  We are interested in shape sensitivity analysis for an optimization problem
arising in medical applications of high intensity focused ultrasound. The goal
is to find the optimal shape of a focusing acoustic lens so that the desired
acoustic pressure at a kidney stone is achieved. Coupling of the silicone
acoustic lens and nonlinearly acoustic fluid region is modeled by the
Westervelt equation with nonlinear strong damping and piecewise constant
coefficients. We follow the variational approach to calculating the shape
derivative of the cost functional which does not require computing the shape
derivative of the state variable; however assumptions of certain spatial
regularity of the primal and the adjoint state are needed to obtain the
derivative, in particular for its strong form according to the
Delfour-Hadamard-Zol\' esio Structure Theorem.
