Study Of Key Technical Issues In Nonlinear Acoustic Simulations

Ultrasonic imaging has been an irreplaceable technique in modern clinical medical imaging. Currently, ultrasonic imaging based on fundamental waves has already been well-studied and applied, nonlinear imaging such as harmonic imaging has become the intensively researched topic. Numerical simulation is an important way to study nonlinear ultrasonic imaging due to its merits of high controllability of parameters, low-cost, high repeatability etc. Generally, simulations involve some key technical issues such as tissue modeling, acoustic equation, numerical discretization methods, boundary conditions and signal processing etc.This article focuses on the study of boundary conditions and numeric discretization method in simulations. It aims to provide a foundation for the building of an effective nonlinear acoustic simulation platform. Specifically, it includes the following parts.Firstly, Perfectly Matched Layer (PML) is currently the most widely-used and effective absorbing boundary conditions (ABC). However, classical PML is originally designed for first-order equations only and it cannot be applied to second-order directly. Although some scholars have extended the PML to second-order equations, these methods are not convenient to implement and require heavy computation costs. The article presents two novel unsplit PML for second-order equations. Based on the complex coordinate-stretching technique, the article derives the PML equation in frequency domain by direct differential operation. Easy time-domain PML equations are derived by equation transformation and an auxiliary differential equation (ADE) scheme. Theoretical analysis and FDTD simulations confirm that, compared with the existing split-filed PMLs, the proposed methods are much easier to implement and require less extra storage and have the same absorbing efficiency as previous split-field methods.Secondly, Convolutional PML (C-PML) can perform better in eliminating artificial reflections and are more stable than PML. However, its usage is mainly limited to first-order equations. The existing C-PMLs are inconvenient to apply and have high computation cost. They are mostly suitable for simulations using Finite Element Method. The article presents a novel, efficient auxiliary-differential equation form of the C-PML for the second-order equation system. Based on the complex coordinate-stretching technique and an efficient ADE scheme, the article proposed a general way to derive the C-PML of second-order equations. The proposed unsplit C-PML scheme requires only three ADEs in each axis. It has the advantage of simpler implementation and very suitable for FDTD simulations. It is also an unsplit-field scheme that can be extended to higher order discretization schemes conveniently. Numerical results confirm that the C-PML can better attenuate artificial reflections than classical PML.Thirdly, Partial Fraction Expansion (PFE) is an important numerical method when deriving the PML of second-order equations. It is also widely used when one performs inverse Laplace Transformation or tackles rational functions. The article presents several simple and recursive methods for rational functions in both factorized and expanded form. The proposed PFE methods require only simple pure-algebraic operations in the whole computation process without involving derivative. They also don’t involve polynomial division when dealing with improper functions. They are more suitable for large scale problems than classic methods. The methods are efficient, easy to apply for both computer and manual calculation. Various numerical experiments confirm that the proposed methods can achieve quite desirable accuracy even for PFE of rational functions with multiple high-order poles or some tricky ill-conditioned poles.Lastly, the article extends the PSTD (Pseudo Spectral Time Domain) method to second-order equation system. The wraparound effect of Pseudo-spectral methods is eliminated using an un-split Perfectly Matched Layer for the second-order wave equations. The method proposed here allows one to use PSTD for large-scale simulations based on well-established second-order equations. As the method has an infinite order of accuracy in the spatial derivatives, it requires much fewer grids than the conventional FDTD method. Numerical simulations based on Westervelt equation are conducted. The results confirm that the proposed PSTD method can maintain high-accuracy and reduce much calculation costs compared with the FDTD method in large-scale simulations.