Finite element discretization of nonlinear models of ultrasound heating

TL;DR

This work develops and analyzes a conforming finite element discretization for a nonlinear, temperature-dependent wave-heat model arising in HIFU heating, incorporating Westervelt and Kuznetsov nonlinearities coupled to a heat equation with a pressure-dependent source. By leveraging the strong acoustic damping and recasting the system into a parabolic framework, the authors derive energy-based a priori error estimates and show optimal convergence in the energy norm for polynomial degrees η ≥ 1, relying on Ritz projections and careful control of temperature-dependent coefficients. The main result proves the existence and uniqueness of the semi-discrete solution on [0,T] and establishes an O($h^{η}$) convergence rate between the exact and discrete solutions, with a detailed analysis of defects and nonlinear terms. Numerical experiments, including liver-tissue simulations and various excitation scenarios, validate the theoretical rates and demonstrate accurate wave propagation and heating patterns, providing a rigorous numerical foundation for nonlinear wave-heat coupling in biomedical applications.

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 \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.

Figures (5)

• Figure 1: Total error \ref{['def:total:error']} (first row) and $L^2$ error \ref{['def:total:error:Ltwo']} (second row) computed from the numerical scheme making use of the implicit Euler method with $\eta=1$, and BDF2 method with $\eta=2$, and $\eta=3$, respectively. The errors are ploted against the meshsize $h$, for the common final time $T=1\,\rm s$ and timestep $\tau = 1/128\,{\rm s}=0.0078125\,{\rm s}$. The manufactured solutions are $u_{\rm ex}$ (cf. \ref{['def:uex']}) and $\theta_{\rm ex}$ (cf. \ref{['def:thetaex']}) with $A_1 = 1$, $A_2 = 10^{-4}$, $\lambda_1 = 1$, and $\lambda_2 = 1/2$.
• Figure 2: (a) Initial pressure $u_0=u_0(x_1,x_2)$ used in Example 2, and (b) source function $f = f(x_1,x_2,t)$ at $t=0$ employed in Example 3. The continuous red lines correspond to the respective functions at $x_2 = 0$ for $0\leq x_1\leq r_0$, and the red-dashed lines stand for the functions' plot at the angles $-\pi/4$ and $\pi/4$ respectively. The black-dashed contour coincide with the boundary $\partial \Omega$.
• Figure 3: Example 2: Snapshots of the discrete pressure $u_h = u_h(x_1,x_2)$ computed with the Westervelt wave equation in \ref{['coupled_problem']}, initial coditions \ref{['eq:IC:example2']} and $\theta_0 \equiv 0$, and setting the source term $f = 0$. The time step is $\tau = 10^{-7}\,\rm s$ and $\eta = 1$, and the time approximation is by the BDF2 method.
• Figure 4: Example 2: Snapshots of the discrete temperature $\theta_h = \theta_h(x_1,x_2)$ computed with the Westervelt wave equation in \ref{['coupled_problem']}, initial coditions \ref{['eq:IC:example2']} and $\theta_0 \equiv 0$, and setting the source term $f = 0$. The time step is $\tau = 10^{-7}\,\rm s$ and $\eta = 1$, and the time approximation is by the BDF2 method.
• Figure 5: Example 3: Snapshots of the discrete acoustic velocity potential $u_h = u_h(x_1,x_2)$ (first row) and discrete temperature $\theta_h = \theta_h(x_1,x_2)$ (second row), at three time points, computed with the Kuznetsov wave equation in \ref{['coupled_problem']} with zero initial conditions and source term $f$ given by \ref{['eq:source:ex3']}. The time step is $\tau = 10^{-7}\,\rm s$ and $\eta = 1$, and the time approximation is by the BDF2 method.

Theorems & Definitions (34)

• Remark 1.1
• Theorem 1.2: A priori error estimate
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof