Advanced Search
Article Contents
Article Contents

Isogeometric shape optimization for nonlinear ultrasound focusing

  • * Corresponding author: Vanja Nikolić

    * Corresponding author: Vanja Nikolić 
Abstract Full Text(HTML) Figure(17) / Table(5) Related Papers Cited by
  • 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 $ 2 $D setting illustrate our findings.

    Mathematics Subject Classification: Primary: 35, 49; Secondary: 49Q10, 35L05.


    \begin{equation} \\ \end{equation}
  • 加载中
  • Figure 1.  Sketches of different ultrasound focusing approaches; left: Focusing by a lens, right: Focusing by an array of transducers placed on a curved surface

    Figure 2.  Formation of a saw-tooth pattern of a nonlinear wave in a channel setting with a sinusoidal excitation signal, compared to a linear wave propagation.

    Figure 3.  Domain $ \Omega $, consisting of lens $ \Omega_l $ and fluid $ \Omega_f $

    Figure 4.  Patches used for the discretization. The lens domain is given by patch 3

    Figure 5.  Nonlinear wave propagation in the lens setting with steepening toward the end of the geometry. The black lines in the picture show the position of the lens

    Figure 6.  Final lens shape together with initial and goal shapes

    Figure 7.  left: Relative cost change versus the number of gradient steps. right: Norm of the shape gradient

    Figure 8.  L2-error over the course of optimization

    Figure 9.  Final lens shape, together with initial and target shape. left: Linear NURBS. right: Quadratic NURBS

    Figure 10.  Relative change of the cost over the course of optimization. left: Linear NURBS. right: Quadratic NURBS

    Figure 11.  Norm of the shape gradient over the course of optimization. left: Linear NURBS. right: Quadratic NURBS

    Figure 12.  L2 shape error over the course of optimization. left: Linear NURBS, right: Quadratic NURBS

    Figure 13.  left: Initial, final and goal shape; the final shape is a local, but not a global minimum. right: Relative cost decay

    Figure 14.  Reference lens that marks the minimal thickness possible to manufacture

    Figure 15.  left: Initial and final lens shape. right: Relative cost decay versus the number of gradient steps

    Figure 16.  Pressure-wave propagation with the final lens shape using quadratic NURBS

    Figure 17.  left: No visible changes between the initial and final lens in the post-processing optimization. right: Cost decay by 0:0729 % of its starting value

    Table 1.  Physical parameter values

    fluid lens
    $c_f=1500 \, \frac{m}{s}$ $c_l=1100 \, \frac{m}{s}$
    $b_f=6\cdot 10^{-9} \, \frac{m^2}{s}$ $ b_l=4\cdot 10^{-9} \, \frac{m^2}{s}$
    $\varrho_f=1000 \, \frac{kg}{m^3}$ $\varrho_l=1250 \, \frac{kg}{m^3}$
    $B/A=5$ $B/A=4$
     | Show Table
    DownLoad: CSV

    Table 2.  Lens measurements

    $\Omega_{l, \text{upper straight}}$ $\Omega_{l, \text{upper curved}}$ $\Omega_{l, \text{both perturbed}}$ $\Omega_{l, \text{both down}}$ $\Omega_{l, \text{Gauss}}$
    $L=0.12$m $L=0.12$m $L=0.12$m $L=0.12$m $L=0.12$m
    $B=0.05$m $B=0.05$m $B=0.05$m $B=0.05$m $B=0.05$m
    $K=0.06$m $K=0.06$m $K=0.06$m $K=0.06$m $K=0.06$m
    $W=0.04$m $W=0.04$m $W=0.04$m $W=0.04$m $W=0.04$m
    $P=0.02$m $P=0.015$m $P = 0.016$m $P=0.021$m $P=0.025$m
    $S=0.09$m $S=0.09$m $S=0.09 $m $S=0.09$m $S=0.09$m
    $R=0.04$m $R=0.04$m $R= 0.042$m $R=0.037$m $R=0.035$m
     | Show Table
    DownLoad: CSV

    Table 3.  Spatial grid sizes

    Spatial degrees of freedom
    Linear NURBS Quadratic NURBS
    $\text{ndof}_x = 46$ $\text{ndof}_x = 48$
    $\text{ndof}_y = 181$ $\text{ndof}_y = 185$
    $\text{ndof} = 7976$ $\text{ndof} = 8484$
     | Show Table
    DownLoad: CSV

    Table 4.  Time integration and numerical parameter values

    Time discretization Method parameter Tolerances
    Final time $T=90 \, \mu$s $\gamma = 0.75, \beta = 0.45$ $\text{TOL}_u=10^{-6}$
    $\text{ndof}_t=3801$ $\gamma_p=0.5, \beta_p = 0.25$ $\text{TOL}_p=10^{-8}$
    $\Delta t=23.684\, $ns $\alpha_m =1/2, \alpha_f = 1/3$ $\text{TOL}_\text{grad}=10^{-4}$
     | Show Table
    DownLoad: CSV

    Table 5.  Cost comparison with final lens shapes

    Interpolated into the
    Optimization with linear NURBS space quadratic NURBS space
    linear NURBS $J=2.937255\cdot 10^7$ $J=2.933371\cdot 10^7$
    quadratic NURBS $J= 2.930353\cdot 10^7$ $J=2.926472\cdot 10^7$
     | Show Table
    DownLoad: CSV
  • [1] L. Beirão da VeigaA. BuffaG. Sangalli and R. Vázquez, Mathematical analysis of variational isogeometric methods, Acta Numerica, 23 (2014), 157-287.  doi: 10.1017/S096249291400004X.
    [2] A. BlanaN. WalterS. Rogenhofer and W. F. Wieland, High-intensity focused ultrasound for the treatment of localized prostate cancer: 5-year experience, Urology, 63 (2004), 297-300.  doi: 10.1016/j.urology.2003.09.020.
    [3] C. BrandenburgF. LindemannM. Ulbrich and 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 (2012), 257-275.  doi: 10.1007/978-3-0348-0133-1_14.
    [4] G. J. Brereton and B. A. Bruno, Particle removal by focused ultrasound, Journal of Sound and Vibration, 173 (1994), 683-698.  doi: 10.1006/jsvi.1994.1253.
    [5] M. S. CanneyBaileyL. A. CrumV. A. Khokhlova and 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 (2008), 2406-2420.  doi: 10.1121/1.2967836.
    [6] S. Cho and S.-H. Ha, Isogeometric shape design optimization: Exact geometry and enhanced sensitivity, Structural and Multidisciplinary Optimization, 38 (2009), 53-70.  doi: 10.1007/s00158-008-0266-z.
    [7] J. Chung and G. M. Hulbert, A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method, Journal of Applied Mechanics, 60 (1993), 371-375.  doi: 10.1115/1.2900803.
    [8] D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983.
    [9] J. A. Cottrell, T. J. R. Hughes and Y. Bazilevs, Isogeometric Analysis: Toward Integration of CAD and FEA, John Wiley & Sons, 2009. doi: 10.1002/9780470749081.
    [10] D. G. Crighton, Model equations of nonlinear acoustics, Annual Review of Fluid Mechanics, 11 (1979), 11-33. 
    [11] F. Demengel, G. Demengel and R. Ern, Functional Spaces for the Theory of Elliptic Partial Differential Equations, London, UK: Springer, 2012. doi: 10.1007/978-1-4471-2807-6.
    [12] G. DoganP. MorinR. H. Nochetto and M. Verani, Discrete gradient flows for shape optimization and applications, Computer Methods in Applied Mechanics and Engineering, 196 (2007), 3898-3914.  doi: 10.1016/j.cma.2006.10.046.
    [13] B. Engquist and A. Majda, Absorbing boundary conditions for the numerical evaluation of waves, Proceedings of the National Academy of Sciences, 74 (1977), 1765-1766.  doi: 10.1073/pnas.74.5.1765.
    [14] K. Eppler and H. Harbrecht, Coupling of FEM and BEM in shape optimization, Numerische Mathematik, 104 (2006), 47-68.  doi: 10.1007/s00211-006-0005-6.
    [15] K. EpplerH. Harbrecht and R. Schneider, On convergence in elliptic shape optimization, SIAM Journal on Control and Optimization, 46 (2007), 61-83.  doi: 10.1137/05062679X.
    [16] S. ErlicherL. Bonaventura and O. S. Bursi, The analysis of the Generalized-α method for nonlinear dynamic problems, Computational Mechanics, 28 (2002), 83-104.  doi: 10.1007/s00466-001-0273-z.
    [17] C. de FalcoA. Reali and R. Vázquez, GeoPDEs: A research tool for Isogeometric Analysis of PDEs, Advances in Engineering Software, 42 (2011), 1020-1034.  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 (1974), 1295-1296.  doi: 10.1121/1.1903422.
    [19] D. FußederA.-V. Vuong and B. Simeon, Fundamental aspects of shape optimization in the context of isogeometric analysis, Computer Methods in Applied Mechanics and Engineering, 286 (2015), 313-331.  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.
