非线性超声仿真中的关键技术问题研究
发布时间：2019-05-22 18:45
【摘要】：超声成像已成为临床应用中不可替代的医学影像技术之一。目前,超声基波成像技术已相对成熟,以谐波成像为代表的非线性成像技术成为研究热点。数值仿真具有参数高度可控,经济快速,可重复性强等优点,是研究超声非线性特性的有效手段。数值仿真涉及组织建模、声传播方程、数值算法、边界条件、信号提取及分析等关键技术。本论文主要对仿真中的边界条件及仿真数值算法进行研究,为建立有效的非线性仿真平台打下基础。主要工作包括以下几个部分：第一,完全匹配层(Perfectly Matched Layer, PML)是目前应用最广泛最有效的吸收边界条件之一,然而经典的PML只适用于一阶方程,不能直接应用于二阶方程。虽然已有少数学者将PML扩展到了二阶方程,但已有方法执行不便,计算代价较高。本文提出了两种适用于二阶波动方程的非分裂PML。基于复坐标伸缩变换(Complex coordinate-stretching),提出了通过微分运算直接得到高阶方程的PML频域方程的方法。利用方程变形及构造辅助微分方程,给出了便于求解时域PML方程的方法。理论分析和FDTD的仿真结果表明,相比于已有的PML方法,本文提出的非分裂方法吸收效果相同,但编程更简单,可较大地降低存储量和计算量,同时便于采用高阶数值方法离散。第二,卷积完全匹配层(Convolutional PML, C-PML)比PML能更好地消除边界反射,稳定性更佳。然而,目前C-PML主要应用于一阶方程,已有的二阶方程的C-PML主要适用于有限元的仿真中,执行复杂,计算代价很高。本文提出了一种新的二阶波动方程的C-PML。通过采用复坐标伸缩变换和构造辅助微分方程,给出了推导二阶波动方程的C-PML的一般方法。与已有的方法相比,本文提出的C-PML无需声压分裂,在一个坐标方向上只需引入三个一阶辅助微分方程,更易执行, 尤其适合FDTD仿真,且适用高阶离散方法。仿真结果表明,本文提出的C-PML能较好地消除边界反射波,优于传统的PML。第三,部分分式分解(Partial Fraction Expansion, PFE)是本文推导二阶方程的PML和C-PML用到的重要数学方法。PFE也在Laplace变换及有理函数的微积分求解等领域有广泛的运用。本文提出了多种直接适用于因式形式(Factorized Form)及展开形式(Expanded Form)的有理函数的PFE方法。这些PFE方法只涉及简单的代数运算,不涉及微分运算,无需求解线性方程组。处理假有理分式时,无需进行长除运算。与经典PFE方法(如微分法,待定系数法)相比,本文的方法更适于处理含高阶极点的大规模问题,更便于计算机编程和手算。数值测试结果表明,这些PFE方法即使在处理大型的、含有高阶极点及病态极点(ill-conditioned poles)的有理分式,也能取得较好的分解效果。最后,本文将时域伪谱法(Pseudo Spectral Time Domain, PSTD)推广到了高阶方程的数值求解中。通过本文提出的二阶波动方程的PML,解决了PSTD存在的周期混叠问题,从而使得PSTD算法可用于高阶方程的数值仿真中。由于PSTD算法空间上的精度可到达无限阶,它所需的采样点要远少于FDTD算法。采用超声仿真中常用的FDTD算法和PSTD对Westervelt方程分别进行数值求解,进行声场仿真。仿真结果表明,PSTD算法在大规模的仿真中,能较大程度地节约存储空间,同时保持较高的仿真精度。
[Abstract]:Ultrasound imaging has become one of the most irreplaceable medical image techniques in clinical application. At present, the ultrasonic fundamental wave imaging technology has been relatively mature, and the non-linear imaging technology represented by harmonic imaging has become the focus of the research. The numerical simulation has the advantages of controllable parameter, fast economy, strong repeatability and so on. The numerical simulation involves the key technologies such as organization modeling, acoustic propagation equation, numerical algorithm, boundary condition, signal extraction and analysis. The paper mainly studies the boundary condition and the simulation numerical algorithm in the simulation, and lays the foundation for the establishment of an effective non-linear simulation platform. The main work includes the following parts: First, the Perfectly Matched Layer (PML) is one of the most widely used and most effective absorption boundary conditions. However, the classical PML is only applicable to the first-order equation and cannot be applied directly to the second-order equation. Although a few scholars have extended PML to the second order equation, the existing method is inconvenient and the calculation cost is high. In this paper, two non-split PML, which are suitable for the second-order wave equation, are proposed. In this paper, a method for obtaining the PML frequency domain equation of high-order equation by differential operation is proposed based on complex-coordinate transformation. The method of solving the time domain PML equation is given by using the equation deformation and the construction of the auxiliary differential equation. The theoretical analysis and the simulation results of the FDTD method show that the non-splitting method has the same absorption effect compared with the existing PML method, but the programming is simpler, the storage amount and the calculation amount can be greatly reduced, and meanwhile, the high-order numerical method is convenient to be discretized. Second, the Convolutive PML (C-PML) can better eliminate the boundary reflection and the stability is better than the PML. However, at present, the C-PML is mainly applied to the first-order equation, and the C-PML of the existing second-order equation is mainly applied to the simulation of the finite element, and the implementation is complex and the calculation cost is high. The C-PML of a new second-order wave equation is presented in this paper. In this paper, a general method for deriving the C-PML of the second-order wave equation is given by using the complex-coordinate telescopic transformation and the construction of the auxiliary differential equation. Compared with the existing method, the C-PML proposed in this paper does not need sound pressure splitting, and only three first-order auxiliary differential equations are introduced in one coordinate direction, which is easier to carry out, and is especially suitable for FDTD simulation, and the high-order discrete method is applied. The simulation results show that C-PML is better than the traditional PML. Third, partial fractional decomposition (PFE) is an important mathematical method for deriving the PML and C-PML of the second order equation. The PFE is also widely used in the fields of the Laplace transform and the calculus of rational functions. This paper presents a variety of PFE methods that are directly applicable to the rational function of the Factorized Form and the expanded form. These PFE methods only involve simple algebraic operation, do not involve the differential operation, do not need to solve the system of linear equations. When the pseudo-rational fraction is processed, no long-addition operation is required. Compared with the classical PFE method (such as the differential method and the undetermined coefficient method), the method is more suitable for the large-scale problem with high-order pole, and is more convenient for computer programming and hand calculation. The results of the numerical test show that these PFE methods can achieve better decomposition effect even when large-scale, high-order pole and ill-conditioned poles are processed. Finally, the time-domain pseudo-spectrum (PSTD) is extended to the numerical solution of higher-order equations. The PML of the second-order wave equation presented in this paper solves the problem of periodic aliasing in PSTD, so that the PSTD algorithm can be used in the numerical simulation of higher-order equations. Because the accuracy in the space of the PSTD algorithm can reach the infinite order, the required sampling point is far less than that of the FDTD algorithm. The numerical solution of the Westvelt equation is carried out by using the FDTD method and the PSTD, which are commonly used in the ultrasonic simulation, and the sound field simulation is carried out. The simulation results show that the PSTD algorithm can save the storage space to a large extent in the large-scale simulation, while keeping the higher simulation precision.
