×

Isogeometric shape optimization for nonlinear ultrasound focusing. (English) Zbl 1423.35315

Summary: The goal of this work is to improve focusing of high-intensity ultrasound by modifying the geometry of acoustic lenses through shape optimization. We formulate the shape optimization problem by introducing a tracking-type cost functional to match a desired pressure distribution in the focal region. Westervelt’s equation, a nonlinear acoustic wave equation, is used to model the pressure field. We apply the optimize first, then discretize approach, where we first rigorously compute the shape derivative of our cost functional. A gradient-based optimization algorithm is then developed within the concept of isogeometric analysis, where the geometry is exactly represented by splines at every gradient step and the same basis is used to approximate the equations. Numerical experiments in a 2D setting illustrate our findings.

MSC:

35Q35 PDEs in connection with fluid mechanics
49Q10 Optimization of shapes other than minimal surfaces
76Q05 Hydro- and aero-acoustics

Software:

GeoPDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] L. Beirão da Veiga; A. Buffa; G. Sangalli; R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numerica, 23, 157-287 (2014) · Zbl 1398.65287 · doi:10.1017/S096249291400004X
[2] A. Blana; N. Walter; S. Rogenhofer; W. F. Wieland, High-intensity focused ultrasound for the treatment of localized prostate cancer: 5-year experience, Urology, 63, 297-300 (2004) · doi:10.1016/j.urology.2003.09.020
[3] C. Brandenburg; F. Lindemann; M. Ulbrich; S. Ulbrich, Advanced numerical methods for PDE constrained optimization with application to optimal design in Navier Stokes flow, Constrained Optimization and Optimal Control for Partial Differential Equations, Springer Basel, 160, 257-275 (2012) · Zbl 1356.49019 · doi:10.1007/978-3-0348-0133-1_14
[4] G. J. Brereton; B. A. Bruno, Particle removal by focused ultrasound, Journal of Sound and Vibration, 173, 683-698 (1994) · doi:10.1006/jsvi.1994.1253
[5] M. S. Canney; Bailey; L. A. Crum; V. A. Khokhlova; O. A. Sapozhnikov, Acoustic characterization of high intensity focused ultrasound fields: A combined measurement and modeling approach, The Journal of the Acoustical Society of America, 124, 2406-2420 (2008) · doi:10.1121/1.2967836
[6] S. Cho; S.-H. Ha, Isogeometric shape design optimization: Exact geometry and enhanced sensitivity, Structural and Multidisciplinary Optimization, 38, 53-70 (2009) · Zbl 1274.74221 · doi:10.1007/s00158-008-0266-z
[7] J. Chung; G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, Journal of Applied Mechanics, 60, 371-375 (1993) · Zbl 0775.73337 · doi:10.1115/1.2900803
[8] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. · Zbl 0522.35001
[9] J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009. · Zbl 1378.65009
[10] D. G. Crighton, Model equations of nonlinear acoustics, Annual Review of Fluid Mechanics, 11, 11-33 (1979) · Zbl 0443.76074
[11] F. Demengel, G. Demengel and R. Ern, Functional Spaces for the Theory of Elliptic Partial Differential Equations, London, UK: Springer, 2012. · Zbl 1239.46001
[12] G. Dogan; P. Morin; R. H. Nochetto; M. Verani, Discrete gradient flows for shape optimization and applications, Computer Methods in Applied Mechanics and Engineering, 196, 3898-3914 (2007) · Zbl 1173.49307 · doi:10.1016/j.cma.2006.10.046
[13] B. Engquist; A. Majda, Absorbing boundary conditions for the numerical evaluation of waves, Proceedings of the National Academy of Sciences, 74, 1765-1766 (1977) · Zbl 0378.76018 · doi:10.1073/pnas.74.5.1765
[14] K. Eppler; H. Harbrecht, Coupling of FEM and BEM in shape optimization, Numerische Mathematik, 104, 47-68 (2006) · Zbl 1104.65065 · doi:10.1007/s00211-006-0005-6
