Asymptotic analysis of shape functionals. (English) Zbl 1031.35020
Shape analysis goes back to early 1970-s, when sensitivity of fracture dependent on shape was studied by Novozhilov and others, who introduced the idea of shape functionals. Banichuk and the Danish school (Bensoe, Olhoff,…), J. P. Zolesio, and his coauthors, studied sensitivity with respect to changes of shape, making the shape of the boundary depend on a “small” parameter. Here a crucial problem arises of sensitivity of functionals to small perturbations of the boundary, when the boundaries are singularly perturbed. The domain contains small cavities, contributing to the complexity of the optimization problem. The present authors expand ideas considering asymptotic expansions in a topological space, in particular of matched asymptotic expansions. The novel feature of a series of papers of the authors and their associates is their use of the concept of topological derivative which in the understanding of the reviewer is a variation, or perhaps a generalization of the Gâteaux derivative.
Several ideas of this rather lengthy paper are continuation of a series of many previous papers of the authors. The past coauthors in papers on related topics include I. S. Zorin, A. Zochowski, W. G. Mazya, J. P. Zolesio.
Several ideas of this rather lengthy paper are continuation of a series of many previous papers of the authors. The past coauthors in papers on related topics include I. S. Zorin, A. Zochowski, W. G. Mazya, J. P. Zolesio.
Reviewer: Vadim Komkov (Florida)
MSC:
| 35B27 | Homogenization in context of PDEs; PDEs in media with periodic structure |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 74P15 | Topological methods for optimization problems in solid mechanics |
Keywords:
shape optimization; weighted function spaces; asymptotic behaviour; small perturbations of the boundary; matched asymptotic expansions; topological derivativeReferences:
| [1] | Allaire, G.; Bonnetier, E.; Francfort, G.; Jouve, F., Shape optimization by the homogenization method, Numer. Math., 76, 27-68 (1997) · Zbl 0889.73051 |
| [2] | Argatov, I. I., Integral characteristics of rigid inclusions and cavities in the two-dimensional theory of elasticity, Prikl. Mat. Mekh.. Prikl. Mat. Mekh., J. Appl. Math. Mech., 62, 263-268 (1998), English transl. · Zbl 1050.74534 |
| [3] | Argatov, I. I., Refinement of the asymptotic solution obtained by the method of matched expansions in contact problem of elasticity theory, Zh. Vychisl. Mat. Mat. Fiz.. Zh. Vychisl. Mat. Mat. Fiz., Comput. Math. Math. Phys., 40, 594-603 (2000), English transl. · Zbl 0998.74050 |
| [4] | Bendsoe, M. Ph., Optimization of Structural Topology, Shape and Material (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0822.73001 |
| [5] | Berezin, F. A.; Faddeev, L. D., Remark on the Schrödinger equation with singular potential, Dokl. Akad. Nauk SSSR. Dokl. Akad. Nauk SSSR, Soviet Math. Dokl., 2, 372-375 (1961), English transl. · Zbl 0117.06601 |
| [6] | Borchers, W.; Pileckas, K. I., Existence,uniqueness and asymptotics of steady jets, Arch. Rat. Mech. Anal., 120, 1-49 (1992) · Zbl 0772.76021 |
| [7] | Campbell, A.; Nazarov, S. A., Une justification de la méthode de raccordement des développements asymptotiques appliquée à un problème de plaque en flexion. Estimation de la matrice d’impédance, J. Math. Pures Appl., 76, 15-54 (1997) · Zbl 0877.35125 |
| [8] | Campbell, A.; Nazarov, S. A., Comportement d’une plaque élastique dont une petite région est rigide et animée d’un mouvement vibratoire. Étude asymptotique de la matrice d’impédance, Ann. Fac. Sci. Toulouse Math. Sér. (6), 4, 2, 211-242 (1995) · Zbl 0834.73041 |
