Topology optimization method with respect to the insertion of small coated inclusion

Abstract

We consider the two-dimensional steady-state heat conduction on a bounded domain Ω containing a heat source at an unknown location $ω \subset Ω$ . We are interested in determining the locations ω allowing a suitable thermal environment. The resulting shape optimization problem consists of a geometric control of either the maximum of the temperature or its $L^{2}$ mean oscillations in Ω. We derive the topological asymptotic expansion of the considered shape functionals with respect to the insertion of small circular coated inclusion characterized by a discontinuous thermal conductivity and their radius. We propose a reconstruction algorithm based on this topological gradient to identify the locations. Some numerical simulations are presented to show the efficiency of the algorithm.

Keywords

Topological optimization asymptotic analysis coated inclusion heat conduction

1. Introduction

We consider the problem of locating circular coated inclusions characterized by discontinuous conductivity in order to obtain an efficient cooling temperature in a conductor material. Such problems are usually encountered in the design of optimal heat conduction structures, like electrical multi-cables for electric vehicles or some electronic devices, where the heat distribution depends on the composition and the shape of the single cables. An important issue in such problems is the position of the single cables to ensure that the final multi-cable produces a suitable thermal environment. The major challenge in the design of these electric devices is the ability to produce a high electrical conductor network with suitable resistance properties, geometric form and a controlled thermal effect to prevent the overheating phenomena.

The main purpose of this article is to develop a shape optimization reconstruction method, based on the topological derivatives concept, of coated inclusions with a discontinuous thermal conductivity embedded in a bounded two-dimensional domain. The mathematical problem is similar to the mixture of materials with different conduction properties extensively studied in the case of two materials [1,8,15,16,18,19].

The main idea in this approach is to perform a sensitivity analysis of a given shape functional with respect to the insertion of a small hole inside the domain. More precisely, we consider a domain $Ω \subset R^{2}$ and a cost functional $J (Ω) = j (Ω, u_{Ω})$ , where $u_{Ω}$ is the state variable, i.e. a solution of a given partial differential equation in Ω. For $ε > 0$ , let $Ω_{ε} = Ω ∖ \overline{(x_{0} + ε B)}$ be the domain obtained by removing a small part $\overline{(x_{0} + ε B)}$ from Ω, at a location $x_{0} \in Ω$ , and B is a fixed bounded domain containing the origin (e.g. the unit ball in $R^{2}$ ). Then, the asymptotic expansion of the cost function $J$ with respect to ε, takes the form $\begin{matrix} (1) & J (Ω ∖ \overline{(x_{0} + ε B)}) - J (Ω) = f (ε) G (x_{0}) + o (f (ε)), \end{matrix}$ where $f (ε)$ is a positive function going to zero with ε and G is the so called topological gradient. Therefore, to minimize the cost functional $J$ , one has to create small holes at the locations x where $G (x)$ is the most negative. For detailed description of this approach we refer the reader to [10,11,14,17] and the references therein. A similar approach for sensibility analysis, considered by Cedio-Fengya, Moskow and Vogelius [2,9,20], consists in perturbing the domain by inserting a small inclusion with a different material property from the background medium. In the case of two conductor materials, a lot of techniques have been designed that encountered a successful application, we refer also to the recent works on shape reconstruction and stability analysis for some imaging problems with the topological derivative [3,4,7,12,13].

In this article, the discontinuous conductivity in the inclusion itself prevents from a direct transposition of such methods based on a local perturbation of the material properties. Therefore, we extend the sensibility analysis to the problem under consideration and we perform the asymptotic expansion of the proposed shape functionals. The asymptotic allows us to deduce a gradient algorithm for the reconstruction.

The outline of the article is as follows. In Section 2 we present the model problem and the shape optimization formulation. In Sections 3 and 4 we derive the topological derivative expression of the shape functional. In Section 5 we give some numerical results as a proof of the concept and in agreement with the theoretical analysis.

2. The model problem

Let Ω be a bounded domain in $R^{2}$ with Lipschitz boundary $\partial Ω$ and let ω be an open subset of Ω composed of two different subdomains $Ω_{1}$ and $Ω_{2}$ where the subset $Ω_{2}$ is surrounded by the subset $Ω_{1}$ . We denote by $Γ_{2} : = \partial Ω_{2}$ and $Γ_{1} \cup Γ_{2} : = \partial Ω_{1}$ as depicted in Fig. 1 and we set $Ω_{0} : = Ω ∖ \overline{Ω_{1} \cup Ω_{2}}$ .

Throughout the article, we consider piecewise constant thermal conductivity $\begin{matrix} σ = σ_{0} 1_{Ω_{0}} + σ_{1} 1_{Ω_{1}} + σ_{2} 1_{Ω_{2}}, \end{matrix}$ where $σ_{0}, σ_{1}, σ_{2} \in R_{+}^{*}$ and $1_{E}$ denotes the indicator function of the set E. We assume further that there exists two constants $c_{0}$ , $c_{1}$ such that $\begin{matrix} 0 < c_{0} ⩽ σ_{0}, σ_{1}, σ_{2} ⩽ c_{1} . \end{matrix}$ For a given source term $f \in L^{2} (Ω)$ and the Dirichlet data $g \in H^{1 / 2} (\partial Ω)$ , the temperature $u_{ω}$ satisfies the following problem $\begin{matrix} (2) & \{\begin{matrix} - div (σ \nabla u_{ω}) = f & in Ω, \\ [[u_{ω}]] = 0 & on Γ_{i}, i = 1, 2, \\ [[σ \partial_{n} u_{ω}]] = 0 & on Γ_{i}, i = 1, 2, \\ u_{ω} = g & on \partial Ω . \end{matrix} \end{matrix}$ For simplicity we take $g = 0$ by choosing a lifting function $G \in H^{2} (Ω)$ , $G = g$ in $\partial Ω$ and modifying the left hand side that we still denote f. Then, the weak solution to problem (2) is defined by: $\begin{matrix} (3) & Find u_{ω} \in H_{0}^{1} (Ω) such that a (u_{ω}, v) = l (v), \forall v \in H_{0}^{1} (Ω), \end{matrix}$ where $\begin{matrix} a (u_{ω}, v) = \int_{Ω} σ \nabla u_{ω} \cdot \nabla v d x, \end{matrix}$ and $\begin{matrix} l (v) = \int_{Ω} f v d x . \end{matrix}$ The existence and uniqueness of the weak solution $u_{ω}$ follows from the Lax–Milgram lemma.

