{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:25:05Z","timestamp":1787336705176,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[1998,1]]},"abstract":"<jats:p>This paper treats the problem of computing the collapse state in limit analysis for a solid with a quadratic yield condition, such as, for example, the von Mises condition. After discretization with the finite element method, using divergence-free elements for the plastic flow, the kinematic formulation reduces to the problem of minimizing a sum of Euclidean vector norms, subject to a single linear constraint. This is a nonsmooth minimization problem, since many of the norms in the sum may vanish at the optimal point. Recently an efficient solution algorithm has been developed for this particular convex optimization problem in large sparse form.<\/jats:p>\n                  <jats:p>\n                    The approach is appliedto test problems in limit analysis in two different plane models: plane strain and plates. In the first case more than 80% of the terms in the objective function are zero in the optimal solution, causing extreme ill conditioning. In the second case all terms are nonzero. In both cases the method works very well, and problems are solved which are larger by at least an order of magnitude than previously reported. The relative accuracy for the solution of the discrete problems, measured by duality gap and feasibility, is typically of the order 10\n                    <jats:sup>-8<\/jats:sup>\n                    .\n                  <\/jats:p>","DOI":"10.1137\/s1064827594275303","type":"journal-article","created":{"date-parts":[[2003,6,11]],"date-time":"2003-06-11T11:12:06Z","timestamp":1055329926000},"page":"1046-1062","source":"Crossref","is-referenced-by-count":69,"title":["Computing Limit Loads by Minimizing a Sum of Norms"],"prefix":"10.1137","volume":"19","author":[{"given":"Knud D.","family":"Andersen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Edmund","family":"Christiansen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael L.","family":"Overton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,25]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1994.0041"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0020-7683(72)90088-1"},{"key":"R3","first-page":"20","volume":"22","author":"Andersen K. D.","year":"1993","journal-title":"Committee on Algorithms Bulletin (COAL)"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/0806006"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(94)00031-U"},{"key":"R6","unstructured":"K. D. Andersen, E. Christiansen, and M. L. Overton,\n                      Computing Limit Loads by Minimizing a Sum of Norms\n                      , Preprint 30, Dept. Mathematics and Computer Science, Odense University, Denmark, 1994."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/0901037"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02591853"},{"key":"R9","first-page":"53","volume":"6","author":"Capurso M.","year":"1971","journal-title":"Meccanica\u2014J. Ital. Assoc. Theoret. Appl. Mech."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/0511049"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02575862"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620171009"},{"key":"R13","first-page":"77","volume":"22","author":"Christiansen E.","year":"1982","journal-title":"Utilitas Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0315-3681","issn-type":"print"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1007\/BF00276859"},{"key":"R15","unstructured":"E. Christiansen,\n                      Limit analysis with unbounded convex yield condition\n                      , in IMACS \u201991: Proceedings of the 13th World Congress on Computation and Applied Mathematics, R. Vichnevetsky and J. J. H. Miller, eds., Criterion Press, Dublin, 1991, pp. 129\u2013130."},{"key":"R16","doi-asserted-by":"crossref","unstructured":"E. Christiansen,\n                      Limit analysis of collapse states\n                      , in Handbook of Numerical Analysis, Vol. 4, P. G. Ciarlet and J. L. Lions, eds., North\u2013Holland, Amsterdam, 1996, pp. 193\u2013312.","DOI":"10.1016\/S1570-8659(96)80004-4"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/BF02023060"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(91)90147-C"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620190203"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1007\/BF01932705"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North\u2013Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R22","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      Basic error estimates for elliptic problems\n                      , in Handbook of Numerical Analysis, Vol. 2, P. G. Ciarlet and J. L. Lions, eds., North\u2013Holland, Amsterdam, 1991.","DOI":"10.1016\/S1570-8659(05)80039-0"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(91)90255-5"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/0117067"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"B. A. Murtagh and M. A. Saunders,\n                      A projected lagrangian algorithm and its implementation for sparse nonlinear constraints\n                      , in Algorithms for Constrained Minimization of Smooth Nonlinear Functions, A. G. Buckley and J.\u2010L. Goffin, eds., Mathematical Programming Study 16, North\u2013Holland, Amsterdam, 1982, pp. 84\u2013117.","DOI":"10.1007\/BFb0120949"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/BF02591963"},{"key":"R27","unstructured":"M. L. Overton,\n                      Numerical solution of a model problem from collapse load analysis\n                      , in Proceedings of the Sixth International Symposium on Computer Methods in Engineering and Applied Sciences, R. Glowinski and J. L. Lions, eds., Institut National de Recherche en Informatique et en Automatique (Versailles, France), North\u2013Holland, Amsterdam, 1984, pp. 421\u2013437."},{"key":"R28","unstructured":"R. Temam,\n                      Navier\u2010Stokes Equations\n                      , North\u2013Holland, Amsterdam, 1977."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/S1064827594275303","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:26:30Z","timestamp":1787333190000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/S1064827594275303"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998,1]]},"references-count":28,"journal-issue":{"issue":"3","published-print":{"date-parts":[[1998,1]]}},"alternative-id":["10.1137\/S1064827594275303"],"URL":"https:\/\/doi.org\/10.1137\/s1064827594275303","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[1998,1]]}}}