Finite element simulation of nonlinear wave propagation in thermoviscous fluids including dissipation

Nonlinear ultrasound is important in medical diagnostics because imaging of the higher harmonics improves resolution and reduces scattering artifacts. Second harmonic imaging is currently standard, and higher harmonic imaging is under investigation. The efficient development of novel imaging modalities and equipment requires accurate simulations of nonlinear wave fields in large volumes of realistic (lossy, inhomogeneous) media. The Iterative Nonlinear Contrast Source (INCS) method has been developed to deal with spatiotemporal domains measuring hundreds of wavelengths and periods. This full wave method considers the nonlinear term of the Westervelt equation as a nonlinear contrast source, and solves the equivalent integral equation via the Neumann iterative solution. Recently, the method has been extended with a contrast source that accounts for spatially varying attenuation. The current paper addresses the problem that the Neumann iterative solution converges badly for strong contrast sources. The remedy is linearization of the nonlinear contrast source, combined with application of more advanced methods for solving the resulting integral equation. Numerical results show that linearization in combination with a Bi-Conjugate Gradient Stabilized method allows the INCS method to deal with fairly strong, inhomogeneous attenuation, while the error due to the linearization can be eliminated by restarting the iterative scheme.

Section: Introductionmentioning

confidence: 99%

Computation of nonlinear ultrasound fields using a linearized contrast source method

Verweij

Demi

Dongen

2013

“…Even for a strongly nonlinear case (z ¼ 0.88r), the error generated by using a step size ¼ r/16 may be considered tolerable relative to the error typically introduced in experimental measurement. In contrast, the finite element method, 3 which solves the time-domain nonlinear wave equation using a predictor/multi-corrector algorithm in combination with standard time-stepping procedures, requires up to 80 elements per wavelength to obtain the Fubini solution. It is noted that the projection size determined here is based on plane-wave propagation in homogeneous media.…”

Section: Error Studymentioning

confidence: 99%

“…These approaches are computationally advantageous compared to space-time methods, 3 as they represent the wave equation in the form of an ordinary differential equation (ODE) as opposed to its partial differential (PDE) form in space-time. If the medium is linear and homogeneous, the ODE will have a known solution, 1 and a projection to any new plane requires only a single operation.…”

Section: Introductionmentioning

confidence: 99%

“…We test whether propagation steps in the order of the fundamental wavelength are possible, as opposed to the 10-20 steps per wavelength required in finite-difference time-domain or finite element methods. 3 The optimal step size for a given problem is found by a study that measures error as a function of step size and propagation distance. A two-dimensional (2-D) steered transducer is then studied to demonstrate the directional independence of this approach when calculating the nonlinear acoustic field.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Evaluation of a wave vector frequency domain method for nonlinear wave propagation.

Jing

Clement

2009

A wave-vector-frequency-domain method is presented to describe one-directional forward or backward acoustic wave propagation in a nonlinear homogeneous medium. Starting from a frequencydomain representation of the second-order nonlinear acoustic wave equation, an implicit solution for the nonlinear term is proposed by employing the Green's function. Its approximation, which is more suitable for numerical implementation, is used. An error study is carried out to test the efficiency of the model by comparing the results with the Fubini solution. It is shown that the error grows as the propagation distance and step-size increase. However, for the specific case tested, even at a step size as large as one wavelength, sufficient accuracy for plane-wave propagation is observed. A two-dimensional steered transducer problem is explored to verify the nonlinear acoustic field directional independence of the model. A three-dimensional single-element transducer problem is solved to verify the forward model by comparing it with an existing nonlinear wave propagation code. Finally, backwardprojection behavior is examined. The sound field over a plane in an absorptive medium is backward projected to the source and compared with the initial field, where good agreement is observed.

“…20,21 Finite element methods (FEM) have also been adopted to simulate nonlinear wave propagation and they are well suited to handling complex geometries such as occur in the body. 22,23 The commercial FEM software PZFLEX (Weidlinger Associates, Inc., Mountain View, CA) is one such full solver that is widely used in industrial and biomedical applications of ultrasound, because it provides a convenient platform with a graphical user interface and optimised computing kernel. The developers demonstrated that the solution for a focused high intensity focused ultrasound (HIFU) source matches the growth of harmonics to within 10%, 24 and that an alternative pseudo-spectral method can model nonlinear pulses from imaging transducers.…”

Section: Introductionmentioning

confidence: 99%

Simulation of nonlinear propagation of biomedical ultrasound using pzflex and the Khokhlov-Zabolotskaya-Kuznetsov Texas code

Qiao¹,

Jackson²,

Coussios³

et al. 2016

Nonlinear acoustics plays an important role in both diagnostic and therapeutic applications of biomedical ultrasound and a number of research and commercial software packages are available. In this manuscript, predictions of two solvers available in a commercial software package, pzflex, one using the finite-element-method (FEM) and the other a pseudo-spectral method, spectralflex, are compared with measurements and the Khokhlov-Zabolotskaya-Kuznetsov (KZK) Texas code (a finite-difference time-domain algorithm). The pzflex methods solve the continuity equation, momentum equation and equation of state where they account for nonlinearity to second order whereas the KZK code solves a nonlinear wave equation with a paraxial approximation for diffraction. Measurements of the field from a single element 3.3 MHz focused transducer were compared with the simulations and there was good agreement for the fundamental frequency and the harmonics; however the FEM pzflex solver incurred a high computational cost to achieve equivalent accuracy. In addition, pzflex results exhibited non-physical oscillations in the spatial distribution of harmonics when the amplitudes were relatively low. It was found that spectralflex was able to accurately capture the nonlinear fields at reasonable computational cost. These results emphasize the need to benchmark nonlinear simulations before using codes as predictive tools.