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.
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References
Delfour, M.C., Zolésio, J.P.: Shapes and Geometries. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001)
Plotnikov, P., Sokołowski, J.: Compressible Navier–Stokes Equations. Theory and Shape Optimization. Springer, Basel (2012)
Sokołowski, J., Zolésio, J.P.: Introduction to Shape Optimization—Shape Sensitivity Analysis. Springer, Berlin (1992)
Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations and their Applications, vol. 65. Birkhäuser Boston, Inc., Boston, MA (2005)
Henrot, A., Pierre, M.: Variation et optimisation de formes, Mathématiques et applications, vol. 48. Springer, Heidelberg (2005)
Allaire, G.: Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, vol. 146. Springer, New York (2002)
Bendsøe, M.P.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin (1995)
Kogut, P.I., Leugering, G.: Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis. Springer, Berlin (2011)
Bendsøe, M.P., Sigmund, O.: Topology Optimization. Theory, Methods and Applications. Springer, Berlin (2003)
Aage, N., Andreassen, E., Lazarov, B.S., Sigmund, O.: Giga-voxel computational morphogenesis for structural design. Nature (2017). https://doi.org/10.1038/nature23911
Nazarov, S.A., Sokołowski, J.: Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82(2), 125–196 (2003)
Nazarov, S.A.: Asymptotic conditions at a point, selfadjoint extensions of operators, and the method of matched asymptotic expansions. Am. Math. Soc. Transl. 198, 77–125 (1999)
Nazarov, S.A.: Elasticity polarization tensor, surface enthalpy and Eshelby theorem. Probl. Mat. Anal. 41, 3–35 (2009). (English transl.: J. Math. Sci. 159(1–2), 133–167, (2009))
Nazarov, S.A.: The Eshelby theorem and a problem on an optimal patch. Algebra Anal. 21(5), 155–195 (2009). (English transl.: St. Petersburg Math. 21(5):791–818, (2009))
Amstutz, S.: Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49(1–2), 87–108 (2006)
Amstutz, S., Novotny, A.A.: Topological asymptotic analysis of the Kirchhoff plate bending problem. ESAIM: Control Optim. Calc. Var. 17(3), 705–721 (2011)
Amstutz, S., Novotny, A.A., Van Goethem, N.: Topological sensitivity analysis for elliptic differential operators of order \(2m\). J. Differ. Equ. 256, 1735–1770 (2014)
Feijóo, R.A., Novotny, A.A., Taroco, E., Padra, C.: The topological derivative for the Poisson’s problem. Math. Models Methods Appl. Sci. 13(12), 1825–1844 (2003)
Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39(6), 1756–1778 (2001)
Khludnev, A.M., Novotny, A.A., Sokołowski, J., Żochowski, A.: Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions. J. Mech. Phys. Solids 57(10), 1718–1732 (2009)
Lewinski, T., Sokołowski, J.: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40(7), 1765–1803 (2003)
Nazarov, S.A., Sokołowski, J.: Self-adjoint extensions for the Neumann Laplacian and applications. Acta Math. Sin. (Engl. Ser.) 22(3), 879–906 (2006)
Novotny, A.A.: Sensitivity of a general class of shape functional to topological changes. Mech. Res. Commun. 51, 1–7 (2013)
Novotny, A.A., Sales, V.: Energy change to insertion of inclusions associated with a diffusive/convective steady-state heat conduction problem. Math. Methods Appl. Sci. 39(5), 1233–1240 (2016)
Sales, V., Novotny, A.A., Rivera, J.E.M.: Energy change to insertion of inclusions associated with the Reissner–Mindlin plate bending model. Int. J. Solids Struct. 59, 132–139 (2013)
Sokołowski, J., Żochowski, A.: Optimality conditions for simultaneous topology and shape optimization. SIAM J. Control Optim. 42(4), 1198–1221 (2003)
Sokołowski, J., Żochowski, A.: Modelling of topological derivatives for contact problems. Numer. Math. 102(1), 145–179 (2005)
Novotny, A.A., Sokołowski, J.: Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer, Berlin (2013)
Amstutz, S.: Analysis of a level set method for topology optimization. Optim. Methods Softw. 26(4–5), 555–573 (2011)
Amstutz, S., Andrä, H.: A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216(2), 573–588 (2006)
Hintermüller, M.: Fast level set based algorithms using shape and topological sensitivity. Control Cybern. 34(1), 305–324 (2005)
Hintermüller, M., Laurain, A.: A shape and topology optimization technique for solving a class of linear complementarity problems in function space. Comput. Optim. Appl. 46(3), 535–569 (2010)
Il’in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. Translations of Mathematical Monographs, vol. 102. American Mathematical Society, Providence, RI (1992). (Translated from the Russian by V. V. Minachin)
Maz’ya, V.G., Nazarov, S.A., Plamenevskij, B.A.: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. 1, Operator Theory: Advances and Applications, vol. 111. Birkhäuser Verlag, Basel (2000). (Translated from the German by Georg Heinig and Christian Posthoff)
Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods. Vol. 1: Dimension Reduction and Integral Estimates. Nauchnaya Kniga, Novosibirsk (2001)
Nazarov, S.A., Plamenevskij, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries, de Gruyter Expositions in Mathematics, vol. 13. Walter de Gruyter & Co., Berlin (1994)
Sokołowski, J., Żochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)
Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals for elasticity systems. Mech. Struct. Mach. 29(3), 333–351 (2001)
Argatov, I.I., Sokolowski, J.: Asymptotics of the energy functional of the Signorini problem under a small singular perturbation of the domain. Comput. Math. Math. Phys. 43, 710–724 (2003)
Novotny, A.A., Feijóo, R.A., Padra, C., Taroco, E.: Topological sensitivity analysis. Comput. Methods Appl. Mech. Eng. 192(7–8), 803–829 (2003)
Samet, B., Amstutz, S., Masmoudi, M.: The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42(5), 1523–1544 (2003)
Guzina, B., Bonnet, M.: Topological derivative for the inverse scattering of elastic waves. Q. J. Mech. Appl. Math. 57(2), 161–179 (2004)
Ammari, H., Garnier, J., Jugnon, V., Kang, H.: Stability and resolution analysis for a topological derivative based imaging functional. SIAM J. Control Optim. 50(1), 48–76 (2012)
Ammari, H., Bretin, E., Garnier, J., Kang, H., Lee, H., Wahab, A.: Mathematical Methods in Elasticity Imaging. Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2015)
Amigo, R.C.R., Giusti, S., Novotny, A.A., Silva, E.C.N., Sokolowski, J.: Optimum design of flextensional piezoelectric actuators into two spatial dimensions. SIAM J. Control Optim. 52(2), 760–789 (2016)
Giusti, S., Mróz, Z., Sokolowski, J., Novotny, A.: Topology design of thermomechanical actuators. Struct. Multidiscip. Optim. 55, 1575–1587 (2017)
Nazarov, S., Sokolowski, J., Specovius-Neugebauer, M.: Polarization matrices in anisotropic heterogeneous elasticity. Asymptot. Anal. 68(4), 189–221 (2010)
Delfour, M.: Topological derivative: a semidifferential via the Minkowski content. J. Convex Anal. 25(3), 957–982 (2018)
Cardone, G., Nazarov, S., Sokolowski, J.: Asymptotic analysis, polarization matrices, and topological derivatives for piezoelectric materials with small voids. SIAM J. Control Optim. 48(6), 3925–3961 (2010)
Laurain, A., Nazarov, S., Sokolowski, J.: Singular perturbations of curved boundaries in three dimensions. The spectrum of the Neumann Laplacian. Z. Anal. Anwend. 30(2), 145–180 (2011)
Nazarov, S.A., Sokolowski, J.: Spectral problems in the shape optimisation. Singular boundary perturbations. Asymptot. Anal. 56(3–4), 159–204 (2008)
Nazarov, S., Sokołowski, J.: Selfadjoint extensions for the elasticity system in shape optimization. Bull. Pol. Acad. Sci. Math. 52(3), 237–248 (2004)
Nazarov, S., Sokolowski, J.: Modeling of topology variations in elasticity. In: System Modeling and Optimization, IFIP International Federation for Information Processing, vol. 166, pp. 147–158. Kluwer Academic Publishers, Boston, MA (2005)
Nazarov, S., Sokołowski, J.: Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization. Control Cybern. 34(3), 903–925 (2005)
Nazarov, S., Sokolowski, J.: Shape sensitivity analysis of eigenvalues revisited. Control Cybern. 37(4), 999–1012 (2008)
Argatov, I.I.: Asymptotic models for the topological sensitivity versus the topological derivative. Open Appl. Math. J. 2, 20–25 (2008)
Pavlov, B.S.: The theory of extensions, and explicitly solvable models. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 42(6(258)), 99–131, 247 (1987)
Novotny, A.A., Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals. Part II: first order method and applications. J. Optim. Theory Appl. 180(3), 1–30 (2019)
Novotny, A.A., Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals. Part III: second order method and applications. J. Optim. Theory Appl. 181, 1–22 (2019)
Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005)
Ammari, H., Kang, H., Nakamura, G., Tanuma, K.: Complete asymptotic expansions of solutions of the system of elastostatics in the presence of inhomogeneities of small diameter. J. Elast. 67, 97–129 (2002)
Beretta, E., Bonnetier, E., Francini, E., Mazzucato, A.L.: Small volume asymptotics for anisotropic elastic inclusions. Inverse Probl. Imaging 6(1), 1–23 (2012)
Schneider, M., Andrä, H.: The topological gradient in anisotropic elasticity with an eye towards lightweight design. Math. Methods Appl. Sci. 37, 1624–1641 (2014)
Khludnev, A.M., Sokołowski, J.: Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. Eur. J. Mech. A/Solids 19, 105–119 (2000)
Khludnev, A.M., Sokołowski, J.: On differentation of energy functionals in the crack theory with possible contact between crack faces. J. Appl. Math. Mech. 64(3), 464–475 (2000)
Khludnev, A.M., Kovtunenko, V.A.: Analysis of Cracks in Solids. WIT Press, Southampton (2000)
Khludnev, A.M., Sokołowski, J.: Modelling and Control in Solid Mechanics. Birkhauser, Basel (1997)
Leugering, G., Sokołowski, J., Żochowski, A.: Control of crack propagation by shape-topological optimization. Discrete Contin. Dyn. Syst. Ser. A 35(6), 2625–2657 (2015)
Bouchitté, G., Fragalà, I., Lucardesi, I.: A variational method for second order shape derivatives. SIAM J. Control Optim. 54(2), 1056–1084 (2016). https://doi.org/10.1137/15100494X
Bouchitté, G., Fragalà, I., Lucardesi, I.: Shape derivatives for minima of integral functionals. Math. Program. 148(1–2, Ser. B), 111–142 (2014). https://doi.org/10.1007/s10107-013-0712-6
Haraux, A.: How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities. J. Math. Soc. Jpn. 29(4), 615–631 (1977)
Mignot, F.: Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22(2), 130–185 (1976)
Sokołowski, J., Zolésio, J.P.: Dérivée par rapport au domaine de la solution d’un problème unilatéral [shape derivative for the solutions of variational inequalities]. C. R. Acad. Sci. Paris Sér. I Math. 301(4), 103–106 (1985)
Frémiot, G., Horn, W., Laurain, A., Rao, M., Sokołowski, J.: On the analysis of boundary value problems in nonsmooth domains. Dissertationes Mathematicae (Rozprawy Matematyczne) vol. 462, p. 149 (2009)
Ammari, H., Khelifi, A.: Electromagnetic scattering by small dielectric inhomogeneities. J. Math. Pures Appl. 82, 749–842 (2003)
Bellis, C., Bonnet, M., Cakoni, F.: Acoustic inverse scattering using topological derivative of far-field measurements-based \(L^2\) cost functionals. Inverse Probl. 29, 075012 (2013)
Vogelius, M.S., Volkov, D.: Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter. ESAIM Math. Model. Numer. Anal. 37, 723–748 (2000)
Kurasov, P., Posilicano, A.: Finite speed of propagation and local boundary conditions for wave equations with point interactions. Proc. Am. Math. Soc. 133(10), 3071–3078 (2005). https://doi.org/10.1090/S0002-9939-05-08063-9
Amstutz, S., Bonnafé, A.: Topological derivatives for a class of quasilinear elliptic equations. J. Math. Pures Appl. 107, 367–408 (2017)
Larnier, S., Masmoudi, M.: The extended adjoint method. ESAIM Math. Model. Numer. Anal. 47, 83–108 (2013)
Buscaglia, G.C., Ciuperca, I., Jai, M.: Topological asymptotic expansions for the generalized poisson problem with small inclusions and applications in lubrication. Inverse Probl. 23(2), 695–711 (2007)
Céa, J., Garreau, S., Guillaume, P., Masmoudi, M.: The shape and topological optimizations connection. Comput. Methods Appl. Mech. Eng. 188(4), 713–726 (2000)
Acknowledgements
This research was partly supported by CNPq (Brazilian Research Council), CAPES (Brazilian Higher Education Staff Training Agency) and FAPERJ (Research Foundation of the State of Rio de Janeiro). These supports are gratefully acknowledged. The authors are indebted to the referee and the editors of JOTA for constructive criticism which allowed them to improve the presentation of this difficult subject.
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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 \(h\in H^{1/2}(\varGamma )\),
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 \(H^{1/2}(\varGamma )\),
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 \(H^{-1/2}(\varGamma )\)):
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 \(\omega _\varepsilon \),
and the inner expansion, valid in a small neighborhood of \(\omega _\varepsilon \),
where the fast variable \(\xi \) is defined by
and
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:
Its regular part is denoted by
and we denote its higher order term \(O(\Vert \xi \Vert ^{-2})\) by
Taking into account this expansion, we get
We denote by \({\mathcal {G}}^{(k)}\) the singular solutions to the problem posed in the punctured domain,
We set
where \({\mathcal {G}}_0^{(k)}\) stands for the regular part. Therefore, far from the cavity \(\omega _\varepsilon \), we have
Substituting this representation into formula (30), we obtain the one-term expansion
Here, \(\alpha \in ]0,1[\) and
If we combine this with the integral equality on the sphere of radius \(\delta >0\),
we get
Since
it follows that
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 \(\varepsilon \rightarrow 0\):
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 \(\varepsilon \rightarrow 0\),
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
We use the method of matched asymptotic expansions and set
hence
Taking into account that
it follows that
In order to replace the integrals over \(\varOmega _\varepsilon \) by integrals over \(\varOmega \), we use the estimates
and
Finally,
1.4 Energy Functional of the Nonhomogeneous Dirichlet Problem in a Perturbed Domain
The energy functional
depends on solutions to the boundary value problem
with the associated Green formula
For completeness of our analysis, we assume that the Dirichlet boundary datum also depends on the small parameter,
and that there is a source term inside the perturbed domain \(\varOmega _\varepsilon \).
The approximation of solutions takes the form
where
The approximation of the normal derivatives is
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
Thus, in the approximation of \(u_\varepsilon (x)\), the terms of order \(O(\varepsilon ^3)\),
can be neglected. Therefore, from the formula
we deduce
and it follows that
since the second-order term
vanishes, by taking into account that \({\mathcal {G}}^{(k)}(x)=0\) on the boundary \(\varGamma \). We return to the shape functional,
and find approximations for the integrals
and
which can be written as
or as
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,
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 \(\delta \rightarrow 0\),
Since \(\varDelta v_0=-f\), we find
Passage to the limit \(\delta \rightarrow 0\) leads to
Finally, we arrive at the expression
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.
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Novotny, A.A., Sokołowski, J. & Żochowski, A. Topological Derivatives of Shape Functionals. Part I: Theory in Singularly Perturbed Geometrical Domains. J Optim Theory Appl 180, 341–373 (2019). https://doi.org/10.1007/s10957-018-1417-z
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DOI: https://doi.org/10.1007/s10957-018-1417-z