Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains

Abstract

Mathematical analysis and numerical solutions of problems with unknown shapes or geometrical domains is a challenging and rich research field in the modern theory of the calculus of variations, partial differential equations, differential geometry as well as in numerical analysis. In this series of three review papers, we describe some aspects of numerical solution for problems with unknown shapes, which use tools of asymptotic analysis with respect to small defects or imperfections to obtain sensitivity of shape functionals. In classical numerical shape optimization, the boundary variation technique is used with a view to applying the gradient or Newton-type algorithms. Shape sensitivity analysis is performed by using the velocity method. In general, the continuous shape gradient and the symmetric part of the shape Hessian are discretized. Such an approach leads to local solutions, which satisfy the necessary optimality conditions in a class of domains defined in fact by the initial guess. A more general framework of shape sensitivity analysis is required when solving topology optimization problems. A possible approach is asymptotic analysis in singularly perturbed geometrical domains. In such a framework, approximations of solutions to boundary value problems (BVPs) in domains with small defects or imperfections are constructed, for instance by the method of matched asymptotic expansions. The approximate solutions are employed to evaluate shape functionals, and as a result topological derivatives of functionals are obtained. In particular, the topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, cavities, inclusions, defects, source terms and cracks. This new concept of variation has applications in many related fields, such as shape and topology optimization, inverse problems, image processing, multiscale material design and mechanical modeling involving damage and fracture evolution phenomena. In the first part of this review, the topological derivative concept is presented in detail within the framework of the domain decomposition technique. Such an approach is constructive, for example, for coupled models in multiphysics as well as for contact problems in elasticity. In the second and third parts, we describe the first- and second-order numerical methods of shape and topology optimization for elliptic BVPs, together with a portfolio of applications and numerical examples in all the above-mentioned areas.

Appendices

Appendices

Formal asymptotic analysis for a scalar elliptic equation is presented. The same method is used for linear elasticity in [11].

1.1 Asymptotic Expansions of Solutions and Functionals

In previous sections, basic derivations are conducted for the simplest case of circular or ball-shaped voids, which nevertheless illustrate the most important features of the approach. In a similar way, ball-shaped penetrable inclusions with contrast $0< \gamma < \infty$ can be obtained [28].

If more general shapes of voids (inclusions) are required, the asymptotic analysis approach applies as well, as is shown below. We refer the reader to [34] for the general theory and examples of asymptotic expansions.

For the convenience of the reader, a two-scale asymptotic analysis of a nonhomogeneous boundary value problem is performed, for a simple model problem. The small cavity $\omega _\varepsilon :=\varepsilon \omega$ with center at the origin ${\mathcal {O}}\in \omega _\varepsilon \subset \omega$ can be considered without loss of generality. We denote by the same symbol $\omega _\varepsilon ({\widehat{x}}):={\widehat{x}}+\omega _\varepsilon$ the cavity with center at ${\widehat{x}}\in \varOmega$. Matched asymptotic expansions are used in two spatial dimensions for scalar problems with the Laplacian, where we consider singular perturbations of the principal part of the elliptic operator. In the case of a penetrable inclusion with contrast parameter $0< \gamma < \infty$, the results are obtained in a similar way, since it is a regular perturbation of the main part of the elliptic operator.

1.2 Asymptotic Expansions of Steklov–Poincaré Operators

We consider a smooth domain $\varOmega _\varepsilon :=\varOmega {\setminus }\overline{\omega _\varepsilon }$ for $\varepsilon \rightarrow 0$ and the nonhomogeneous Dirichlet problem with $h\in H^{1/2}(\varGamma )$,

$$\begin{aligned} \varDelta w_\varepsilon = 0&\quad \text {in} \,\, \varOmega _\varepsilon , \nonumber \\ w_\varepsilon = h&\quad \text {on} \,\, \varGamma ,\\ \displaystyle {\frac{\partial w_\varepsilon }{\partial n}} = 0&\quad \text {on} \,\, \partial \omega _\varepsilon ,\nonumber \end{aligned}$$

(29)

where $\varGamma = \partial \varOmega$ is the boundary of $\varOmega$.

