{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T18:56:45Z","timestamp":1782932205566,"version":"3.54.5"},"publisher-location":"Basel","reference-count":33,"publisher":"Springer Basel","isbn-type":[{"value":"9783034801324","type":"print"},{"value":"9783034801331","type":"electronic"}],"license":[{"start":{"date-parts":[[2011,10,28]],"date-time":"2011-10-28T00:00:00Z","timestamp":1319760000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2011,10,28]],"date-time":"2011-10-28T00:00:00Z","timestamp":1319760000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2012]]},"DOI":"10.1007\/978-3-0348-0133-1_13","type":"book-chapter","created":{"date-parts":[[2012,1,2]],"date-time":"2012-01-02T10:34:06Z","timestamp":1325500446000},"page":"245-256","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":44,"title":["Phase-field Approaches to Structural Topology Optimization"],"prefix":"10.1007","author":[{"given":"Luise","family":"Blank","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Harald","family":"Garcke","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lavinia","family":"Sarbu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Tarin","family":"Srisupattarawanit","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Vanessa","family":"Styles","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Axel","family":"Voigt","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2011,10,28]]},"reference":[{"key":"13_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9286-6","volume-title":"Optimization by the Homogenization Method","author":"G Allaire","year":"2002","unstructured":"G. Allaire, Optimization by the Homogenization Method. Springer, Berlin 2002."},{"key":"13_CR2","doi-asserted-by":"publisher","first-page":"363","DOI":"10.1016\/j.jcp.2003.09.032","volume":"194","author":"G Allaire","year":"2004","unstructured":"G. Allaire, F. Jouve, A.-M. Toader, Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194 (2004), 363\u2013393.","journal-title":"J. Comput. Phys"},{"key":"13_CR3","doi-asserted-by":"publisher","first-page":"738","DOI":"10.1137\/S0036142902413421","volume":"46","author":"JW Barrett","year":"2004","unstructured":"J.W. Barrett, R. N\u00fcrnberg, V. Styles, Finite element approximation of a phase field model for void electromigration. SIAM J. Numer. Anal. 46 (2004), 738\u2013772.","journal-title":"SIAM J. Numer. Anal."},{"key":"13_CR4","volume-title":"Topology Optimization","author":"MP Bends\u00f8e","year":"2003","unstructured":"M.P. Bends\u00f8e, O. Sigmund, Topology Optimization. Springer, Berlin 2003."},{"key":"13_CR5","unstructured":"L. Blank, M. Butz, H. Garcke, Solving the Cahn-Hilliard variational inequality with a semi-smooth Newton method. To appear in ESAIM Control Optim. Calc. Var., online: DOI 10.1051\/cocv\/2010032."},{"key":"13_CR6","unstructured":"L. Blank, H. Garcke, L. Sarbu, V. Styles, Primal-dual active set methods for Allen-Cahn variational inequalities with non-local constraints. DFG priority program 1253 \u201cOptimization with PDEs\u201d, preprint no. 1253-09-01."},{"key":"13_CR7","unstructured":"L. Blank, H. Garcke, L. Sarbu, V. Styles, Non-local Allen-Cahn systems: Analysis and primal dual active set methods. Preprint No. 02\/2011, Universtit\u00a8at Regensburg."},{"key":"13_CR8","unstructured":"L. Blank, L. Sarbu, M. Stoll, Preconditioning for Allen-Cahn variational inequalities with non-local constraints. Preprint Nr.11\/2010, Universit\u00e4t Regensburg, Mathematik."},{"key":"13_CR9","first-page":"19","volume":"9","author":"B Bourdin","year":"2003","unstructured":"B. Bourdin, A. Chambolle, Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9 (2003), 19\u201348.","journal-title":"Calc. Var"},{"key":"13_CR10","doi-asserted-by":"crossref","unstructured":"B. Bourdin, A. Chambolle, The phase-field method in optimal design. IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials (M.P. Bends\u00f8e, N.Olhoff, and O. Sigmund eds.), Solid Mechanics and its Applications 137 (2006), 207\u2013216.","DOI":"10.1007\/1-4020-4752-5_21"},{"key":"13_CR11","doi-asserted-by":"publisher","first-page":"1447","DOI":"10.1137\/05062723X","volume":"45","author":"M Burger","year":"2006","unstructured":"M. Burger, R. Stainko, Phase-field relaxation of topology optimization with local stress constraints. SIAM J. Control Optim. 45 (2006), 1447\u20131466.","journal-title":"SIAM J. Control Optim"},{"key":"13_CR12","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1093\/oso\/9780198500148.003.0016","volume-title":"State of the art in Numerical Analysis","author":"CM Elliott","year":"1997","unstructured":"C.M. Elliott, Approximation of curvature dependent interface motion. State of the art in Numerical Analysis, IMA Conference Proceedings 63, Clarendon Press, Oxford, 1997, 407\u2013440."},{"key":"13_CR13","doi-asserted-by":"publisher","first-page":"524","DOI":"10.1016\/S0021-9991(03)00122-0","volume":"187","author":"CM Elliott","year":"2003","unstructured":"C.M. Elliott, V. Styles, Computations of bi-directional grain boundary dynamics in thin films. J. Comput. Phys. 187 (2003), 524\u2013543.","journal-title":"J. Comput. Phys"},{"issue":"3","key":"13_CR14","doi-asserted-by":"publisher","first-page":"786","DOI":"10.1093\/imanum\/drq005","volume":"31","author":"CM Elliott","year":"2011","unstructured":"C.M. Elliott, B. Stinner, V. Styles, R. Welford, Numerical computation of advection and diffusion on evolving diffuse interfaces. IMA J. Num. Anal. 31(3) (2011), 786\u2013812, doi:10.1093\/imanum\/drq005.","journal-title":"J. Num. Anal"},{"issue":"2","key":"13_CR15","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1017\/S0308210500002419","volume":"133","author":"H Garcke","year":"2003","unstructured":"H. Garcke, On Cahn-Hilliard systems with elasticity. Proc. Roy. Soc. Edinburgh Sect. A 133 (2003), no. 2, 307\u2013331.","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A"},{"key":"13_CR16","unstructured":"H. Garcke, Mechanical effects in the Cahn-Hilliard model: A review on mathematical results. In \u201cMathematical Methods and Models in phase transitions\u201d, ed.: Alain Miranville, Nova Science Publ. (2005), 43\u201377."},{"key":"13_CR17","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1017\/S095679250700719X","volume":"18","author":"H Garcke","year":"2007","unstructured":"H. Garcke, R. N\u00fcrnberg, V. Styles, Stress and diffusion induced interface motion: Modelling and numerical simulations. European Journal of Applied Math. 18 (2007), 631\u2013657.","journal-title":"European Journal of Applied Math"},{"key":"13_CR18","doi-asserted-by":"publisher","first-page":"639","DOI":"10.1007\/s00211-004-0578-x","volume":"100","author":"H Garcke","year":"2005","unstructured":"H. Garcke, U. Weikard, Numerical approximation of the Cahn-Larch\u00e9 equation. Numer. Math. 100 (2005) 639\u2013662.","journal-title":"Numer. Math"},{"key":"13_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/BF01279647","volume":"11","author":"RB Haber","year":"1996","unstructured":"R.B. Haber, C.S. Jog, M.P. Bends\u00f8e, A new approach to variable topology shape design using a constraint on perimeter. Struct. Multidisc. Optim. 11 (1996), 1\u201312.","journal-title":"Struct. Multidisc. Optim"},{"key":"13_CR20","unstructured":"M. Hinze, R. Pinnau, M. Ulbrich, S. Ulbrich Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications 23, Springer (2008)."},{"key":"13_CR21","doi-asserted-by":"publisher","first-page":"1835","DOI":"10.1016\/0001-6160(82)90023-2","volume":"30","author":"FC Larch\u00e9","year":"1982","unstructured":"F.C. Larch\u00e9, J.W. Cahn, The effect of self-stress on diffusion in solids, Acta Metall. 30 (1982) 1835\u20131845.","journal-title":"Acta Metall"},{"key":"13_CR22","doi-asserted-by":"publisher","first-page":"81","DOI":"10.4310\/CMS.2009.v7.n1.a4","volume":"7","author":"X Li","year":"2009","unstructured":"X. Li, J. Lowengrub, A. R\u00e4tz, A. Voigt, Solving PDEs in complex geometries- a diffuse domain approach. Comm. Math. Sci. 7 (2009), 81\u2013107.","journal-title":"Comm. Math. Sci"},{"key":"13_CR23","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/BF00251230","volume":"98","author":"L Modica","year":"1987","unstructured":"L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rat. Mech. Anal. 98 (1987), 123\u2013142.","journal-title":"Arch. Rat. Mech. Anal"},{"key":"13_CR24","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/S0045-7825(98)00248-5","volume":"171","author":"J