应用时空守恒元和解元方法数值研究溃坝洪水波和ESWL中的聚焦冲击波
|
摘要
本文主要应用一种全新的数值方法(即:时空守恒元和解元方法)来数值研究溃坝洪水波和水下冲击波在其最成功的医学应用ESWL(Extracorporeal Shock Wave Lithotripsy)中的聚焦行为。它们分别属于两类不同的波:浅水波和水下冲击波。虽然它们物理形态不同,但是都属于双曲波的范畴,因此可用统一的数学形式来描述它们并进行数值研究。
横断河流的拦河大坝有蓄水、防洪、灌溉、发电等造福于人类的功能,河流两岸的堤防保护人类免于洪水灭顶之灾,但是,它们又对人类生命和财产具有潜在的巨大危险。当人为原因或自然力造成大坝或堤防溃坝时,大量水体突然释放而形成的狂暴的洪水会对下游滞洪区造成毁灭性的灾难。因此,预测和确定溃坝坝址的流量和水位过程线,以及洪水向下游演进时沿程各处的流量、水位、流速、波前和洪峰到达的时间等具有重要的意义,长久以来一直受到国内外学者研究的关注。
因其突发性、暴烈性和灾害性,难以获得真实溃坝水流的直接观测数据。数学上,可采用浅水方程来描述溃坝洪水波的流动。而无论浅水方程的初始值是否光滑,其解都可能产生间断,即流场中出现涌波。对于溃坝洪水,其初始条件本身就是间断的,构成了浅水方程的Riemann问题。所以数值研究溃坝水流具有特定的困难,尤其是在超临界流情况,多数算法常常失效。由重力引起的浅水运动与无粘可压缩流体的不定常流动在数学上是同类的,因此求解含冲击波的空气动力学问题的数值方法经过适当改造,可用于求解含溃坝波的浅水运动问题。
体外冲击波聚焦粉碎结石术(ESWL)因无手术的侵入性创伤又能有效地粉碎人体结石,目前已成为治疗尿路结石的一种常用临床手段。ESWL通过水下放电或爆炸产生冲击波,利用聚焦冲击波的高能量密度和空化作用而粉碎结石。一般认为ESWL冲击波的负压波段是引起水空化的主要原因。然而,临床和实验研究均发现伴随着ESWL的不同形式损伤,例如血尿、慢性出血、多重肾软组织内的血肿、以及肾水肿等。因此,在冲击波医疗技术中,医疗效率和安全性十分重要。为了ESWL碎石效率最优化和组织损伤最小化,确定聚焦冲击波的动力学焦点和研究负压的产生和演化过程具有很重要的意义。
冲击波是一种非线性波,对于它线性波的Snell反射定律等不再成立。水下冲击波聚焦的数值研究,必须考虑水的可压缩性,本文采用Tait状态方程来描述水的特性,并假设水是无粘的、运动中无热传导和热辐射。这样,在数学形式上,可以借助空气动力学中研究冲击波的一系列方法,来研究ESWL中的水下冲击波传播、反射和聚焦。
浅水方程和理想可压缩流体的不定常运动的Euler方程可以统一写成一个守恒形式的双曲型一阶拟线性偏微分方程组。溃坝洪水涌波和水下冲击波,在数学上分别表现为浅水方程和Euler方程的广义解(也称为物理解)。由于双曲波在科学问题中的广泛性、多样性和重要性,发展求解双曲型一阶拟线性偏微分方程组广义解的数值方法,一直是非线性问题研究的一个热点,尽管已经出现了大量的数值格式,新方法依然层出不穷。
其中值得注意的是Chang S.C在1995年提出了一种求解双曲型方程的崭新的数值方法,即时空守恒元和解元(CE/SE)方法(space-time Conservation Element and Solution Element method),目前,已被美国NASA列为第二代CFD程序中的主要算法之一。
CE/SE方法不是对以前方法的改进和优化,而是一种全新的数值方法,无论从概念上还是从构造方法上都与传统的数值方法(如:有限差分法、有限元法、有限体积法、特征线法等)有所不同,具有独特的优点:首先,该方法把时间与空间完全统一起来同等对待,从守恒积分型方程出发,通过设立守恒元和解元,使格式局部和全局都严格保证其物理意义上的守恒律;其次,它把流场基本变量及其对空间偏导数都作为独立变量,同时进行求解。与传统方法相比,在相同网格点数的情况下,格式的精度可以达到更高;第三,它除利用了简单的Taylor级数展开式外,无需任何其它的数值逼近技术,无需求解Riemann问题,也不需要任何单调性限制或特征技术,因此该方法格式构造思想非常简单,物理意义清晰,通用性好;最后,该方法可直接推广到多维情形,无需采用维数分解或交替方向技术。该方法不仅可用于求解连续流动问题,而且可用于求解含有冲击波等不连续流动问题,其高分辨率的数值结果甚至比目前广泛流行的某些高分辨率格式的计算结果还好。目前,CE/SE方法已经用于一些复杂流场的计算,如内爆和外爆问题、声/波及波/涡干扰问题、ZND爆轰波问题等,但运用该方法来计算本文关注的溃坝洪水波问题和ESWL中的水下冲击波聚焦问题迄今未见先例。
本文首先经过适当改造张增产等改进的CE/SE方法来离散浅水方程,建立一维数学模型和二维基于结构网格的数学模型来分别研究理想条件下和简单计算区域的溃坝洪水波的运动特性。其次,进一步改进和构造了CE/SE方法求解双曲型方程的数值格式,建立了基于非结构混合网格的二维数值格式,使之既能够满足精度的要求,又能很好的拟和复杂的计算边界。第三,采用本文建立的新的CE/SE数值格式来离散浅水方程,建立了基于非结构混合网格的二维数学模型来研究复杂边界下的溃坝洪水波的运动特性。第四,应用本文建立的新的CE/SE数值格式来离散Euler方程,建立了研究ESWL中的水下聚焦冲击波的二维数学模型,以球面压电陶瓷型ESWL为例,研究负压的产生及其演化的过程。第五,为了数值追踪ESWL中的水下冲击波的波阵面的传播,本文从另外一个角度出发,采用CCW几何冲击波动力学方法来建立数学模型,研究ESWL中的冲击波的波阵面的演化、聚焦的问题。
以上建立的数学模型都是本人采用FORTREAN语言来编写代码,实现功能。本文成功应用CE/SE方法用来计算溃坝洪水波问题和ESWL中的水下冲击波聚焦的问题,使得此新方法的应用面进一步的拓宽。
溃坝洪水波的计算表明:本文建立的基于时空守恒元和解元方法的溃坝洪水波数学模型格式简单,稳定性好,通用性更好,计算量小,计算结果精度高,对溃坝波间断具有很高的分辨率,能够很好的展示溃坝波的运动特性,是研究溃坝洪水波一种高精度的新的数值方法,为预估和应对溃坝洪水的毁灭性灾难提供一种有效的新的研究手段。ESWL的计算算例表明:本文建立的基于时空守恒元和解元方法的求解ESWL中的水下冲击波的数学模型,第一次数值模拟了球面压电陶瓷型ESWL的冲击波传播的压力场,从计算结果可知,由于在边缘衍射产生的负压,使得在焦点附近的压力在冲击波的正压作用以后紧跟着负压的作用,说明球面压电陶瓷型ESWL水下冲击波聚焦时出现空化是不可避免的,所以在球面压电陶瓷型EWSL机的设计制造及临床应用中,都不可忽视空蚀。其次,建立的基于CCW几何冲击波动力学方法的数学模型能够很好的追踪冲击波波阵面传播的情况,数值解展示了球面压电陶瓷型EWSL几何中心即球心附近的冲击波聚焦特性,计算结果说明实际的动力学焦点未必就是几何中心,有可能偏离球心,因此,在临床应用球面压电陶瓷型ESWL粉碎结石时,实际焦点应当规定为压力最大的点,而不是简单地就将球心选为焦点——轰击的靶点。
In this Dissertation, dam-break flood wave and the focusing of underwater shock wave in ESWL (Extracorporeal Shock Wave Lithotripsy) where the underwater shock wave is the most successful medical application are studied by using a novel numerical scheme (space-time Conservation Element and Solution Element method). They belong to two different waves respectively: shallow water wave and underwater shock wave. Their physical forms are different, but all of them belong to the hyperbolic wave. Therefore, they can be mathematically described by uniform formal equations and are numerical simulated.
