω}}_{z, ε})}) . \end{matrix}$ Combining the estimates (2.9) and (2.10), one can derive $\begin{matrix} ‖ \nabla L . n ‖_{L^{2} (0, T; H^{- 1 / 2} (\partial H_{z, ε}))} ⩽ c ε . \end{matrix}$ The smoothness of the functions $v_{0}$ and γ implies $\begin{matrix} {‖ γ (x) v_{0} (x, t) - γ (z) v_{0} (z, t) ‖}_{L^{2} (0, T; H^{1} (H_{z, ε}))} ⩽ c ε^{1 / 2} . \end{matrix}$ Then, it follows $\begin{matrix} \int_{0}^{T} \int_{\partial H_{z, ε}} \nabla L . n [γ (x) v_{0} (x, t) - γ (z) v_{0} (z, t)] d s (x) d t = o (ε) . \end{matrix}$ Consequently, one can conclude $\begin{matrix} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L . n v_{0} d s (x) d t = ε γ (z) (- \int_{\partial H} \nabla_{y} ξ (y) . n d s (y)) \int_{0}^{T} Φ_{0} (z, t) v_{0} (z, t) d t + o (ε) . \end{matrix}$ The last estimate is established using the relation (2.24) and the change of variable $x = z + ε y$ . Then, the variation reads $\begin{matrix} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) = ε γ (z) (- \int_{\partial H} \nabla_{y} ξ (y) . n d s (y)) \int_{0}^{T} Φ_{0} (z, t) v_{0} (z, t) d t + ε δ G (z) + o (ε) . \end{matrix}$ In the last expansion, the function ξ, which is the solution of the exterior problem (2.5), admits the following integral representation $\begin{matrix} ξ (y) = \int_{\partial H} Γ (x - y) q (x) d s (x), \forall y \in R^{3} ∖ \overline{H}, \end{matrix}$ where $q \in H^{- 1 / 2} (\partial H)$ is the associated density solution of the integral equation $\begin{matrix} \int_{\partial H} Γ (x - y) q (x) d s (x) = - 1, \forall y \in \partial H . \end{matrix}$ Using the jump condition of the single layer potential (see [12]), the density q can be characterized as $\begin{matrix} q = {(\nabla ξ . n - \nabla h^{ξ} . n)}_{| \partial H}, \end{matrix}$ where $h^{ξ}$ is an extension of ξ in the domain $H$ satisfying the following interior problem $\begin{matrix} (3.17) & \{\begin{matrix} - Δ h^{ξ} = 0 & in H, \\ h^{ξ} = ξ & on \partial H . \end{matrix} \end{matrix}$ From the uniqueness result and the fact that $ξ = - 1$ on $\partial H$ , one can easily check that $\begin{matrix} h^{ξ} (y) = - 1, \forall y \in H . \end{matrix}$ Then, it follows that $\begin{matrix} q (y) = \nabla ξ (y) . n, \forall y \in \partial H . \end{matrix}$ The last relation leads to the desired asymptotic expansion $\begin{matrix} G (Ω ∖ {\overline{H}}_{z, ε}) = G (Ω) + ε T (z) + o (ε), \end{matrix}$ with $\begin{matrix} T (x) = - γ (x) \int_{0}^{T} Φ_{0} (x, t) v_{0} (x, t) d t [\int_{\partial H} q (y) d s (y)] + δ G (x), \forall x \in Ω . \end{matrix}$ □

As one can remark that the expression of the topological sensitivity function $T$ depends on the shape (described by $H$ ) of the geometric perturbation. In the following corollary, we consider the case of a spherical geometric perturbation (i.e. $H = B (0, 1)$ ). In this particular case, the obtained sensitivity function admits an explicit expression.
Corollary 3.4.
In the case of a spherical perturbation, the topological sensitivity function $T$ is given by $\begin{matrix} (3.18) & T (x) = 4 π γ (x) \int_{0}^{T} Φ_{0} (x, t) v_{0} (x, t) d t + δ G (x), \forall x \in Ω . \end{matrix}$
Proof.
If $H$ is the unit ball centred at the origin, $H = B (0, 1)$ , the solution ξ can be calculated explicitly and one can easily show that $\begin{matrix} \int_{\partial H} q (y) d s (y) = - 4 π . \end{matrix}$ □

The expression of the scalar function $δ G$ depends on $g_{ε}$ . In the next section, we will present some standard examples.
4. Objective function examples

We present some useful examples of objective functions and we calculate their variations $δ G$ .