[15] K. Eppler; H. Harbrecht; R. Schneider, On convergence in elliptic shape optimization, SIAM Journal on Control and Optimization, 46, 61-83 (2007) · Zbl 1354.49093 · doi:10.1137/05062679X
[16] S. Erlicher; L. Bonaventura; O. S. Bursi, The analysis of the Generalized-α method for nonlinear dynamic problems, Computational Mechanics, 28, 83-104 (2002) · Zbl 1146.74327 · doi:10.1007/s00466-001-0273-z
[17] C. de Falco; A. Reali; R. Vázquez, GeoPDEs: A research tool for Isogeometric Analysis of PDEs, Advances in Engineering Software, 42, 1020-1034 (2011) · Zbl 1246.35010 · doi:10.1016/j.advengsoft.2011.06.010
[18] D. L. Folds, Speed of sound and transmission loss in silicone rubbers at ultrasonic frequencies, The Journal of the Acoustical Society of America, 56, 1295-1296 (1974) · doi:10.1121/1.1903422
[19] D. Fußeder; A.-V. Vuong; B. Simeon, Fundamental aspects of shape optimization in the context of isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 286, 313-331 (2015) · Zbl 1425.65159 · doi:10.1016/j.cma.2014.12.028
[20] D. Fußeder and B. Simeon, Algorithmic aspects of isogeometric shape optimization, In B. Jüttler and B. Simeon (editors), Isogeometric Analysis and Applications 2014, 183-207, Lect. Notes Comput. Sci. Eng., 107, Springer, Cham, 2015. · Zbl 1334.65109
[21] P. Gangl; U. Langer; A. Laurain; H. Meftahi; K. Sturm, Shape optimization of an electric motor subject to nonlinear magnetostatics, SIAM Journal on Scientific Computing, 37, B1002-B1025 (2015) · Zbl 1330.49042 · doi:10.1137/15100477X
[22] H. Harbrecht, Analytical and numerical methods in shape optimization, Mathematical Methods in the Applied Sciences, 31, 2095-2114 (2008) · Zbl 1153.49039 · doi:10.1002/mma.1008
[23] J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material, and Topology Design, John Wiley & Sons, 1996. · Zbl 0845.73001
[24] A. Henrot and M. Pierre, Variation et Optimisation de Formes: Une Analyse Géométrique, Springer Science & Business Media, 48, Springer, Berlin, 2005. · Zbl 1098.49001
[25] J. Hoffelner; H. Landes; M. Kaltenbacher; R. Lerch, Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 48, 779-786 (2001) · doi:10.1109/58.920712
[26] S. Hofmann; M. Mitrea; M. Taylor, Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro Domains, and other classes of finite perimeter domains, The Journal of Geometric Analysis, 17, 593-647 (2007) · Zbl 1142.49021 · doi:10.1007/BF02937431
[27] K. Höllig, Finite Element Methods with B-Splines, SIAM, 2003. · Zbl 1020.65085
[28] T. J. R. Hughes; J. A. Cottrell; Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194, 4135-4195 (2005) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008
[29] K. Ito; K. Kunisch; G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems, Journal of Mathematical Analysis and Applications, 314, 126-149 (2006) · Zbl 1088.49028 · doi:10.1016/j.jmaa.2005.03.100
[30] K. Ito; K. Kunisch; G. Peichl, Variational approach to shape derivatives, ESAIM: Control, Optimisation and Calculus of Variations, 14, 517-539 (2008) · Zbl 1357.49148 · doi:10.1051/cocv:2008002
[31] B. Kaltenbacher; I. Lasiecka, Well-posedness of the Westervelt and the Kuznetsov equation with nonhomogeneous Neumann boundary conditions, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications. 8th AIMS Conference. Suppl., 2, 763-773 (2011) · Zbl 1306.35075
[32] B. Kaltenbacher; I. Lasiecka; S. Veljović, Well-posedness and exponential decay for the Westervelt equation with inhomogeneous Dirichlet boundary data, Progress in Nonlinear Differential Equations and Their Applications, 80, 357-387 (2011) · Zbl 1250.35145 · doi:10.1007/978-3-0348-0075-4_19
[33] B. Kaltenbacher; G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy, Evolution Equations and Control Theory(EECT), 5, 399-429 (2016) · Zbl 1350.49064 · doi:10.3934/eect.2016011
[34] B. Kaltenbacher; S. Veljović, Sensitivity analysis of linear and nonlinear lithotripter models, European Journal of Applied Mathematics, 22, 21-43 (2011) · Zbl 1215.35170 · doi:10.1017/S0956792510000276
[35] M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuators, Springer, Berlin, 2004. · Zbl 1072.78001
[36] J. E. Kennedy, High-intensity focused ultrasound in the treatment of solid tumors, Nature Reviews Cancer, 5, 321-327 (2005)
[37] J. E. Kennedy; F. Wu; G. R. Ter Haar; F. V. Gleeson; R. R. Phillips; M. R. Middleton; D. Cranston, High-intensity focused ultrasound for the treatment of liver tumors, Ultrasonics, 42, 931-935 (2004)
[38] J. Kiendl; R. Schmidt; R. Wüchner; K.-U. Bletzinger, Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting, Computer Methods in Applied Mechanics and Engineering, 274, 148-167 (2014) · Zbl 1296.74082 · doi:10.1016/j.cma.2014.02.001
[39] D. Kuhl; M. A. Crisfield, Energy-conserving and decaying algorithms in nonlinear structural mechanics, International Journal for Numerical Methods in Engineering, 45, 569-599 (1999) · Zbl 0946.74078 · doi:10.1002/(SICI)1097-0207(19990620)45:53.0.CO;2-A
[40] E. Laporte and P. Le Tallec, Numerical Methods in Sensitivity Analysis and Shape Optimization, Birkh’auser Boston, Inc., Boston, MA, 2003. · Zbl 1013.74002
[41] Y.-S. Lee, Numerical Solution of the KZK Equation for Pulsed Finite Amplitude Sound Beams in Thermoviscous Fluids, PhD Thesis, The University of Texas at Austin, 1993.
[42] D. Lee; N. Koizumi; K. Ota; S. Yoshizawa; A. Ito; Y. Kaneko; Y. Matsumoto; M. Mitsuishi, Ultrasound-based visual serving system for lithotripsy, Intelligent Robots and Systems, 877-882 (2007)
[43] F. Maestre; A. Münch; P. Pedregal, A spatio-temporal design problem for a damped wave equation, SIAM Journal on Applied Mathematics, 68, 109-132 (2007) · Zbl 1147.35052 · doi:10.1137/07067965X
[44] E. Maloney; J. H. Hwang, Emerging HIFU applications in cancer therapy, International Journal of Hyperthermia, 31, 302-309 (2015) · doi:10.3109/02656736.2014.969789
[45] J. G. Mancini; A. Neisius; N. Smith; G. Sankin; G. M. Astroza; M. E. Lipkin; W. N. Simmons; G. M. Preminger; P. Zhong, Assessment of a modified acoustic lens for electromagnetic shock wave lithotripters in a swine model, The Journal of Urology, 190, 1096-1101 (2013) · doi:10.1016/j.juro.2013.02.074
[46] S. Meyer; M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Applied Mathematics & Optimization, 64, 257-271 (2011) · Zbl 1233.35061 · doi:10.1007/s00245-011-9138-9
[47] A. Münch, Optimal design of the support of the control for the 2-D wave equation: A numerical method, International Journal of Numerical Analysis and Modeling, 5, 331-351 (2008) · Zbl 1242.49091
[48] F. Murat; S. Simon, Etudes de problems d’optimal design, Lecture Notes in Computer Science, 41, 54-62 (1976) · Zbl 0334.49013
[49] A. Neisius; N. B. Smith; G. Sankin; N. J. Kuntz; J. F. Madden; D. E. Fovargue; S. Mitran; M. E. Lipkin; W. N. Simmons; G. M. Preminger; P. Zhong, Improving the lens design and performance of a contemporary electromagnetic shock wave lithotripter, Proceedings of the National Academy of Sciences, 111, E1167-E1175 (2014) · doi:10.1073/pnas.1319203111
[50] N. M. Newmark, A method of computation for structural dynamics, Journal of Engineering Mechanics, ASCE, 85, 67-94 (1959)
[51] D. M. Nguyen; A. Evgrafov; J. Gravesen, Isogeometric shape optimization for scattering problems, Progress In Electromagnetics Research B, 45, 117-146 (2012)
[52] V. Nikolić; B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy, Applied Mathematics and Optimization, 76, 261-301 (2017) · Zbl 1378.49051 · doi:10.1007/s00245-016-9340-x
[53] P. Nortoft; J. Gravesen, Isogeometric shape optimization in fluid mechanics, Struct Multidisc Optim, 48, 909-925 (2013) · doi:10.1007/s00158-013-0931-8
[54] A. Paganini, Numerical Shape Optimization with Finite Elements, PhD Thesis, ETH Zürich, 2016.
