A modified phase field approximation for mean curvature flow with conservation of the volume
This paper is concerned with the motion of a time‐dependent hypersurface ∂Ω(t) in ℝd that evolves with a normal velocity where κ is the mean curvature of ∂Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation where ε is an approximation pa...
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| Veröffentlicht in: | Mathematical methods in the applied sciences 2011-07, Vol.34 (10), p.1157-1180 |
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| creator | Brassel, M. Bretin, E. |
| description | This paper is concerned with the motion of a time‐dependent hypersurface ∂Ω(t) in ℝd that evolves with a normal velocity
where κ is the mean curvature of ∂Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation
where ε is an approximation parameter, W a double well potential and cW a constant that depends only on W. We study here a modified version of this equation
and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd. |
| doi_str_mv | 10.1002/mma.1426 |
| format | Article |
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where κ is the mean curvature of ∂Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation
where ε is an approximation parameter, W a double well potential and cW a constant that depends only on W. We study here a modified version of this equation
and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd.</description><identifier>ISSN: 0170-4214</identifier><identifier>ISSN: 1099-1476</identifier><identifier>EISSN: 1099-1476</identifier><identifier>DOI: 10.1002/mma.1426</identifier><identifier>CODEN: MMSCDB</identifier><language>eng</language><publisher>Chichester, UK: John Wiley & Sons, Ltd</publisher><subject>Algebra ; Algebraic geometry ; Allen-Cahn equation ; Approximation ; asymptotic analysis ; Conservation ; Convergence ; Curvature ; Differential geometry ; Exact sciences and technology ; Geometry ; Mathematical analysis ; Mathematics ; mean curvature flow ; Numerical Analysis ; Sciences and techniques of general use ; volume conservation</subject><ispartof>Mathematical methods in the applied sciences, 2011-07, Vol.34 (10), p.1157-1180</ispartof><rights>Copyright © 2011 John Wiley & Sons, Ltd.</rights><rights>2015 INIST-CNRS</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c4336-a0e5d98946f833e4407fe286cb1ab3573ee27e1612ca1d13003f5e8e7ef637b63</citedby><cites>FETCH-LOGICAL-c4336-a0e5d98946f833e4407fe286cb1ab3573ee27e1612ca1d13003f5e8e7ef637b63</cites><orcidid>0000-0003-1319-7538</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://onlinelibrary.wiley.com/doi/pdf/10.1002%2Fmma.1426$$EPDF$$P50$$Gwiley$$H</linktopdf><linktohtml>$$Uhttps://onlinelibrary.wiley.com/doi/full/10.1002%2Fmma.1426$$EHTML$$P50$$Gwiley$$H</linktohtml><link.rule.ids>230,314,780,784,885,1417,27924,27925,45574,45575</link.rule.ids><backlink>$$Uhttp://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=24212856$$DView record in Pascal Francis$$Hfree_for_read</backlink><backlink>$$Uhttps://hal.science/hal-00749638$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Brassel, M.</creatorcontrib><creatorcontrib>Bretin, E.</creatorcontrib><title>A modified phase field approximation for mean curvature flow with conservation of the volume</title><title>Mathematical methods in the applied sciences</title><addtitle>Math. Meth. Appl. Sci</addtitle><description>This paper is concerned with the motion of a time‐dependent hypersurface ∂Ω(t) in ℝd that evolves with a normal velocity
where κ is the mean curvature of ∂Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation
where ε is an approximation parameter, W a double well potential and cW a constant that depends only on W. We study here a modified version of this equation
and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd.</description><subject>Algebra</subject><subject>Algebraic geometry</subject><subject>Allen-Cahn equation</subject><subject>Approximation</subject><subject>asymptotic analysis</subject><subject>Conservation</subject><subject>Convergence</subject><subject>Curvature</subject><subject>Differential geometry</subject><subject>Exact sciences and technology</subject><subject>Geometry</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>mean curvature flow</subject><subject>Numerical Analysis</subject><subject>Sciences and techniques of general use</subject><subject>volume conservation</subject><issn>0170-4214</issn><issn>1099-1476</issn><issn>1099-1476</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2011</creationdate><recordtype>article</recordtype><recordid>eNqF0cuKFDEUBuAgCrbjgI-QjaCLGnOrpGrZjnMRerzAiBshpFMndDRVaZOq7pm3N0U3jRtxlXD4-DmHH6FXlFxQQti7vjcXVDD5BC0oaduKCiWfogWhilSCUfEcvcj5JyGkoZQt0I8l7mPnnYcObzcmAy7f0GGz3ab44Hsz-jhgFxPuwQzYTmlnxikVFuIe7_24wTYOGebxLKPD4wbwLoaph5fomTMhw_nxPUPfrq_uL2-r1eebj5fLVWUF57IyBOqubVohXcM5CEGUA9ZIu6ZmzWvFAZgCKimzhnaUE8JdDQ0ocJKrteRn6O0hd2OC3qaydXrU0Xh9u1zpeUaIEq3kzY4V--Zgy32_J8ij7n22EIIZIE5ZU6mo4KqW7f8p4ZQ1SpG_qE0x5wTutAYleu5Fl1703Euhr4-pJlsTXDKD9fnkWSmJNfXsqoPb-wCP_8zTd3fLY-7R-zzCw8mb9EtLVQ7S3z_d6A9cfn3_hbWa8T-pxqjw</recordid><startdate>20110715</startdate><enddate>20110715</enddate><creator>Brassel, M.