Isogeometric shape optimization for nonlinear ultrasound focusing

Markus Muhr; Vanja Nikolić; Barbara Wohlmuth; Linus Wunderlich

doi:10.3934/eect.2019010

Article Contents

2019, Volume 8, Issue 1: 163-202. Doi: 10.3934/eect.2019010

This issue Previous Article Nonlinear waves in thermoelastic dielectrics Next Article Optimal scalar products in the Moore-Gibson-Thompson equation

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
* Corresponding author: Vanja Nikolić

Received: December 2017

Revised: January 2018

Early access: January 2019

Published: January 2019

Abstract / Introduction Full Text(HTML) Figure(17) / Table(5) Related Papers Cited by

Abstract

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.

Keywords:

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

Citation:

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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

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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.

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

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

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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

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

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

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Figure 8. L²-error over the course of optimization

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

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

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

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

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

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

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

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

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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

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

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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

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Table 3. Spatial grid sizes

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

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

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

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