Fig. 1.

The domain $Ω = Ω_{0} \cup Ω_{1} \cup Γ_{1} \cup Γ_{2} \cup Ω_{2}$ .

The shape optimization problem under consideration may be formulated as: $\begin{matrix} (4) & Find the location of the inclusion ω \subset Ω to get an efficient cooling temperature. \end{matrix}$ To solve effectively problem (4), we consider two objective functions.The first one corresponds to the minimization of the maximum temperature $\begin{matrix} J_{\infty} (ω, u_{ω}) = ‖ u_{ω} ‖_{L^{\infty} (Ω)} . \end{matrix}$ As the functional $J_{\infty}$ is not differentiable and thus the topological approach would not be applicable, we replace $J_{\infty}$ by the functional $J_{p}$ , for large $p ⩾ 2$ : $\begin{matrix} J_{p} (ω, u_{ω}) = \frac{1}{p} \int_{Ω} | u_{ω} |^{p} d x . \end{matrix}$ The second objective function may appear as a particular case of $J_{p}$ but has its own physical and mathematical interests, corresponds to the minimization of the $L^{2}$ mean oscillations of the temperature $\begin{matrix} K (ω, u_{ω}) : = \int_{Ω} {(u_{ω} - \frac{1}{| Ω |} \int_{Ω} u_{ω} d x)}^{2} d x . \end{matrix}$ Then, the optimization problems read: $\begin{matrix} (5) & \{\begin{matrix} minimize J_{p} (ω, u_{ω}) : = \frac{1}{p} \int_{Ω} | u_{ω} |^{p} d x \\ subject to ω \in A and u_{ω} the solution of (2), \end{matrix} \end{matrix}$ and $\begin{matrix} (6) & \{\begin{matrix} minimize K (ω, u_{ω}) : = \int_{Ω} {(u_{ω} - \frac{1}{| Ω |} \int_{Ω} u_{ω} d x)}^{2} d x \\ subject to ω \in A and u_{ω} the solution of (2) . \end{matrix} \end{matrix}$ Here $\begin{matrix} A = {ω \subset Ω : P (ω, Ω) < \infty}, \end{matrix}$ where $P (ω, Ω)$ , is the relative perimeter of ω in Ω. For the sake of completeness, we give the definition of the relative perimeter. $\begin{matrix} P (ω, Ω) : = sup {\int_{ω} div ϕ d x : ϕ \in C_{c}^{1} (Ω, R^{2}), ‖ ϕ ‖_{\infty} ⩽ 1} . \end{matrix}$

3. Topological derivatives

Now, we assume that the domain $Ω_{2} : = x_{0} + α ε B$ and $Ω_{1}$ is such that $Ω_{1} \cup Ω_{2} = x_{0} + ε B$ , where B is the unit ball in $R^{2}$ , $ε > 0$ and $0 < α < 1$ . We rewrite $Ω_{1}^{ε}$ and $Ω_{2}^{ε}$ instead of $Ω_{1}$ and $Ω_{2}$ . This allows to perform an asymptotic expansion of the shape functional $J_{p} (ω_{ε})$ where $ω_{ε} : = Ω_{2}^{ε} \cup Ω_{1}^{ε}$ . We also introduce $Γ_{2}^{ε} : = \partial Ω_{2}^{ε}$ and $Γ_{1}^{ε}$ the outer boundary of $Ω_{1}^{ε}$ .

In the perturbed domain, the state $u_{ε}$ is solution to the following problem: $\begin{matrix} (7) & \{\begin{matrix} - div (σ_{ε} \nabla u_{ε}) = f_{ε} & in Ω \\ u_{ε} = 0 & on \partial Ω \end{matrix} \end{matrix}$ where $\begin{matrix} σ_{ε} = σ_{2} 1_{Ω_{2}^{ϵ}} + σ_{1} 1_{Ω_{1}^{ϵ}} + σ_{0} 1_{Ω ∖ \overline{Ω_{1}^{ϵ} \cup Ω_{2}^{ϵ}}}, \end{matrix}$ and $\begin{matrix} f_{ε} = f_{2} 1_{Ω_{2}^{ϵ}} + f_{1} 1_{Ω_{1}^{ϵ}} + f_{0} 1_{Ω ∖ \overline{Ω_{1}^{ϵ} \cup Ω_{2}^{ϵ}}} . \end{matrix}$ The functions $f_{i}$ are in $L^{2}$ . The variational formulation associated with the problem (7) is defined by: $\begin{matrix} (8) & \{\begin{matrix} find u_{ε} \in H_{0}^{1} (Ω), such that \\ a_{ε} (u_{ε}, v) = l_{ε} (v), \forall v \in H_{0}^{1} (Ω), \end{matrix} \end{matrix}$ where $\begin{matrix} a_{ε} (u_{ε}, v) = \int_{Ω} σ_{ε} \nabla u_{ε} \cdot \nabla v d x, \end{matrix}$ and $\begin{matrix} l_{ε} (v) = \int_{Ω} f_{ε} v d x . \end{matrix}$ We denote by $u_{0}$ the background solution of the following problem: $\begin{matrix} (9) & - div (σ_{0} \nabla u_{0}) = f_{0}, in Ω, u_{0} = 0 on \partial Ω . \end{matrix}$ We introduce the following proposition which describes an adjoint method for the computation of the first variation of a given cost functional. For further details the reader is referred to [5].