| [9] | Cherkaev, A. V.; Grabovsky, Y.; Movchan, A. B.; Serkov, S. K., The cavity of the optimal shape under the shear stresses, Internat. J. Solids Structures, 25, 4391-4410 (1999) · Zbl 0917.73051 |
| [10] | Delfour, M. C.; Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization, Adv. Design Control (2001), SIAM: SIAM Philadelphia · Zbl 1002.49029 |
| [11] | Eschenauer, H. A.; Kobelev, V. V.; Schumacher, A., Bubble method for topology and shape optimization of structures, Struct. Optim., 8, 42-51 (1994) |
| [12] | Gadyl’shin, R. R., Existence and asymptotics of poles with a small imaginary part for the Helmholtz resonator, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Russian Math. Surveys, 52, 1, 1-72 (1997), in Russian; English transl. · Zbl 0902.35025 |
| [13] | Garreau, S.; Guillaume, Ph.; Masmoudi, M., The topological asymptotic for PDE systems: the elasticity case, SIAM J. Control Optim., 39, 6, 1756-1778 (2001) · Zbl 0990.49028 |
| [14] | Il’in, A. M., A boundary value problem for the elliptic equation of second order in a domain with a narrow slit. I. The two-dimensional case, Mat. Sb.. Mat. Sb., Math. USSR-Sb., 28, 141, 514-537 (1976), English transl. · Zbl 0333.35033 |
| [15] | Il’in, A. M., Study of the asymptotic behavior of the solution of an elliptic boundary value problem in a domain with a small hole, Trudy Sem. Petrovsk., 6, 57-82 (1981), in Russian · Zbl 0461.35032 |
| [16] | Il’in, A. M., Matching of Asymptotic Expansions of Solutions of Boundary Value Problems, Transl. Math. Monogr., 102 (1992), Amer. Math. Society · Zbl 0754.34002 |
| [17] | L. Jackowska-Strumiłło, J. Sokołowski, A. Żochowski, The topological derivative method and artificial neural networks for numerical solution of shape inverse problems, RR-3739, INRIA-Lorraine, 1999; L. Jackowska-Strumiłło, J. Sokołowski, A. Żochowski, The topological derivative method and artificial neural networks for numerical solution of shape inverse problems, RR-3739, INRIA-Lorraine, 1999 · Zbl 1108.74378 |
| [18] | Jackowska-Strumiłło, L.; Sokołowski, J.; Żochowski, A.; Henrot, A., On numerical solution of shape inverse problems, Comput. Optim. Appl., 23, 231-255 (2002) · Zbl 1033.65048 |
| [19] | Kachanov, M.; Tsukrov, I.; Shafiro, B., Effective moduli of solids with cavities of various shapes, Appl. Mech. Rev., 47, 1, S151-S174 (1994) |
| [20] | Kamotski, I. V.; Nazarov, S. A., Spectral problems in singular perturbed domains and selfadjoint extensions of differential operators, Trudy St.-Petersburg Mat. Obshch.. Trudy St.-Petersburg Mat. Obshch., Proc. St. Petersburg Math. Soc.. (Amer. Math. Soc. Transl. Ser., 2 (1999), Amer. Math. Society: Amer. Math. Society Providence, RI), 6, 127-181 (2000), English transl. · Zbl 1288.35363 |
| [21] | Kanaun, S. K.; Levin, V. M., The Effective Field Method in the Mechanics of Composite Materials (1993), Izdatel’stvo Petrozavodskogo Universiteta: Izdatel’stvo Petrozavodskogo Universiteta Petrozavodsk, in Russian · Zbl 0879.73001 |
| [22] | Kondrat’ev, V. A., Boundary problems for elliptic equations in domains with conical or angular points, Trudy Moskov. Mat. Obshch.. Trudy Moskov. Mat. Obshch., Trans. Moscow Math. Soc., 16, 227-313 (1967), English transl. · Zbl 0194.13405 |
| [23] | Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., Elliptic Boundary Value Problems in Domains with Point Singularities (1997), Amer. Math. Society: Amer. Math. Society Providence, RI · Zbl 0947.35004 |
| [24] | Kupradze, V.; Gegelia, T.; Basheleishvili, M.; Burchuladze, T., Three-Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity (1979), North-Holland: North-Holland Amsterdam · Zbl 0406.73001 |
