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The shape derivative for an optimization problem in lithotripsy
1. | Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt |
2. | University of Graz, Heinrichstraße 36, 8010 Graz, Austria |
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics. Elsevier Science, 2003. |
[2] |
K. Atkinson and W. Han, Theoretical Numerical Analysis, Number 39 in Texts in Applied Mathematics. Springer, 2009.
doi: 10.1007/978-1-4419-0458-4. |
[3] |
A. Ben Abda, F. Bouchon, G. Peichl, M. Sayeh and R. Touzani, A Dirichlet-Neumann cost functional approach for the Bernoulli problem, Journal of Engineering Mathematics, 81 (2013), 157-176.
doi: 10.1007/s10665-012-9608-3. |
[4] |
C. Clason and B. Kaltenbacher, Avoiding degeneracy in the Westervelt equation by state constrained optimal control, Evolution Equations and Control Theory, 2 (2013), 281-300.
doi: 10.3934/eect.2013.2.281. |
[5] |
C. Clason, B. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equation, Journal of Mathematical Analysis and Applications, 356 (2009), 738-751.
doi: 10.1016/j.jmaa.2009.03.043. |
[6] |
C. Delfour and J. P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 2011.
doi: 10.1137/1.9780898719826. |
[7] |
L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. |
[8] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub. Program Boston, 1985. |
[9] |
K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 517-539.
doi: 10.1051/cocv:2008002. |
[10] |
V. Nikolić and B. Kaltenbacher, Sensitivity analysis for an optimal shape of a focusing lens in lithotripsy, Applied Mathematics and Optimization, pages 1-41, 2016. and arXiv:1506.02781. |
[11] |
I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, J. Comp. Phys., 302 (2015), 200-221.
doi: 10.1016/j.jcp.2015.08.051. |
[12] |
J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58106-9. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure and Applied Mathematics. Elsevier Science, 2003. |
[2] |
K. Atkinson and W. Han, Theoretical Numerical Analysis, Number 39 in Texts in Applied Mathematics. Springer, 2009.
doi: 10.1007/978-1-4419-0458-4. |
[3] |
A. Ben Abda, F. Bouchon, G. Peichl, M. Sayeh and R. Touzani, A Dirichlet-Neumann cost functional approach for the Bernoulli problem, Journal of Engineering Mathematics, 81 (2013), 157-176.
doi: 10.1007/s10665-012-9608-3. |
[4] |
C. Clason and B. Kaltenbacher, Avoiding degeneracy in the Westervelt equation by state constrained optimal control, Evolution Equations and Control Theory, 2 (2013), 281-300.
doi: 10.3934/eect.2013.2.281. |
[5] |
C. Clason, B. Kaltenbacher and S. Veljović, Boundary optimal control of the Westervelt and the Kuznetsov equation, Journal of Mathematical Analysis and Applications, 356 (2009), 738-751.
doi: 10.1016/j.jmaa.2009.03.043. |
[6] |
C. Delfour and J. P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition, Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104), 2011.
doi: 10.1137/1.9780898719826. |
[7] |
L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998. |
[8] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman Advanced Pub. Program Boston, 1985. |
[9] |
K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives, ESAIM: Control, Optimisation and Calculus of Variations, 14 (2008), 517-539.
doi: 10.1051/cocv:2008002. |
[10] |
V. Nikolić and B. Kaltenbacher, Sensitivity analysis for an optimal shape of a focusing lens in lithotripsy, Applied Mathematics and Optimization, pages 1-41, 2016. and arXiv:1506.02781. |
[11] |
I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation, J. Comp. Phys., 302 (2015), 200-221.
doi: 10.1016/j.jcp.2015.08.051. |
[12] |
J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer Series in Computational Mathematics, 16. Springer-Verlag, Berlin, 1992.
doi: 10.1007/978-3-642-58106-9. |
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