    [21] P. GanglU. LangerA. LaurainH. Meftahi and K. Sturm, Shape optimization of an electric motor subject to nonlinear magnetostatics, SIAM Journal on Scientific Computing, 37 (2015), B1002-B1025.  doi: 10.1137/15100477X.
    [22] H. Harbrecht, Analytical and numerical methods in shape optimization, Mathematical Methods in the Applied Sciences, 31 (2008), 2095-2114.  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.
    [24] A. Henrot and M. Pierre, Variation et Optimisation de Formes: Une Analyse Géométrique, Springer Science & Business Media, 48, Springer, Berlin, 2005. doi: 10.1007/3-540-37689-5.
    [25] J. HoffelnerH. LandesM. Kaltenbacher and R. Lerch, Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 48 (2001), 779-786.  doi: 10.1109/58.920712.
    [26] S. HofmannM. Mitrea and 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 (2007), 593-647.  doi: 10.1007/BF02937431.
    [27] K. Höllig, Finite Element Methods with B-Splines, SIAM, 2003. doi: 10.1137/1.9780898717532.
    [28] T. J. R. HughesJ. A. Cottrell and Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Computer Methods in Applied Mechanics and Engineering, 194 (2005), 4135-4195.  doi: 10.1016/j.cma.2004.10.008.
    [29] K. ItoK. Kunisch and G. Peichl, Variational approach to shape derivatives for a class of Bernoulli problems, Journal of Mathematical Analysis and Applications, 314 (2006), 126-149.  doi: 10.1016/j.jmaa.2005.03.100.
    [30] K. ItoK. 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.
    [31] B. Kaltenbacher and 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 (2011), 763-773. 
    [32] B. KaltenbacherI. Lasiecka and 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 (2011), 357-387.  doi: 10.1007/978-3-0348-0075-4_19.
    [33] B. Kaltenbacher and G. Peichl, Sensitivity analysis for a shape optimization problem in lithotripsy, Evolution Equations and Control Theory(EECT), 5 (2016), 399-429.  doi: 10.3934/eect.2016011.
    [34] B. Kaltenbacher and S. Veljović, Sensitivity analysis of linear and nonlinear lithotripter models, European Journal of Applied Mathematics, 22 (2011), 21-43.  doi: 10.1017/S0956792510000276.
    [35] M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuators, Springer, Berlin, 2004.
    [36] J. E. Kennedy, High-intensity focused ultrasound in the treatment of solid tumors, Nature Reviews Cancer, 5 (2005), 321-327. 
    [37] J. E. KennedyF. WuG. R. Ter HaarF. V. GleesonR. R. PhillipsM. R. Middleton and D. Cranston, High-intensity focused ultrasound for the treatment of liver tumors, Ultrasonics, 42 (2004), 931-935. 
    [38] J. KiendlR. SchmidtR. Wüchner and K.-U. Bletzinger, Isogeometric shape optimization of shells using semi-analytical sensitivity analysis and sensitivity weighting, Computer Methods in Applied Mechanics and Engineering, 274 (2014), 148-167.  doi: 10.1016/j.cma.2014.02.001.
    [39] D. Kuhl and M. A. Crisfield, Energy-conserving and decaying algorithms in nonlinear structural mechanics, International Journal for Numerical Methods in Engineering, 45 (1999), 569-599.  doi: 10.1002/(SICI)1097-0207(19990620)45:5<569::AID-NME595>3.0.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. doi: 10.1007/978-1-4612-0069-7.
    [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. LeeN. KoizumiK. OtaS. YoshizawaA. ItoY. KanekoY. Matsumoto and M. Mitsuishi, Ultrasound-based visual serving system for lithotripsy, Intelligent Robots and Systems, (2007), 877-882. 
    [43] F. MaestreA. Münch and P. Pedregal, A spatio-temporal design problem for a damped wave equation, SIAM Journal on Applied Mathematics, 68 (2007), 109-132.  doi: 10.1137/07067965X.
    [44] E. Maloney and J. H. Hwang, Emerging HIFU applications in cancer therapy, International Journal of Hyperthermia, 31 (2015), 302-309.  doi: 10.3109/02656736.2014.969789.
