# American Institute of Mathematical Sciences

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
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##### References:
Sketches of different ultrasound focusing approaches; left: Focusing by a lens, right: Focusing by an array of transducers placed on a curved surface
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.
Domain $\Omega$, consisting of lens $\Omega_l$ and fluid $\Omega_f$
Patches used for the discretization. The lens domain is given by patch 3
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
Final lens shape together with initial and goal shapes
left: Relative cost change versus the number of gradient steps. right: Norm of the shape gradient
L2-error over the course of optimization
Final lens shape, together with initial and target shape. left: Linear NURBS. right: Quadratic NURBS
Relative change of the cost over the course of optimization. left: Linear NURBS. right: Quadratic NURBS
Norm of the shape gradient over the course of optimization. left: Linear NURBS. right: Quadratic NURBS
L2 shape error over the course of optimization. left: Linear NURBS, right: Quadratic NURBS
left: Initial, final and goal shape; the final shape is a local, but not a global minimum. right: Relative cost decay
Reference lens that marks the minimal thickness possible to manufacture
left: Initial and final lens shape. right: Relative cost decay versus the number of gradient steps
Pressure-wave propagation with the final lens shape using quadratic NURBS
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
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 $c_f=1500 \, \frac{m}{s}$ $c_l=1100 \, \frac{m}{s}$ $b_f=6\cdot 10^{-9} \, \frac{m^2}{s}$ $b_l=4\cdot 10^{-9} \, \frac{m^2}{s}$ $\varrho_f=1000 \, \frac{kg}{m^3}$ $\varrho_l=1250 \, \frac{kg}{m^3}$ $B/A=5$ $B/A=4$
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
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 $\text{ndof}_x = 46$ $\text{ndof}_x = 48$ $\text{ndof}_y = 181$ $\text{ndof}_y = 185$ $\text{ndof} = 7976$ $\text{ndof} = 8484$
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 $\alpha_m =1/2, \alpha_f = 1/3$ $\text{TOL}_\text{grad}=10^{-4}$
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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