The energy associated with (29) is given by a symmetric bilinear form on the fractional Sobolev space $H^{1/2}(\varGamma )$,

$$\begin{aligned} a_\varepsilon (h,h)=\displaystyle \int _{\varOmega _\varepsilon } \Vert \nabla w_\varepsilon \Vert ^2 \mathrm{d}x. \end{aligned}$$

We are interested in the asymptotic expansion of this quadratic functional for $\varepsilon \rightarrow 0$. To this end, the technique of matched asymptotic expansions [33, 34] is used.

Using Green’s formula, we derive equivalent forms of the energy (here, the boundary integrals stand for the duality pairing between $H^{1/2}(\varGamma )$ and its dual $H^{-1/2}(\varGamma )$):

$$\begin{aligned} a_\varepsilon (h,h)= \int _{\varGamma }w_\varepsilon \frac{\partial w_\varepsilon }{\partial n}\mathrm{d}s= \int _{\varGamma }h \frac{\partial w_\varepsilon }{\partial n}\mathrm{d}s. \end{aligned}$$

(30)

Now, we introduce two-scale asymptotic approximation of solutions. We use the method of matched asymptotic expansions and look for two types of expansions, the outer expansion valid far from the cavity $\omega _\varepsilon$,

$$\begin{aligned} w_\varepsilon (x)=w_0(x)+ \varepsilon ^2 w_1(x)+\varepsilon ^3w_2(x)+\cdots , \end{aligned}$$

and the inner expansion, valid in a small neighborhood of $\omega _\varepsilon$,

$$\begin{aligned} w_\varepsilon (x)=W_0(\xi )+ \varepsilon W_1(\xi )+\varepsilon ^2W(\xi )+\cdots , \end{aligned}$$

where the fast variable $\xi$ is defined by

$$\begin{aligned} \xi := \displaystyle {\frac{x}{\varepsilon }}. \end{aligned}$$

Following [33, 34], we obtain

$$\begin{aligned} W_0(\xi )\equiv w_0(0), \end{aligned}$$

and

$$\begin{aligned} W_1(\xi )=\sum _{j=1}^2{\mathcal {Y}}^j(\xi )\displaystyle {\frac{\partial w_0}{\partial x_j}(0)}, \end{aligned}$$

where ${\mathcal {Y}}^j$ is harmonic in ${\mathbb {R}}^2{\setminus }{\overline{\omega }}$ and $\omega :=\omega _1$. In addition, ${\mathcal {Y}}^j$ satisfies the homogeneous Neumann boundary conditions on $\partial \omega$ and enjoys the following behavior at infinity:

$$\begin{aligned} {\mathcal {Y}}^j(\xi )=\xi _j+ \displaystyle {\frac{1}{2\pi \Vert \xi \Vert ^2}}\sum _{k=1}^2m_{kj}^\omega \xi _k+O(\Vert \xi \Vert ^{-2}),\quad \Vert \xi \Vert \rightarrow \infty . \end{aligned}$$

Its regular part is denoted by

$$\begin{aligned} {\mathcal {Y}}_0^j(\xi ):= \displaystyle {\frac{1}{2\pi \Vert \xi \Vert ^2}}\sum _{k=1}^2m_{kj}^\omega \xi _k+O(\Vert \xi \Vert ^{-2}), \end{aligned}$$

and we denote its higher order term $O(\Vert \xi \Vert ^{-2})$ by

$$\begin{aligned} {{\mathfrak {y}}}_0^j(\xi ):={\mathcal {Y}}_0^j(\xi )- \displaystyle {\frac{1}{2\pi \Vert \xi \Vert ^2}}\sum _{k=1}^2m_{kj}^\omega \xi _k. \end{aligned}$$

Taking into account this expansion, we get

$$\begin{aligned} w_1(x)=\sum _{j,k=1}^2 \frac{\partial w_0}{\partial x_j}(0)m_{kj}^\omega {\mathcal {G}}^{(k)}(x). \end{aligned}$$

We denote by ${\mathcal {G}}^{(k)}$ the singular solutions to the problem posed in the punctured domain,

$$\begin{aligned} \begin{array}{ll} \varDelta _x {\mathcal {G}}^{(k)}(x) = 0 &{} \quad \text {in}\,\, \varOmega {\setminus }{{\mathcal {O}}}, \\ {\mathcal {G}}^{(k)}(x) = 0 &{}\quad \text {on}\,\, \varGamma , \\ {\mathcal {G}}^{(k)}(x) = \displaystyle {\frac{x_k}{2\pi \Vert x\Vert ^2}} + O(1)&{}\quad \Vert x\Vert \rightarrow 0. \end{array} \end{aligned}$$

(31)

We set

$$\begin{aligned} {\mathcal {G}}^{(k)}(x)= \displaystyle {\dfrac{x_k}{2\pi \Vert x\Vert ^2}} + {\mathcal {G}}_0^{(k)}(x), \end{aligned}$$

where ${\mathcal {G}}_0^{(k)}$ stands for the regular part. Therefore, far from the cavity $\omega _\varepsilon$, we have

$$\begin{aligned} w_\varepsilon (x)=w_0(x)+\varepsilon ^2\sum _{j,k=1}^2\dfrac{\partial w_0}{\partial x_j}(0)m_{kj}^\omega {\mathcal {G}}^{(k)}(x) +O(\varepsilon ^2). \end{aligned}$$

Substituting this representation into formula (30), we obtain the one-term expansion

$$\begin{aligned} a_\varepsilon (h,h)=a(h,h)+\varepsilon ^2b(h,h) + O(\varepsilon ^{3-\alpha }). \end{aligned}$$

Here, $\alpha \in ]0,1[$ and

$$\begin{aligned} b(h,h)=\displaystyle \int _\varGamma h(x)\sum _{j,k=1}^2\dfrac{\partial w_0}{\partial x_j}(0)m_{kj}^\omega \dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}(x)\mathrm{d}s. \end{aligned}$$

If we combine this with the integral equality on the sphere of radius $\delta >0$,

$$\begin{aligned} \displaystyle \int _{{\mathbb {S}}_\delta ({\mathcal {O}})} \left( x_j\dfrac{\partial }{\partial n} \displaystyle {\frac{x_k}{2\pi \Vert x\Vert ^2}}- \displaystyle {\frac{x_k}{2\pi \Vert x\Vert ^2}}\dfrac{\partial x_j}{\partial n} \right) \mathrm{d}s=\delta _{jk}, \end{aligned}$$

we get

$$\begin{aligned} b(h,h)=- m^\omega \nabla w_0(0) \cdot \nabla w_0(0). \end{aligned}$$

Since

$$\begin{aligned} \displaystyle \int _{\varGamma }w_0(x)\partial _n{\mathcal {G}}^{(k)}(x) \mathrm{d}s=-\dfrac{\partial w_0}{\partial x_k}(0), \end{aligned}$$

it follows that

$$\begin{aligned} b(h,h)=-\left( \displaystyle \int _{\varGamma } h(x)\partial _n{\mathcal {G}}^{(j)}(x) \mathrm{d}s \right) m^{\omega }_{jk} \left( \displaystyle \int _{\varGamma }h(x)\partial _n{\mathcal {G}}^{(k)}(x) \mathrm{d}s\right) \mathrm{d}s. \end{aligned}$$

Remark A.1

It can be shown that the following supremum taken with respect to the $H^{1/2}(\varGamma )$-norm is bounded with respect to $\varepsilon \rightarrow 0$:

$$\begin{aligned} \sup _{\Vert h \Vert \le 1}\left| a_\varepsilon (h,h)-a(h,h)-\varepsilon ^2b(h,h)\right| \le C_\alpha \varepsilon ^{3-\alpha }. \end{aligned}$$