4.1. Example 1

We consider the objective function $g_{ε}$ defined by $\begin{matrix} (4.1) & g_{ε} (Φ_{ε}) = \int_{Σ} | Φ_{ε} - Φ_{mes} |^{2} d s, \forall Φ_{ε} \in H^{1} (Ω ∖ {\overline{H}}_{z, ε}), \end{matrix}$ where $Φ_{mes} \in L^{2} (0, T; L^{2} (\partial Ω))$ is a given measured datum on Σ.

Proposition 4.1.
The objective function ( 4.1 ) verifies Assumption ( $A$ ) with $\begin{array}{l} D g_{0} (Φ_{0} (\cdot, t)) h = 2 \int_{Σ} [Φ_{0} (\cdot, t) - Φ_{mes} (\cdot, t)] h d s, \forall h \in H^{1} (Ω), \\ δ G (x) = 0, \forall x \in Ω . \end{array}$
Proof.
From the fact that $Φ_{mes} \in L^{2} (0, T, H^{1 / 2} (Σ))$ and $Φ_{ε} \in L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))$ , it follows that the function $t \mapsto g_{ε} (Φ_{ε} (\cdot, t))$ belongs to $L^{1} (0, T)$ . Moreover, one can easily check that the function $g_{0}$ is differentiable at $Φ_{0} (\cdot, t)$ and we have $\begin{matrix} D g_{0} (Φ_{0} (\cdot, t)) h = 2 \int_{Σ} (Φ_{0} (\cdot, t) - Φ_{mes} (\cdot, t)) h d x, \forall h \in H^{1} (Ω) . \end{matrix}$ The variation of the shape function $G$ with respect to the presence of the perturbation $H_{z, ε}$ in Ω reads $\begin{array}{l} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = \int_{0}^{T} \int_{Σ} | Φ_{ε} - Φ_{mes} |^{2} v d s d t - \int_{0}^{T} \int_{Σ} | Φ_{0} - Φ_{mes} |^{2} d t \\ = \int_{0}^{T} D g_{0} (Φ_{0} (\cdot, t)) (Φ_{ε} - Φ_{0}) d t + \int_{0}^{T} \int_{Σ} | Φ_{ε} - Φ_{0} |^{2} d t . \end{array}$ Due to trace theorem and the fact that $Ω_{ϱ} \subset Ω ∖ {\overline{H}}_{z, ε}$ , we get $\begin{matrix} \int_{0}^{T} \int_{Σ} | Φ_{ε} - Φ_{0} |^{2} d s (x) d t ⩽ c ‖ Φ_{ε} - Φ_{0} ‖_{L^{2} (0, T, H^{1} (Ω_{ϱ}))}^{2} . \end{matrix}$ By the triangle inequality and the identity ${(a + b)}^{2} ⩽ 2 (a^{2} + b^{2})$ , we have $\begin{matrix} \int_{0}^{T} \int_{Σ} | Φ_{ε} - Φ_{0} |^{2} d s (x) d t ⩽ 2 (‖ Φ_{ε} - Φ_{0} - L ‖_{L^{2} (0, T, H^{1} (Ω_{ϱ}))}^{2} + ‖ L ‖_{L^{2} (0, T, H^{1} (Ω_{ϱ}))}^{2}) . \end{matrix}$ From the definition (2.24) and the estimates (2.7) and (2.8), we get $\begin{matrix} ‖ L ‖_{L^{2} (0, T, H^{1} (Ω_{ϱ}))}^{2} = {‖ Φ_{0} (z, t) ‖}_{L^{2} (0, T)}^{2} {‖ ξ ((x - z) / ε) ‖}_{H^{1} (Ω_{ϱ})}^{2} = o (ε) . \end{matrix}$ Besides, using the fact that $Ω_{ϱ} \subset Ω ∖ {\overline{H}}_{z, ε}$ and Proposition 2.3, it follows $\begin{matrix} ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{2} (0, T, H^{1} (Ω_{ϱ}))}^{2} = o (ε) . \end{matrix}$ Then, we deduce that $\begin{matrix} \int_{0}^{T} \int_{Σ} | Φ_{ε} - Φ_{0} |^{2} d s (x) d t = o (ε), \end{matrix}$ which implies that the chosen function $g_{ε}$ satisfies Assumption ( A ) with $\begin{matrix} δ G (x) = 0, \forall x \in Ω . \end{matrix}$ □

