Heating generated by high-intensity focused ultrasound waves is central to many emerging medical applications, including non-invasive cancer therapy and targeted drug delivery. In this study, we aim to gain a fundamental understanding of numerical simulations in this context by analyzing conforming finite element approximations of the underlying nonlinear models that describe ultrasound-heat interactions. These models are based on a coupling of a nonlinear Westervelt--Kuznetsov acoustic wave equation to the heat equation with a pressure-dependent source term. A particular challenging feature of the system is that the acoustic medium parameters may depend on the temperature. The core of our new arguments in the \emph{a prior} error analysis lies in devising energy estimates for the coupled semi-discrete system that can accommodate the nonlinearities present in the model. To derive them, we exploit the parabolic nature of the system thanks to the strong damping present in the acoustic component. Theoretically obtained optimal convergence rates in the energy norm are confirmed by the numerical experiments. In addition, we conduct a further numerical study of the problem, where we simulate the propagation of acoustic waves in liver tissue for an initially excited profile and under high-frequency sources.

翻译：高强度聚焦超声波产生的加热是许多新兴医学应用的核心，包括无创癌症治疗和靶向药物递送。在本研究中，我们旨在通过分析描述超声-热相互作用的基础非线性模型的协调有限元近似，从而在此背景下获得对数值模拟的基本理解。这些模型基于非线性韦斯特维尔特-库兹涅佐夫声波方程与具有压力相关源项的热方程的耦合。该系统一个特别具有挑战性的特征是声学介质参数可能依赖于温度。我们在先验误差分析中新论证的核心在于为耦合半离散系统设计能量估计，该估计能够适应模型中存在的非线性。为了推导这些估计，我们利用了声学分量中存在的强阻尼所带来的系统抛物特性。理论上获得的能量范数下的最优收敛速率得到了数值实验的确认。此外，我们对问题进行了进一步的数值研究，模拟了在初始激发轮廓和高频源作用下声波在肝组织中的传播。