{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,24]],"date-time":"2026-08-24T09:37:42Z","timestamp":1787564262078,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>In this paper, we consider cracks with Dirichlet boundary conditions. We first derive an asymptotic expansion of the boundary perturbations that are due to the presence of a small crack. Based on this formula, we design a noniterative approach for locating a collection of small cracks. In order to do so, we construct a response matrix from the boundary measurements. The location and the length of the crack are estimated, respectively, from the projection onto the noise space and the first significant singular value of the response matrix. Indeed, the direction of the crack is estimated from the second singular vector. We then consider an extended crack with Dirichlet boundary conditions. We rigorously derive an asymptotic expansion for the boundary perturbations that are due to a shape deformation of the crack. To reconstruct an extended crack from many boundary measurements, we develop two methods for obtaining a good guess. Several numerical experiments show how the proposed techniques for imaging small cracks as well as those for obtaining good initial guesses toward reconstructing an extended crack behave.<\/jats:p>","DOI":"10.1137\/090749013","type":"journal-article","created":{"date-parts":[[2010,3,19]],"date-time":"2010-03-19T18:22:02Z","timestamp":1269022922000},"page":"894-922","source":"Crossref","is-referenced-by-count":78,"title":["Asymptotic Imaging of Perfectly Conducting Cracks"],"prefix":"10.1137","volume":"32","author":[{"given":"Habib","family":"Ammari","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hyeonbae","family":"Kang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hyundae","family":"Lee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Won-Kwang","family":"Park","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,3,19]]},"reference":[{"key":"R1","unstructured":"H. Ammari,\n                      An Introduction to Mathematics of Emerging Biomedical Imaging\n                      , Math. Appl., Vol. 62, Springer-Verlag, Berlin, 2008."},{"key":"R2","unstructured":"H. Ammari, P. Garapon, F. Jouve, and H. Kang,\n                      A New Optimal Control Approach Toward Reconstruction of Acoustic and Elastic Inclusions\n                      , in preparation."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/040612518"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0130-x"},{"key":"R5","doi-asserted-by":"crossref","unstructured":"H. Ammari, H. Kang, and H. Lee,\n                      Layer Potential Techniques in Spectral Analysis\n                      , Math. Surveys Monogr., Vol. 153, AMS, Providence, RI, 2009.","DOI":"10.1090\/surv\/153\/10"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1007\/s00033-005-0018-5"},{"key":"R7","first-page":"81","volume":"34","author":"Amstutz S.","year":"2005","journal-title":"Control Cybernet.","ISSN":"https:\/\/id.crossref.org\/issn\/0324-8569","issn-type":"print"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/12\/5\/002"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1051\/proc\/2009003"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-008-0121-2"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012904441179"},{"key":"R12","first-page":"63","volume":"29","author":"Berenstein C.","year":"1997","journal-title":"Rend. Istit. Mat. Univ. Trieste"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"K. Bryan and M. S. Vogelius,\n                      A review of selected works on crack identification\n                      , in Geometric Methods in Inverse Problems and PDE Control, IMA Vol. 137, Springer-Verlag, Berlin, 2004.","DOI":"10.1007\/978-1-4684-9375-7_3"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2001128"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/17\/4\/301"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/11\/2\/004"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF01742933"},{"key":"R18","first-page":"115","volume":"12","author":"Fremiot G.","year":"2002","journal-title":"Adv. Math. Sci. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1343-4373","issn-type":"print"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"G. Fremiot and J. Soko\u0142owski,\n                      Hadamard formula in nonsmooth domains and applications\n                      , in Partial Differential Equations on Multistructures, Lecture Notes in Pure and Appl. Math. 219, Dekker, New York, 2001, pp. 99\u2013120.","DOI":"10.1201\/9780203902196.ch6"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"G. Fremiot and J. Soko\u0142owski,\n                      Shape sensitivity analysis of problems with singularities\n                      , in Shape Optimization and Optimal Design, Lecture Notes in Pure and Appl. Math. 216, Dekker, New York, 2001, pp. 255\u2013276.","DOI":"10.1201\/9780203904169.ch9"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012900369538"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2004.02.010"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/19\/4\/308"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/16\/1\/308"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1090\/qam\/1878261"},{"key":"R26","first-page":"135","volume":"43","author":"Kovtunenko V. A.","year":"2002","journal-title":"Prikl. Mekh. Tekhn. Fiz.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-4626","issn-type":"print"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1002\/mma.341"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1002\/zamm.200510354"},{"key":"R29","unstructured":"A. Laurain,\n                      Domaines Singuli\u00e8rements Perturb\u00e9s en Optimisation de Forme\n                      , Ph.D. thesis, Institut \u00c9lie Cartan, Universit\u00e9 Henri Poincar\u00e9 Nancy I, France, 2006."},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1137\/060674430"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1093\/imamat\/47.2.127"},{"key":"R32","unstructured":"F. Murat and J. Simon,\n                      Sur le contr\u00f4le par un domaine g\u00e9om\u00e9trique\n                      , Pr\u00e9-publication du Laboratoire d'Analyse Num\u00e9rique, Universit\u00e9 de Paris 6, 1976."},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620320702"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/25\/7\/075002"},{"key":"R35","doi-asserted-by":"publisher","DOI":"10.1016\/0165-2125(94)90039-6"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2007.01.007"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012997323230"},{"key":"R38","unstructured":"J. Soko\u0142owski and J. P. Zol\u00e9sio,\n                      Introduction to Shape Optimization\n                      , Springer Ser. Comput. 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