{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T09:12:57Z","timestamp":1787389977522,"version":"build-2736575974"},"reference-count":20,"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":[[2009,1]]},"abstract":"<jats:p>This paper discusses the symbol of the Hessian of a shape optimization problem in a viscid, incompressible flow. The symbol of the Hessian for the Stokes shape optimization problem is analytically approximated by tracking a Fourier mode through the operators involved. We propose a discrete method for finding the symbol and confirm the correctness by comparison with the analytical data. A preconditioner is constructed for both the Stokes and Navier\u2013Stokes equation, which greatly accelerates the optimization.<\/jats:p>","DOI":"10.1137\/080719844","type":"journal-article","created":{"date-parts":[[2009,9,4]],"date-time":"2009-09-04T18:16:11Z","timestamp":1252088171000},"page":"2562-2580","source":"Crossref","is-referenced-by-count":28,"title":["Impulse Response Approximations of Discrete Shape Hessians with Application in CFD"],"prefix":"10.1137","volume":"48","author":[{"given":"Stephan","family":"Schmidt","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Volker","family":"Schulz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,9,4]]},"reference":[{"key":"R1","unstructured":"E. Arian,\n                      Analysis of the Hessian for aeroelastic optimization\n                      , Technical report 95-84, ICASE, NASA Langley Research Center, Hampton, VA, 1995."},{"key":"R2","unstructured":"E. Arian and S. Ta'asan,\n                      Analysis of the Hessian for aerodynamic optimization: Inviscid flow\n                      , Technical report 96-28, ICASE, NASA Langley Research Center, Hampton, VA, 1996."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1080\/10618569908940813"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.2514\/1.24859"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1080\/10618560600839415"},{"key":"R6","unstructured":"M. Delfour and J.P. Zol\u00e9sio,\n                      Shapes and Geometries\n                      , SIAM, Philadelphia, 2001."},{"key":"R7","first-page":"203","volume":"34","author":"Eppler K.","year":"2005","journal-title":"Control Cybernet.","ISSN":"https:\/\/id.crossref.org\/issn\/0324-8569","issn-type":"print"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/s10957-008-9507-y"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1002\/pamm.200510345"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/BF01061285"},{"key":"R11","doi-asserted-by":"crossref","unstructured":"A. Jameson,\n                      Future research avenues in computational engineering and design\n                      , in Proceedings of the Fourth International Congress on Industrial and Applied Mathematics, J. M. Ball, J. C. R. Hunt, eds., Oxford University Press, New York, 2000.","DOI":"10.1093\/oso\/9780198505143.003.0008"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"S. Kim, K. Hosseini, K. Leoviriyakit, and A. Jameson,\n                      Enhancement of adjoint design methods via optimization of adjoint parameters\n                      , 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 10\u201313, 2005.","DOI":"10.2514\/6.2005-448"},{"key":"R13","first-page":"199","volume":"1","author":"Laurain A.","year":"2005","journal-title":"Adv. Math. Sci. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1343-4373","issn-type":"print"},{"key":"R14","unstructured":"B. Mohammadi and O. Pironneau,\n                      Applied Shape Optimization for Fluids\n                      , Oxford University Press, Oxford, 2001."},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1017\/S002211207300145X"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112074002023"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/070694272"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"P. I. Plotnikov and J. Sokolowski,\n                      Stationary boundary value problems for compressible Navier\u2013Stokes equations\n                      , in Handbook of Differential Equations 6, Elsevier, New York, 2008, pp. 313\u2013410.","DOI":"10.1016\/S1874-5733(08)80022-8"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/s00021-004-0134-6"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"J. Sokolowski and J.P. Zol\u00e9sio,\n                      Introduction to Shape Optimization\n                      , Springer, Berlin, 1992.","DOI":"10.1007\/978-3-642-58106-9"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/080719844","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:52:07Z","timestamp":1787316727000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/080719844"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":20,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/080719844"],"URL":"https:\/\/doi.org\/10.1137\/080719844","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}