| [25] | Leguillon, D.; Sanches-Palencia, E., Computation of Singular Solutions in Elliptic Problems and Elasticity (1987), Masson: Masson Paris · Zbl 0647.73010 |
| [26] | T. Lewinski, J. Sokolowski, Optimal shells formed on a sphere. The topological derivative method. RR-3495, INRIA-Lorraine, 1998; T. Lewinski, J. Sokolowski, Optimal shells formed on a sphere. The topological derivative method. RR-3495, INRIA-Lorraine, 1998 |
| [27] | Lewinski, T.; Sokolowski, J., Topological derivative for nucleation of non-circular voids, (Contemp. Math., 268 (2000), Amer. Math. Society), 341-361 · Zbl 1050.49028 |
| [28] | T. Lewinski, J. Sokolowski, Energy change due to appearing of cavities in elastic solids, Les prépublications de l’Institut Élie Cartan 23/2001, Internat. J. Solids Structures, submitted; T. Lewinski, J. Sokolowski, Energy change due to appearing of cavities in elastic solids, Les prépublications de l’Institut Élie Cartan 23/2001, Internat. J. Solids Structures, submitted |
| [29] | T. Lewinski, J. Sokolowski, A. Żochowski, Justification of the bubble method for the compliance minimization problems of plates and spherical shells, CD-Rom, 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) Buffalo/Niagara Falls, New York, May 17-21, 1999; T. Lewinski, J. Sokolowski, A. Żochowski, Justification of the bubble method for the compliance minimization problems of plates and spherical shells, CD-Rom, 3rd World Congress of Structural and Multidisciplinary Optimization (WCSMO-3) Buffalo/Niagara Falls, New York, May 17-21, 1999 |
| [30] | Lions, J. L.; Magenes, E., Problèmes aux Limites non Homogènes (1968), Dunod: Dunod Paris · Zbl 0165.10801 |
| [31] | Maz’ya, V. G.; Plamenevskii, B. A., Estimates in \(L_p\) and in Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary, Math. Nachr.. (Amer. Math. Soc. Transl. Ser. 2, 123 (1984)), 81, 1-56 (1978), English transl. · Zbl 0554.35035 |
| [32] | Maz’ya, V. G.; Morozov, N. F.; Plamenevskii, B. A., On nonlinear bending of a plate with a crack, (Differential and Integral Equations. Boundary Value Problems (I.N. Vekua Memorial collection) (1979), Tbilisi). (Amer. Math. Soc. Transl. Ser. 2, 123 (1984)), 125-139, English transl. · Zbl 0552.73043 |
| [33] | Maz’ya, V. G.; Nazarov, S. A., The asymptotic behavior of energy integrals under small perturbations of the boundary near corner points and conical points, Trudy Moskov. Mat. Obshch.. Trudy Moskov. Mat. Obshch., Trans. Moscow Math. Soc., 50, 77-127 (1988), English transl. · Zbl 0699.35072 |
| [34] | Maz’ya, V. G.; Nazarov, S. A.; Plamenevskii, B. A., Asymptotic behavior of solutions of a quasi-linear equation in nonregular perturbed domains, Differentsial’nye Uravnenia i Primenen., 27, 17-50 (1980), in Russian · Zbl 0463.35038 |
| [35] | Maz’ya, V. G.; Nazarov, S. A.; Plamenevskii, B. A., On the asymptotic behavior of solutions of elliptic boundary value problems with irregular perturbations of the domain, Probl. Mat. Anal., 8, 72-153 (1981), in Russian · Zbl 0491.35013 |
| [36] | Maz’ya, V. G.; Nazarov, S. A.; Plamenevskii, B. A., Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes, Izv. Akad. Nauk SSSR. Ser. Mat.. Izv. Akad. Nauk SSSR. Ser. Mat., Math. USSR-Izv., 24, 2, 321-345 (1985), English transl. · Zbl 0566.35031 |
| [37] | Maz’ya, V. G.; Nazarov, S. A.; Plamenevskii, B. A., Asymptotics of Solutions to Elliptic Boundary-Value Problems Under a Singular Perturbation of the Domain (1981), Tbilisi University: Tbilisi University Tbilisi, in Russian · Zbl 0462.35001 |