4.2. Example 2

We consider an objective function, already used in geometric control problems, defined by $\begin{matrix} (4.2) & g_{ε} (Φ_{ε}) = \int_{Ω ∖ {\overline{H}}_{z, ε}} | Φ_{ε} - Φ_{des} |^{2} d x, \forall Φ \in H^{1} (Ω ∖ {\overline{H}}_{z, ε}), \end{matrix}$ where $Φ_{des} \in L^{2} (0, T; H^{1} (Ω))$ is a given desired temperature state.

Proposition 4.2.
The considered function ( 4.2 ) verifies Assumption ( A ) with $\begin{array}{l} D g_{0} (Φ_{0} (\cdot, t)) h = 2 \int_{Ω} [Φ_{0} (\cdot, t) - Φ_{des} (\cdot, t)] h d x, \forall h \in H^{1} (Ω), \\ δ G (x) = 0, \forall x \in Ω . \end{array}$
Proof.
Using the fact that $Φ_{ε}, Φ_{des} \in L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))$ , it follows that $g_{ε} (Φ_{ε} (\cdot, t)) \in L^{1} (0, T)$ . It is easy to show that the function $g_{0}$ is differentiable at $Φ_{0} (\cdot, t)$ and we have $\begin{matrix} D g_{0} (Φ_{0} (\cdot, t)) h = 2 \int_{Ω} (Φ_{0} (\cdot, t) - Φ_{des}) h d x, \forall h \in H^{1} (Ω) . \end{matrix}$ The shape function variation reads $\begin{array}{l} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = 2 \int_{0}^{T} \int_{Ω} (Φ_{0} - Φ_{des}) (Φ_{ε} - Φ_{0}) d x d t \\ (4.3) & + \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} | Φ_{ε} - Φ_{0} |^{2} d x d t \\ (4.4) & + \int_{0}^{T} \int_{H_{z, ε}} | Φ_{0} |^{2} d x d t - \int_{0}^{T} \int_{H_{z, ε}} | Φ_{des} |^{2} d x d t . \end{array}$ The smoothness of $Φ_{0}$ and $Φ_{des}$ involve $\begin{matrix} \int_{0}^{T} \int_{H_{z, ε}} | Φ_{0} |^{2} d x d t - \int_{0}^{T} \int_{H_{z, ε}} | Φ_{des} |^{2} d x d t = O (| H_{z, ε} |) = o (ε) . \end{matrix}$ Moreover, we have $\begin{matrix} (4.5) & \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} | Φ_{ε} - Φ_{0} |^{2} d x d t ⩽ 2 [\int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} | Φ_{ε} - Φ_{0} - L |^{2} d x d t + \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} | L |^{2} d x d t] . \end{matrix}$ Using Propositions 2.3, we have $\begin{matrix} ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{2} (0, T; H^{1} (Ω ∖ {\overline{H}}_{z, ε}))}^{2} = o (ε) . \end{matrix}$ Using the smoothness of $Φ_{0}$ and (2.9), we get $\begin{matrix} ‖ L ‖_{L^{2} (0, T; L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}^{2} = 2 π {‖ Φ_{0} (z, t) ‖}_{L^{2} (0, T)}^{2} {‖ ξ ((x - z) / ε) ‖}_{L^{2} (Ω ∖ {\overline{H}}_{z, ε})}^{2} = o (ε) \end{matrix}$ and then $\begin{matrix} (4.6) & \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} | Φ_{ε} - Φ_{0} |^{2} d x d t = o (ε) . \end{matrix}$ By combining the above estimates, we deduce that $g_{ε}$ verifies Assumption ( A ) with $\begin{matrix} δ G (x) = 0, \forall x \in Ω . \end{matrix}$ □