Since the operators associated with the bilinear forms $(h,h) \mapsto a_\varepsilon (h,h)$ are positive and self-adjoint, the one-term expansion of Steklov–Poincaré operators is obtained for $\varepsilon \rightarrow 0$,

$$\begin{aligned} {\mathcal {A}}_\varepsilon = {\mathcal {A}} - \varepsilon ^2 {\mathcal {B}} + O(\varepsilon ^{3-\alpha }), \end{aligned}$$

with the remainder bounded in the operator norm $H^{1/2}(\varGamma ) \rightarrow H^{-1/2}(\varGamma )$. The self-adjoint positive linear operators ${\mathcal {A}}_\varepsilon$ are uniquely determined by the symmetric and coercive bilinear forms $h \mapsto a_\varepsilon (h,h)$. The operator ${\mathcal {B}}$ is determined by $h \mapsto b(h,h)$.

1.3 Asymptotic Expansion of a Linear Form

Let us now consider the linear form

$$\begin{aligned} L_\varepsilon (h)=\displaystyle \int _{\varOmega _\varepsilon } f(x)w_\varepsilon (x) \mathrm{d}x. \end{aligned}$$

We use the method of matched asymptotic expansions and set

$$\begin{aligned} w_\varepsilon (w)=w_0(x)+ \varepsilon \sum _{j=1}^{2}\dfrac{\partial w_0 }{\partial x_j}(0) {\mathcal {Y}}_0^j(\xi )+\varepsilon ^2\sum _{j,k=1}^{2} \dfrac{\partial w_0 }{\partial x_j}(0)m^{\omega }_{jk}{\mathcal {G}}_0^{(k)}(x) +\cdots , \end{aligned}$$

hence

$$\begin{aligned} L_\varepsilon (h)= & {} \displaystyle \int _{\varOmega _\varepsilon } f(x)w_0(x) \mathrm{d}x + \varepsilon \displaystyle \int _{\varOmega _\varepsilon } f(x)\sum _{j=1}^{2} \dfrac{\partial w_0 }{\partial x_j}(0) {\mathcal {Y}}_0^j(\xi ) \\&+\,\varepsilon ^2\displaystyle \int _{\varOmega _\varepsilon } f(x)\sum _{j,k=1}^{2} \dfrac{\partial w_0 }{\partial x_j}(0)m^{\omega }_{jk}{\mathcal {G}}_0^{(k)}(x)\mathrm{d}x. \end{aligned}$$

Taking into account that

$$\begin{aligned} \left| {\mathcal {Y}}_0^j(\xi )\right| \le C_0\dfrac{1}{\Vert \xi \Vert }=C_0\dfrac{\varepsilon }{\Vert x\Vert }\quad \text {in}\quad \varOmega _\varepsilon , \end{aligned}$$

it follows that

$$\begin{aligned} L_\varepsilon (h)= & {} \displaystyle \int _{\varOmega } f(x)w_0(x) \mathrm{d}x - \displaystyle \int _{\omega _\varepsilon } f(x)w_0(x) \mathrm{d}x \\&+\, \varepsilon ^2\displaystyle \int _{\varOmega } f(x)\sum _{j,k=1}^{2} \dfrac{\partial w_0 }{\partial x_j}(0)m^{\omega }_{jk}{\mathcal {G}}^{(k)}(x)\mathrm{d}x+\cdots \end{aligned}$$