Dams in the rivers are benefited to human, such as sluice, prevent and control flood, irrigation, generate electricity and so on. Dikes in the rivers control river flowing and prevent human from flooding. Flooding due to the failure of a dam or dike with man-made or natural causes has potentially disastrous consequences to human life and property. It can reasonably determine the standards of preventing or controlling flood and measures of avoiding danger in the dam or dyke. It is important to the forecast and determination of the dam-break flood flux, water level, wave height, velocity, the front of dam-break and the arrived time at different locations along the channel. This problem has occupied the attention of researchers as well as practicing engineers for several decades.
The observational data of natural dam-break flows are difficult to obtain because of its sudden, violent and disastrous event. Mathematically, the dam-break problem is commonly described by the shallow water equations. One feature of these equations is the formation of surge bore which is the rapidly varying discontinuous flow whether the initial values are continuous. In the dam-break flood wave problem, the initial values are discontinuous, which constitutes the Riemann problem of shallow water equations. It is an important basis for validating the numerical method which can capture the dam-break bore waves accurately without nonphysical distortion and numerical oscillation, specifically, in the supercritical-flow problem. Because the movement of free surface flows is similar the propagation of compressible fluids flows, the methods of simulating the problems of gas dynamics with shock wave can simulate the problems of hydrodynamics with dam-break bore waves.
Extracorporeal shock wave lithotripsy (ESWL) is the most common treatment of kidney stone disease because of the treatment is non-invasive and the stone can be effectively broken down into small fragments. Hundreds of underwater shock waves are generated outside the patient’s body and focused on the kidney stone. The stone are broken down into small fragments because of the high energy of the focusing shock waves and the cavitations. Generally, the water generates a cloud of cavitating bubbles because of the negative pressures component of an ESWL shock wave. Although effective in breaking kidney stones, ESWL can also cause significant short- and long-term damage to the kidneys. Damage has been observed on both cellular and systemic level. A common side effect of a lithotripsy treatment is the presence of blood in the urine (hematuria), chronic hemorrhage, hematoma in kidney parenchyma, kidney edema and so on. It is important that various solutions designed to maximize stone comminution and minimize tissue damage in extracorporeal shock wave lithotripsy. Both the determination of the dynamical focus and the generation and the evolvement of the negative pressures play very important roles in design and clinical ESWL.
Shock wave is a nonlinear wave, which is not valid to the linear wave law of Snell refection. In numerical simulating the evolution of underwater shock wave, water is compressible, inviscid and non-heat exchange. The characteristic of water can be represented by Tait’s state equation. Mathematically, the focusing shock wave problem in ESWL can be described by the Euler equations with Tait’s state equation, the methods of simulating the problems of gas dynamics with shock wave can simulate the problems of underwater focusing shock wave.
Shallow water equations and Euler equations can be mathematically described by a uniform formal time-dependent set of nonlinear partial differential conservation equations of hyperbolic type. Dam-break flood wave and underwater shock wave are the physical solution of shallow water equations and Euler equations respectively. The development of the numerical scheme to solve the nonlinear partial differential hyperbolic equations is a hot topic in studying the nonlinear problems.
The space-time conservation element and solution element (CE/SE) method, originally proposed by Chang in 1995, is a novel numerical framework for solving the problem of hyperbolic conservation laws. Now this new method is already become to one of primary methods in the second CFD scheme in NASA.
The CE/SE method is not an incremental improvement of a previously existing CFD method, and it is different in both concept and methodology from well-established traditional numerical methods (such as finite difference method, finite element method, finite volume method etc.). The CE/SE method has many nontraditional features. Firstly, space and time are unified and treated on the same footing, and by the introduction of conservation element and solution element, both local and global flux conservations in space and time in stead of in space only are enforced. Secondly, all flow variables and their spatial derivatives are considered as individual unknowns to be solved for simultaneously at each grid point, its accuracy is higher than well-established traditional numerical methods in the same grids. Thirdly, it is conceptually simple and robust, neither Riemann solver nor technique based on characteristics are involved. Last, this new scheme is easily extended to solve multidimensional problem without flux-splitting. The CE/SE method can be used to simulate the general problem, it also solve the discontinuous problem with shock wave or dam-break and so on. Until now, this new scheme doesn’t use to numerical study the problem of dam-break flood wave and the problem of the focusing of underwater shock wave.
Firstly, the shallow water equations are discretized by the Zhang’s modified CE/SE method. One-dimensional numerical model and two-dimensional numerical model on structured meshes are constructed. On this basis, the dynamic characteristic of dam-break flood waves is studied in the ideal conditions and the 2-D simple computational regions. Secondly, the CE/SE method is modified and a new constructing method on the 2-D unstructured mixed meshes is developed by author. The new modified CE/SE method is used to solve the hyperbolic conservation equations. The reconstructing method not only maintains all the features, but also is easy to study the problems on the complicated computational regions. Thirdly, the shallow water equations are discretized by the new modified CE/SE method. Two-dimensional numerical model on the 2-D unstructured mixed meshes are constructed. On this basis, the dynamic characteristic of dam-break flood waves is studied on the complicated computational regions. Fourthly, the 2-D axisymmetrical Euler equations are discretized by the new modified CE/SE method and numerical model on the 2-D unstructured mixed meshes are constructed to study the generation and the evolvement of the negative pressures in spherical piezoelectric emitters ESWL. Last, using Chisnell-Chester-Whitham (CCW) geometric shock wave dynamic method, the evolution and focusing of the fronts of underwater shock wave in ESWL are simulated numerically.
All codes of the numerical models are written by author with FORTRAN90 to implement function. It is successful that the problem of dam-break flood wave and the problem of the focusing of underwater shock wave in ESWL are simulated by using the CE/SE method.
The computational results of dam-break flood wave’s examples show that many advantages of the CE/SE method such as simple, robust, high efficiency, high accuracy and high resolution of dam-break. It shows that the proposed model can satisfactorily describe the whole process and flow characteristics of the dam-break wave and possesses high computation accuracy. It is proved that the CE/SE method is a new and high accuracy numerical method for studying dam-break flood wave.
In study of the ESWL, a robust model for the solution of focusing of underwater shock wave in ESWL has been described. Using the modified space-time Conservation Element and Solution Element method on unstructured meshes, the two-dimensional compressible inviscid Euler equations with Tait’s equation of state for water are solved. The pressure field of underwater shock wave in spherical piezoelectric emitters ESWL is simulated successfully in the first. The computational results show that the negative pressure is generated because of the diffraction in the edge of ESWL. The pressure near of the focus of ESWL is the negative pressure after the positive pressure, which shows that the cavitations are inevitable for the focusing of underwater shock wave in spherical piezoelectric emitters ESWL. The cavitations must be considered in clinical and designing ESWL. Secondly, another model for the track of the front of the underwater shock wave has been described, using CCW geometric shock wave dynamic method, the evolution and focusing of the fronts of underwater shock wave in ESWL are simulated numerically. The computational results show that the dynamic focus in spherical piezoelectric emitters ESWL is different from geometric focus in general. The correct location of the factual focus plays very important roles in clinical ESWL.
横断河流的拦河大坝有蓄水、防洪、灌溉、发电等造福于人类的功能,河流两岸的堤防保护人类免于洪水灭顶之灾,但是,它们又对人类生命和财产具有潜在的巨大危险。当人为原因或自然力造成大坝或堤防溃坝时,大量水体突然释放而形成的狂暴的洪水会对下游滞洪区造成毁灭性的灾难。因此,预测和确定溃坝坝址的流量和水位过程线,以及洪水向下游演进时沿程各处的流量、水位、流速、波前和洪峰到达的时间等具有重要的意义,长久以来一直受到国内外学者研究的关注。
因其突发性、暴烈性和灾害性,难以获得真实溃坝水流的直接观测数据。数学上,可采用浅水方程来描述溃坝洪水波的流动。而无论浅水方程的初始值是否光滑,其解都可能产生间断,即流场中出现涌波。对于溃坝洪水,其初始条件本身就是间断的,构成了浅水方程的Riemann问题。所以数值研究溃坝水流具有特定的困难,尤其是在超临界流情况,多数算法常常失效。由重力引起的浅水运动与无粘可压缩流体的不定常流动在数学上是同类的,因此求解含冲击波的空气动力学问题的数值方法经过适当改造,可用于求解含溃坝波的浅水运动问题。
体外冲击波聚焦粉碎结石术(ESWL)因无手术的侵入性创伤又能有效地粉碎人体结石,目前已成为治疗尿路结石的一种常用临床手段。ESWL通过水下放电或爆炸产生冲击波,利用聚焦冲击波的高能量密度和空化作用而粉碎结石。一般认为ESWL冲击波的负压波段是引起水空化的主要原因。然而,临床和实验研究均发现伴随着ESWL的不同形式损伤,例如血尿、慢性出血、多重肾软组织内的血肿、以及肾水肿等。因此,在冲击波医疗技术中,医疗效率和安全性十分重要。为了ESWL碎石效率最优化和组织损伤最小化,确定聚焦冲击波的动力学焦点和研究负压的产生和演化过程具有很重要的意义。
冲击波是一种非线性波,对于它线性波的Snell反射定律等不再成立。水下冲击波聚焦的数值研究,必须考虑水的可压缩性,本文采用Tait状态方程来描述水的特性,并假设水是无粘的、运动中无热传导和热辐射。这样,在数学形式上,可以借助空气动力学中研究冲击波的一系列方法,来研究ESWL中的水下冲击波传播、反射和聚焦。
浅水方程和理想可压缩流体的不定常运动的Euler方程可以统一写成一个守恒形式的双曲型一阶拟线性偏微分方程组。溃坝洪水涌波和水下冲击波,在数学上分别表现为浅水方程和Euler方程的广义解(也称为物理解)。由于双曲波在科学问题中的广泛性、多样性和重要性,发展求解双曲型一阶拟线性偏微分方程组广义解的数值方法,一直是非线性问题研究的一个热点,尽管已经出现了大量的数值格式,新方法依然层出不穷。
其中值得注意的是Chang S.C在1995年提出了一种求解双曲型方程的崭新的数值方法,即时空守恒元和解元(CE/SE)方法(space-time Conservation Element and Solution Element method),目前,已被美国NASA列为第二代CFD程序中的主要算法之一。
CE/SE方法不是对以前方法的改进和优化,而是一种全新的数值方法,无论从概念上还是从构造方法上都与传统的数值方法(如:有限差分法、有限元法、有限体积法、特征线法等)有所不同,具有独特的优点:首先,该方法把时间与空间完全统一起来同等对待,从守恒积分型方程出发,通过设立守恒元和解元,使格式局部和全局都严格保证其物理意义上的守恒律;其次,它把流场基本变量及其对空间偏导数都作为独立变量,同时进行求解。与传统方法相比,在相同网格点数的情况下,格式的精度可以达到更高;第三,它除利用了简单的Taylor级数展开式外,无需任何其它的数值逼近技术,无需求解Riemann问题,也不需要任何单调性限制或特征技术,因此该方法格式构造思想非常简单,物理意义清晰,通用性好;最后,该方法可直接推广到多维情形,无需采用维数分解或交替方向技术。该方法不仅可用于求解连续流动问题,而且可用于求解含有冲击波等不连续流动问题,其高分辨率的数值结果甚至比目前广泛流行的某些高分辨率格式的计算结果还好。目前,CE/SE方法已经用于一些复杂流场的计算,如内爆和外爆问题、声/波及波/涡干扰问题、ZND爆轰波问题等,但运用该方法来计算本文关注的溃坝洪水波问题和ESWL中的水下冲击波聚焦问题迄今未见先例。
本文首先经过适当改造张增产等改进的CE/SE方法来离散浅水方程,建立一维数学模型和二维基于结构网格的数学模型来分别研究理想条件下和简单计算区域的溃坝洪水波的运动特性。其次,进一步改进和构造了CE/SE方法求解双曲型方程的数值格式,建立了基于非结构混合网格的二维数值格式,使之既能够满足精度的要求,又能很好的拟和复杂的计算边界。第三,采用本文建立的新的CE/SE数值格式来离散浅水方程,建立了基于非结构混合网格的二维数学模型来研究复杂边界下的溃坝洪水波的运动特性。第四,应用本文建立的新的CE/SE数值格式来离散Euler方程,建立了研究ESWL中的水下聚焦冲击波的二维数学模型,以球面压电陶瓷型ESWL为例,研究负压的产生及其演化的过程。第五,为了数值追踪ESWL中的水下冲击波的波阵面的传播,本文从另外一个角度出发,采用CCW几何冲击波动力学方法来建立数学模型,研究ESWL中的冲击波的波阵面的演化、聚焦的问题。
以上建立的数学模型都是本人采用FORTREAN语言来编写代码,实现功能。本文成功应用CE/SE方法用来计算溃坝洪水波问题和ESWL中的水下冲击波聚焦的问题,使得此新方法的应用面进一步的拓宽。
溃坝洪水波的计算表明:本文建立的基于时空守恒元和解元方法的溃坝洪水波数学模型格式简单,稳定性好,通用性更好,计算量小,计算结果精度高,对溃坝波间断具有很高的分辨率,能够很好的展示溃坝波的运动特性,是研究溃坝洪水波一种高精度的新的数值方法,为预估和应对溃坝洪水的毁灭性灾难提供一种有效的新的研究手段。ESWL的计算算例表明:本文建立的基于时空守恒元和解元方法的求解ESWL中的水下冲击波的数学模型,第一次数值模拟了球面压电陶瓷型ESWL的冲击波传播的压力场,从计算结果可知,由于在边缘衍射产生的负压,使得在焦点附近的压力在冲击波的正压作用以后紧跟着负压的作用,说明球面压电陶瓷型ESWL水下冲击波聚焦时出现空化是不可避免的,所以在球面压电陶瓷型EWSL机的设计制造及临床应用中,都不可忽视空蚀。其次,建立的基于CCW几何冲击波动力学方法的数学模型能够很好的追踪冲击波波阵面传播的情况,数值解展示了球面压电陶瓷型EWSL几何中心即球心附近的冲击波聚焦特性,计算结果说明实际的动力学焦点未必就是几何中心,有可能偏离球心,因此,在临床应用球面压电陶瓷型ESWL粉碎结石时,实际焦点应当规定为压力最大的点,而不是简单地就将球心选为焦点——轰击的靶点。
In this Dissertation, dam-break flood wave and the focusing of underwater shock wave in ESWL (Extracorporeal Shock Wave Lithotripsy) where the underwater shock wave is the most successful medical application are studied by using a novel numerical scheme (space-time Conservation Element and Solution Element method). They belong to two different waves respectively: shallow water wave and underwater shock wave. Their physical forms are different, but all of them belong to the hyperbolic wave. Therefore, they can be mathematically described by uniform formal equations and are numerical simulated.
