{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:40:55Z","timestamp":1787323255915,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2002,1]]},"abstract":"<jats:p>The topological sensitivity analysis provides an asymptotic expansion of a shape function when creating a small hole inside a domain. This expansion yields a descent direction which can be used for shape optimization if one wishes to keep a classical domain throughout the optimization process. In this paper, such an expansion is obtained for the Poisson equation for a large class of cost functions and arbitrarily shaped holes. In the three-dimensional case, this expansion depends on the shape of the hole but not on its orientation if the cost function involves only the solution u to the underlying partial differential equation, whereas it may also depend on its orientation if the cost function involves the gradient $\\nabla u$. In contrast, the asymptotic expansion is independent of the shape in the two-dimensional case. A numerical example illustrates the use of the asymptotic expansion, which yields a minimizing sequence of classical domains in a case where no classical solution exists.<\/jats:p>","DOI":"10.1137\/s0363012901384193","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1042-1072","source":"Crossref","is-referenced-by-count":66,"title":["The Topological Asymptotic Expansion for the Dirichlet Problem"],"prefix":"10.1137","volume":"41","author":[{"given":"Ph.","family":"Guillaume","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K. Sid","family":"Idris","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,26]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/1247433"},{"key":"R2","unstructured":"M. Becker,\n                      Optimisation topologique de structure en variables discr\u00e8\n                      tes, Thesis, Universit\u00e9 de Li\u00e8ges, 1996."},{"key":"R3","unstructured":"M. Bends\u00f8e,\n                      Optimal Topology Design of Continuum Structure: An Introduction,\n                      Technical report, Department of Mathematics, Technical University of Denmark, Lyngby, Denmark, 1996."},{"key":"R4","volume-title":"Analyse fonctionnelle","author":"Brezis Ha\u00efm","year":"1983"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/BF01442391"},{"key":"R6","first-page":"47","volume":"1","author":"Chipot M.","year":"1992","journal-title":"Adv. Math. Sci. Appl."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1986200303711"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02575843"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(99)00357-6"},{"key":"R10","unstructured":"D. J. Cedio\u2010Fengya, S. Moskow, and M. Vogelius,\n                      Identification of Conductivity Imperfections of Small Diameter by Boundary Measurements. Continuous Dependence and Computational Reconstruction\n                      , Preprint 1502, Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, MN, 1997."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396415"},{"key":"R12","volume-title":"Analyse math\u00e9matique et calcul num\u00e9rique pour les sciences et les techniques. Vol. 1","author":"Dautray Robert","year":"1987"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF00281494"},{"key":"R14","unstructured":"J. Giroire,\n                      Formulations variationnelles par \u00e9quations int\u00e9grales de probl\u00e8mes aux limites ext\u00e9rieurs\n                      , Thesis, Ecole Polytechnique, Palaiseau, France, 1976."},{"key":"R15","unstructured":"S. Garreau, Ph. Guillaume, and M. 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Ural\u2019ceva,\n                      \u00c9quations aux d\u00e9riv\u00e9es partielles de type elliptique\n                      , Dunod, Paris, 1968."},{"key":"R21","volume-title":"Probl\u00e8mes aux limites non homog\u00e8nes et applications. Vol. 1","author":"Lions J.\u2010L.","year":"1968"},{"key":"R22","unstructured":"M. Masmoudi,\n                      Outils pour la conception optimale de formes\n                      , th\u00e8se d\u2019\u00e9tat, Universit\u00e9 de Nice, Nice, France, 1987."},{"key":"R23","unstructured":"M. Masmoudi,\n                      The topological asymptotic\n                      , in Computational Methods for Control Applications, International Series GAKUTO, H. Kawarada and J. Periaux, eds., to appear."},{"key":"R24","unstructured":"F. Murat and J. Simon,\n                      Sur le controle par un domaine g\u00e9om\u00e9trique\n                      , th\u00e8se d\u2019\u00e9tat, Universit\u00e9 Pierre et Marie Curie, Paris, 1976."},{"key":"R25","unstructured":"F. Murat and L. Tartar,\n                      Calcul des variations et homog\u00e9\n                      n\u00e9isation, in Les m\u00e9thodes de l\u2019homog\u00e9n\u00e9isation: Th\u00e9orie et Applications en Physique, Eyrolles, 1985, pp. 319\u2013369."},{"key":"R26","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1080\/12506559.1996.10511240","volume":"5","author":"Schoenauer Marc","year":"1996","journal-title":"Rev. Europ\u00e9enne \u00c9l\u00e9m. Finis"},{"key":"R27","unstructured":"A. Schumacher,\n                      Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien\n                      , Thesis, Universit\u00e4t\u2010Gesamthochschule\u2010Siegen, 1995."},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012997323230"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S0363012901384193","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:51:14Z","timestamp":1787320274000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S0363012901384193"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,1]]},"references-count":28,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2002,1]]}},"alternative-id":["10.1137\/S0363012901384193"],"URL":"https:\/\/doi.org\/10.1137\/s0363012901384193","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,1]]}}}