    [45] J.G. ManciniA. NeisiusN. SmithG. SankinG. M. AstrozaM. E. LipkinW. N. SimmonsG. M. Preminger and P. Zhong, Assessment of a modified acoustic lens for electromagnetic shock wave lithotripters in a swine model, The Journal of Urology, 190 (2013), 1096-1101.  doi: 10.1016/j.juro.2013.02.074.
    [46] S. Meyer and M. Wilke, Optimal regularity and long-time behavior of solutions for the Westervelt equation, Applied Mathematics & Optimization, 64 (2011), 257-271.  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 (2008), 331-351. 
    [48] F. Murat and S. Simon, Etudes de problems d'optimal design, Lecture Notes in Computer Science, 41 (1976), 54-62. 
    [49] A. NeisiusN. B. SmithG. SankinN. J. KuntzJ. F. MaddenD. E. FovargueS. MitranM. E. LipkinW. N. SimmonsG. M. Preminger and P. Zhong, Improving the lens design and performance of a contemporary electromagnetic shock wave lithotripter, Proceedings of the National Academy of Sciences, 111 (2014), E1167-E1175.  doi: 10.1073/pnas.1319203111.
    [50] N. M. Newmark, A method of computation for structural dynamics, Journal of Engineering Mechanics, ASCE, 85 (1959), 67-94. 
    [51] D. M. NguyenA. Evgrafov and J. Gravesen, Isogeometric shape optimization for scattering problems, Progress In Electromagnetics Research B, 45 (2012), 117-146. 
    [52] V. Nikolić and B. Kaltenbacher, Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy, Applied Mathematics and Optimization, 76 (2017), 261-301.  doi: 10.1007/s00245-016-9340-x.
    [53] P. Nortoft and J. Gravesen, Isogeometric shape optimization in fluid mechanics, Struct Multidisc Optim, 48 (2013), 909-925.  doi: 10.1007/s00158-013-0931-8.
    [54] A. Paganini, Numerical Shape Optimization with Finite Elements, PhD Thesis, ETH Zürich, 2016.
    [55] S. H. ParkJ. W. YoonD. Y. Yang and Y. H. Kim, Optimum blank design in sheet metal forming by the deformation path iteration method, International Journal of Mechanical Sciences, 41 (1999), 1217-1232.  doi: 10.1016/S0020-7403(98)00084-8.
    [56] R. F. PatersonE. BarretT. M. SiqueiraT. A. GardnerJ. TavakkoliV. V. RaoN. T. SanghviL. Cheng and A. L. Shalhav, Laparoscopic partial kidney ablation with high intensity focused ultrasound, The Journal of Urology, 169 (2003), 347-351. 
    [57] L. Piegl and W. Tiller, The NURBS Book, Springer, 1997.
    [58] X. Qian and O. Sigmund, Isogeometric shape optimization of photonic crystals via Coons patches, Computer Methods in Applied Mechanics and Engineering, 200 (2011), 2237-2255.  doi: 10.1016/j.cma.2011.03.007.
    [59] T. D. Rossing (Ed.), Springer Handbook of Acoustics, Springer, 2014.
    [60] L. Schumaker, Spline Functions: Basic Theory, 3rd Edition, Cambridge University Press, Cambridge, 2007. doi: 10.1017/CBO9780511618994.
    [61] S. Veljović, Shape Optimization and Optimal Boundary Control for High Intensity Focused Ultrasound (HIFU), PhD Thesis, University of Erlangen-Nuremberg, 2009.
    [62] W. A. WallM. A. Frenzel and C. Cyron, Isogeometric structural shape optimization, Computer Methods in Applied Mechanics and Engineering, 197 (2008), 2976-2988.  doi: 10.1016/j.cma.2008.01.025.
    [63] F. WuW. Z. ChenJ. BaiJ. Z. ZouZ. L. WangH. Zhu and Z. B. Wang, Pathological changes in human malignant carcinoma treated with high-intensity focused ultrasound, Ultrasound in Medicine & Biology, 27 (2001), 1099-1106.  doi: 10.1016/S0301-5629(01)00389-1.
    [64] S. YoshizawaT. IkedaA. ItoR. OtaS. Takagi and Y. Matsumoto, High intensity focused ultrasound lithotripsy with cavitating microbubbles, Medical & Biological Engineering & Computing, 47 (2009), 851-860.  doi: 10.1007/s11517-009-0471-y.
    [65] P. ZhongN. SmithN. W. Simmons and G. Sankin, A new acoustic lens design for electromagnetic shock wave lithotripters, AIP Conference Proceedings, 1359 (2011), 42-47.  doi: 10.1063/1.3607880.
    [66] J.-P. Zolésio and M. C. Delfour, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, SIAM, 2011. doi: 10.1137/1.9780898719826.
  • 加载中




Article Metrics

HTML views(656) PDF downloads(417) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint