{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:39:54Z","timestamp":1760243994122,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2009,8,18]],"date-time":"2009-08-18T00:00:00Z","timestamp":1250553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The automated approximation of solutions to differential equations which involve discontinuities across evolving surfaces is addressed. Finite element technology has developed to the point where it is now possible to model evolving discontinuities independently of the underlying mesh, which is particularly useful in simulating failure of solids. However, the approach remains tedious to program, particularly in the case of coupled problems where a variety of finite element bases are employed and where a mixture of continuous and discontinuous fields may be used. We tackle this point by exploring the scope for employing automated code generation techniques for modelling discontinuities. Function spaces and variational forms are defined in a language that resembles mathematical notation, and computer code for modelling discontinuities is automatically generated. Principles underlying the approach are elucidated and a number of two- and three-dimensional examples for different equations are presented.<\/jats:p>","DOI":"10.3390\/a2031008","type":"journal-article","created":{"date-parts":[[2009,8,18]],"date-time":"2009-08-18T13:51:01Z","timestamp":1250603461000},"page":"1008-1030","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Automated Modelling of Evolving Discontinuities"],"prefix":"10.3390","volume":"2","author":[{"given":"Mehdi","family":"Nikbakht","sequence":"first","affiliation":[{"name":"Faculty of Civil Engineering and Geosciences, Delft University of Technology, Stevinweg 1, 2628 CN Delft, The Netherlands"}]},{"given":"Garth N.","family":"Wells","sequence":"additional","affiliation":[{"name":"Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK"}]}],"member":"1968","published-online":{"date-parts":[[2009,8,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/S0045-7825(96)01087-0","article-title":"The partition of unity finite element method: Basic theory and applications","volume":"139","author":"Melenk","year":"1996","journal-title":"Comput. Method. Appl. Mech. Eng."},{"key":"ref_2","first-page":"231","article-title":"A finite element method for crack growth without remeshing","volume":"46","author":"Dolbow","year":"1999","journal-title":"Int. J. Numer. Method. Eng."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"993","DOI":"10.1002\/1097-0207(20010210)50:4<993::AID-NME164>3.0.CO;2-M","article-title":"Arbitrary discontinuities in finite elements","volume":"50","author":"Belytschko","year":"2001","journal-title":"Int. J. Numer. Method. Eng."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"4081","DOI":"10.1016\/S0045-7825(01)00188-8","article-title":"The generalized finite element method","volume":"190","author":"Strouboulis","year":"2001","journal-title":"Comput. Method. Appl. Mech. Eng."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1002\/nme.1966","article-title":"An extended finite element library","volume":"71","author":"Bordas","year":"2007","journal-title":"Int. J. Numer. Method. Eng."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1145\/1163641.1163644","article-title":"A compiler for variational forms","volume":"32","author":"Kirby","year":"2006","journal-title":"ACM Trans. Math. Software"},{"key":"ref_7","unstructured":"Logg, A., and Wells, G. N. DOLFIN: Automated finite element modelling,. Available online: http:\/\/www.dspace.cam.ac.uk\/handle\/1810\/214787."},{"key":"ref_8","first-page":"849","article-title":"Automated code generation for discontinuous Galerkin methods","volume":"31","author":"Logg","year":"2008","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Rognes, M. E., Kirby, R. C., and Logg, A. (2009). Efficient assembly of H(div) and H(curl) conforming finite elements, Submitted.","DOI":"10.1137\/08073901X"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"\u00d8lgaard, K. B., and Wells, G. N. (2010). 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Available online: http:\/\/simula.no\/research\/scientific\/publications\/Simula.SC.96\/simula_pdf_file."},{"key":"ref_16","unstructured":"Logg, A., Wells, G. N. DOLFIN 2009. Available online: http:\/\/www.fenics.org\/dolfin."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1002\/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N","article-title":"The Partition of Unity Method","volume":"40","author":"Melenk","year":"1997","journal-title":"Int. J. Numer. Method. Eng."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1002\/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S","article-title":"Elastic crack growth in finite elements with minimal remeshing","volume":"45","author":"Belytschko","year":"1999","journal-title":"Int. J. Numer. Method. Eng."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"2667","DOI":"10.1002\/nme.143","article-title":"A new method for modelling cohesive cracks using finite elements","volume":"50","author":"Wells","year":"2001","journal-title":"Int. J. Numer. Method. Eng."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/2\/3\/1008\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:10:57Z","timestamp":1760220657000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/2\/3\/1008"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,18]]},"references-count":19,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2009,9]]}},"alternative-id":["a2031008"],"URL":"https:\/\/doi.org\/10.3390\/a2031008","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2009,8,18]]}}}