{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,3]],"date-time":"2026-10-03T17:14:42Z","timestamp":1791047682358,"version":"4.1.0"},"reference-count":57,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>The purpose of this work is the development of a fully monolithic solution algorithm for quasi-static phase-field fracture propagation. Phase-field fracture consists of two coupled partial differential equations, and it is well known that the underlying energy functional is nonconvex and sophisticated algorithms are required. For the incremental, spatially discretized problem, we employ an adaptive error-oriented Newton algorithm which works as an inner loop within an inexact augmented Lagrangian iteration. The latter approach relaxes the crack irreversibility constraint, which is an inequality constraint in time. Six numerical tests and benchmarks are consulted to demonstrate the performance of the algorithmic techniques. Specifically, the fully monolithic approach is compared to a quasi-monolithic approach in which the phase-field is approximated through extrapolation in the displacement equation. These comparisons are done in terms of certain quantities of interest and computational cost. Moreover, features such as crack nucleation, joining, branching, and fracture networks are addressed. Most examples are in two dimensions, but three-dimensional (3D) testing is provided as well. All findings are critically analyzed and point to open questions and future improvements.<\/jats:p>","DOI":"10.1137\/16m1063873","type":"journal-article","created":{"date-parts":[[2017,7,6]],"date-time":"2017-07-06T12:08:25Z","timestamp":1499342905000},"page":"B589-B617","source":"Crossref","is-referenced-by-count":64,"title":["An Error-Oriented Newton\/Inexact Augmented Lagrangian Approach for Fully Monolithic Phase-Field Fracture Propagation"],"prefix":"10.1137","volume":"39","author":[{"given":"Thomas","family":"Wick","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,7,6]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2011.03.024"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2014.07.009"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-014-1109-y"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160430805"},{"key":"atypb5","first-page":"105","volume":"6","author":"Ambrosio L.","year":"1992","journal-title":"Boll. Un. Mat. Ital. B (7)"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmps.2009.04.011"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/140970495"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1145\/1268776.1268779"},{"key":"atypb9","volume-title":"deal. II","author":"Bangerth W.","year":"2012"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1080\/10556780701228532"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492901000010"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2012.01.008"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.4171\/ifb\/171"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5096(99)00028-9"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1007\/s10659-007-9107-3"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.112.014301"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1137\/080741033"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1142\/S021820251350019X"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/S106482750037620X"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894246905"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/0719025"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23899-4"},{"key":"atypb23","volume-title":"Stud. Math. Appl. 15","author":"Fortin M.","year":"1983"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-5096(98)00034-9"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.12.017"},{"key":"atypb26","volume-title":"Braunschweig","author":"Gerasimov T.","year":"2016"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970838"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1921.0006"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechrescom.2016.07.002"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.03.009"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1007\/BF00927673"},{"key":"atypb32","first-page":"1115","volume":"49","author":"Kuhn C.","year":"2011","journal-title":"J. Theoret. Appl. Mech."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2016.02.037"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2016.10.022"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1016\/j.mechrescom.2016.04.002"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2014.10.052"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2014.11.017"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2010.04.011"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2015.09.021"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2014.11.016"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2861"},{"key":"atypb42","unstructured":"A. Mikeli\u0107, M. F. Wheeler, and T. Wick,\n                      A Phase-Field Approach to the Fluid Filled Fracture Surrounded by a Poroelastic Medium\n                      , ICES Report 13-15, ICES, The University of Texas at Austin, Austin, TX, 2013."},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/140967118"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1007\/s10596-015-9532-5"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/28\/5\/1371"},{"key":"atypb46","volume-title":"Numerical Optimization","author":"Nocedal J.","year":"2006","edition":"2"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-008-9216-3"},{"key":"atypb48","first-page":"283","volume-title":"Optimization (Sympos.","author":"Powell M. J. D.","year":"1968"},{"key":"atypb49","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-014-1045-x"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1946.0077"},{"key":"atypb51","volume-title":"Crack Problems in the Classical Theory of Elasticity","author":"Sneddon I. N.","year":"1969"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1007\/s11012-013-9862-0"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2013.12.005"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1016\/j.commatsci.2015.07.034"},{"key":"atypb55","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.09.024"},{"key":"atypb56","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-016-1275-1"},{"key":"atypb57","unstructured":"B. J. Winkler,\n                      Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes fuer Beton\n                      , Ph.D. thesis, University of Innsbruck, Innsbruck, Austria, 2001."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1063873","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:58:35Z","timestamp":1787335115000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1063873"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":57,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1063873"],"URL":"https:\/\/doi.org\/10.1137\/16m1063873","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}