Self-adaptive absorbing boundary conditions for quasilinear acoustic wave propagation

Abstract

We propose a self-adaptive absorbing technique for quasilinear ultrasound waves in twoand three-dimensional computational domains. As a model for the nonlinear ultrasound propagation in thermoviscous fluids, we employ Westervelt's wave equation solved for the acoustic velocity potential. The angle of incidence of the wave is computed based on the information provided by the wave-field gradient which is readily available in the finite element framework. The absorbing boundary conditions are then updated with the angle values in real time. Numerical experiments illustrate the accuracy and efficiency of the proposed method.

A Priori Error Estimates for the Finite Element Approximation of Westervelt's Quasi-linear Acoustic Wave Equation

“…We mention that nonlinear absorbing conditions for the Westervelt equation have also been derived and investigated in [35,45]. Discretization in time is performed with 3500 time steps for the final time T = 40 µs and we again employ the Newmark scheme for time stepping with the same parameters as before.…”

“…In the present experiment, our computational domain is a rectangle [0, 0.04] m × [0, 0.05] m with a curved bottom side that belongs to the circle centered at (0.02 m, 0.04 m) with radius R 2 = 0.002 m 2 . On this bottom side we impose Neu-mann boundary conditions as a modulated sinusoidal wave: We mention that nonlinear absorbing conditions for the Westervelt equation have also been derived and investigated in [35,45]. Discretization in time is performed with 3500 time steps for the final time T = 40 µs and we again employ the Newmark scheme for time stepping with the same parameters as before.…”

“…Efficient simulation of the Westervelt equation and, in general, nonlinear sound propagation by the finite element method has been an active area of research. We refer to, e.g., [16,22,24,29,35,36,50,52], which all focus on algorithmic aspects of finite element discretizations without any a priori analysis.…”

Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

“…Alonso-Mallo and Reguera [65] computed the approximation of the solution near the boundary as the sum of plane waves, whereas Xu et al [66,67] proposed effective wavenumber computation using the Fourier-Gabor transform and used it for the construction of adaptive ABCs. A similar technique was developed by Muhr et al [68] for the nonlinear ultrasound propagation problem described by Westervelt's equation. In that scenario, the principle of self-adaptive ABCs consists of computing local wave vectors and updating ABCs with angle information in real time.…”

Exponential Integrators for Quasilinear Wave-Type Equations

“…These bound were shown for a trigonometric integrator by a sophisticated stability analysis. In [2,27] continuous and discontinuous Galerkin (dG) methods were used for the space discretization of the Westervelt equation in two and three dimensions and absorbing boundary conditions for this equation were treated in [25]. Concerning full discretization, we further mention the work [5], where error bounds for linear finite elements in space combined with a dG method of order 0 in time for parabolic problems were derived under low regularity assumptions.…”