Dams in the rivers are benefited to human, such as sluice, prevent and control flood, irrigation, generate electricity and so on. Dikes in the rivers control river flowing and prevent human from flooding. Flooding due to the failure of a dam or dike with man-made or natural causes has potentially disastrous consequences to human life and property. It can reasonably determine the standards of preventing or controlling flood and measures of avoiding danger in the dam or dyke. It is important to the forecast and determination of the dam-break flood flux, water level, wave height, velocity, the front of dam-break and the arrived time at different locations along the channel. This problem has occupied the attention of researchers as well as practicing engineers for several decades.
The observational data of natural dam-break flows are difficult to obtain because of its sudden, violent and disastrous event. Mathematically, the dam-break problem is commonly described by the shallow water equations. One feature of these equations is the formation of surge bore which is the rapidly varying discontinuous flow whether the initial values are continuous. In the dam-break flood wave problem, the initial values are discontinuous, which constitutes the Riemann problem of shallow water equations. It is an important basis for validating the numerical method which can capture the dam-break bore waves accurately without nonphysical distortion and numerical oscillation, specifically, in the supercritical-flow problem. Because the movement of free surface flows is similar the propagation of compressible fluids flows, the methods of simulating the problems of gas dynamics with shock wave can simulate the problems of hydrodynamics with dam-break bore waves.
Extracorporeal shock wave lithotripsy (ESWL) is the most common treatment of kidney stone disease because of the treatment is non-invasive and the stone can be effectively broken down into small fragments. Hundreds of underwater shock waves are generated outside the patient’s body and focused on the kidney stone. The stone are broken down into small fragments because of the high energy of the focusing shock waves and the cavitations. Generally, the water generates a cloud of cavitating bubbles because of the negative pressures component of an ESWL shock wave. Although effective in breaking kidney stones, ESWL can also cause significant short- and long-term damage to the kidneys. Damage has been observed on both cellular and systemic level. A common side effect of a lithotripsy treatment is the presence of blood in the urine (hematuria), chronic hemorrhage, hematoma in kidney parenchyma, kidney edema and so on. It is important that various solutions designed to maximize stone comminution and minimize tissue damage in extracorporeal shock wave lithotripsy. Both the determination of the dynamical focus and the generation and the evolvement of the negative pressures play very important roles in design and clinical ESWL.
Shock wave is a nonlinear wave, which is not valid to the linear wave law of Snell refection. In numerical simulating the evolution of underwater shock wave, water is compressible, inviscid and non-heat exchange. The characteristic of water can be represented by Tait’s state equation. Mathematically, the focusing shock wave problem in ESWL can be described by the Euler equations with Tait’s state equation, the methods of simulating the problems of gas dynamics with shock wave can simulate the problems of underwater focusing shock wave.
Shallow water equations and Euler equations can be mathematically described by a uniform formal time-dependent set of nonlinear partial differential conservation equations of hyperbolic type. Dam-break flood wave and underwater shock wave are the physical solution of shallow water equations and Euler equations respectively. The development of the numerical scheme to solve the nonlinear partial differential hyperbolic equations is a hot topic in studying the nonlinear problems.
The space-time conservation element and solution element (CE/SE) method, originally proposed by Chang in 1995, is a novel numerical framework for solving the problem of hyperbolic conservation laws. Now this new method is already become to one of primary methods in the second CFD scheme in NASA.
The CE/SE method is not an incremental improvement of a previously existing CFD method, and it is different in both concept and methodology from well-established traditional numerical methods (such as finite difference method, finite element method, finite volume method etc.). The CE/SE method has many nontraditional features. Firstly, space and time are unified and treated on the same footing, and by the introduction of conservation element and solution element, both local and global flux conservations in space and time in stead of in space only are enforced. Secondly, all flow variables and their spatial derivatives are considered as individual unknowns to be solved for simultaneously at each grid point, its accuracy is higher than well-established traditional numerical methods in the same grids. Thirdly, it is conceptually simple and robust, neither Riemann solver nor technique based on characteristics are involved. Last, this new scheme is easily extended to solve multidimensional problem without flux-splitting. The CE/SE method can be used to simulate the general problem, it also solve the discontinuous problem with shock wave or dam-break and so on. Until now, this new scheme doesn’t use to numerical study the problem of dam-break flood wave and the problem of the focusing of underwater shock wave.
Firstly, the shallow water equations are discretized by the Zhang’s modified CE/SE method. One-dimensional numerical model and two-dimensional numerical model on structured meshes are constructed. On this basis, the dynamic characteristic of dam-break flood waves is studied in the ideal conditions and the 2-D simple computational regions. Secondly, the CE/SE method is modified and a new constructing method on the 2-D unstructured mixed meshes is developed by author. The new modified CE/SE method is used to solve the hyperbolic conservation equations. The reconstructing method not only maintains all the features, but also is easy to study the problems on the complicated computational regions. Thirdly, the shallow water equations are discretized by the new modified CE/SE method. Two-dimensional numerical model on the 2-D unstructured mixed meshes are constructed. On this basis, the dynamic characteristic of dam-break flood waves is studied on the complicated computational regions. Fourthly, the 2-D axisymmetrical Euler equations are discretized by the new modified CE/SE method and numerical model on the 2-D unstructured mixed meshes are constructed to study the generation and the evolvement of the negative pressures in spherical piezoelectric emitters ESWL. Last, using Chisnell-Chester-Whitham (CCW) geometric shock wave dynamic method, the evolution and focusing of the fronts of underwater shock wave in ESWL are simulated numerically.
All codes of the numerical models are written by author with FORTRAN90 to implement function. It is successful that the problem of dam-break flood wave and the problem of the focusing of underwater shock wave in ESWL are simulated by using the CE/SE method.
The computational results of dam-break flood wave’s examples show that many advantages of the CE/SE method such as simple, robust, high efficiency, high accuracy and high resolution of dam-break. It shows that the proposed model can satisfactorily describe the whole process and flow characteristics of the dam-break wave and possesses high computation accuracy. It is proved that the CE/SE method is a new and high accuracy numerical method for studying dam-break flood wave.
In study of the ESWL, a robust model for the solution of focusing of underwater shock wave in ESWL has been described. Using the modified space-time Conservation Element and Solution Element method on unstructured meshes, the two-dimensional compressible inviscid Euler equations with Tait’s equation of state for water are solved. The pressure field of underwater shock wave in spherical piezoelectric emitters ESWL is simulated successfully in the first. The computational results show that the negative pressure is generated because of the diffraction in the edge of ESWL. The pressure near of the focus of ESWL is the negative pressure after the positive pressure, which shows that the cavitations are inevitable for the focusing of underwater shock wave in spherical piezoelectric emitters ESWL. The cavitations must be considered in clinical and designing ESWL. Secondly, another model for the track of the front of the underwater shock wave has been described, using CCW geometric shock wave dynamic method, the evolution and focusing of the fronts of underwater shock wave in ESWL are simulated numerically. The computational results show that the dynamic focus in spherical piezoelectric emitters ESWL is different from geometric focus in general. The correct location of the factual focus plays very important roles in clinical ESWL.
引文
[1] G.B.Whitham 编.庄峰青,岳曾元译.线性与非线性波[M].北京:科学出版社,1986
[2] 李德元,徐国荣,水鸿寿等.二维非定常流体力学数值方法.北京:科学出版社,1998
[3] R.Courant, K.O.Friedrichs 编.李维新等译.超声速流与冲击波[M].北京:科学出版社,1986
[4] M.J.Zucrow,J.D.Hoffman 编.王汝涌等译.气体动力学[M].北京:国防工业出版社,1984
[5] 谭维炎.计算浅水动力学——有限体积法的应用[M].北京:清华大学出版社,1998
[6] 王保国,刘淑艳,黄伟光.气体动力学[M].北京:北京理工大学出版社,2005
[7] 刘儒勋,舒其望.计算流体力学的若干新方法[M].北京:科学出版社,2003
[8] 陶文铨.计算传热学的近代进展[M].北京:科学出版社,2000
[9] 陶文铨.数值传热学[M].第 2 版.西安:西安交通大学出版社,2001
[10] 吴望一.流体力学[M].北京:北京大学出版社,2000
[11] 张兆顺,崔桂香.流体力学[M].北京:清华大学出版社,1999
[12] 周光炯,严宗毅等.流体力学[M].北京:高等教育出版社,2003
[13] 汪德爟.计算水力学理论与应用[M].南京:河海大学出版社,1989
[14] 陈景秋,王宗立.多维双曲波问题的双特征方法[M].重庆:重庆大学出版社,2001
[15] Chen Jingqiu. Ein Bicharakteristikenverfahren zur Berechnung der Druckwellenfocusie- rung in kompressiblen Fluiden[D]. Dissertation der RWTH Aachen. Germany. 1988:29-63
[16] 孙西钊.医用冲击波[M].北京:中国科学技术出版社,2006
[17] 陈景秋,王宗笠.冲击波聚焦粉碎人体结石过程中的空化现象的数值模拟[J]. 中国生物医学工程学报,2001, 20(1):53-55
[18] 陈景秋,赵万星,曾忠. ESWL 实际焦点位置的理论和数值分析[J]. 中国生物医学工程学报, 2004,23(3):247-251
[19] 张永祥,陈景秋,韦春霞等.球面压电陶瓷型 ESWL 的冲击波聚焦点对球心的偏移[J].中国生物医学工程学报,2007, 26(2):177-181
[20] Z.JIANG, K. TAKAYAMA. Reflection and focusing of toroidal shock waves from coaxial annular shock tubes[J]. computer & fluids, 1998,27:553-562
[21] HWAN-SEOK CHOI, JE-HYUN BAEK. Computations of nonlinear wave interaction in shock-wave focusing process using finite volume TVD schemes[J]. computer & fluids, 1996,25:509-525
[22] 谭炜.水激波聚焦数值计算[D].绵阳:中国空气动力学研究与发展中心,1991
[23] 韩见知,吴开俊. 体外冲击波碎石技术[M]. 人民卫生出版社,2004
[24] Cole R.H. Underwater Explosions[M]. Dover Publication Inc.1965.
[25] T G Liu, B C Khoo, W F Xie. Isentropic one-fluid modeling of unsteady cavitating flow[J]. Journal of Computational Physics, 2004,201:80-108
[26] W.F. Xie, T.G. Liu, B.C. Khoo. Application of a one-fluid model for large scale homogeneous unsteady cavitation: The modified Schmidt model[J]. Computers & Fluids, 2006, 35:1177–1192
[27] Michel Tanguay. Computation of bubbly cavitating flow in shock wave lithotripsy[D]. California: California Institute of Technology, 2004
[28] Holl R. Wellenfokussierung in Fluiden [D]. Dissertation der RWTH Aachen.Germany: bewahrt in der Bibliothek der RWTH Aachen. Germany , 1982
[29] 王如云,方国洪.溃坝洪水长波与地面障碍物作用的数值模拟[J]. 计算物理,1997,14:556-557
[30] C. Zoppou, S. Roberts. Numerical solution of the two-dimensional unsteady dam break[J]. Applied Math Modeling, 2000, 24:457-475
[31] 谢任之.溃坝水力学[M].济南:山东出版社,1992
[32] 吴小川.溃坝过程及洪水波演进数值模拟研究[D].南京:南京水利科学研究院,2004
[33] 韦春霞.溃坝二维问题的数值模拟——利用算子分裂的特征线法[D].重庆:重庆大学,2002
[34] Ritter. Die Forpflanzung der Wasserwellen ver[J]. Deutsch Ingenieure Zeitschr, 1892, 33(2):947-954
[35] Stoker JJ. Water waves: Mathematical theory with applications[M]. Singapore: Wiley-Interscience; 1958.
[36] 谢任之. 溃坝坝址流量计算[J]. 水利水运科学研究,1982,1:1-5
[37] Waterways Experiment Station (WES). Flood resulting from suddenly breached dams, Miscellaneous paper, 2 (374), Report 1. US Army Engineer Waterways Experiment Station, Corps of Engineers, Vicksburg, Mississippi, 1960.
[38] Chih-Tsung Hsu,Keh-chia Yeh. Iterative explicit simulation of 1D surges and dam-break flows[J]. Int. J. Numer. Meth.Fluids, 2002, 38:647-675
[39] Sky Miller. M.Hanif Chauhdry. Dam-break flows in curved channel[J]. Journal of Hydraulic Engineering, 1989,115(1):1465-1478
[40] S.Soares Fraz?o, Y.Zech. Dam break in channels with 90 degree bend[J]. Journal of Hydraulic Engineering, 2002,128(11):956-968
[41] S. Soares Fraz?o, X.Sillen, Y. Zech. Dam-break Flow through Sharp Bends Physical Modeland 2D Boltzmann Model Validation[J]. Proceedings of the CADAM meeting. HR Wallingford, U.K., 2–3 March, 1998
[42] Katopodes N, Strelkoff T. Computing two-dimensional dam-break flood waves[J]. Journal of Hydraulic Engineering-ASCE, 1978, 104(9):1269–88
[43] Katopodes N, Strelkoff T. Two-dimensional shallow water-wave models[J]. J Mechanic Eng ASCE, 1979, 105(2):317–34
[44] 韦春霞,张永祥,陈景秋.溃坝洪水的二维算子分裂——特征线模拟[J].重庆大学学报,2003, 23(9):18-21.
[45] 陈景秋,张永祥,韦春霞.一维溃坝涌波的特征线——激波装配法[J].重庆大学学报,2004, 24(5):99-102
[46] Cunge JA. Rapidly Varied Flow in Power and Pumping Canals in Unsteady Flow in Open Channel[M]. Water Resources Publications: Fort Collins, CO, 1975
[47] Yang JC, Chen KN, Lee HY. An accurate computation for rapidly varied flow in an open channel[J]. International Journal for Numerical Methods in Fluids, 1992,14:361- 374
[48] Garcia P, Kahawita RA. Numerical solution of the St. Venant equations with the MacCormack finite-difference scheme[J]. Int. J. Numer. Meth. Fluids, 1986,6: 259-274
[49] 魏文礼,沈永明. 二维溃坝洪水波演进的数值模拟[J]. 水利学报,2003,9:43-47
[50] Fennema RJ, Chaudhry MH. Explicit numerical schemes for unsteady free-surface flows with shocks. Water Resources Research 1986; 22(13):1923–1930
[51] Jha AK, Akiyama J, Ura M. Modeling unsteady open-channel flows-modification to Beam and Warming scheme[J]. J. Hydr. Engrg-ASCE, 1994, 120(4): 461-476
[52] Jha AK, Akiyama J, Ura M. A fully conservative Beam and Warming scheme for transient open channel flows[J]. J. Hydr. Res., 1996, 34(5):605-621
[53] Jha AK, Akiyama J, Ura M. An implicit model based on conservative flux splitting technique for one dimensional unsteady flow[J]. Journal of Hydroscience Hydraulics Engineering-ASCE, 1994, 11(2):69–82
[54] Alcrudo F, Garcia-Vavarro P, Saviron JM. Flux difference splitting 1D open channel flow equations[J]. International Journal for Numerical Methods in Fluids, 1992, 14:1009–1018
[55] Glaister P. Flux difference splitting for open-channel flows[J]. International Journal for Numerical Methods in Fluids, 1993, 16:629–654
[56] Gottardi G, Venutelli M. Central scheme for two-dimensional dam-break flow simulation[J]. Advances in Water Resources, 2004,27:259-268
[57] Louaked M, Hanich L. TVD schemes for the shallow water equations[J]. J. Hydr. Res., 1998, 36(3), 363-378
[58] Wang JS, Ni HG, He YS. Finite-difference TVD scheme for computation of dam-break problems[J]. J. Hydr. Engrg-ASCE. 2000,126(4), 253-262
[59] 胡四一,谭维炎. 用 TVD 格式预测溃坝洪水波的演进[J]. 水利学报,1989(7):1-11
[60] 王嘉松, 倪汉根. 用 TVD 显隐格式摸拟一维溃坝洪水波的演进与反射[J]. 水利学报, 1998(5):7-11
[61] 王嘉松,倪汉根. 二维溃坝波传播和绕流特性的高精度数值模拟[J]. 水利学报,1998(10):1-6
[62] 王嘉松,倪汉根. 二维溃坝问题的高分辨率数值模拟[J]. 上海交通大学学报,1999(10):1213-1216
[63] 王嘉松,倪汉根. 瞬间全溃溃坝波的传播、反射和绕射的数值模拟[J]. 水动力学研究与进展,2000, 15(3):1-7
[64] 潘存鸿,林炳尧. 一维浅水流动方程的 Godunov 格式求解[J]. 水科学进展,2003, 14(4):430-436
[65] 王志峰,李宏. 基于模板选择方法的 Godunov 型差分格式[J]. 中国科学技术大学学报,2001, 31(8):429-435
[66] Savic L, Holly FM. Dambreak flood waves computed by modifed Godunov method[J]. Journal of Hydraulics Research, 1993,31(2):187-204
[67] Garcia-Navarro P, Priestley A. A conservative and shape-preserving semi-Lagrangian method for the solution of the shallow water equations[J]. International Journal for Numerical Methods in Fluids, 1994, 18:273-294
[68] Moin SMA, Lam DCL, Smith AA. Eularian–Lagrangian linked algorithm for simulating discontinuous open channel flows[R]. Proceedings of the VII International Conference on Computer Methods in Water Resources, MIT, USA, 1988, 2:363–368
[69] 张修忠,王光谦. 浅水流动有限元分析及其高分辨率格式[J]. 长江科学院院报,2001, (2):13-15
[70] 汪继文,刘儒勋. 间断解问题的有限体积法[J]. 计算物理,2001, (3):97-105
[71] Hicks FE, SteVer PM. Characteristic dissipative Galerkin scheme for open-channel flow[J]. Journal of Hydraulics Engineering-ASCE, 1992, 118(2):337-352
[72] Hicks FE, SteVer PM, Yasmin N. One-dimensional dam-break solutions for variable width channels[J]. Journal of Hydraulics Engineering-ASCE, 1997, 123(5):464-468
[73] Katopodes ND. A dissipative Galerkin scheme for open-channel flow[J]. Journal of Hydraulics Engineering-ASCE, 1984, 110(4):450-466
[74] Berger RC, Stockstill RL. Finite-element model for high-velocity channels[J]. Journal of Hydraulics Engineering-ASCE, 1995, 121(10):710-716
[75] Katopodes ND, Wu CT. Explicit computation of discontinuous channel flow[J]. Journal of Hydraulics Engineering-ASCE, 1986, 112(6):456-475
[76] Alcrudo F, Garcia NP. A high resolution Godunov type scheme in finite volumes for the 2D shallow water equations[J]. Int. J. Numer. Meth. Fluids., 1993, 16: 489-505
[77] Rao VS, Latha G. A slope modification method for shallow water equations[J]. Int J Numer Meth Fluids, 1992, 14:189-96
[78] Nujic M. Efficient implementation of non-oscillatory schemes for the computation of free-surface flows[J]. J Hydr Res, 1995,3:101-111
[79] Jiasong Wang, Yousheng He, Hangen, Ni. Two-dimensional free surface flow in branch channels by a finite-volum TVD scheme[J]. Advances in Water Resource, 2003,26:623-633
[80] Gwo-Fong Lin, Jihn-Sung Lai, Wen-Dar Guo. Finite-volume component-wise TVD schemes for 2D shallow water equations[J]. Advances in Water Resource, 2003,26:861-873
[81] Ji-Wen Wang, Ru-Xun Liu. A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problems[J]. Mathematics and Computers in Simulation, 2000, 53:171-184
[82] Louaked M, Hanich L. TVD scheme for the shallow water equations[J]. J Hydraulic Res, 1998, 36(3):363-378
[83] Van Leer B. Towards the ultimate conservative difference scheme IV: a new approach to numerical convection[J]. Journal of Computational Physics, 1977, 23:276
[84] Van Leer B. Towards the ultimate conservative difference scheme V: a second order sequel to Godunov's method[J]. Journal of Computational Physics, 1979, 32:101-111
[85] DH Zhao, HW Shen et al. Finite-volume two-dimensional unsteady flow model for river basins[J]. Journal of Hydraulics Engineering-ASCE, 1986, 112(6): 456-475
[86] Anastasiou K, Chan CT. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes[J]. International Journal for Numerical Methods in Fluids, 1997,24:1225 –1245
[87] 李玉兰,陈景秋. 现代医学手段诱发空蚀的形成及其影响[J]. 重庆大学学报, 2002,25(12):114-118
[88] Chaussy C., Brendel W., Schmiedt E. Extracorporesally induced destruction of kidney stones by shock waves[J]. Lancet 1980,8207,2:1265-1268
[89] Chaussy C., and Fuchs G.J., Current state and future developments of noninvasive treatment of human urinary stones with extracorporeal shock wave lithotripsy[J]. J.Urol.,1989,141:782-792
[90] Chaussy C., Extracorporeal Shock Wave Lithotripsy: New Aspect in the Treatment of KidneyStone Disease[M]. 1982
[91] Gavestein D. Extracorporeal shock wave lithotripsy and percutaneous nephrolithotomy[J]. Anesthesiol Clin North America 2000,18(4):953-971
[92] Delius M. Lithotripsy[J]. Ultrasound Med Biol 2000,26 Suppl 1:S55-8
[93] Delius M., Medical applications and bioeffects of extracorporeal shock waves[J]. Shock Waves, 1994,4:55-72
[94] Delius M., Enders G.,Xuan Z., Liebich H., and Brendel W. Biological effects of shock waves: Kindey damage by shock waves in dogs-Dose dependence[J]. Ultrasound in Med. & Biol., 1988,14:117-122
[95] Blomgren P., Connors B.A., Lingeman J.E., Willis L.R., and Evan A.P., Quantitation of shock wave lithotripsy-induced lesion in small and large pig kidneys[J]. Aatomical Rec., 1997,249:341-348
[96] Evan A., Willis L.R., Lingeman J.E. and McAteer J.A., Renal trauma and the risk of long-term complications in shock wave lithotripsy[J]. Nephron, 1998,78:1-8
[97] Teichman J.M.H., Portis A.J., Cecconi P.P., Bub W.L., Endicott R.C., Denes B., Pearle M.S. and Clayman R.V., In vitro comparison of shock wave lithotripsy machines[J]. J. Urol., 2000,164:1259-1264
[98] Mueller M, Stosswellenfokusierung in Wasser[D], Dissertation der RWTH Aachen. Germany. 1988
