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Isogeometric shape optimization for nonlinear ultrasound focusing

  • * Corresponding author: Vanja Nikolić

    * Corresponding author: Vanja Nikolić 
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  • 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.


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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}$
    $\varrho_f=1000 \, \frac{kg}{m^3}$ $\varrho_l=1250 \, \frac{kg}{m^3}$
    $B/A=5$ $B/A=4$
     | 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
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