4.3. Example 3

We consider the objective function $\begin{matrix} (4.7) & g_{ε} (Φ) = \int_{Ω ∖ {\overline{H}}_{z, ε}} γ (x) | \nabla Φ - \nabla Φ_{w} |^{2} d x, \forall Φ \in H^{1} (Ω ∖ {\overline{H}}_{z, ε}), \end{matrix}$ where $Φ_{w} \in L^{2} (0, T; H^{2} (Ω))$ is a given wanted state. It has been used in various shape optimization problems.

Proposition 4.3.
The function ( 4.7 ) verifies Assumption ( A ) with $\begin{array}{l} D g_{0} (Φ_{0} (\cdot, t)) h = 2 \int_{Ω} γ (x) [\nabla Φ_{0} (\cdot, t) - \nabla Φ_{w} (\cdot, t)] \nabla h d x, \forall h \in H^{1} (Ω), \\ δ G (x) = - γ (x) \int_{0}^{T} {| Φ_{0} (x, t) |}^{2} d t [\int_{\partial H} q (y) d s (y)], \forall x \in Ω . \end{array}$
Proof.
Similarly to the previous case, one can easily see that the function $g_{0}$ is differentiable at $Φ_{0} (\cdot, t)$ and we have $\begin{matrix} D g_{0} (Φ_{0} (\cdot, t)) h = 2 \int_{Ω} γ (x) (\nabla Φ_{0} - \nabla Φ_{w}) \nabla h d x, \forall h \in H^{1} (Ω) . \end{matrix}$ The variation of the shape function $G$ is given by $\begin{array}{l} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = 2 \int_{0}^{T} \int_{Ω} γ (x) (\nabla Φ_{0} - \nabla Φ_{w}) (\nabla Φ_{ε} - \nabla Φ_{0}) d x d t \\ + \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} γ (x) | \nabla Φ_{ε} - \nabla Φ_{0} |^{2} d x d t \\ + \int_{0}^{T} \int_{H_{z, ε}} γ (x) | \nabla Φ_{0} |^{2} d x d t - \int_{0}^{T} \int_{H_{z, ε}} γ (x) | \nabla Φ_{w} |^{2} d x d t . \end{array}$ Using the regularity of $\nabla Φ_{0}$ and $\nabla Φ_{w}$ , we obtain $\begin{matrix} \int_{0}^{T} \int_{H_{z, ε}} γ (x) | \nabla Φ_{0} |^{2} d x d t - \int_{0}^{T} \int_{H_{z, ε}} γ (x) | \nabla Φ_{w} |^{2} d x d t = o (ε) . \end{matrix}$ Moreover, by the weak formulation of (2.3), we get $\begin{array}{l} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = \int_{0}^{T} D g_{0} (Φ_{0} (\cdot, t)) (Φ_{ε} - Φ_{0}) d t \\ - \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} \frac{\partial (Φ_{ε} - Φ_{0})}{\partial t} (Φ_{ε} - Φ_{0}) d x d t \\ (4.8) & - \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n Φ_{0} d x d t + o (ε) . \end{array}$

