A LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEM

Handle URI:
http://hdl.handle.net/10754/597294
Title:
A LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEM
Authors:
Burger, Martin; Matevosyan, Norayr; Wolfram, Marie-Therese
Abstract:
In this paper, we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities, we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments. © 2011 World Scientific Publishing Company.
Citation:
Burger M, Matevosyan N, Wolfram M-T (2011) A LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEM. Mathematical Models and Methods in Applied Sciences 21: 619–649. Available: http://dx.doi.org/10.1142/S0218202511005155.
Publisher:
World Scientific Pub Co Pte Lt
Journal:
Mathematical Models and Methods in Applied Sciences
KAUST Grant Number:
KUK-I1-007-43
Issue Date:
Apr-2011
DOI:
10.1142/S0218202511005155
Type:
Article
ISSN:
0218-2025; 1793-6314
Sponsors:
Part of this work was carried out when the authors were with the Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz, and the Johannes Kepler University Linz, respectively. The authors thank Heinz Engl (RICAM and University of Vienna) and Peter Markowich (Cambridge University and RICAM) for stimulating this joint research. M.B. acknowledges financial support by the Austrian Science Foundation FWF through project SFB F 013/08, and the German Research Foundation DFG through the project Regularization with Singular Energies. Work of N.M. was partially supported by the WWTF (Wiener Wissenschafts, Forschungs und Technologiefonds) project No. CI06 003, while he was at the University of Vienna. Also the work of N.M. and M.T.W. supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).
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Full metadata record

DC FieldValue Language
dc.contributor.authorBurger, Martinen
dc.contributor.authorMatevosyan, Norayren
dc.contributor.authorWolfram, Marie-Thereseen
dc.date.accessioned2016-02-25T12:30:02Zen
dc.date.available2016-02-25T12:30:02Zen
dc.date.issued2011-04en
dc.identifier.citationBurger M, Matevosyan N, Wolfram M-T (2011) A LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEM. Mathematical Models and Methods in Applied Sciences 21: 619–649. Available: http://dx.doi.org/10.1142/S0218202511005155.en
dc.identifier.issn0218-2025en
dc.identifier.issn1793-6314en
dc.identifier.doi10.1142/S0218202511005155en
dc.identifier.urihttp://hdl.handle.net/10754/597294en
dc.description.abstractIn this paper, we construct a level set method for an elliptic obstacle problem, which can be reformulated as a shape optimization problem. We provide a detailed shape sensitivity analysis for this reformulation and a stability result for the shape Hessian at the optimal shape. Using the shape sensitivities, we construct a geometric gradient flow, which can be realized in the context of level set methods. We prove the convergence of the gradient flow to an optimal shape and provide a complete analysis of the level set method in terms of viscosity solutions. To our knowledge this is the first complete analysis of a level set method for a nonlocal shape optimization problem. Finally, we discuss the implementation of the methods and illustrate its behavior through several computational experiments. © 2011 World Scientific Publishing Company.en
dc.description.sponsorshipPart of this work was carried out when the authors were with the Johann Radon Institute for Computational and Applied Mathematics (RICAM) Linz, and the Johannes Kepler University Linz, respectively. The authors thank Heinz Engl (RICAM and University of Vienna) and Peter Markowich (Cambridge University and RICAM) for stimulating this joint research. M.B. acknowledges financial support by the Austrian Science Foundation FWF through project SFB F 013/08, and the German Research Foundation DFG through the project Regularization with Singular Energies. Work of N.M. was partially supported by the WWTF (Wiener Wissenschafts, Forschungs und Technologiefonds) project No. CI06 003, while he was at the University of Vienna. Also the work of N.M. and M.T.W. supported by Award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherWorld Scientific Pub Co Pte Lten
dc.subjectfree boundary problemsen
dc.subjectLevel set methodsen
dc.subjectobstacle problemen
dc.titleA LEVEL SET BASED SHAPE OPTIMIZATION METHOD FOR AN ELLIPTIC OBSTACLE PROBLEMen
dc.typeArticleen
dc.identifier.journalMathematical Models and Methods in Applied Sciencesen
dc.contributor.institutionWestfalische Wilhelms-Universitat Munster, Munster, Germanyen
dc.contributor.institutionUniversity of Cambridge, Cambridge, United Kingdomen
kaust.grant.numberKUK-I1-007-43en
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