Proposition 1.
Let $V$ be a Hilbert space. For $ε \in [0, ξ), ξ ⩾ 0$ , consider a function $u_{ε} \in V$ that’s solution of a variational problem of the form $\begin{array}{l} (10) & a_{ε} (u_{ε}, v) = l_{ε} (v) \forall v \in V, \end{array}$ where $a_{ε}$ and $l_{ε}$ are a bilinear form and a linear form on $V$ , respectively. For all $ε \in [0, ξ)$ , consider a functional $J_{ε} : V ⟶ R$ which is Frêchet differentiable at the point $u_{0}$ . Suppose that the following hypotheses hold.

$H_{1}$
There exist two numbers $δ a$ , $δ l$ and a function $f (ε) ⩾ 0$ such that $\begin{array}{l} (11) & (a_{ε} - a_{0}) (u_{0}, v_{ε}) = f (ε) δ a + o (f (ε)), \\ (12) & (l_{ε} - l_{0}) (v_{ε}) = f (ε) δ l + o (f (ε)), \\ (13) & lim_{ε \to 0} f (ε) = 0, \end{array}$ where $v_{ε} \in V$ is an adjoint state satisfying $\begin{array}{l} (14) & a_{ε} (φ, v_{ε}) = - D J_{ε} (u_{0}) φ, \forall φ \in V . \end{array}$
$H_{2}$
There exist two numbers $δ J_{1}$ and $δ J_{2}$ such that $\begin{array}{l} (15) & J_{ε} (u_{ε}) = J_{ε} (u_{0}) + D J_{ε} (u_{0}) (u_{ε} - u_{0}) + f (ε) δ J_{1} + o (f (ε)), \\ (16) & J_{ε} (u_{0}) = J_{0} (u_{0}) + f (ε) δ J_{2} + o (f (ε)) . \end{array}$ Then the first variation of the cost function with respect to ε is given by $\begin{array}{l} (17) & J_{ε} (u_{ε}) - J_{0} (u_{0}) = f (ε) (δ a - δ l + δ J_{1} + δ J_{2}) + o (f (ε)) . \end{array}$

3.1. Application to the model problem

In this subsection, we will give explicitly the variations $δ a, δ l, δ J_{1}, δ J_{2}$ and we derive the topological derivative for the functional $J_{p}$ . The analysis is the same for the functional K and we will not repeat it here.

3.1.1. Variation of the bilinear form

We are interested in the asymptotic analysis of the variation $\begin{matrix} (18) & (a_{ε} - a_{0}) (u_{0}, v_{ε}) = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) \nabla u_{0} \cdot \nabla v_{ε} d x + \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) \nabla u_{0} \cdot \nabla v_{ε} d x . \end{matrix}$ Let us first look at the behavior of the adjoint state $v_{ε}$ solution of the following boundary value problem $\begin{matrix} (19) & \{\begin{matrix} div (σ_{ε} \nabla v_{ε}) = u_{ω} | u_{ω} |^{p - 2} & in Ω, \\ v_{ε} = 0 & on \partial Ω . \end{matrix} \end{matrix}$ Since $u_{ω}$ is Hölder continuous, $u_{ω} | u_{ω} |^{p - 2}$ is at least in $L^{2} (Ω)$ . Therefore problem (19) has a unique solution $v_{ε} \in H_{0}^{1} (Ω)$ .

We split in (18) $v_{ε}$ into $v_{0} + (v_{ε} - v_{0})$ and introducing the “small” terms (which will be checked later) $\begin{array}{l} (20) & E_{1} (ε) : = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) (\nabla u_{0} \cdot \nabla v_{0} - \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0})) d x, \\ (21) & E_{2} (ε) : = \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) (\nabla u_{0} \cdot \nabla v_{0} - \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0})) d x, \end{array}$ we obtain $\begin{array}{l} (a_{ε} - a_{0}) (u_{0}, v_{ε}) = & π ε^{2} ((1 - α^{2}) (σ_{1} - σ_{0}) + α^{2} (σ_{2} - σ_{0})) \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0}) \\ (22) & + F_{1} (ε) + F_{2} (ε) + E_{1} (ε) + E_{2} (ε), \end{array}$ where $\begin{array}{l} (23) & F_{1} (ε) : = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) \nabla u_{0} \cdot \nabla (v_{ε} - v_{0}) d x, \\ (24) & F_{2} (ε) : = \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) \nabla u_{0} \cdot \nabla (v_{ε} - v_{0}) d x . \end{array}$ We will now study the asymptotic of $F_{1}$ and $F_{2}$ . Introducing the variation ${\tilde{v}}_{ε} : = v_{ε} - v_{0}$ , we obtain from (19) that ${\tilde{v}}_{ε}$ solves $\begin{matrix} (25) & \{\begin{matrix} Δ {\tilde{v}}_{ε} = 0 & in Ω_{1}^{ε} \cup Ω_{2}^{ε} \cup (Ω ∖ \overline{Ω_{1}^{ε} \cup Ω_{2}^{ε}}), \\ [[σ \partial_{ν} {\tilde{v}}_{ε,}]] = - (σ_{1} - σ_{0}) \nabla v_{0} \cdot ν & on Γ_{1}^{ε}, \\ [[σ \partial_{ν} {\tilde{v}}_{ε}]] = - (σ_{2} - σ_{1}) \nabla v_{0} \cdot ν & on Γ_{2}^{ε}, \\ {\tilde{v}}_{ε} = 0 & on \partial Ω . \end{matrix} \end{matrix}$ We set $V : = \nabla v_{0} (x_{0})$ and we approximate ${\tilde{v}}_{ε}$ by the solution $h_{ε}^{V}$ of the auxiliary problem $\begin{matrix} (26) & \{\begin{matrix} Δ h_{ε}^{V} = 0 & in Ω_{1}^{ε} \cup Ω_{2}^{ε} \cup (R^{2} ∖ \overline{Ω_{1}^{ε} \cup Ω_{2}^{ε}}), \\ [[σ \partial_{ν} h_{ε}^{V}]] = (σ_{0} - σ_{1}) V \cdot ν & on Γ_{1}^{ε}, \\ [[σ \partial_{ν} h_{ε}^{V}]] = (σ_{1} - σ_{2}) V \cdot ν & on Γ_{2}^{ε}, \\ h_{ε}^{V} ⟶ 0 & at \infty . \end{matrix} \end{matrix}$ By shifting the coordinate system, we can assume for simplicity that $x_{0} = 0$ . For our case, we can compute explicitly the function $h_{ε}^{V}$ using polar coordinates: $\begin{matrix} (27) & h_{ε}^{V} (x) = \{\begin{matrix} (β + γ) V \cdot x & if x \in Ω_{2}^{ε}, \\ β V \cdot x + γ {(α ε)}^{2} \frac{V \cdot x}{| x |^{2}} & if x \in Ω_{1}^{ε}, \\ (β ε^{2} + γ {(α ε)}^{2}) \frac{V \cdot x}{| x |^{2}} & if x \in R^{2} ∖ \overline{Ω_{1}^{ε} \cup Ω_{2}^{ε}}, \end{matrix} \end{matrix}$ where $\begin{matrix} β : = \frac{(σ_{1} + σ_{2}) (σ_{0} - σ_{1}) - α^{2} (σ_{1} - σ_{2}) (σ_{0} - σ_{1})}{(σ_{1} + σ_{2}) (σ_{1} + σ_{0}) + α^{2} (σ_{1} - σ_{2}) (σ_{0} - σ_{1})}, \end{matrix}$ and $\begin{matrix} γ : = \frac{2 σ_{0} (σ_{1} - σ_{2})}{(σ_{1} + σ_{2}) (σ_{1} + σ_{0}) + α^{2} (σ_{1} - σ_{2}) (σ_{0} - σ_{1})} . \end{matrix}$ Its gradient is given by $\begin{matrix} (28) & \nabla h_{ε}^{V} = \{\begin{matrix} (β + γ) V & if x \in Ω_{2}^{ε}, \\ β V + γ {(α ε)}^{2} [\frac{V}{| x |^{2}} - 2 V \cdot x \frac{x}{| x |^{4}}] & if x \in Ω_{1}^{ε}, \\ (β ε^{2} + γ {(α ε)}^{2}) [\frac{V}{| x |^{2}} - 2 V \cdot x \frac{x}{| x |^{4}}] & if x \in R^{2} ∖ \overline{Ω_{1}^{ε} \cup Ω_{2}^{ε}} . \end{matrix} \end{matrix}$ Denoting $\begin{array}{l} (29) & E_{3} (ε) : = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) \nabla u_{0} \cdot \nabla ({\tilde{v}}_{ε} - h_{ε}^{V}) d x, \\ (30) & E_{4} (ε) : = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) (\nabla u_{0} - \nabla u_{0} (x_{0})) \cdot \nabla h_{ε}^{V} d x, \\ (31) & E_{5} (ε) : = \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) \nabla u_{0} \cdot \nabla ({\tilde{v}}_{ε} - h_{ε,}^{V}) d x, \end{array}$ and $\begin{matrix} (32) & E_{6} (ε) : = \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) (\nabla u_{0} - \nabla u_{0} (x_{0})) \cdot \nabla h_{ε}^{V} d x . \end{matrix}$ Then we obtain $\begin{matrix} F_{1} (ε) = & \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) \nabla u_{0} (x_{0}) \cdot \nabla h_{ε}^{V} d x + E_{3} (ε) + E_{4} (ε) \\ = & (σ_{1} - σ_{0}) \nabla u_{0} (x_{0}) \cdot \int_{Ω_{1}^{ε}} \nabla h_{ε}^{V} d x + E_{3} (ε) + E_{4} (ε) . \end{matrix}$ Using polar coordinates and integrating by parts, yields $\begin{array}{l} F_{1} (ε) = π (1 - α^{2}) ε^{2} β (σ_{1} - σ_{0}) \nabla u_{0} (x_{0}) \cdot V + E_{3} (ε) + E_{4} (ε), \\ F_{2} (ε) = \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) \nabla u_{0} (x_{0}) \cdot \nabla h_{ε}^{V} d x + E_{5} (ε) + E_{6} (ε) \\ = π α^{2} ε^{2} (β + γ) (σ_{2} - σ_{0}) \nabla u_{0} (x_{0}) \cdot V + E_{5} (ε) + E_{6} (ε) . \end{array}$ After rearrangement, we get $\begin{matrix} (a_{ε} - a_{0}) (u_{0}, v_{ε}) = π ε^{2} Λ \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0}) + \sum_{i = 1}^{6} E_{i}, \end{matrix}$ where $\begin{array}{l} Λ & : = (1 - α^{2}) (1 + β) (σ_{1} - σ_{0}) + α^{2} (σ_{2} - σ_{0}) (1 + β + γ) \\ (33) & = \frac{2 (1 - α^{2}) σ_{0} (σ_{1} - σ_{0}) (σ_{1} + σ_{2}) + 4 α^{2} σ_{0} σ_{1} (σ_{2} - σ_{0})}{(σ_{1} + σ_{2}) (σ_{1} + σ_{0}) - α^{2} (σ_{1} - σ_{2}) (σ_{1} - σ_{0})} . \end{array}$

3.1.2. Variation of the linear form

Let us now turn to the asymptotic analysis of the variation $\begin{matrix} (34) & (l_{ε} - l_{0}) (v_{ε}) = \int_{Ω_{1}^{ε}} (f_{1} - f_{0}) v_{ε} d x + \int_{Ω_{2}^{ε}} (f_{2} - f_{0}) v_{ε} d x . \end{matrix}$ We can rewrite (34) as $\begin{matrix} (l_{ε} - l_{0}) (v_{ε}) = π ε^{2} [(1 - α^{2}) (f_{1} - f_{0}) + α^{2} (f_{2} - f_{0})] v_{0} (x_{0}) + E_{7} (ε) + E_{8} (ε), \end{matrix}$ where $\begin{array}{l} E_{7} (ε) = \int_{Ω_{1}^{ε}} (f_{1} - f_{0}) {\tilde{v}}_{ε} d x, \\ E_{8} (ε) = \int_{Ω_{2}^{ε}} (f_{2} - f_{0}) {\tilde{v}}_{ε} d x . \end{array}$ Again, it will be proved that $E_{7}$ and $E_{8}$ are small terms. Consequently we set $\begin{matrix} δ l = π [(1 - α^{2}) (f_{1} - f_{0}) + α^{2} (f_{2} - f_{0})] v_{0} (x_{0}) . \end{matrix}$