In order to replace the integrals over $\varOmega _\varepsilon$ by integrals over $\varOmega$, we use the estimates

$$\begin{aligned} \varepsilon \displaystyle \int _{\omega _\varepsilon } f(x)\sum _{j,k=1}^{2}\dfrac{\partial w_0 }{\partial x_j}(0){m^{\omega }_{jk}}\dfrac{\varepsilon x_k}{2\pi \Vert x\Vert ^2}\mathrm{d}x\le C_0\varepsilon ^2 \sup _{x\in {\overline{\varOmega }}}|f(x)|\,\displaystyle \int _0^\varepsilon \dfrac{1}{r}r\mathrm{d}r \end{aligned}$$

and

$$\begin{aligned} \displaystyle \int _{\omega _\varepsilon } f(x)\sum _{j,k=1}^{2}\dfrac{\partial w_0 }{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}_0^{(k)}(x)\le C_0\varepsilon ^2. \end{aligned}$$

Finally,

$$\begin{aligned} L_\varepsilon (h)=L_0(h)-\varepsilon ^2 f(0)w_0(0)|\omega | + \varepsilon ^2\displaystyle \int _{\varOmega } f(x)\sum _{j,k=1}^{2} \dfrac{\partial w_0 }{\partial x_j}(0)m^{\omega }_{jk}{\mathcal {G}}^{(k)}(x)\mathrm{d}x+\cdots \end{aligned}$$

1.4 Energy Functional of the Nonhomogeneous Dirichlet Problem in a Perturbed Domain

The energy functional

$$\begin{aligned} {\mathcal {J}}_{\varOmega _\varepsilon }(u_\varepsilon )= \dfrac{1}{2}\displaystyle \int _{\varOmega _\varepsilon }\Vert \nabla u_\varepsilon \Vert ^2\mathrm{d}x - \displaystyle \int _{\varOmega _\varepsilon }fu_\varepsilon \mathrm{d}x = \dfrac{1}{2}\displaystyle \int _{\varGamma } u_\varepsilon \dfrac{\partial u_\varepsilon }{\partial n}\mathrm{d}s - \dfrac{1}{2}\displaystyle \int _{\varOmega _\varepsilon }fu_\varepsilon \mathrm{d}x \end{aligned}$$

depends on solutions to the boundary value problem

$$\begin{aligned} \begin{array}{ll} -\varDelta u_\varepsilon = f &{}\quad \text {in} \,\, \varOmega _\varepsilon ,\\ u_\varepsilon = h_\varepsilon &{}\quad \text {on} \,\, \varGamma , \\ \displaystyle {\frac{\partial u_\varepsilon }{\partial n}} = 0 &{}\quad \text {on} \,\, \partial \omega _\varepsilon . \end{array} \end{aligned}$$

(32)

with the associated Green formula

$$\begin{aligned} \displaystyle \int _{\varOmega _\varepsilon }\Vert \nabla u_\varepsilon \Vert ^2\mathrm{d}x = \displaystyle \int _{\varGamma } h_\varepsilon \dfrac{\partial u_\varepsilon }{\partial n}\mathrm{d}s + \displaystyle \int _{\varOmega _\varepsilon }fu_\varepsilon \mathrm{d}x. \end{aligned}$$

For completeness of our analysis, we assume that the Dirichlet boundary datum also depends on the small parameter,

$$\begin{aligned} h_\varepsilon =h_0+\varepsilon ^2 h_1+ o(\varepsilon ^2)\quad \text {in}\quad H^{1/2}(\varGamma ), \end{aligned}$$

and that there is a source term inside the perturbed domain $\varOmega _\varepsilon$.

The approximation of solutions takes the form

$$\begin{aligned} u_\varepsilon (x) = v_0(x)+\varepsilon ^2v_1(x)+\varepsilon \sum _{j=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){\mathcal {Y}}_0^j(\xi )+ \varepsilon ^2\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}_0^{(k)}(x)+\cdots \end{aligned}$$

where

$$\begin{aligned} \begin{array}{ll} -\varDelta v_0 = f &{}\quad \text {in} \,\, \varOmega , \\ v_0 = h_0 &{} \quad \text {on} \,\, \varGamma , \end{array} \quad \text {and} \quad \begin{array}{ll} -\varDelta v_1 = 0 &{}\quad \text {in} \,\, \varOmega , \\ v_1 = h_1 &{}\quad \text {on} \,\, \varGamma . \end{array} \end{aligned}$$