[99] Bailey M.R., Blackstock D.T., Cleveland R.O., and Crum L.A., Comparison of electrohydraulic lithotripters with rigid and pressure-release ellipsoidal reflectors[J]. II. Acoustic fields, J. Acoust. Soc. Am., 1999,104:2517-2524
[100] 中华人民共和国行业标准 YY0001-90 体外冲击波碎石机通用技术条件.国家医药管理局发布.1990:1-2
[101] Wolfgang Eisenmenger. The mechanisms of stone fragmentation in ESWL[J]. Ultrasound in Medicine & Biology, 2001,27(5): 683-693
[102] Lokhandwall M, Sturtevant B. Fracture mechanics model of stone comminution in ESWL and implications for tissue damage[J]. Phys Med Biol, 2000,45:1923-1940
[103] Andreas Skolarikos, Gerasimos Alivizatos, Jean de la Rosette. Extracorpeal shock wave lithotripsy 25 year later: Complications and their prevention[J]. European Urology, 2006,1207:1-10
[104] Christopher E.Brennen. Cavitation and Bubble Dynamics[M].Oxford University Press,1995:18-25
[105] Jens J.Rassweiler, Geert G.Tailly, Christian Chaussy. Progress in Lithotriptor Technology[J].EAU Update Series, 2005,3: 17-36
[106] Songlin Zhu, Franklin H.Cocks. The role of stress waves and cavitation in stone comminution in shock wave lithotripsy[J]. Ultrasound in Med.& Biol., 2002, 28(5):661-671
[107] Bailey M.R., Blackstock D.T., Cleveland R.O., and Crum L.A., Comparison of electrohydraulic lithotripters with rigid and pressure-release ellipsoidal reflectors. II. Acoustic fields[J]. J. Acoust. Soc. Am., 1999,104:2517-2524
[108] Xi X, Zhong P. Dynamic photoelastic study of transient stress field in solids during shock wave lithotripsy[J]. J Acoust Soc Am 2001;109(3):1226-39
[109] Sapozhnikov OA, Khoklova VA, Bailey MR, et al. Effect of overpressure and pulse repetition frequency on cavitation in shock wave lithotripsy[J]. J Acoust Soc Am 2002, 3:45-82
[110] Huber P, Debus J, Jochle K, et al. Control of cavitation activity by different shockwave pulsing regimes[J]. Phys Med and Biol 1999; 44(6): 1427-37
[111] Sturtevant B, Kulkarny V A. The focusing of weak shock waves[J]. Journal of Fluid mechanics, 1976,73(4):651-671
[112] Sommerfeld M, Mueller M. Experimental and numerical studies of shock wave focusing in water [J]. Experiments in Fluids, 1988,6:209-216
[113] Hamilton, M. Transient axial solution for the reflection of a spherical wave from a concave ellipsoidal mirror[J]. Journal of the Acoustical Society of America, 1993, 93(3):1256-1266
[114] Christopher T. Modeling the Dornier HM3 lithotripter[J]. Journal of the Acoustical Society of America, 1994,96(5):3088-3095
[115] Coleman A M, Choi, J Saunders. Theoretical predictions of the acoustic pressure generated by a shock wave lithotripter[J]. Ultrasound in Medicine and Biology, 1991,17:245-255
[116] Averkiou M, R Cleveland. Moldeling of an electrohydraulic lithotripter with the KZK equation[J]. Journal of the Acoustical Society of America, 1999, 106(1):102-112
[117] Cates J. B Sturtevant. Shock wave focusing using geometrical shock dynamics[J]. Physics of Fluids, 1997,9(10), 3058-3068
[118] Sokolov DL, Bailey MR, Crum LA. Use of a dual-pulse lithotripter to generate a localized and intensified cavitation field[J]. J Acoust Soc Am, 2001,110: 1685-95
[119] Zhong P., Cioanta I., Cocks F.H., Preminger G.M. Inertial cavitation and associated acoustic emission produced during electrohydraulic shock wave lithotripsy[J]. J Acoust. Soc. Am., 1997,2940-2950
[120] A J Coleman. M J Choi, J E Sauders . and T G Leighton. Acoustic emission and sonoluminescence due to cavitation at the beam focus of an electrohydraulic shock wave lithotripter[J]. Ultrasound Med. Biol., 1992,18:267-281
[121] S.C.Chang. The method of space-time conservation element and solution element-a new approach for solving the Navier-Stokes and Euler equations[J]. J Comput Phys 1995; 119:295-324
[122] ZC Zhang, ST John Yu, SC Chang. A space-time conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes[J]. J Comput Phys 2002; 175:168-199
[123] XY Wang, SC Chang. A 2D non-splitting unstructured triangular mesh Euler solver based on the space-time conservation element and solution element metho[J]d, Comput. Fluid Dyn. J 1999; 8:309-315
[124] S Jerez, JV Romero. A Semi-Implicit space-time CE-SE method to improve mass conservation through tapered ducts in internal combustion engines[J]. Mathematical and computer modeling 2004; 40:941-951
[125] Y Guo, AT Hsu, J Wu. Extension of CE/SE method to 2D viscous flows[J]. computers and fluids 2004; 33:1349-1361
[126] XY Wang, SC Chang. Accuracy Study of the Space-Time CE/SE Method for Computational Aeroacoustics Problems Involving Shock Waves[J]. AIAA, 2000,0474:1-4
[127] YH.Guo, Andrew T. Hsu. Extension of CE/SE method to 2D viscous flows[J]. Computers and Fliods, 2004, 33:1349-1361
[128] Zeng-chan Zhang, S.T. John Yu. Direct calculations of two and three- dimensional detonations by an extended CE/SE method[J]. AIAA, 2001-0476
[129] S.C. Chang, X.Y.Wang, W.M. To. Application of the space-time conservation element and solution element method to one-dimensional convection-diffusion problems[J]. J. Com. Phy. 2000, 165:189-215
[130] Ching Y. Loh, Cleveland OH. Noise computation of a shock-containing supersonic axisymmetric jet by the CE/SE method[J]. AIAA, 2000-0475
[131] Ching Y. Loh, Lennart S. Hultgren. Near field screech noise computation for an underexpanded supersonic jet by the CE/SE method[J]. AIAA, 2001-2252
[132] Philip C.E. Jorgenson, Ching Y.Loh. computing axsymmetric jet screech tones using unstructured grids[J]. AIAA, 2000-3889
[133] 张增产,沈孟育.改进德时空守恒元和解元方法[J].清华大学学报,1997,37(8):65-68
[134] 张增产,沈孟育.用时空守恒方法求带源项及刚性源项德守恒律方程[J].清华大学学报,1998,38(11):87-90
[135] 张增产,沈孟育.一种严格保证时-空守恒律德数值方法[J].计算物理,1997,14(6):835-840
[136] 张增产,沈孟育.求解二维 Euler 方程的时-空守恒格式[J].力学学报,1999,31(2):152-158
[137] 张增产,沈孟育.一般坐标系下的二维 Euler 方程时-空守恒格式[J].应用力学学报,1999,16(1):10-14
[138] 张增产.一般形式的二维时-空守恒格式[J].计算力学学报,1997,14(4):377-381
[139] 张增产.二维时-空守恒格式及其在激波传播计算中的应用.爆炸与冲击,1999,19(1):7-12
[2] 李德元,徐国荣,水鸿寿等.二维非定常流体力学数值方法.北京:科学出版社,1998
[3] R.Courant, K.O.Friedrichs 编.李维新等译.超声速流与冲击波[M].北京:科学出版社,1986
[4] M.J.Zucrow,J.D.Hoffman 编.王汝涌等译.气体动力学[M].北京:国防工业出版社,1984
[5] 谭维炎.计算浅水动力学——有限体积法的应用[M].北京:清华大学出版社,1998
[6] 王保国,刘淑艳,黄伟光.气体动力学[M].北京:北京理工大学出版社,2005
[7] 刘儒勋,舒其望.计算流体力学的若干新方法[M].北京:科学出版社,2003
[8] 陶文铨.计算传热学的近代进展[M].北京:科学出版社,2000
[9] 陶文铨.数值传热学[M].第 2 版.西安:西安交通大学出版社,2001
[10] 吴望一.流体力学[M].北京:北京大学出版社,2000
[11] 张兆顺,崔桂香.流体力学[M].北京:清华大学出版社,1999
[12] 周光炯,严宗毅等.流体力学[M].北京:高等教育出版社,2003
[13] 汪德爟.计算水力学理论与应用[M].南京:河海大学出版社,1989
[14] 陈景秋,王宗立.多维双曲波问题的双特征方法[M].重庆:重庆大学出版社,2001
[15] Chen Jingqiu. Ein Bicharakteristikenverfahren zur Berechnung der Druckwellenfocusie- rung in kompressiblen Fluiden[D]. Dissertation der RWTH Aachen. Germany. 1988:29-63
[16] 孙西钊.医用冲击波[M].北京:中国科学技术出版社,2006
[17] 陈景秋,王宗笠.冲击波聚焦粉碎人体结石过程中的空化现象的数值模拟[J]. 中国生物医学工程学报,2001, 20(1):53-55
[18] 陈景秋,赵万星,曾忠. ESWL 实际焦点位置的理论和数值分析[J]. 中国生物医学工程学报, 2004,23(3):247-251
[19] 张永祥,陈景秋,韦春霞等.球面压电陶瓷型 ESWL 的冲击波聚焦点对球心的偏移[J].中国生物医学工程学报,2007, 26(2):177-181
[20] Z.JIANG, K. TAKAYAMA. Reflection and focusing of toroidal shock waves from coaxial annular shock tubes[J]. computer & fluids, 1998,27:553-562
[21] HWAN-SEOK CHOI, JE-HYUN BAEK. Computations of nonlinear wave interaction in shock-wave focusing process using finite volume TVD schemes[J]. computer & fluids, 1996,25:509-525
[22] 谭炜.水激波聚焦数值计算[D].绵阳:中国空气动力学研究与发展中心,1991
[23] 韩见知,吴开俊. 体外冲击波碎石技术[M]. 人民卫生出版社,2004
[24] Cole R.H. Underwater Explosions[M]. Dover Publication Inc.1965.