3.1.3. Variation of the cost function

Expression of $J_{p, ε} (u_{ε}) - J_{p, ε} (u_{0})$ . For simplicity of the calculus, we assume that p is even, then we have $\begin{matrix} J_{p, ε} (u_{ε}) - J_{p, ε} (u_{0}) & = \frac{1}{p} \int_{Ω} | u_{ε} |^{p} d x - \frac{1}{p} \int_{Ω} | u_{0} |^{p} d x \\ = \frac{1}{p} \int_{Ω} {| (u_{ε} - u_{0}) + u_{0} |}^{p} - \frac{1}{p} \int_{Ω} | u_{0} |^{p} d x \\ = \frac{1}{p} \sum_{k = 0}^{p} (\binom{p}{k}) \int_{Ω} u_{0}^{p - k} {(u_{ε} - u_{0})}^{k} d x - \frac{1}{p} \int_{Ω} | u_{0} |^{p} d x \\ = \frac{1}{p} \sum_{k = 2}^{p} (\binom{p}{k}) \int_{Ω} u_{0}^{p - k} {(u_{ε} - u_{0})}^{k} d x + \int_{Ω} u_{0}^{p - 1} (u_{ε} - u_{0}) d x . \end{matrix}$ Therefore $\begin{matrix} J_{p, ε} (u_{ε}) - J_{p, ε} (u_{0}) - D J_{p, ε} (u_{0}) (u_{ε} - u_{0}) = E_{9} (ε), \end{matrix}$ where $\begin{matrix} E_{9} (ε) : = \frac{1}{p} \sum_{k = 2}^{p} (\binom{p}{k}) \int_{Ω} u_{0}^{p - k} {(u_{ε} - u_{0})}^{k} d x . \end{matrix}$ We will prove in the next section that $\begin{matrix} E_{9} (ε) = o (ε^{2}), \end{matrix}$ and thus $δ J_{1} = 0$ .

Expression of $J_{p, ε} (u_{0}) - J_{p, 0} (u_{0})$ . We have $\begin{matrix} J_{p, ε} (u_{0}) - J_{p, 0} (u_{0}) = \frac{1}{p} \int_{Ω} | u_{0} |^{p} d x - \frac{1}{p} \int_{Ω} | u_{0} |^{p} d x = 0 . \end{matrix}$ Consequently $δ J_{2} = 0$ .

Now, we are ready to state the main result of this paper.

Theorem 1.
The topological asymptotic expansion of the functional J with respect to the insertion of small coated inclusion $ω_{ε}$ is given by $\begin{matrix} J_{p, ε} (u_{ε}) - J_{p, 0} (u_{0}) = ε^{2} G (x_{0}) + o (ε^{2}), \end{matrix}$ where $\begin{matrix} (35) & G (x_{0}) = π Λ \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0}) - π [(1 - α^{2}) (f_{1} - f_{0}) + α^{2} (f_{2} - f_{0})] v_{0} (x_{0}), \end{matrix}$ and $\begin{array}{l} Λ : = \frac{2 (1 - α^{2}) σ_{0} (σ_{1} - σ_{0}) (σ_{1} + σ_{2}) + 4 α^{2} σ_{0} σ_{1} (σ_{2} - σ_{0})}{(σ_{1} + σ_{2}) (σ_{1} + σ_{0}) - α^{2} (σ_{1} - σ_{2}) (σ_{1} - σ_{0})} . \end{array}$
Remark 1.
When $σ_{1} = σ_{2}$ and $α = 0$ , the topological derivative defined in (35) becomes $\begin{matrix} (36) & G (x_{0}) = 2 π ρ σ_{0} \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0}) + π (f_{1} - f_{0}) v_{0} (x_{0}), ρ = \frac{σ_{1} - σ_{0}}{σ_{1} + σ_{0}} . \end{matrix}$ Expression (36) is known in the literature when the inclusion ω is an homogenous disk; see for instance [6, Thm 4.3].

