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# The shape derivative for an optimization problem in lithotripsy

• In this paper we consider a shape optimization problem motivated by the use of high intensity focused ultrasound in lithotripsy. This leads to the problem of designing a Neumann boudary part in the context of the Westervelt equation, which is a common model in nonlinear acoustics. Based on regularity results for solutions of this equation and its linearization, we rigorously compute the shape derivative for this problem, relying on the variational framework from [9].
Mathematics Subject Classification: Primary: 49Q10, 35L72; Secondary: 35L80, 49K20.

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•  [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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