The approximation of the normal derivatives is

$$\begin{aligned} \begin{aligned} \dfrac{\partial u_\varepsilon }{\partial n}(x)=&\dfrac{\partial v_0}{\partial n}(x)+ \varepsilon ^2\dfrac{\partial v_1}{\partial n}(x)+ \varepsilon \sum _{j=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0)\dfrac{\partial {\mathcal {Y}}_0^j}{\partial \nu }(\xi )\\&+ \varepsilon ^2\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}\dfrac{\partial {\mathcal {G}}_0^{(k)}}{\partial n}(x)+\cdots \end{aligned} \end{aligned}$$

where $\dfrac{\partial }{\partial n}=n\cdot \nabla _x$, $\dfrac{\partial }{\partial \nu }=n\cdot \nabla _{\xi }$ and $\xi =x/\varepsilon$. We recall that the higher order term of ${\mathcal {Y}}_0^j(\xi )$ satisfies

$$\begin{aligned} \left| {\mathfrak {y}}_0^j(\xi )\right| \le C_0\dfrac{1}{\Vert \xi \Vert ^2}=C_0\dfrac{\varepsilon ^2}{\Vert x\Vert ^2} \quad \text {for}\quad \xi =\dfrac{x}{\varepsilon }\in {\mathbb {R}}^2{\setminus }\omega \quad \text {or for}\quad x\in \varOmega _\varepsilon . \end{aligned}$$

Thus, in the approximation of $u_\varepsilon (x)$, the terms of order $O(\varepsilon ^3)$,

$$\begin{aligned} \varepsilon \sum _{j=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){\mathfrak {y}}_0^j(\xi ) \end{aligned}$$

can be neglected. Therefore, from the formula

$$\begin{aligned} u_\varepsilon (x)=v_0(x)+\varepsilon ^2v_1(x)+\varepsilon \sum _{j=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){\mathfrak {y}}_0^j(\xi )+\varepsilon ^2\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x)+\cdots \end{aligned}$$

we deduce

$$\begin{aligned} \dfrac{\partial u_\varepsilon }{\partial n}(x)=\dfrac{\partial v_0}{\partial n}(x)+ \varepsilon ^2\dfrac{\partial v_1}{\partial n}(x)+ \varepsilon ^2\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}\dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}(x)+\cdots , \end{aligned}$$

and it follows that

$$\begin{aligned} u_\varepsilon (x)\dfrac{\partial u_\varepsilon }{\partial n}(x)= & {} v_0(x)\dfrac{\partial v_0}{\partial n}(x) \\&+\,\varepsilon ^2\left( v_1(x)\dfrac{\partial v_0}{\partial n}(x)+v_0(x)\dfrac{\partial v_1}{\partial n}(x) \right. \\&\left. +\, v_0(x)\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x)\right) +\cdots \end{aligned}$$

since the second-order term

$$\begin{aligned} \dfrac{\partial v_0}{\partial n}(x)\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x) \end{aligned}$$

vanishes, by taking into account that ${\mathcal {G}}^{(k)}(x)=0$ on the boundary $\varGamma$. We return to the shape functional,

$$\begin{aligned} {\mathcal {J}}_{\varOmega _\varepsilon }(u_\varepsilon ) = \dfrac{1}{2}\int _\varGamma u_\varepsilon (x)\dfrac{\partial u_\varepsilon }{\partial n}(x)\mathrm{d}s- \dfrac{1}{2}\int _{\varOmega _\varepsilon }fu_\varepsilon \mathrm{d}x, \end{aligned}$$