4. Estimates of the remainders

4.1. Preliminary lemmas

Lemma 1.
(i) For any vector $V \in R^{2}$ , $x_{0} \in Ω$ and positive radius R, we have $\begin{array}{l} {‖ h_{ε}^{V} ‖}_{L^{2} (Ω)} = O (ε^{3 / 2}), \\ {‖ \nabla h_{ε}^{V} ‖}_{L^{p} (Ω)} = O (ε^{2 / p}) \forall p > 1, \\ {‖ h_{ε}^{V} ‖}_{L^{p} (Ω ∖ B (x_{0}, R))} + {‖ \nabla h_{ε}^{V} ‖}_{L^{p} (Ω ∖ B (x_{0}, R))} = O (ε^{2}) \forall p ⩾ 1 . \end{array}$ (ii) Given a function $ψ : Ω \to R^{2}$ which is θ Hölder continuous ( $0 < θ < 1$ ) in a neighborhood of $x_{0}$ and consider the solution $w_{ε}$ of the system: $\begin{matrix} (37) & \{\begin{matrix} - div (σ_{ε} \nabla w_{ε}) = 0 & in Ω_{1}^{ε} \cup Ω_{2}^{ε} \cup (Ω ∖ \overline{Ω_{1}^{ε} \cup Ω_{2}^{ε}}), \\ [[σ \partial_{ν} w_{ε,}]] = (σ_{0} - σ_{1}) ψ \cdot ν & on Γ_{1}^{ε}, \\ [[σ \partial_{ν} w_{ε}]] = (σ_{1} - σ_{2}) ψ \cdot ν & on Γ_{2}^{ε}, \\ w_{ε} = 0 & on \partial Ω . \end{matrix} \end{matrix}$ Then, we have $\begin{array}{l} (38) & {‖ w_{ε} - h_{ε}^{ψ (x_{0})} ‖}_{H^{1} (Ω)} = o (ε) . \end{array}$
Proof.
The estimates i) in Lemma 1 follow directly from (27) and (28). Now, we prove the second part. Let $φ \in H_{0}^{1} (Ω)$ be an arbitrary test function, then from (37), we have $\begin{matrix} \int_{Ω} σ_{ε} \nabla w_{ε} \cdot \nabla φ d x = (σ_{0} - σ_{1}) \int_{Γ_{1}^{ε}} ψ \cdot ν φ d s + (σ_{1} - σ_{2}) \int_{Γ_{2}^{ε}} ψ \cdot ν φ d s . \end{matrix}$ Using Green’s formula together with (26), we obtain $\begin{matrix} \int_{Ω} σ_{ε} \nabla h_{ε}^{ψ (x_{0})} \cdot \nabla φ d x = (σ_{0} - σ_{1}) \int_{Γ_{1}^{ε}} ψ (x_{0}) \cdot ν φ d s + (σ_{1} - σ_{2}) \int_{Γ_{2}^{ε}} ψ (x_{0}) \cdot ν φ d s . \end{matrix}$ Denote $Θ_{ε} : = w_{ε} - h_{ε}^{ψ (x_{0})}$ . It follows that $\begin{matrix} (39) & \int_{Ω} σ_{ε} \nabla Θ_{ε} \cdot \nabla φ d x = (σ_{0} - σ_{1}) \int_{Γ_{1}^{ε}} (ψ - ψ (x_{0})) \cdot ν φ d s + (σ_{1} - σ_{2}) \int_{Γ_{2}^{ε}} (ψ - ψ (x_{0})) \cdot ν φ d s . \end{matrix}$ Using the change for variable, the θ-Hölder continuity of ψ in the vicinity of $x_{0}$ and the trace theorem, we get for ε small enough $\begin{matrix} | \int_{Γ_{2}^{ε}} (ψ - ψ (x_{0})) \cdot ν φ d s | & = ε | \int_{Γ_{2}} (ψ (ε x) - ψ (x_{0})) \cdot ν φ (ε x) d s | \\ ⩽ c ε^{1 + θ} {‖ φ (ε x) ‖}_{H^{1 / 2} (Γ_{2})} \\ ⩽ c ε^{1 + θ} {‖ φ (ε x) ‖}_{H^{1} (Ω_{2})} \\ ⩽ c ε^{θ} ‖ φ ‖_{L^{2} (Ω_{2}^{ε})} + c ε^{θ + 1} ‖ \nabla φ ‖_{L^{2} (Ω_{2}^{ε})} . \end{matrix}$ From the Hölder inequality and the Sobolev imbedding theorem, we obtain $\begin{matrix} ‖ φ ‖_{L^{2} (Ω_{2}^{ε})} ⩽ c ε^{\frac{1}{p}} ‖ φ ‖_{L^{\frac{2 p}{p - 1}} (Ω_{2}^{ε})} ⩽ c ε^{1 / p} ‖ φ ‖_{H^{1} (Ω)}, for any p > 1 . \end{matrix}$ Therefore $\begin{matrix} | \int_{Γ_{2}^{ε}} (ψ - ψ (x_{0})) \cdot ν φ d s | ⩽ c (ε^{θ + 1 / p} + ε^{θ + 1}) ‖ φ ‖_{H^{1} (Ω)} . \end{matrix}$ Analogously, we can prove that $\begin{matrix} | \int_{Γ_{1}^{ε}} (ψ - ψ (x_{0})) \cdot ν φ d s | ⩽ c^{'} (ε^{θ + 1 / p} + ε^{θ + 1}) ‖ φ ‖_{H^{1} (Ω)} . \end{matrix}$ From (39) and the first part of Lemma 1, we deduce that $\begin{matrix} (40) & | \int_{Ω} σ_{ε} \nabla Θ_{ε} \cdot \nabla φ d x | ⩽ c^{″} (ε^{θ + 1 / p} + ε^{θ + 1} + ε^{2}) ‖ φ ‖_{H^{1} (Ω)} . \end{matrix}$ Choosing $φ = Θ$ and $p \in (1, \frac{1}{1 - θ})$ in (40), yield $\begin{matrix} ‖ Θ ‖_{H^{1} (Ω)} = o (ε), \end{matrix}$ and the proof is completed. □
Lemma 2.
We have $\begin{array}{l} (41) & ‖ u_{ε} - u_{0} ‖_{H^{1} (Ω)} = O (ε), \\ (42) & ‖ v_{ε} - v_{0} ‖_{H^{1} (Ω)} = O (ε), \\ (43) & ‖ u_{ε} - u_{0} ‖_{L^{2} (Ω)} = o (ε), \\ (44) & ‖ v_{ε} - v_{0} ‖_{L^{2} (Ω)} = o (ε) . \end{array}$
Proof.
From the Poincary inequality, we deduce that $\begin{matrix} (45) & \int_{Ω} | u_{ε} - u_{0} |^{2} d x ⩽ C (\int_{Ω} {| \nabla (u_{ε} - u_{0}) |}^{2} d x), \end{matrix}$ for some constant C independent of ε. Then, it suffices to show that $\begin{matrix} \int_{Ω} {| \nabla (u_{ε} - u_{0}) |}^{2} d x ⩽ C ε^{2} . \end{matrix}$ From (8), we obtain immediately that $\begin{matrix} (46) & a_{ε} (u_{ε} - u_{0}, v) = - (a_{ε} - a_{0}) (u_{0}, v) + (l_{ε} - l_{0}) (v), \forall v \in H_{0}^{1} (Ω) . \end{matrix}$ According to (18), we get $\begin{matrix} (a_{ε} - a_{0}) (u_{0}, v) = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) \nabla u_{0} \cdot \nabla v d x + \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) \nabla u_{0} \cdot \nabla v d x . \end{matrix}$ Using the fact that $\nabla u_{0}$ is uniformly bounded on $Ω_{1}^{ε}$ and $Ω_{2}^{ε}$ , we obtain $\begin{matrix} | (a_{ε} - a_{0}) (u_{0}, v) | ⩽ & | σ_{1} - σ_{0} | sup_{Ω_{1}^{ε}} | \nabla u_{0} | {| Ω_{1}^{ε} |}^{1 / 2} ‖ \nabla v ‖_{L^{2} (Ω)} \\ + | σ_{2} - σ_{0} | sup_{Ω_{2}^{ε}} | \nabla u_{0} | {| Ω_{2}^{ε} |}^{1 / 2} ‖ \nabla v ‖_{L^{2} (Ω)} \\ ⩽ & C ε ‖ \nabla v ‖_{L^{2} (Ω)} . \end{matrix}$ Now, we estimate the term $(l_{ε} - l_{0}) (v)$ . We have $\begin{matrix} | (l_{ε} - l_{0}) (v) | & = | \int_{Ω_{1}^{ϵ}} (f_{1} - f_{0}) v d x + \int_{Ω_{2}^{ε}} (f_{2} - f_{0}) v d x | \\ ⩽ (| f_{1} - f_{0} | \sqrt{π (1 - α^{2})} + | f_{2} - f_{0} | \sqrt{π}) ε ‖ v ‖_{L^{2} (Ω)} . \end{matrix}$ From (46), the above estimates and Poincary inequality, we obtain $\begin{matrix} \int_{Ω} {| \nabla (u_{ε} - u_{0}) |}^{2} d x ⩽ C^{'} ε^{2} . \end{matrix}$ This proves the asymptotic formula (41). Analogously we derive the estimate (42). The proof of (43) and (44) follows straightforwardly from [5, Lemma 9.3]. □
4.2. Asymptotic behavior of the remainders