| [38] | Mazja, W. G.; Nazarov, S. A.; Plamenevskii, B. A., Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. Band 1, Asymptotic Theory of elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. 1 (2000), Akademie-Verlag: Akademie-Verlag Berlin: Birkhäuser: Akademie-Verlag: Akademie-Verlag Berlin: Birkhäuser Basel, English transl. |
| [39] | Mazja, W. G.; Nazarov, S. A.; Plamenevskii, B. A., Asymptotische Theorie elliptischer Randwertaufgaben in singulär gestörten Gebieten. Band 2, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. 2 (2000), Akademie-Verlag: Akademie-Verlag Berlin: Birkhäuser: Akademie-Verlag: Akademie-Verlag Berlin: Birkhäuser Basel, English transl. · Zbl 1127.35301 |
| [40] | Mazja, W. G.; Nazarov, S. A.; Plamenevskii, B. A., On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone, Mat. Sb.. Mat. Sb., Math. USSR-Sb., 50, 4, 415-437 (1985), English transl. · Zbl 0599.35056 |
| [41] | Movchan, A. B., Wiener polarization and capacity matrices for the operator of the theory of elasticity in doubly connected regions, Mat. Zametki, 47, 2, 151-153 (1990) · Zbl 0712.73012 |
| [42] | Movchan, A. B., Integral characteristics of elastic inclusions and cavities in the two-dimensional theory of elasticity, European J. Appl. Math., 3, 21-30 (1992) · Zbl 0753.73010 |
| [43] | Nazarov, S. A.; Plamenevsky, B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, De Gruyter Exp. Math., 13 (1994), de Gruyter · Zbl 0806.35001 |
| [44] | Nazarov, S. A.; Polyakova, O. R., Rupture criteria, asymptotic conditions at crack tips, and selfadjoint extensions of the Lamé operator, Trudy Moskov. Mat. Obshch.. Trudy Moskov. Mat. Obshch., Trans. Moscow Math. Soc., 57, 13-66 (1996), English transl. · Zbl 0906.73048 |
| [45] | Nazarov, S. A., Self-adjoint boundary value problems. The polynomial property and formal positive operators, Probl. Mat. Anal., 16, 167-192 (1997), in Russian |
| [46] | Nazarov, S. A., Asymptotic solution to a problem with small obstacles, Differentsial’nye Uravneniya.. Differentsial’nye Uravneniya., Differential Equations, 31, 6, 965-974 (1995), English transl. · Zbl 0862.35018 |
| [47] | Nazarov, S. A., Asymptotic Expansions of Eigenvalues (1987), Leningrad University, in Russian · Zbl 0618.35005 |
| [48] | Nazarov, S. A., Two-term asymptotics of solutions of spectral problems with singular perturbations, Mat. Sb.. Mat. Sb., Math. USSR-Sb., 69, 2, 307-340 (1991), English transl. · Zbl 0732.35004 |
| [49] | Nazarov, S. A.; Romashev, Yu. A., Variation of the intensity factor under rupture of the ligament between two collinear cracks, Izv. Akad. Nauk Armenian SSR. Mekh., 4, 30-40 (1982), in Russian · Zbl 0538.73126 |
| [50] | Nazarov, S. A., Asymptotic conditions at points, selfadjoint extensions of operators and the method of matched asymptotic expansions, Trudy St.-Petersburg Mat. Obshch.. Trudy St.-Petersburg Mat. Obshch., Trans. Amer. Math. Soc. Ser. 2, 193, 77-126 (1999), in Russian; English transl. |
| [51] | Nazarov, S. A., Korn’s inequalities for junctions of spatial bodies and thin rods, Math. Methods Appl. Sci., 20, 3, 219-243 (1997) · Zbl 0880.35040 |
| [52] | Nazarov, S. A., The polynomial property of self-adjoint elliptic boundary-value problems and the algebraic description of their attributes, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Russian Math. Surveys, 54, 5, 947-1014 (1999), in Russian; English transl. · Zbl 0970.35026 |
| [53] | Nazarov, S. A., Selfadjoint extensions of the Dirichlet problem operator in weighted function spaces, Mat. Sb.. Mat. Sb., Math. USSR-Sb., 65, 1, 229-247 (1990), English transl. · Zbl 0683.35033 |
| [54] | Nazarov, S. A., Concentrated masses problems for a spatial elastic body, C. R. Acad. Sci. Paris. Sér. 1, 316, 12, 1329-1334 (1993) · Zbl 0785.35021 |