and find approximations for the integrals

$$\begin{aligned} \dfrac{1}{2}\int _\varGamma u_\varepsilon (x)\dfrac{\partial u_\varepsilon }{\partial n}(x)\mathrm{d}s= & {} \dfrac{1}{2}\int _\varGamma v_0(x)\dfrac{\partial v_0}{\partial n}(x)\mathrm{d}s \\&+\, \dfrac{\varepsilon ^2}{2}\int _\varGamma \left( v_1(x)\dfrac{\partial v_0}{\partial n}(x) + v_0(x)\dfrac{\partial v_1}{\partial n}(x)\right) \mathrm{d}s \\&+\, \dfrac{\varepsilon ^2}{2}\int _\varGamma v_0(x)\sum _{j,k=1}^{2} \dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}} \dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}(x)\mathrm{d}s +\cdots \end{aligned}$$

and

$$\begin{aligned} -\dfrac{1}{2}\int _{\varOmega _\varepsilon }f u_\varepsilon \mathrm{d}x= & {} -\dfrac{1}{2}\int _{\varOmega _\varepsilon }f v_0 \mathrm{d}x -\dfrac{\varepsilon ^2}{2}\int _{\varOmega _\varepsilon }f v_1 \mathrm{d}x \\&-\,\dfrac{\varepsilon ^2}{2}\int _{\varOmega _\varepsilon }f\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x)+\cdots , \end{aligned}$$

which can be written as

$$\begin{aligned} -\dfrac{1}{2}\int _{\varOmega _\varepsilon }f u_\varepsilon \mathrm{d}x= & {} -\dfrac{1}{2}\int _{\varOmega }f v_0 \mathrm{d}x+\dfrac{1}{2}\int _{\omega _\varepsilon }f v_0 \mathrm{d}x -\dfrac{\varepsilon ^2}{2}\int _{\varOmega }f v_1 \mathrm{d}x+\dfrac{\varepsilon ^2}{2}\int _{\omega _\varepsilon }f v_1 \mathrm{d}x \\&-\,\dfrac{\varepsilon ^2}{2}\int _{\varOmega }f\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x)\mathrm{d}x \\&+\,\dfrac{\varepsilon ^2}{2}\int _{\omega _\varepsilon }f\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x)\mathrm{d}x +\cdots \end{aligned}$$

or as

$$\begin{aligned} -\dfrac{1}{2}\int _{\varOmega _\varepsilon }f u_\varepsilon \mathrm{d}x= & {} -\dfrac{1}{2}\int _{\varOmega }f v_0 \mathrm{d}x+\dfrac{\varepsilon ^2}{2}f(0)v_0(0)|\omega | -\dfrac{\varepsilon ^2}{2}\int _{\varOmega }f v_1 \mathrm{d}x \\&-\,\dfrac{\varepsilon ^2}{2}\int _{\varOmega }f\sum _{j,k=1}^{2} \dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}{\mathcal {G}}^{(k)}(x)\mathrm{d}x +O(\varepsilon ^3). \end{aligned}$$

Here, we take into account that the Taylor formula $\dfrac{1}{2}\int _{\omega _\varepsilon }f v_0 \mathrm{d}x$ is replaced by $\dfrac{\varepsilon ^2}{2}f(0)v_0(0)|\omega |$. In the same way, it follows that $\dfrac{\varepsilon ^2}{2}\int _{\omega _\varepsilon }f v_1 \mathrm{d}x$ is $O(\varepsilon ^4)$; finally, the latter integral over $\omega _\varepsilon$ is $O\left( \int _0^\varepsilon \dfrac{1}{r} r\right)$. As a result,