In this subsection, we shall prove that $E_{i} (ε) = o (ε^{2})$ for $i = 1, \dots, 9$ . We have $\begin{matrix} E_{2} (ε) = \int_{Ω_{2}^{ε}} (\nabla u_{0} \cdot \nabla v_{0} - \nabla u_{0} (x_{0}) \cdot \nabla v_{0} (x_{0})) d x . \end{matrix}$ Using the regularity of $u_{0}$ and $v_{0}$ near $x_{0}$ and Taylor–Lagrange expansion, we straightforwardly obtain $\begin{matrix} E_{2} (ε) ⩽ c ε^{3}, \end{matrix}$ and thus $E_{2} (ε) = o (ε^{2})$ . Similarly, we can prove that $\begin{matrix} E_{1} (ε) = o (ε^{2}) . \end{matrix}$ Let’s now prove that $E_{4} (ε) = o (ε^{2})$ and $E_{6} (ε) = o (ε^{2})$ . We have $\begin{matrix} E_{4} (ε) : = \int_{Ω_{1}^{ε}} (σ_{1} - σ_{0}) (\nabla u_{0} - \nabla u_{0} (x_{0})) \cdot \nabla h_{ε}^{V} d x, \end{matrix}$ and $\begin{matrix} E_{6} (ε) : = \int_{Ω_{2}^{ε}} (σ_{2} - σ_{0}) (\nabla u_{0} - \nabla u_{0} (x_{0})) \cdot \nabla h_{ε}^{V} d x . \end{matrix}$ Using Cauchy–Schwarz inequality, yields $\begin{matrix} | E_{6} (ε) | & ⩽ | σ_{2} - σ_{0} | {(\int_{Ω_{2}^{ε}} {| (\nabla u_{0} - \nabla u_{0} (x_{0})) |}^{2} d x)}^{1 / 2} {(\int_{Ω_{2}^{ε}} {| \nabla h_{ε}^{V} |}^{2} d x)}^{1 / 2} \\ ⩽ | (β + γ) (σ_{2} - σ_{0}) | | V | α \sqrt{π} ε {(\int_{Ω_{2}^{ε}} {| (\nabla u_{0} - \nabla u_{0} (x_{0})) |}^{2} d x)}^{1 / 2} . \end{matrix}$ From the regularity of $u_{0}$ near $x_{0}$ and Taylor–Lagrange expansion, we obtain the bound $\begin{matrix} | E_{6} (ε) | ⩽ c ε^{5 / 2} . \end{matrix}$ Analogously, we can show that $\begin{matrix} E_{4} (ε) = o (ε^{2}) . \end{matrix}$ Let’s now focus on $E_{3}$ and $E_{5} (ε)$ . Using Hölder inequality, we obtain $\begin{array}{l} | E_{3} (ε) | & = | \int_{Ω_{1}^{ε}} (σ_{0} - σ_{1}) \nabla u_{0} \cdot \nabla ({\tilde{v}}_{ε} - h_{ε}^{V}) d x | \\ ⩽ | σ_{0} - σ_{1} | sup_{x \in Ω} | \nabla u_{0} (x) | {| Ω_{1}^{ε} |}^{1 / 2} {‖ \nabla ({\tilde{v}}_{ε} - h_{ε}^{V}) ‖}_{L^{2} (Ω_{1}^{ε})} \\ ⩽ C ε {‖ \nabla ({\tilde{v}}_{ε} - h_{ε}^{V}) ‖}_{L^{2} (Ω_{1}^{ε})} . \end{array}$ From Lemma 1, we deduce that $\begin{matrix} {‖ \nabla ({\tilde{v}}_{ε} - h_{ε}^{V}) ‖}_{L^{2} (Ω_{1}^{ε})} ⩽ {‖ \nabla ({\tilde{v}}_{ε} - h_{ε}^{V}) ‖}_{L^{2} (Ω)} = o (ε) . \end{matrix}$ Therefore, we conclude that $E_{3} (ε) = o (ε^{2})$ . By the same techniques, we prove that $\begin{matrix} E_{5} (ε) = o (ε^{2}) . \end{matrix}$ Let’s now check that $E_{7} (ε) = o (ε^{2})$ and $E_{8} (ε) = o (ε^{2})$ . We have $\begin{array}{l} | E_{7} (ε) | ⩽ c \int_{Ω_{1}^{ε}} | v_{ε} - v_{0} | d x . \end{array}$ From the Hölder inequality, we obtain $\begin{matrix} | E_{7} (ε) | ⩽ c ε^{2 / r} ‖ v_{ε} - v_{0} ‖_{L^{s} (Ω_{1}^{ε})} ⩽ c ε^{2 / r} ‖ v_{ε} - v_{0} ‖_{L^{s} (Ω)}, \end{matrix}$ for all $r, s \in [1, + \infty]$ satisfying $\frac{1}{r} + \frac{1}{s} = 1$ . Due to Lemma 2, we conclude that $E_{7} (ε) = o (ε^{2})$ . Similarly, we obtain $\begin{matrix} E_{8} (ε) = o (ε^{2}) . \end{matrix}$

