非線性超聲仿真中的關(guān)鍵技術(shù)問(wèn)題研究
發(fā)布時(shí)間：2019-05-22 18:45
【摘要】：超聲成像已成為臨床應(yīng)用中不可替代的醫(yī)學(xué)影像技術(shù)之一。目前,超聲基波成像技術(shù)已相對(duì)成熟,以諧波成像為代表的非線性成像技術(shù)成為研究熱點(diǎn)。數(shù)值仿真具有參數(shù)高度可控,經(jīng)濟(jì)快速,可重復(fù)性強(qiáng)等優(yōu)點(diǎn),是研究超聲非線性特性的有效手段。數(shù)值仿真涉及組織建模、聲傳播方程、數(shù)值算法、邊界條件、信號(hào)提取及分析等關(guān)鍵技術(shù)。本論文主要對(duì)仿真中的邊界條件及仿真數(shù)值算法進(jìn)行研究,為建立有效的非線性仿真平臺(tái)打下基礎(chǔ)。主要工作包括以下幾個(gè)部分：第一,完全匹配層(Perfectly Matched Layer, PML)是目前應(yīng)用最廣泛最有效的吸收邊界條件之一,然而經(jīng)典的PML只適用于一階方程,不能直接應(yīng)用于二階方程。雖然已有少數(shù)學(xué)者將PML擴(kuò)展到了二階方程,但已有方法執(zhí)行不便,計(jì)算代價(jià)較高。本文提出了兩種適用于二階波動(dòng)方程的非分裂PML�；趶�(fù)坐標(biāo)伸縮變換(Complex coordinate-stretching),提出了通過(guò)微分運(yùn)算直接得到高階方程的PML頻域方程的方法。利用方程變形及構(gòu)造輔助微分方程,給出了便于求解時(shí)域PML方程的方法。理論分析和FDTD的仿真結(jié)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,但編程更簡(jiǎn)單,可較大地降低存儲(chǔ)量和計(jì)算量,同時(shí)便于采用高階數(shù)值方法離散。第二,卷積完全匹配層(Convolutional PML, C-PML)比PML能更好地消除邊界反射,穩(wěn)定性更佳。然而,目前C-PML主要應(yīng)用于一階方程,已有的二階方程的C-PML主要適用于有限元的仿真中,執(zhí)行復(fù)雜,計(jì)算代價(jià)很高。本文提出了一種新的二階波動(dòng)方程的C-PML。通過(guò)采用復(fù)坐標(biāo)伸縮變換和構(gòu)造輔助微分方程,給出了推導(dǎo)二階波動(dòng)方程的C-PML的一般方法。與已有的方法相比,本文提出的C-PML無(wú)需聲壓分裂,在一個(gè)坐標(biāo)方向上只需引入三個(gè)一階輔助微分方程,更易執(zhí)行, 尤其適合FDTD仿真,且適用高階離散方法。仿真結(jié)果表明,本文提出的C-PML能較好地消除邊界反射波,優(yōu)于傳統(tǒng)的PML。第三,部分分式分解(Partial Fraction Expansion, PFE)是本文推導(dǎo)二階方程的PML和C-PML用到的重要數(shù)學(xué)方法。PFE也在Laplace變換及有理函數(shù)的微積分求解等領(lǐng)域有廣泛的運(yùn)用。本文提出了多種直接適用于因式形式(Factorized Form)及展開(kāi)形式(Expanded Form)的有理函數(shù)的PFE方法。這些PFE方法只涉及簡(jiǎn)單的代數(shù)運(yùn)算,不涉及微分運(yùn)算,無(wú)需求解線性方程組。處理假有理分式時(shí),無(wú)需進(jìn)行長(zhǎng)除運(yùn)算。與經(jīng)典PFE方法(如微分法,待定系數(shù)法)相比,本文的方法更適于處理含高階極點(diǎn)的大規(guī)模問(wèn)題,更便于計(jì)算機(jī)編程和手算。數(shù)值測(cè)試結(jié)果表明,這些PFE方法即使在處理大型的、含有高階極點(diǎn)及病態(tài)極點(diǎn)(ill-conditioned poles)的有理分式,也能取得較好的分解效果。最后,本文將時(shí)域偽譜法(Pseudo Spectral Time Domain, PSTD)推廣到了高階方程的數(shù)值求解中。通過(guò)本文提出的二階波動(dòng)方程的PML,解決了PSTD存在的周期混疊問(wèn)題,從而使得PSTD算法可用于高階方程的數(shù)值仿真中。由于PSTD算法空間上的精度可到達(dá)無(wú)限階,它所需的采樣點(diǎn)要遠(yuǎn)少于FDTD算法。采用超聲仿真中常用的FDTD算法和PSTD對(duì)Westervelt方程分別進(jìn)行數(shù)值求解,進(jìn)行聲場(chǎng)仿真。仿真結(jié)果表明,PSTD算法在大規(guī)模的仿真中,能較大程度地節(jié)約存儲(chǔ)空間,同時(shí)保持較高的仿真精度。
[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.
【學(xué)位授予單位】：復(fù)旦大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：TP391.41;R445.1
本文編號(hào)：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.
【學(xué)位授予單位】：復(fù)旦大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：TP391.41;R445.1
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