[55] S. H. Park; J. W. Yoon; D. Y. Yang; Y. H. Kim, Optimum blank design in sheet metal forming by the deformation path iteration method, International Journal of Mechanical Sciences, 41, 1217-1232 (1999) · Zbl 0976.74049 · doi:10.1016/S0020-7403(98)00084-8
[56] R. F. Paterson; E. Barret; T. M. Siqueira; T. A. Gardner; J. Tavakkoli; V. V. Rao; N. T. Sanghvi; L. Cheng; A. L. Shalhav, Laparoscopic partial kidney ablation with high intensity focused ultrasound, The Journal of Urology, 169, 347-351 (2003)
[57] L. Piegl and W. Tiller, The NURBS Book, Springer, 1997. · Zbl 0868.68106
[58] X. Qian; O. Sigmund, Isogeometric shape optimization of photonic crystals via Coons patches, Computer Methods in Applied Mechanics and Engineering, 200, 2237-2255 (2011) · Zbl 1230.74149 · doi:10.1016/j.cma.2011.03.007
[59] T. D. Rossing (Ed.), Springer Handbook of Acoustics, Springer, 2014. · Zbl 1304.92001
[60] L. Schumaker, Spline Functions: Basic Theory, 3rd Edition, Cambridge University Press, Cambridge, 2007. · Zbl 1123.41008
[61] S. Veljović, Shape Optimization and Optimal Boundary Control for High Intensity Focused Ultrasound (HIFU), PhD Thesis, University of Erlangen-Nuremberg, 2009. · Zbl 1281.76003
[62] W. A. Wall; M. A. Frenzel; C. Cyron, Isogeometric structural shape optimization, Computer Methods in Applied Mechanics and Engineering, 197, 2976-2988 (2008) · Zbl 1194.74263 · doi:10.1016/j.cma.2008.01.025
[63] F. Wu; W. Z. Chen; J. Bai; J. Z. Zou; Z. L. Wang; H. Zhu; Z. B. Wang, Pathological changes in human malignant carcinoma treated with high-intensity focused ultrasound, Ultrasound in Medicine & Biology, 27, 1099-1106 (2001) · doi:10.1016/S0301-5629(01)00389-1
[64] S. Yoshizawa; T. Ikeda; A. Ito; R. Ota; S. Takagi; Y. Matsumoto, High intensity focused ultrasound lithotripsy with cavitating microbubbles, Medical & Biological Engineering & Computing, 47, 851-860 (2009) · doi:10.1007/s11517-009-0471-y
[65] P. Zhong; N. Smith; N. W. Simmons; G. Sankin, A new acoustic lens design for electromagnetic shock wave lithotripters, AIP Conference Proceedings, 1359, 42-47 (2011) · doi:10.1063/1.3607880
[66] J.-P. Zolésio and M. C. Delfour, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, SIAM, 2011. · Zbl 1251.49001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.