Petersson","year":"1999","unstructured":"J. Petersson, Some convergence results in perimeter-controlled topology optimization. Computer Meth. Appl. Mech. Eng. 171 (1999), 123\u2013140.","journal-title":"Appl. Mech. Eng"},{"key":"13_CR25","doi-asserted-by":"publisher","first-page":"575","DOI":"10.4310\/CMS.2006.v4.n3.a5","volume":"4","author":"A R\u00e4tz","year":"2006","unstructured":"A. R\u00e4tz, A. Voigt, PDEs on surfaces \u2013 a diffuse interface approach. Comm. Math. Sci. 4 (2006), 575\u2013590.","journal-title":"Comm. Math. Sci"},{"key":"13_CR26","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1016\/j.jcp.2005.09.013","volume":"214","author":"A R\u00e4tz","year":"2006","unstructured":"A. R\u00e4tz, A. Ribalta, A. Voigt, Surface evolution of elastically stressed films under deposition by a diffuse interface model. J. Comput. Phys. 214 (2006), 187\u2013208.","journal-title":"J. Comput. Phys"},{"key":"13_CR27","volume-title":"Lecture Notes in Computational Science and Engineering","author":"A Schmidt","year":"2005","unstructured":"A. Schmidt, K.G. Siebert, Design of adaptive finite element software. The finite element toolbox ALBERTA, Lecture Notes in Computational Science and Engineering 42. Springer-Verlag, Berlin, 2005."},{"key":"13_CR28","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1007\/BF01214002","volume":"16","author":"O Sigmund","year":"1998","unstructured":"O. Sigmund, J. Petersson, Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Struct. Multidisc. Optim. 16 (1998), 68\u201375.","journal-title":"Struct. Multidisc. Optim."},{"issue":"7","key":"13_CR29","doi-asserted-by":"publisher","first-page":"2697","DOI":"10.1016\/j.jcp.2009.12.017","volume":"229","author":"A Takezawa","year":"2010","unstructured":"A. Takezawa, S. Nishiwaki, M. Kitamura, Shape and topology optimization based on the phase field method and sensitivity analysis. Journal of Computational Physics 229 (7) (2010), 2697\u20132718.","journal-title":"Journal of Computational Physics"},{"key":"13_CR30","doi-asserted-by":"publisher","first-page":"183","DOI":"10.1007\/BF02186838","volume":"1\u20132","author":"JE Taylor","year":"1994","unstructured":"J.E. Taylor, J.W. Cahn, Linking anisotropic sharp and diffuse surface motion laws via gradient flows. J. Statist. Phys. 77 (1994), no. 1\u20132, 183\u2013197.","journal-title":"J. Statist. Phys. 77"},{"key":"13_CR31","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.4310\/CMS.2009.v7.n4.a10","volume":"7","author":"KE Teigen","year":"2009","unstructured":"K.E. Teigen, X. Li, J. Lowengrub, F. Wang, A. Voigt, A diffuse-interface approach for modeling transport, diffusion and adsorption\/desorption of material quantities on a deformable interface. Comm. Math. Sci. 7 (2009), 1009\u20131037.","journal-title":"Comm. Math. Sci"},{"key":"13_CR32","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1007\/s00791-006-0048-3","volume":"10","author":"S Vey","year":"2007","unstructured":"S. Vey, A. Voigt, AMDiS \u2013 adaptive multidimensional simulations. Comput. Vis. Sci. 10 (2007), 57\u201367.","journal-title":"Comput. Vis. Sci"},{"key":"13_CR33","first-page":"547","volume":"6","author":"MY Wang","year":"2004","unstructured":"M.Y. Wang, S.W. Zhou, Phase field: A variational method for structural topology optimization. Comput. Model. Eng. Sci. 6 (2004), 547\u2013566.","journal-title":"Comput. Model. Eng. Sci"}],"container-title":["International Series of Numerical Mathematics","Constrained Optimization and Optimal Control for Partial Differential Equations"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-0348-0133-1_13","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,16]],"date-time":"2024-04-16T22:46:30Z","timestamp":1713307590000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-0348-0133-1_13"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,10,28]]},"ISBN":["9783034801324","9783034801331"],"references-count":33,"URL":"https:\/\/doi.org\/10.1007\/978-3-0348-0133-1_13","relation":{},"ISSN":["0373-3149","2296-6072"],"issn-type":[{"value":"0373-3149","type":"print"},{"value":"2296-6072","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,10,28]]},"assertion":[{"value":"28 October 2011","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}}]}}