Taking the supremum for all $t \in [0, T]$ , we have $\begin{array}{l} \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} \frac{\partial (Φ_{ε} - Φ_{0})}{\partial t} (Φ_{ε} - Φ_{0}) d x d t & = \frac{1}{2} \int_{Ω ∖ {\overline{H}}_{z, ε}} {| Φ_{ε} (\cdot, T) - Φ_{0} (\cdot, T) |}^{2} d t \\ ⩽ c ‖ Φ_{ε} - Φ_{0} ‖_{L^{\infty} (0, T, L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}^{2} . \end{array}$ From the triangular inequality and the inequality ${(a + b)}^{2} ⩽ 2 (a^{2} + b^{2})$ , we obtain $\begin{array}{l} \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} \frac{\partial (Φ_{ε} - Φ_{0})}{\partial t} (Φ_{ε} - Φ_{0}) d x d t & ⩽ c [2 ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{\infty} (0, T, L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}^{2} \\ + 2 ‖ U ‖_{L^{\infty} (0, T, L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}^{2}] . \end{array}$ Using Proposition 2.3, we get $\begin{matrix} (4.9) & ‖ Φ_{ε} - Φ_{0} - L ‖_{L^{\infty} (0, T, L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}^{2} = o (ε) . \end{matrix}$ We recall that the function $Φ_{0} (\cdot, T)$ is sufficiently smooth, then the estimate (2.9) implies $\begin{matrix} ‖ L ‖_{L^{\infty} (0, T, L^{2} (Ω ∖ {\overline{H}}_{z, ε}))}^{2} = {‖ Φ_{0} (z, t) ‖}_{L^{\infty} (0, T)}^{2} {‖ ξ ((x - z) / ε) ‖}_{L^{2} (Ω ∖ {\overline{H}}_{z, ε})}^{2} = o (ε) . \end{matrix}$ Then $\begin{matrix} (4.10) & \int_{0}^{T} \int_{Ω ∖ {\overline{H}}_{z, ε}} \frac{\partial (Φ_{ε} - Φ_{0})}{\partial t} (Φ_{ε} - Φ_{0}) d x d t = o (ε) . \end{matrix}$ Consequently, $\begin{array}{rcl} G (Ω ∖ {\overline{H}}_{z, ε}) - G (Ω) & = & \int_{0}^{T} D g_{0} (Φ_{0} (\cdot, t)) (Φ_{ε} - Φ_{0}) d t \\ (4.11) & - \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n Φ_{0} d x d t + o (ε) . \end{array}$ Furthermore, the second integral term in (4.11) can be written $\begin{array}{rcl} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n Φ_{0} d x d t & = & \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0} - L) . n Φ_{0} d s d t \\ + \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla L . n Φ_{0} d s d t . \end{array}$ Using the same technique developed to compute the integral term in (3.5), we get $\begin{matrix} \int_{0}^{T} \int_{\partial H_{z, ε}} γ (x) \nabla (Φ_{ε} - Φ_{0}) . n Φ_{0} d x d t = - γ (x) \int_{0}^{T} {| Φ_{0} (x, t) |}^{2} d t [\int_{\partial H} q (y) d s (y)] + o (ε) . \end{matrix}$ Then, we deduce that the design function $g_{ε}$ satisfies Assumption ( A ) with $\begin{matrix} δ G (x) = - γ (x) \int_{0}^{T} {| Φ_{0} (x, t) |}^{2} d t [\int_{\partial H} q (y) d s (y)], \forall x \in Ω . \end{matrix}$ □

5. Conclusion

A topological sensitivity analysis is derived for a parabolic type problem in the three-dimensional case. The obtained topological asymptotic expansion is valid for a large class of shape functions and arbitrary shaped geometric perturbations. The established theoretical results are based on a preliminary estimate describing the asymptotic behavior of the perturbed heat equation solution. The employed mathematical analysis is general and can be adapted for other parabolic problems. The obtained theoretical results can be used in different numerical applications such as the topological optimization problem related to the turbine blade cooling [24].

Footnotes

Acknowledgements

Researchers Supporting Project number (RSP-2019/136), King Saud University, Riyadh, Saudi Arabia.

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