Evaluation of an Analytical Volterra Series Solution to the Burgers Equation
Comparison of a Volterra series solution to the Burgers equation to a fractional steps method.
Background
The Burgers equation models the nonlinear propagation of plane ultrasonic waves in homogeneous viscous fluids.
Its incorporation into fast tissue harmonic imaging or the detection of ultrasound contrast agents potentially improves these imaging modes. Moreover, the decompositions of arbitrary types of waves into steered plane waves permit the application of this model the Burgers equation to other types of waves.
Content
The script "eval_methods.m" compares both methods for a simple example by evaluating various error metrics. It additionally creates a short movie illustrating the wave propagation.
The package +volterra contains the functions for the proposed Volterra polynomial, whereas the package +fractional_steps contains the functions for the fractional steps reference method.
Results
The following animation depicts the waveforms predicted by the 10th-degree Volterra polynomial for distilled water (propagation distance: 25 cm, step length: 0.5 cm). The initial waveform was a modulated Gaussian pulse (center frequency: 3.5 MHz, amplitude: 750 kPa). The deformation of the shape (see left plot) involved the generation of harmonics (see right plot), which were subject to significantly increased absorption.
References đź““
The solution and exemplary applications were published in:
- M. Schiffner, M. Mleczko, and G. Schmitz, "Evaluation of an analytical solution to the Burgers equation based on Volterra series," in 2009 IEEE Int. Ultrasonics Symp. (IUS), Rome, Sep. 2009, pp. 573-576,
- M. Schiffner, M. Mleczko, and G. Schmitz, "Application of Volterra Series to Ultrasound Imaging," in NAG/DAGA 2009 Int. Conf. Acoustics, Rotterdam, Mar. 2009, pp. 301--304
- M. Schiffner, M. Mleczko, and G. Schmitz, "Application of Volterra Series to the Detection of Ultrasound Contrast Agents," in World Congr. Medical Physics and Biomedical Engineering, Sep. 2009, Munich. IFMBE Proceedings, vol. 25/2, pp. 478-481,