$$\begin{aligned} {\mathcal {J}}_{\varOmega _\varepsilon }(u_\varepsilon )= & {} \dfrac{1}{2}\int _\varGamma v_0(x)\dfrac{\partial v_0}{\partial n}(x)\mathrm{d}s - \dfrac{1}{2}\int _{\varOmega }f v_0 \mathrm{d}x \\&+\, \dfrac{\varepsilon ^2}{2}\int _\varGamma \left( v_1(x)\dfrac{\partial v_0}{\partial n}(x)+v_0(x)\dfrac{\partial v_1}{\partial n}(x)\right) \mathrm{d}s\\&+\, \dfrac{\varepsilon ^2}{2}\int _\varGamma v_0(x)\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}} \dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}(x)\mathrm{d}s \\&-\, \dfrac{\varepsilon ^2}{2}\int _{\varOmega }f\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}} {\mathcal {G}}^{(k)}(x)\mathrm{d}x\\&+\,\dfrac{\varepsilon ^2}{2}f(0)v_0(0)|\omega | - \dfrac{\varepsilon ^2}{2}\int _{\varOmega }f v_1 \mathrm{d}x. \end{aligned}$$

We denote by $B_\delta ({\mathcal {O}})$ the ball at the origin of radius $\delta$, with boundary ${\mathbb {S}}_\delta :={\mathbb {S}}_\delta ({\mathcal {O}})$. By the Green formula in the domain $\varOmega _\delta =\varOmega {\setminus }\overline{B_\delta ({\mathcal {O}})}$ with boundary $\partial \varOmega _\delta =\varGamma \cup {\mathbb {S}}_\delta$, for $\delta \rightarrow 0$,

$$\begin{aligned} \int _{\varOmega _\delta }(v_0\varDelta {\mathcal {G}}^{(k)}-{\mathcal {G}}^{(k)}\varDelta v_0)\mathrm{d}x = \int _{\partial \varOmega _\delta }\left( v_0\dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}-{\mathcal {G}}^{(k)}\dfrac{\partial v_0}{\partial n}\right) \mathrm{d}s. \end{aligned}$$

Since $\varDelta v_0=-f$, we find

$$\begin{aligned} \int _{\varOmega _\delta }{\mathcal {G}}^{(k)} f \mathrm{d}x=\int _{\varGamma }v_0\dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}\mathrm{d}s+\int _{{\mathbb {S}}_\delta }\left( v_0\dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}-{\mathcal {G}}^{(k)}\dfrac{\partial v_0}{\partial n}\right) \mathrm{d}s. \end{aligned}$$

Passage to the limit $\delta \rightarrow 0$ leads to

$$\begin{aligned} \int _{\varOmega _\delta }{\mathcal {G}}^{(k)} f \mathrm{d}x-\int _{\varGamma }v_0\dfrac{\partial {\mathcal {G}}^{(k)}}{\partial n}\mathrm{d}s=\dfrac{\partial v_0}{\partial x_k}(0) \end{aligned}$$

Finally, we arrive at the expression

$$\begin{aligned} {\mathcal {J}}_{\varOmega _\varepsilon }(u_\varepsilon )= & {} {\mathcal {J}}_{\varOmega }(v_0)+\dfrac{\varepsilon ^2}{2}f(0)v_0(0)|\omega |- \dfrac{\varepsilon ^2}{2}\sum _{j,k=1}^{2}\dfrac{\partial v_0}{\partial x_j}(0){m^{\omega }_{jk}}\dfrac{\partial v_0}{\partial x_k}(0) \\&+\, \dfrac{\varepsilon ^2}{2}\displaystyle \int _{\varGamma }\left( v_1(x)\dfrac{\partial v_0}{\partial n}(x)+ v_0(x)\dfrac{\partial v_1}{\partial n}(x)\right) \mathrm{d}s\\&-\,\dfrac{\varepsilon ^2}{2}\displaystyle \int _{\varOmega }f(x)v_1(x) \mathrm{d}x +O(\varepsilon ^3). \end{aligned}$$

Remark A.2

The adjoint equations are commonly used in final formulas for topological derivatives. We refer the reader to[80] for related results. Some asymptotic expansions for the Laplace operator are given in [81, 82].

Remark A.3

The case of the elasticity system in the same framework of asymptotic analysis is considered in [11], where the results obtained are given with complete proofs.