Unmitigated numerical solution to the diffraction term in the parabolic nonlinear ultrasound wave equation

Abstract

Various numerical algorithms have been developed to solve the Khokhlov-Kuznetsov-Zabolotskaya (KZK) parabolic nonlinear wave equation. In this work, a generalized time-domain numerical algorithm is proposed to solve the diffraction term of the KZK equation. This algorithm solves the transverse Laplacian operator of the KZK equation in three-dimensional (3D) Cartesian coordinates using a finite-difference method based on the five-point implicit backward finite difference and the five-point Crank-Nicolson finite difference discretization techniques. This leads to a more uniform discretization of the Laplacian operator which in turn results in fewer calculation gridding nodes without compromising accuracy in the diffraction term. In addition, a new empirical algorithm based on the LU decomposition technique is proposed to solve the system of linear equations obtained from this discretization. The proposed empirical algorithm improves the calculation speed and memory usage, while the order of computational complexity remains linear in calculation of the diffraction term in the KZK equation. For evaluating the accuracy of the proposed algorithm, two previously published algorithms are used as comparison references: the conventional 2D Texas code and its generalization for 3D geometries. The results show that the accuracy/efficiency performance of the proposed algorithm is comparable with the established time-domain methods.

Feasibility of noninvasive thermometry in hyperthermia regime using harmonics generated by nonlinear ultrasound wave propagation

“…The finite amplitude method is based on measuring the second harmonic generated in the medium [74]. The thermodynamic method is considered to be the most accurate way of measuring B/A.…”

“…A mathematical model is needed to simulate the propagation of ultrasound wave in a medium in order to predict the acoustic filed generated by a transducer. The model needs to be capable of taking into account the nonlinear propagation of the sound, finite dimensions of the transducer and the dissipating effects of the medium [74].…”

Temperature dependence of acoustic harmonics generated by nonlinear ultrasound wave propagation in water at various frequencies

“…(1) account for diffraction, nonlinearity, and thermoviscous absorption, respectively. 13 The source boundary condition was defined as…”

“…The time window was also large enough to encompass the entire waveform, including delayed edge waves. 13 Nonlinear propagation of 9 and 7 cycle pulses at 3.3 and 13 MHz in water with source pressure amplitudes of 0.26 and 0.05 MPa, respectively, and for the same transducer geometries used in our experiments were simulated. The source pressure amplitudes used in simulations were selected in order to achieve similar pressure and a similar degree of nonlinear waveform distortion as to that obtained in the experiments at the transducer focus and along the acoustic axes when water was at the baseline temperature of 26 C. Simulations for the 1 and 20 MHz pulses were performed using the same parameters as those at 3.3 and 13 MHz, respectively.…”

“…The Khokhlov-Zabolotskaya-Kuznetsov (KZK) nonlinear wave equation is a well-established model for finiteamplitude wave propagation. 3,[12][13][14] The KZK equation is a parabolic approximation of the Westervelt nonlinear wave equation that consists of terms to account for diffraction, absorption, and nonlinearity. 3 The diffraction term accounts for the finite dimensions of the source and the attenuation term considers the heat conduction and viscosity of the medium.…”