American Institute of Mathematical Sciences

Advanced
September  2016, 5(3): 399-430. doi: 10.3934/eect.2016011

The shape derivative for an optimization problem in lithotripsy

Barbara Kaltenbacher 1, and  Gunther Peichl 2,

1. 

Alpen-Adria-Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt
2. 

University of Graz, Heinrichstraße 36, 8010 Graz, Austria

Received  January 2016 Revised  March 2016 Published  August 2016

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].
Keywords: well-posedness, Nonlinear acoustics, partial differential equations, shape optimization..
Mathematics Subject Classification: Primary: 49Q10, 35L72; Secondary: 35L80, 49K2.
Citation: Barbara Kaltenbacher, Gunther Peichl. The shape derivative for an optimization problem in lithotripsy. Evolution Equations & Control Theory, 2016, 5 (3) : 399-430. doi: 10.3934/eect.2016011
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, Pure and Applied Mathematics. Elsevier Science, (2003).   Google Scholar

[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.  Google Scholar

[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.  doi: 10.1007/s10665-012-9608-3.  Google Scholar

[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.  doi: 10.3934/eect.2013.2.281.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[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).  doi: 10.1137/1.9780898719826.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics,, American Mathematical Society, (1998).   Google Scholar

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman Advanced Pub. Program Boston, (1985).   Google Scholar

[9]

K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives,, ESAIM: Control, 14 (2008), 517.  doi: 10.1051/cocv:2008002.  Google Scholar

[10]

V. Nikolić and B. Kaltenbacher, Sensitivity analysis for an optimal shape of a focusing lens in lithotripsy,, Applied Mathematics and Optimization, (2016), 1.   Google Scholar

[11]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation,, J. Comp. Phys., 302 (2015), 200.  doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[12]

J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis,, Springer Series in Computational Mathematics, (1992).  doi: 10.1007/978-3-642-58106-9.  Google Scholar

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces,, Pure and Applied Mathematics. Elsevier Science, (2003).   Google Scholar

[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.  Google Scholar

[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.  doi: 10.1007/s10665-012-9608-3.  Google Scholar

[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.  doi: 10.3934/eect.2013.2.281.  Google Scholar

[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.  doi: 10.1016/j.jmaa.2009.03.043.  Google Scholar

[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).  doi: 10.1137/1.9780898719826.  Google Scholar

[7]

L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics,, American Mathematical Society, (1998).   Google Scholar

[8]

P. Grisvard, Elliptic Problems in Nonsmooth Domains,, Pitman Advanced Pub. Program Boston, (1985).   Google Scholar

[9]

K. Ito, K. Kunisch and G. Peichl, Variational approach to shape derivatives,, ESAIM: Control, 14 (2008), 517.  doi: 10.1051/cocv:2008002.  Google Scholar

[10]

V. Nikolić and B. Kaltenbacher, Sensitivity analysis for an optimal shape of a focusing lens in lithotripsy,, Applied Mathematics and Optimization, (2016), 1.   Google Scholar

[11]

I. Shevchenko and B. Kaltenbacher, Absorbing boundary conditions for nonlinear acoustics: The Westervelt equation,, J. Comp. Phys., 302 (2015), 200.  doi: 10.1016/j.jcp.2015.08.051.  Google Scholar

[12]

J. Sokolowski and J. P. Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis,, Springer Series in Computational Mathematics, (1992).  doi: 10.1007/978-3-642-58106-9.  Google Scholar

[1]

Can Li, Weihua Deng, Lijing Zhao. Well-posedness and numerical algorithm for the tempered fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1989-2015. doi: 10.3934/dcdsb.2019026
[2]

Jerry Bona, Hongqiu Chen. Well-posedness for regularized nonlinear dispersive wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (4) : 1253-1275. doi: 10.3934/dcds.2009.23.1253
[3]

Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455
[4]

Thomas Y. Hou, Congming Li. Global well-posedness of the viscous Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 1-12. doi: 10.3934/dcds.2005.12.1
[5]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[6]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737
[7]

Luciano Abadías, Carlos Lizama, Pedro J. Miana, M. Pilar Velasco. On well-posedness of vector-valued fractional differential-difference equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2679-2708. doi: 10.3934/dcds.2019112
[8]

Hiroyuki Hirayama, Mamoru Okamoto. Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity. Communications on Pure & Applied Analysis, 2016, 15 (3) : 831-851. doi: 10.3934/cpaa.2016.15.831
[9]

Changxing Miao, Bo Zhang. Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 181-200. doi: 10.3934/dcds.2007.17.181
[10]

Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15
[11]

Chengchun Hao. Well-posedness for one-dimensional derivative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2007, 6 (4) : 997-1021. doi: 10.3934/cpaa.2007.6.997
[12]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563
[13]

Yonggeun Cho, Gyeongha Hwang, Tohru Ozawa. Global well-posedness of critical nonlinear Schrödinger equations below $L^2$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1389-1405. doi: 10.3934/dcds.2013.33.1389
[14]

Takeshi Wada. A remark on local well-posedness for nonlinear Schrödinger equations with power nonlinearity-an alternative approach. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1359-1374. doi: 10.3934/cpaa.2019066
[15]

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu. On the probabilistic well-posedness of the nonlinear Schrödinger equations with non-algebraic nonlinearities. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3479-3520. doi: 10.3934/dcds.2019144
[16]

Hartmut Pecher. Local well-posedness for the nonlinear Dirac equation in two space dimensions. Communications on Pure & Applied Analysis, 2014, 13 (2) : 673-685. doi: 10.3934/cpaa.2014.13.673
[17]

Zhaohui Huo, Boling Guo. The well-posedness of Cauchy problem for the generalized nonlinear dispersive equation. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 387-402. doi: 10.3934/dcds.2005.12.387
[18]

David Melching, Ulisse Stefanelli. Well-posedness of a one-dimensional nonlinear kinematic hardening model. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020188
[19]

Hongjun Gao, Chengfeng Sun. Well-posedness of stochastic primitive equations with multiplicative noise in three dimensions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3053-3073. doi: 10.3934/dcdsb.2016087
[20]

Yong Zhou, Jishan Fan. Local well-posedness for the ideal incompressible density dependent magnetohydrodynamic equations. Communications on Pure & Applied Analysis, 2010, 9 (3) : 813-818. doi: 10.3934/cpaa.2010.9.813

2018 Impact Factor: 1.048

Tools

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences