March  2019, 8(1): 163-202. doi: 10.3934/eect.2019010

Isogeometric shape optimization for nonlinear ultrasound focusing

1. 

Technical University of Munich, Department of Mathematics, Chair of Numerical Mathematics, Boltzmannstraẞe 3, 85748 Garching, Germany

2. 

Queen Mary University of London, School of Mathematical Sciences, Mile End Road, London E1 4NS, United Kingdom

* Corresponding author: Vanja Nikolić

Received  December 2017 Revised  January 2018 Published  January 2019

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.

Citation: Markus Muhr, Vanja Nikolić, Barbara Wohlmuth, Linus Wunderlich. Isogeometric shape optimization for nonlinear ultrasound focusing. Evolution Equations & Control Theory, 2019, 8 (1) : 163-202. doi: 10.3934/eect.2019010
References:
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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. Google Scholar

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show all references

References:
[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

D. Colton and R. Kress, Integral Equation Methods in Scattering Theory, Wiley, New York, 1983. Google Scholar

[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. Google Scholar

[10]

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[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[23]

J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape, Material, and Topology Design, John Wiley & Sons, 1996. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[27]

K. Höllig, Finite Element Methods with B-Splines, SIAM, 2003. doi: 10.1137/1.9780898717532. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[35]

M. Kaltenbacher, Numerical Simulations of Mechatronic Sensors and Actuators, Springer, Berlin, 2004.Google Scholar

[36]

J. E. Kennedy, High-intensity focused ultrasound in the treatment of solid tumors, Nature Reviews Cancer, 5 (2005), 321-327. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[45]

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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}$
$\varrho_f=1000 \, \frac{kg}{m^3}$ $\varrho_l=1250 \, \frac{kg}{m^3}$
$B/A=5$ $B/A=4$
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}$
$\varrho_f=1000 \, \frac{kg}{m^3}$ $\varrho_l=1250 \, \frac{kg}{m^3}$
$B/A=5$ $B/A=4$
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
$\Omega_{l, \text{upper straight}}$ $\Omega_{l, \text{upper curved}}$ $\Omega_{l, \text{both perturbed}}$ $\Omega_{l, \text{both down}}$ $\Omega_{l, \text{Gauss}}$
$L=0.12$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
Table 3.  Spatial grid sizes
Spatial degrees of freedom
Linear NURBS Quadratic NURBS
$\text{ndof}_x = 46$ $\text{ndof}_x = 48$
$\text{ndof}_y = 181$ $\text{ndof}_y = 185$
$\text{ndof} = 7976$ $\text{ndof} = 8484$
Spatial degrees of freedom
Linear NURBS Quadratic NURBS
$\text{ndof}_x = 46$ $\text{ndof}_x = 48$
$\text{ndof}_y = 181$ $\text{ndof}_y = 185$
$\text{ndof} = 7976$ $\text{ndof} = 8484$
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 $\alpha_m =1/2, \alpha_f = 1/3$ $\text{TOL}_\text{grad}=10^{-4}$
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 $\alpha_m =1/2, \alpha_f = 1/3$ $\text{TOL}_\text{grad}=10^{-4}$
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 $J= 2.930353\cdot 10^7$ $J=2.926472\cdot 10^7$
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 $J= 2.930353\cdot 10^7$ $J=2.926472\cdot 10^7$
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