Abstract
Acoustic beamforming has applications in medical ultrasonic imaging, water pipe diagnostic, hearing aid, etc. These applications involve body tissues, fluids, and polymers, which are all viscoelastic materials. They have been shown to exhibit power law attenuation in wave propagation. This thesis aims to explore methods which will enable the numerical analysis of beamforming in materials described by wave models which account for such power law attenuation. Wave physics and numerical methods are also reviewed for better understanding of the derivation of the methods.Beamforming methods presented in this thesis are based on (i), solving the Fredholm integral resulting from the superposition of point sources by eigen-decomposition, and (ii), applying Rayleigh integral I where the wave field generated by a point source is assumed to be symmetrical from a plane. Because these methods assume a continuous source, there are artefacts when the solution is discretized. The ultraspherical window has been studied next to attenuate the side lobes of the beampattern introduced by the finite source.
The inverse Fourier transform of the fractional power function leads to fractional derivatives. One type of wave equations is ad hoc designed by directly adding such power term to the wavenumber. For example, Treeby and Cox applied fractional Laplacian to model this. Another family is based on well-known constitutive equations in physics, for instance, the standard linear solid model, which leads to the fractional Zener wave equation.
Fractional operators are non-local and have a long tail. This makes solving equations involving these operators challenging. For the linear solid model, the problem is expressed in terms of the Mittag-Leffler function (MLF). An exponential sum approximation, formulated based on the Gauss-Legendre quadrature, is proposed. Analysis shows that this approximation converges uniformly for all non-positive input, allowing unrestricted adpative time-stepping schemes. This approximation together with the finite element method are applied to solve the fractional Zener wave equation. It is proved that the implicit time stepping scheme based on generalised alpha method is unconditionally stable despite the addition of undetermined amount of sub-feedback loops. Furthermore, the solution converges to the true solution at O(T2) where T is the interval of each time step, provided that the approximation error of the MLF is sufficiently small.
For the fractional Laplacian, two methods to derive the finite difference approximation of the Riesz derivative have been investigated. In contrast to the integer order derivatives, the fractional finite difference operator must be dense regardless of the order of approximation accuracy. Therefore, methods to elevate the convergence order to an arbitrary order are explored. The first approach is based on applying fractional power to the Fourier transform of a difference method. For 1D, the Grünwald-Letnikov central difference method is employed, because the analytic solution is known. A filter is designed to be convolved with the difference stencil to eliminate higher order terms to achieve faster convergence of the approximation. For higher dimensions, the generalised Lattice-Boltzmann method is applied to achieve higher order isotropy. As exact solutions are not known for the integrals involved, numerical integration tools are developed for linear memory efficiency compared to the Fourier transform method.
The aforementioned method is not suitable for inhomogeneous boundary condition. Therefore, an alternative approach based on L2 approximation has been proposed to solve this problem. The modified Chebyshev moments are applied to evaluate the quadrature rule for the left and right fractional derivatives of piecewise Lagrange polynomials to avoid numerical instability in higher order approximations. Analysis and experiment both confirm the improved convergence even with discontinuity at the boundary.
| Date of Award | 14 Oct 2022 |
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| Original language | English |
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| Supervisor | Cheung Fat CHAN (Supervisor) & Hing Cheung SO (Supervisor) |