Finite element analysis of broadband acoustic pulses through inhomogenous media with power law attenuation

Abstract

Acoustic waves in tissues and weakly attenuative fluids often have an attenuation parameter, alpha(omega), satisfying alpha(omega)= alpha0omegay in which alpha0 is a constant, omega is the frequency, and y is between 1 and 2. This power law attenuation is not predicted by the classical thermoviscous wave equation and researchers have proposed different modified viscous wave equations in which the loss term is a convolution operator or a fractional spatial or temporal derivative. In this paper, acoustic waves undergoing power law attenuation are modeled by a modification to the thermoviscous wave equation in which the time derivative of the viscous term is replaced by a fractional time derivative. An explicit time domain, finite element formulation leads to a stable algorithm capable of simulating axisymmetric, broadband acoustic pulses propagating through attenuative and dispersive media. The algorithm does not depend on the Born approximation, long wavelength limit, or plane wave assumptions. The algorithm is validated for planar and focused transducers and results include radiation patterns from a viscous scatterer in a lossless background and signals reflected from a viscous layer. The program can be used to determine scattering parameters for large, strong, possibly viscous scatterers, in either a lossless or viscous background, for which analytic results are scarce.

Nonlinear acoustic wave equations with fractional loss operators

“…Therefore, fractional derivatives offer an alternative to multiple relaxations used for time-domain simulators like the KZKTEXAS code 15 to approximate attenuation and dispersion in biological tissue. The fractional derivative can, for instance, be solved in the time domain by using a backward difference power series 10 or the Grün-wald-Letnikov formulation. 23 Using the constitutive equations as a starting point instead of an approximated wave equation could also be an alternative to simulate sound propagation in non-homogeneous media.…”

“…Different forms of the wave equation have been proposed to reflect this complexity. 4,7,[10][11][12] Nonlinear effects in sound wave propagation, may also be taken into account during numerical simulation. This is the case for the BERGEN code, 13,14 the KZKTEXAS code, [15][16][17] and the angular spectrum method of Christopher and Parker.…”

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

“…[19][20][21] Variations in the time-domain approach center on efficient methods to model the solutions. [22][23][24][25] On the other hand, the general case of anomalous dispersion can be handled with relative ease in wave-vector space. 26 A relevant medical example of anomalous dispersion is trabecular bone.…”

Fractal ladder models and power law wave equations

“…PDE formulations incorporate loss via integer-ordered derivatives, 16,17 whereas FPDEs add loss to the wave equation with a timefractional derivative, 18,19 a space-fractional derivative, 20,21 or the combination of an integer-ordered spatial derivative and a time-fractional derivative. 22,23 Nonlinear dissipative propagation has also been described with fractional spatial derivatives via generalizations of Burgers equation, 24 as well as transient elastic wave propagation in porous 25 and viscoelastic media. 26 These FPDE models build on previous applications of fractional calculus to diffusion processes, 27,28 relaxation processes, 29 viscoelasticity, 30,31 and seismology.…”

“…IV for linear macro-homogeneous media. This FPDE, which was originally proposed within the seismology community 22 and later considered in the biomedical acoustics community, 23 yields both a power law attenuation coefficient and a phase velocity predicted by the Kramers-Kronig relationships. In Sec.…”