An inverse model for a free-boundary problem with a contact line: Steady case
This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamical...
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| Published in | Journal of computational physics Vol. 228; no. 13; pp. 4893 - 4910 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
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20.07.2009
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| ISSN | 0021-9991 1090-2716 |
| DOI | 10.1016/j.jcp.2009.03.042 |
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| Abstract | This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena. |
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| AbstractList | This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with respect to the position of the interface and subject to PDE constraints. An advantage of this formulation is that it allows for a thermodynamically consistent treatment of the interface conditions in the presence of a contact point involving a third phase. It is argued that such an approach in fact represents a closure model for the original system and some of its key properties are investigated. We describe an efficient iterative solution method for the Stefan problem formulated in this way which uses shape differentiation and adjoint equations to determine the gradient of the cost functional. Performance of the proposed approach is illustrated with sample computations concerning 2D steady solidification phenomena. |
| Author | Protas, Bartosz Volkov, Oleg |
| Author_xml | – sequence: 1 givenname: Oleg surname: Volkov fullname: Volkov, Oleg – sequence: 2 givenname: Bartosz surname: Protas fullname: Protas, Bartosz email: bprotas@mcmaster.ca |
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| Cites_doi | 10.1017/S0022112094001333 10.1007/BF01204238 10.1063/1.869345 10.1093/imamat/hxl024 10.1023/A:1026095405906 10.1051/m2an/1997310708051 10.1007/978-3-0348-8627-7_27 10.1080/01630563.1980.10120631 10.1137/05062679X 10.1002/fld.203 10.1016/j.jcp.2003.08.031 10.1209/0295-5075/6/7/007 10.1063/1.869344 10.1090/qam/1359510 10.1093/imanum/14.3.411 10.1017/S0022112006001935 10.1137/S0036139995279901 10.1002/(SICI)1097-0207(19990320)44:8<1079::AID-NME543>3.0.CO;2-I 10.1016/j.apnum.2006.03.017 |
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| Keywords | Sobolev gradients 64.70.D 02.30.Zz Stefan conditions Shape calculus 44.05.+e Free-boundary problem Contact line Partial differential equations Stefan problem Free boundary problem Calculation methods Closure model Inverse problems Functionals Positions Calculation Iterative methods Performance |
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| Snippet | This paper reformulates the two-phase solidification problem (i.e., the Stefan problem) as an inverse problem in which a cost functional is minimized with... |
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| SubjectTerms | BOUNDARY CONDITIONS Computational techniques Contact line Exact sciences and technology Free-boundary problem ITERATIVE METHODS MATHEMATICAL METHODS AND COMPUTING Mathematical methods in physics MATHEMATICAL SOLUTIONS PARTIAL DIFFERENTIAL EQUATIONS Physics Shape calculus Sobolev gradients SOLIDIFICATION Stefan conditions |
| Title | An inverse model for a free-boundary problem with a contact line: Steady case |
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