</creator><creator>Bretin, E.</creator><general>John Wiley & Sons, Ltd</general><general>Wiley</general><scope>BSCLL</scope><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7ST</scope><scope>7U6</scope><scope>C1K</scope><scope>SOI</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>KR7</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0003-1319-7538</orcidid></search><sort><creationdate>20110715</creationdate><title>A modified phase field approximation for mean curvature flow with conservation of the volume</title><author>Brassel, M. ; Bretin, E.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c4336-a0e5d98946f833e4407fe286cb1ab3573ee27e1612ca1d13003f5e8e7ef637b63</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2011</creationdate><topic>Algebra</topic><topic>Algebraic geometry</topic><topic>Allen-Cahn equation</topic><topic>Approximation</topic><topic>asymptotic analysis</topic><topic>Conservation</topic><topic>Convergence</topic><topic>Curvature</topic><topic>Differential geometry</topic><topic>Exact sciences and technology</topic><topic>Geometry</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>mean curvature flow</topic><topic>Numerical Analysis</topic><topic>Sciences and techniques of general use</topic><topic>volume conservation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Brassel, M.</creatorcontrib><creatorcontrib>Bretin, E.</creatorcontrib><collection>Istex</collection><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Environment Abstracts</collection><collection>Sustainability Science Abstracts</collection><collection>Environmental Sciences and Pollution Management</collection><collection>Environment Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>Civil Engineering Abstracts</collection><collection>Hyper Article en Ligne (HAL)</collection><jtitle>Mathematical methods in the applied sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Brassel, M.</au><au>Bretin, E.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>A modified phase field approximation for mean curvature flow with conservation of the volume</atitle><jtitle>Mathematical methods in the applied sciences</jtitle><addtitle>Math. Meth. Appl. Sci</addtitle><date>2011-07-15</date><risdate>2011</risdate><volume>34</volume><issue>10</issue><spage>1157</spage><epage>1180</epage><pages>1157-1180</pages><issn>0170-4214</issn><issn>1099-1476</issn><eissn>1099-1476</eissn><coden>MMSCDB</coden><abstract>This paper is concerned with the motion of a time‐dependent hypersurface ∂Ω(t) in ℝd that evolves with a normal velocity
where κ is the mean curvature of ∂Ω(t), and g is an external forcing term. Phase field approximation of this motion leads to the Allen–Cahn equation
where ε is an approximation parameter, W a double well potential and cW a constant that depends only on W. We study here a modified version of this equation
and we prove its convergence to the same geometric motion. We then make use of this modified equation in the context of mean curvature flow with conservation of the volume, and we show that it has better volume‐preserving properties than the traditional nonlocal Allen–Cahn equation. Copyright © 2011 John Wiley & Sons, Ltd.</abstract><cop>Chichester, UK</cop><pub>John Wiley & Sons, Ltd</pub><doi>10.1002/mma.1426</doi><tpages>24</tpages><orcidid>https://orcid.org/0000-0003-1319-7538</orcidid></addata></record> |
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| subjects | Algebra Algebraic geometry Allen-Cahn equation Approximation asymptotic analysis Conservation Convergence Curvature Differential geometry Exact sciences and technology Geometry Mathematical analysis Mathematics mean curvature flow Numerical Analysis Sciences and techniques of general use volume conservation |
| title | A modified phase field approximation for mean curvature flow with conservation of the volume |
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