Simulation of high-energy ultrasound propagation in heterogeneous medium using k-space method

16/07/2014 3:02pm

Автор: Vladimir Morkun, Natalia Morkun, Andrey Pikilnyak

Категории: automatization

The method and software based on this method for parameters estimation of the ultrasonic waves propagating in random heterogeneous medium are described
Key words: phased array, ultrasound, pulp, control, k-space

Simulation of high-energy ultrasound propagation in heterogeneous medium using k-space method




Vladimir Morkun
Vice-Rector for research, Doctor of Science, professor of Computer Science, Automation and Control Systems department
Kryvyi Rih National University





Morkun N



Natalia Morkun
PhD, Associate professor of Economic Cybernetics and Project Management Department
Kryvyi Rih National University








Andrey Pikilnyak
PhD- student of Computer systems and networks department, research assistant of the Computer Science, Automation and Control Systems department
Kryvyi Rih National University







Control of  mineral beneficiation process requires controlling the parameters of complex heterogeneous mediums, including solid, liquid and gas phases.
Ultrasonic oscillations are periodic disturbances of the elastic medium state, characterized by a change in its physical properties, which occur synchronously with perturbation. During ultrasonic propagation the local medium volume oscillations are transmitted to adjacent areas by means of elastic waves, characterized by a change in medium density in space and which transfer the fluctuations energy [1-3].
The basic relations describing ultrasonic oscillations and waves in the medium, follow from the equation of medium state, Newtonian equations of motion and the continuity equation [4,5]. The results are the wave-type equations that can be solved with appropriate initial and boundary conditions.
The equations of ultrasonic waves nonlinear propagation in heterogeneous medium can be derived from the mass, momentum, and energy conservation laws under the assumption of a quiescent, isotropic, and inviscid medium as follows[6,7]

formula 1, formula 2,     (1)

where p is the acoustic pressure, ρ is the density, u is the particle velocity, ρ0 is the ambient density. The second order terms in (1) can be re-written in terms of the Lagrangian density by the repeated substitution of the acoustic equations in linearized form [8]. The Lagrangian density terms can be neglected as follows.

formula 3, formula 4.                  (2)

Let’s neglect thermoviscous losses and include a phenomenological loss operator to account the arbitrary power law absorption [8], then the equation of state from the total pressure expansion using a Taylor series can be written as follows

formula 5,     (3)

where c0 is the isentropic sound speed, d is the particle displacement vector, and τ and η are the absorption and dispersion proportionality coefficients where formula 6 and formula 7. α is the acoustic absorption where formula 8, α0 is the absorption coefficient prefactor and y is the power law exponent [9].
The expressions given in (2) and (3) are the acoustic particle velocity, density, and pressure coupled set of equations.
Let’s combine these expressions into a single second order wave equation for the acoustic pressure. The modified Westervelt equation by neglecting higher order absorption, nonlinearity, and heterogeneity terms, can be written in the following form

formula 9,             (4)

where formula10 is the nonlinearity coefficient.
By neglecting higher order absorption and nonlinearity effects, the conservation equations in (2) using a k-space method [10] can be written as follows

formula 11
                                            formula 12                                  (5)
formula 13
formula 14

where i is the imaginary unit, kξ is the wavenumber in the  ξ direction, κ is the k-space adjustment where κ =sinc(c0kΔt/2), F and F−1 denote the forward and inverse Fourier transform, Δt is the time step.
The corresponding equation of state in discrete form is

formula 15,                       (6)

where the total density is given by formula 16and the discrete loss term is

formula 17,          (7)

Let's simulate the ultrasonic pressure field propagation in a heterogeneous medium using k-Wave toolbox (Matlab) which is designed for time domain ultrasound simulations in complex media like heterogeneous pulp. The simulation functions of this software are based on the k-space method and are both fast and easy to use [10].
The net pressure of all piezoelectric elements can be obtained by adding the effects of each source and written in the form

formula 18.                             (8)

Due to attenuation, the useful power at the point (x, y, z) is given by [11]

formula 19,                                (9)

The total energy at a point (x, y, z) is given by

formula 20,                         (10)

where formula 21 - intensity at the point (x, y, z), W/m.2
The results of the ultrasonic wave propagation through a heterogeneous medium with density formula 22kg/m3, for source strength of 1MPa and tone burst frequency of 1MHz for 16-element phased array with focus distance of 20mm are shown in Fig. 1. The central slice absorption distribution in grayscale as a background and the square of the pressure distribution on the surface of this background are shown.

Total beam pattern
Fig.1. Total beam pattern using maximum of recorded pressure
The final pressure field (a), the maximum pressure (b) and standard pressure (c) of the beam are shown on Fig. 2. The transducer focus and sidelobes are visible.
 Ultrasonic wave propagation
Fig. 2. Ultrasonic wave propagation in heterogeneous medium: a) the final pressure field, b) the maximum pressure c) the rms pressure
The shape
Fig. 3. The shape of the main wavefront
The linear cross-section of the focus in x direction is shown on Fig. 4: 1) for the single source; 2) simulation by k-space method in the water; 3) in a heterogeneous medium.
 The simulation results comparisonmodeling by k-space method
Fig. 4. The simulation results comparison of the normalized square of pressure for: 1) a simple screened source, 2) modeling by k-space method in a homogeneous medium (water) and 3) in the inhomogeneous medium (pulp) along the axis: a) - z and b) - x.
For the process of energy accumulation and transfer in a certain point of space can be given a numerical estimate by measuring the temperature. The simulation of temperatures was performed using the heat transfer equation [12]. The accumulated power was extracted from the three perpendicular lines that crossed the values of the simulated and measured ultrasound pressure focal area change (Fig. 5).
 The simulation results comparison
Fig. 5. The simulation results comparison of the temperature distribution for: 1) a simple screened source, 2) modeling by k-space method in a homogeneous medium (water) and 3) in the inhomogeneous medium (pulp) along the x axis.


To build a model of the ultrasonic field in a randomly inhomogeneous medium, the fiber spaces method (k-space), which increased the accuracy of parameter estimation field is used.


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