【学位授予单位】：复旦大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TP391.41;R445.1
本文编号：2483159
[Abstract]:Ultrasound imaging has become one of the most irreplaceable medical image techniques in clinical application. At present, the ultrasonic fundamental wave imaging technology has been relatively mature, and the non-linear imaging technology represented by harmonic imaging has become the focus of the research. The numerical simulation has the advantages of controllable parameter, fast economy, strong repeatability and so on. The numerical simulation involves the key technologies such as organization modeling, acoustic propagation equation, numerical algorithm, boundary condition, signal extraction and analysis. The paper mainly studies the boundary condition and the simulation numerical algorithm in the simulation, and lays the foundation for the establishment of an effective non-linear simulation platform. The main work includes the following parts: First, the Perfectly Matched Layer (PML) is one of the most widely used and most effective absorption boundary conditions. However, the classical PML is only applicable to the first-order equation and cannot be applied directly to the second-order equation. Although a few scholars have extended PML to the second order equation, the existing method is inconvenient and the calculation cost is high. In this paper, two non-split PML, which are suitable for the second-order wave equation, are proposed. In this paper, a method for obtaining the PML frequency domain equation of high-order equation by differential operation is proposed based on complex-coordinate transformation. The method of solving the time domain PML equation is given by using the equation deformation and the construction of the auxiliary differential equation. The theoretical analysis and the simulation results of the FDTD method show that the non-splitting method has the same absorption effect compared with the existing PML method, but the programming is simpler, the storage amount and the calculation amount can be greatly reduced, and meanwhile, the high-order numerical method is convenient to be discretized. Second, the Convolutive PML (C-PML) can better eliminate the boundary reflection and the stability is better than the PML. However, at present, the C-PML is mainly applied to the first-order equation, and the C-PML of the existing second-order equation is mainly applied to the simulation of the finite element, and the implementation is complex and the calculation cost is high. The C-PML of a new second-order wave equation is presented in this paper. In this paper, a general method for deriving the C-PML of the second-order wave equation is given by using the complex-coordinate telescopic transformation and the construction of the auxiliary differential equation. Compared with the existing method, the C-PML proposed in this paper does not need sound pressure splitting, and only three first-order auxiliary differential equations are introduced in one coordinate direction, which is easier to carry out, and is especially suitable for FDTD simulation, and the high-order discrete method is applied. The simulation results show that C-PML is better than the traditional PML. Third, partial fractional decomposition (PFE) is an important mathematical method for deriving the PML and C-PML of the second order equation. The PFE is also widely used in the fields of the Laplace transform and the calculus of rational functions. This paper presents a variety of PFE methods that are directly applicable to the rational function of the Factorized Form and the expanded form. These PFE methods only involve simple algebraic operation, do not involve the differential operation, do not need to solve the system of linear equations. When the pseudo-rational fraction is processed, no long-addition operation is required. Compared with the classical PFE method (such as the differential method and the undetermined coefficient method), the method is more suitable for the large-scale problem with high-order pole, and is more convenient for computer programming and hand calculation. The results of the numerical test show that these PFE methods can achieve better decomposition effect even when large-scale, high-order pole and ill-conditioned poles are processed. Finally, the time-domain pseudo-spectrum (PSTD) is extended to the numerical solution of higher-order equations. The PML of the second-order wave equation presented in this paper solves the problem of periodic aliasing in PSTD, so that the PSTD algorithm can be used in the numerical simulation of higher-order equations. Because the accuracy in the space of the PSTD algorithm can reach the infinite order, the required sampling point is far less than that of the FDTD algorithm. The numerical solution of the Westvelt equation is carried out by using the FDTD method and the PSTD, which are commonly used in the ultrasonic simulation, and the sound field simulation is carried out. The simulation results show that the PSTD algorithm can save the storage space to a large extent in the large-scale simulation, while keeping the higher simulation precision.
【学位授予单位】：复旦大学
【学位级别】：硕士
【学位授予年份】：2014
【分类号】：TP391.41;R445.1
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