Let’s now focus on $\begin{matrix} E_{9} (ε) : = \frac{1}{p} \sum_{k = 2}^{p} (\binom{p}{k}) \int_{Ω} u_{0}^{p - k} {(u_{ε} - u_{0})}^{k} d x . \end{matrix}$ Using Hölder inequality and the estimates in Lemma 2, we obtain $\begin{matrix} E_{9} (ε) = o (ε^{2}) . \end{matrix}$

5. Numerical results

In this section, we present some numerical results using the topological gradient derived in (35) in order to get a cooling temperature. For the numerical computations, the domain Ω is the unit disk centered at the origin. We use the Dirichlet data $g = sin (θ), θ \in [0, 2 π]$ on the boundary $\partial Ω$ . The conductivities and the term source are chosen as $σ_{0} = 1, σ_{1} = 1 / 2, σ_{2} = 30, f_{0} = 0, f_{1} = 0$ and $f_{2} = 20$ . The direct and the adjoint problems are solved using the finite element method.

Fig. 2.

On the left the topological derivative of the functional $J_{2}$ and on the right the temperature distribution relative to the position $x_{0} = (- 0.0115, - 0.6915)$ of the coated inclusion given by Algorithm 1.

Fig. 3.

Values of the objective function $J_{2}$ for variation of the $x_{2}$ -coordinate of the center of the inclusion.

In order to find the position of the coated inclusion to minimize the cost function, we use the following algorithm:

Algorithm 1.

Solve the direct problem ( 7 ) and the adjoint problem ( 19 ) in the unperturbed domain ( $ω_{ε} = \emptyset$ ).

Compute the topological derivative G defined in ( 35 ).

Determine the point $x_{0}$ where the topological derivative is the most negative.

Locate the coated inclusion ω at the point $x_{0}$ .

5.1. Example 1

In this example, we present some numerical results using the functional $J_{p}$ for $p = 2, 10, 50$ and the functional K. The coated inclusion ω to be located in order to minimize the objective function, is composed of two concentric disks $Ω_{1}$ and $Ω_{2}$ with radius $r_{1} = 0.2$ and $r_{2} = 0.1$ . We compute the position $x_{0}$ of ω using Algorithm 1.

Fig. 4.

On the left the topological derivative of the functional $J_{10}$ and on the right the temperature distribution. $x_{0} = (- 0.0035, - 0.9109)$ is the position given by Algorithm 1. In order to locate the inclusion far from the boundary $\partial Ω$ , we have taken an approximation of $x_{0}$ , that is $x_{a} = (0, - 0.75)$ for the optimization process.

Fig. 5.

Values of the objective function $J_{10}$ for variation of the $x_{2}$ -coordinate of the center of the inclusion.

In Figs 3, 5, 7 and 9, the $x_{1}$ -coordinate of the center of the inclusion is fixed to zero and the $x_{2}$ -coordinate $⩽ 0$ . We take $x_{2} ⩽ - 0.7$ to prevent the inclusion ω to touch the boundary $\partial Ω$ . In each case, the objective function value is monotonically decreasing with respect to the $x_{2}$ -positions. Figures 2, 4 and 8 show the topological derivative and the temperature distribution after the optimization process. Figure 6 shows the topological derivative for the cost functional $J_{50}$ . In this case and for large p the cost functional goes to zero and the topological derivative is not negative on the boundary $\partial Ω$ , this situation prevents the optimization procedure.

Fig. 6.

The topological derivative of the functional $J_{50}$ .

Fig. 7.

Values of the objective function $J_{50}$ for variation of the $x_{2}$ -coordinate of the center of the inclusion.

5.2. Example 2

In the second example, we present numerical results for the case of two coated inclusions using the topological derivative of the functional K. We have used the same parameters as for example 1, except Dirichlet data g where we have taken $g = sin (2 θ), θ \in [0, 2 π]$ .

Figure 10 show the topological derivative of the functional K and the temperature distribution when the positions are given by Algorithm 1.

Fig. 8.

On the left the topological derivative of the functional K and on the right the temperature distribution with respect the position $x_{0} = (- 0.0046, - 0.5850)$ given by Algorithm 1.

Fig. 9.

Values of the objective function K for variation of the $x_{2}$ -coordinate of the center of the inclusion.

Fig. 10.

On the left the topological derivative of the functional K and on the right the temperature distribution when the positions $x_{0}^{1} = (0.577, - 0.598)$ and $x_{0}^{2} = (- 0.578, 0.598)$ are given by Algorithm 1.

6. Conclusion

In this paper, we have developed the asymptotic expansion of cost functionals with respect to the insertion of small coated inclusion for a heat conduction problem in order to get a suitable thermal environment. The implementation of the obtained topological derivative in a numerical algorithm provides an interesting location of the shape, which allows us to improve the thermal environment. We intended to extend this approach for practical situations like the thermal optimization of electrical multicables.

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