Title: Shape sensitivity analysis for general optimization problems in nonlinear acoustics Show Chinese title

Title: 非线性声学中一般优化问题的形状敏感性分析

Abstract: In various biomedical applications, precise focusing of nonlinear ultrasonic waves is crucial for efficiency and safety. This work examines the shape sensitivity analysis for a class of optimization problems constrained by general quasi-linear acoustic wave equations that arise in high-intensity focused ultrasound (HIFU) applications. Within our theoretical framework, the Westervelt and Kuznetsov equations of nonlinear acoustics are obtained as particular cases. The quadratic gradient nonlinearity, specific to the Kuznetsov equation, requires special attention throughout the paper. To prove the existence of the Eulerian shape derivative, we successively study the local well-posedness and regularity of the forward problem and prove that it does not degenerate under the hypothesis of small initial and boundary data. We then derive and analyze the corresponding adjoint problems for several different cost functionals of practical interest and conclude with the expressions of well-defined shape derivatives. Show Chinese abstract

Abstract: 在各种生物医学应用中，非线性超声波的精确聚焦对于效率和安全性至关重要。 本研究考察了一类由高强聚焦超声（HIFU）应用中出现的一般准线性声学波动方程约束的优化问题的形状敏感性分析。 在我们的理论框架中，非线性声学的Westervelt和Kuznetsov方程作为特例得到。 Kuznetsov方程特有的二次梯度非线性在整个论文中需要特别关注。 为了证明Eulerian形状导数的存在性，我们依次研究了正问题的局部适定性和正则性，并证明在小初始和边界数据假设下它不会退化。 然后，我们推导并分析了几种具有实际兴趣的成本泛函的相应伴随问题，并最终得出明确的形状导数表达式。