Finite element discretization of nonlinear models of ultrasound heating

¹ Bernoulli Institute, University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands
² Institute for Applied and Numerical Mathematics, Karlsruhe Institute of Technology, Englerstraße 2, 76149 Karlsruhe, Germany
³ Department of Mathematics, Radboud University, Heyendaalseweg 135, 6525 AJ Nijmegen, The Netherlands

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 14 February 2025
Accepted: 6 December 2025

Abstract

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 a priori 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.