Article Contents
Article Contents

# Isogeometric shape optimization for nonlinear ultrasound focusing

• * Corresponding author: Vanja Nikolić
• 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.

 Citation:

• 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 $c_f=1500 \, \frac{m}{s}$ $c_l=1100 \, \frac{m}{s}$ $b_f=6\cdot 10^{-9} \, \frac{m^2}{s}$ $b_l=4\cdot 10^{-9} \, \frac{m^2}{s}$ $\varrho_f=1000 \, \frac{kg}{m^3}$ $\varrho_l=1250 \, \frac{kg}{m^3}$ $B/A=5$ $B/A=4$

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

Table 3.  Spatial grid sizes

 Spatial degrees of freedom Linear NURBS Quadratic NURBS $\text{ndof}_x = 46$ $\text{ndof}_x = 48$ $\text{ndof}_y = 181$ $\text{ndof}_y = 185$ $\text{ndof} = 7976$ $\text{ndof} = 8484$

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 $\alpha_m =1/2, \alpha_f = 1/3$ $\text{TOL}_\text{grad}=10^{-4}$

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 $J= 2.930353\cdot 10^7$ $J=2.926472\cdot 10^7$

Figures(17)

Tables(5)