Journal of Optimization Theory and Applications

, Volume 180, Issue 2, pp 341–373

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

• Antonio André Novotny
• Jan Sokołowski
• Antoni Żochowski
Invited Paper

## 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.

## Keywords

Topological derivatives Asymptotic analysis Singular perturbations Domain decomposition

## Mathematics Subject Classification

35C20 35J15 35S05 49J40 49Q12

## Notes

### 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.

## References

1. 1.
Delfour, M.C., Zolésio, J.P.: Shapes and Geometries. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2001)
2. 2.
Plotnikov, P., Sokołowski, J.: Compressible Navier–Stokes Equations. Theory and Shape Optimization. Springer, Basel (2012)
3. 3.
Sokołowski, J., Zolésio, J.P.: Introduction to Shape Optimization—Shape Sensitivity Analysis. Springer, Berlin (1992)
4. 4.
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)
5. 5.
Henrot, A., Pierre, M.: Variation et optimisation de formes, Mathématiques et applications, vol. 48. Springer, Heidelberg (2005)
6. 6.
Allaire, G.: Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, vol. 146. Springer, New York (2002)
7. 7.
Bendsøe, M.P.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin (1995)
8. 8.
Kogut, P.I., Leugering, G.: Optimal Control Problems for Partial Differential Equations on Reticulated Domains: Approximation and Asymptotic Analysis. Springer, Berlin (2011)
9. 9.
Bendsøe, M.P., Sigmund, O.: Topology Optimization. Theory, Methods and Applications. Springer, Berlin (2003)
10. 10.
Aage, N., Andreassen, E., Lazarov, B.S., Sigmund, O.: Giga-voxel computational morphogenesis for structural design. Nature (2017).
11. 11.
Nazarov, S.A., Sokołowski, J.: Asymptotic analysis of shape functionals. J. Math. Pures Appl. 82(2), 125–196 (2003)
12. 12.
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)
13. 13.
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))
14. 14.
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))Google Scholar
15. 15.
Amstutz, S.: Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49(1–2), 87–108 (2006)
16. 16.
Amstutz, S., Novotny, A.A.: Topological asymptotic analysis of the Kirchhoff plate bending problem. ESAIM: Control Optim. Calc. Var. 17(3), 705–721 (2011)
17. 17.
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)
18. 18.
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)
19. 19.
Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39(6), 1756–1778 (2001)
20. 20.
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)
21. 21.
Lewinski, T., Sokołowski, J.: Energy change due to the appearance of cavities in elastic solids. Int. J. Solids Struct. 40(7), 1765–1803 (2003)
22. 22.
Nazarov, S.A., Sokołowski, J.: Self-adjoint extensions for the Neumann Laplacian and applications. Acta Math. Sin. (Engl. Ser.) 22(3), 879–906 (2006)
23. 23.
Novotny, A.A.: Sensitivity of a general class of shape functional to topological changes. Mech. Res. Commun. 51, 1–7 (2013)Google Scholar
24. 24.
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)
25. 25.
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)Google Scholar
26. 26.
Sokołowski, J., Żochowski, A.: Optimality conditions for simultaneous topology and shape optimization. SIAM J. Control Optim. 42(4), 1198–1221 (2003)
27. 27.
Sokołowski, J., Żochowski, A.: Modelling of topological derivatives for contact problems. Numer. Math. 102(1), 145–179 (2005)
28. 28.
Novotny, A.A., Sokołowski, J.: Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer, Berlin (2013)
29. 29.
Amstutz, S.: Analysis of a level set method for topology optimization. Optim. Methods Softw. 26(4–5), 555–573 (2011)
30. 30.
Amstutz, S., Andrä, H.: A new algorithm for topology optimization using a level-set method. J. Comput. Phys. 216(2), 573–588 (2006)
31. 31.
Hintermüller, M.: Fast level set based algorithms using shape and topological sensitivity. Control Cybern. 34(1), 305–324 (2005)
32. 32.
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)
33. 33.
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)
34. 34.
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)Google Scholar
35. 35.
Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods. Vol. 1: Dimension Reduction and Integral Estimates. Nauchnaya Kniga, Novosibirsk (2001)Google Scholar
36. 36.
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)Google Scholar
37. 37.
Sokołowski, J., Żochowski, A.: On the topological derivative in shape optimization. SIAM J. Control Optim. 37(4), 1251–1272 (1999)
38. 38.
Sokołowski, J., Żochowski, A.: Topological derivatives of shape functionals for elasticity systems. Mech. Struct. Mach. 29(3), 333–351 (2001)
39. 39.
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)
40. 40.
Novotny, A.A., Feijóo, R.A., Padra, C., Taroco, E.: Topological sensitivity analysis. Comput. Methods Appl. Mech. Eng. 192(7–8), 803–829 (2003)
41. 41.
Samet, B., Amstutz, S., Masmoudi, M.: The topological asymptotic for the Helmholtz equation. SIAM J. Control Optim. 42(5), 1523–1544 (2003)
42. 42.
Guzina, B., Bonnet, M.: Topological derivative for the inverse scattering of elastic waves. Q. J. Mech. Appl. Math. 57(2), 161–179 (2004)
43. 43.
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)
44. 44.
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)
45. 45.
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)
46. 46.
Giusti, S., Mróz, Z., Sokolowski, J., Novotny, A.: Topology design of thermomechanical actuators. Struct. Multidiscip. Optim. 55, 1575–1587 (2017)
47. 47.
Nazarov, S., Sokolowski, J., Specovius-Neugebauer, M.: Polarization matrices in anisotropic heterogeneous elasticity. Asymptot. Anal. 68(4), 189–221 (2010)
48. 48.
Delfour, M.: Topological derivative: a semidifferential via the Minkowski content. J. Convex Anal. 25(3), 957–982 (2018)
49. 49.
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)
50. 50.
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)
51. 51.
Nazarov, S.A., Sokolowski, J.: Spectral problems in the shape optimisation. Singular boundary perturbations. Asymptot. Anal. 56(3–4), 159–204 (2008)
52. 52.
Nazarov, S., Sokołowski, J.: Selfadjoint extensions for the elasticity system in shape optimization. Bull. Pol. Acad. Sci. Math. 52(3), 237–248 (2004)
53. 53.
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)Google Scholar
54. 54.
Nazarov, S., Sokołowski, J.: Self-adjoint extensions of differential operators and exterior topological derivatives in shape optimization. Control Cybern. 34(3), 903–925 (2005)
55. 55.
Nazarov, S., Sokolowski, J.: Shape sensitivity analysis of eigenvalues revisited. Control Cybern. 37(4), 999–1012 (2008)
56. 56.
Argatov, I.I.: Asymptotic models for the topological sensitivity versus the topological derivative. Open Appl. Math. J. 2, 20–25 (2008)
57. 57.
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)Google Scholar
58. 58.
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)
59. 59.
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)Google Scholar
60. 60.
Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005)Google Scholar
61. 61.
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)
62. 62.
Beretta, E., Bonnetier, E., Francini, E., Mazzucato, A.L.: Small volume asymptotics for anisotropic elastic inclusions. Inverse Probl. Imaging 6(1), 1–23 (2012)
63. 63.
Schneider, M., Andrä, H.: The topological gradient in anisotropic elasticity with an eye towards lightweight design. Math. Methods Appl. Sci. 37, 1624–1641 (2014)
64. 64.
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)
65. 65.
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)Google Scholar
66. 66.
Khludnev, A.M., Kovtunenko, V.A.: Analysis of Cracks in Solids. WIT Press, Southampton (2000)Google Scholar
67. 67.
Khludnev, A.M., Sokołowski, J.: Modelling and Control in Solid Mechanics. Birkhauser, Basel (1997)
68. 68.
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)
69. 69.
Bouchitté, G., Fragalà, I., Lucardesi, I.: A variational method for second order shape derivatives. SIAM J. Control Optim. 54(2), 1056–1084 (2016).
70. 70.
Bouchitté, G., Fragalà, I., Lucardesi, I.: Shape derivatives for minima of integral functionals. Math. Program. 148(1–2, Ser. B), 111–142 (2014).
71. 71.
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)
72. 72.
Mignot, F.: Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22(2), 130–185 (1976)
73. 73.
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)
74. 74.
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)Google Scholar
75. 75.
Ammari, H., Khelifi, A.: Electromagnetic scattering by small dielectric inhomogeneities. J. Math. Pures Appl. 82, 749–842 (2003)
76. 76.
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)
77. 77.
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)
78. 78.
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).
79. 79.
Amstutz, S., Bonnafé, A.: Topological derivatives for a class of quasilinear elliptic equations. J. Math. Pures Appl. 107, 367–408 (2017)
80. 80.
Larnier, S., Masmoudi, M.: The extended adjoint method. ESAIM Math. Model. Numer. Anal. 47, 83–108 (2013)
81. 81.
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)
82. 82.
Céa, J., Garreau, S., Guillaume, P., Masmoudi, M.: The shape and topological optimizations connection. Comput. Methods Appl. Mech. Eng. 188(4), 713–726 (2000)

## Authors and Affiliations

• Antonio André Novotny
• 1
• Jan Sokołowski
• 2
• 3
• Antoni Żochowski
• 3
1. 1.Laboratório Nacional de Computação Científica LNCC/MCTCoordenação de Matemática Aplicada e ComputacionalPetrópolisBrazil
2. 2.UMR 7502 Laboratoire de Mathématiques, Institut Élie CartanUniversité de LorraineVandoeuvre Lès Nancy CedexFrance
3. 3.Systems Research InstitutePolish Academy of SciencesWarsawPoland