| [55] | Nazarov, S. A., The damage tensor and measures. 1. Asymptotic analysis of anisotropic media with defects, Mekh. Tverd. Tela. Mekh. Tverd. Tela, Mech. Solids, 35, 3, 96-105 (2000), in Russian; English transl. |
| [56] | Nazarov, S. A.; Specovius-Neugebauer, M., Approximation of unbounded domains by bounded domains. Boundary-value problems for the Lamé operator, Algebra i Analiz. Algebra i Analiz, St. Petersburg Math. J., 8, 5, 879-912 (1997), English transl. · Zbl 0888.35006 |
| [57] | Nazarov, S. A.; Specovius-Neugebauer, M., Selfadjoint extensions of the Neumann Laplacian in domains with cylindrical outlets, Comm. Math. Phys., 185, 3, 689-707 (1997) · Zbl 0880.35036 |
| [58] | Nazarov, S. A.; Specovius-Neugebauer, M., The errors due to approximating unbounded elastic bodies by bounded ones, Prikl. Mat. Mekh.. Prikl. Mat. Mekh., J. Appl. Math. Mech., 62, 4, 605-616 (1998), English transl. |
| [59] | Nazarov, S. A.; Specovius-Neugebauer, M., Artificial boundary conditions for elliptic systems in domains with conical outlets to infinity, Dokl. Ross. Akad. Nauk, 377, 3, 1-4 (2001), in Russian · Zbl 1046.65104 |
| [60] | Pavlov, B. S., The theory of extension and explicitly soluble models, Uspekhi Mat. Nauk. Uspekhi Mat. Nauk, Soviet Math. Surveys, 42, 6, 127-168 (1987), English transl. · Zbl 0665.47004 |
| [61] | Pólya, G.; Szegö, G., Isoperimetric Inequalities in Mathematical Physics (1951), Princeton Univ. Press: Princeton Univ. Press Pinceton, NJ · Zbl 0044.38301 |
| [62] | Schiffer, M.; Szegö, G., Virtual mass and polarization, Trans. Amer. Math. Soc., 67, 130-205 (1949) · Zbl 0035.11803 |
| [63] | Osava, Shin, Singular Hadamard’s variation of domains and eigenvalues of Laplacian, Parts 1, 2, Proc. Japan Acad. Sci.. Proc. Japan Acad. Sci., Proc. Japan Acad. Sci., 57, 242-246 (1981) · Zbl 0509.35060 |
| [64] | Smirnov, V. I., A Course of Higher Mathematics (1964), Pergamon: Pergamon New York · Zbl 0122.29703 |
| [65] | Sokołowski, J.; Zolesio, J.-P., Introduction to Shape Optimization. Shape Sensitivity Analysis (1992), Springer-Verlag · Zbl 0761.73003 |
| [66] | Sokołowski, J.; Żochowski, A., On topological derivative in shape optimization, SIAM J. Control Optim., 37, 4, 1251-1272 (1999) · Zbl 0940.49026 |
| [67] | Sokołowski, J.; Żochowski, A., Topological derivatives for elliptic problems, Inverse Problems, 15, 1, 123-134 (1999) · Zbl 0926.35165 |
| [68] | Sokołowski, J.; Żochowski, A., Topological derivatives of shape functionals for elasticity systems, Mech. Structures Mach., 29, 3, 333-351 (2001) |
| [69] | J. Sokołowski, A. Żochowski, Optimality conditions for simultaneous topology and shape optimization, Les prépublications de l’Institut Élie Cartan 8/2001, SIAM J. Control Optim., submitted; J. Sokołowski, A. Żochowski, Optimality conditions for simultaneous topology and shape optimization, Les prépublications de l’Institut Élie Cartan 8/2001, SIAM J. Control Optim., submitted |
| [70] | Zorin, I. S.; Movchan, A. B.; Nazarov, S. A., Application of the elastic polarization tensor in the problems of the crack mechanics, Mekh. Tverd. Tela, 6, 128-134 (1988), in Russian |
| [71] | Zorin, I. S.; Movchan, A. B.; Nazarov, S. A., Application of tensors of elastic capacity, polarization and associated deformation, Studies in Elasticity and Plasticity, 16 (1990), Leningrad University: Leningrad University Leningrad, 75-91, in Russian |
| [72] | Van Dyke, M. D., Perturbation Methods in Fluid Mechanics (1964), Academic Press: Academic Press New York · Zbl 0136.45001 |
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