[25] T G Liu, B C Khoo, W F Xie. Isentropic one-fluid modeling of unsteady cavitating flow[J]. Journal of Computational Physics, 2004,201:80-108
[26] W.F. Xie, T.G. Liu, B.C. Khoo. Application of a one-fluid model for large scale homogeneous unsteady cavitation: The modified Schmidt model[J]. Computers & Fluids, 2006, 35:1177–1192
[27] Michel Tanguay. Computation of bubbly cavitating flow in shock wave lithotripsy[D]. California: California Institute of Technology, 2004
[28] Holl R. Wellenfokussierung in Fluiden [D]. Dissertation der RWTH Aachen.Germany: bewahrt in der Bibliothek der RWTH Aachen. Germany , 1982
[29] 王如云,方国洪.溃坝洪水长波与地面障碍物作用的数值模拟[J]. 计算物理,1997,14:556-557
[30] C. Zoppou, S. Roberts. Numerical solution of the two-dimensional unsteady dam break[J]. Applied Math Modeling, 2000, 24:457-475
[31] 谢任之.溃坝水力学[M].济南:山东出版社,1992
[32] 吴小川.溃坝过程及洪水波演进数值模拟研究[D].南京:南京水利科学研究院,2004
[33] 韦春霞.溃坝二维问题的数值模拟——利用算子分裂的特征线法[D].重庆:重庆大学,2002
[34] Ritter. Die Forpflanzung der Wasserwellen ver[J]. Deutsch Ingenieure Zeitschr, 1892, 33(2):947-954
[35] Stoker JJ. Water waves: Mathematical theory with applications[M]. Singapore: Wiley-Interscience; 1958.
[36] 谢任之. 溃坝坝址流量计算[J]. 水利水运科学研究,1982,1:1-5
[37] Waterways Experiment Station (WES). Flood resulting from suddenly breached dams, Miscellaneous paper, 2 (374), Report 1. US Army Engineer Waterways Experiment Station, Corps of Engineers, Vicksburg, Mississippi, 1960.
[38] Chih-Tsung Hsu,Keh-chia Yeh. Iterative explicit simulation of 1D surges and dam-break flows[J]. Int. J. Numer. Meth.Fluids, 2002, 38:647-675
[39] Sky Miller. M.Hanif Chauhdry. Dam-break flows in curved channel[J]. Journal of Hydraulic Engineering, 1989,115(1):1465-1478
[40] S.Soares Fraz?o, Y.Zech. Dam break in channels with 90 degree bend[J]. Journal of Hydraulic Engineering, 2002,128(11):956-968
[41] S. Soares Fraz?o, X.Sillen, Y. Zech. Dam-break Flow through Sharp Bends Physical Modeland 2D Boltzmann Model Validation[J]. Proceedings of the CADAM meeting. HR Wallingford, U.K., 2–3 March, 1998
[42] Katopodes N, Strelkoff T. Computing two-dimensional dam-break flood waves[J]. Journal of Hydraulic Engineering-ASCE, 1978, 104(9):1269–88
[43] Katopodes N, Strelkoff T. Two-dimensional shallow water-wave models[J]. J Mechanic Eng ASCE, 1979, 105(2):317–34
[44] 韦春霞,张永祥,陈景秋.溃坝洪水的二维算子分裂——特征线模拟[J].重庆大学学报,2003, 23(9):18-21.
[45] 陈景秋,张永祥,韦春霞.一维溃坝涌波的特征线——激波装配法[J].重庆大学学报,2004, 24(5):99-102
[46] Cunge JA. Rapidly Varied Flow in Power and Pumping Canals in Unsteady Flow in Open Channel[M]. Water Resources Publications: Fort Collins, CO, 1975
[47] Yang JC, Chen KN, Lee HY. An accurate computation for rapidly varied flow in an open channel[J]. International Journal for Numerical Methods in Fluids, 1992,14:361- 374
[48] Garcia P, Kahawita RA. Numerical solution of the St. Venant equations with the MacCormack finite-difference scheme[J]. Int. J. Numer. Meth. Fluids, 1986,6: 259-274
[49] 魏文礼,沈永明. 二维溃坝洪水波演进的数值模拟[J]. 水利学报,2003,9:43-47
[50] Fennema RJ, Chaudhry MH. Explicit numerical schemes for unsteady free-surface flows with shocks. Water Resources Research 1986; 22(13):1923–1930
[51] Jha AK, Akiyama J, Ura M. Modeling unsteady open-channel flows-modification to Beam and Warming scheme[J]. J. Hydr. Engrg-ASCE, 1994, 120(4): 461-476
[52] Jha AK, Akiyama J, Ura M. A fully conservative Beam and Warming scheme for transient open channel flows[J]. J. Hydr. Res., 1996, 34(5):605-621
[53] Jha AK, Akiyama J, Ura M. An implicit model based on conservative flux splitting technique for one dimensional unsteady flow[J]. Journal of Hydroscience Hydraulics Engineering-ASCE, 1994, 11(2):69–82
[54] Alcrudo F, Garcia-Vavarro P, Saviron JM. Flux difference splitting 1D open channel flow equations[J]. International Journal for Numerical Methods in Fluids, 1992, 14:1009–1018
[55] Glaister P. Flux difference splitting for open-channel flows[J]. International Journal for Numerical Methods in Fluids, 1993, 16:629–654
[56] Gottardi G, Venutelli M. Central scheme for two-dimensional dam-break flow simulation[J]. Advances in Water Resources, 2004,27:259-268
[57] Louaked M, Hanich L. TVD schemes for the shallow water equations[J]. J. Hydr. Res., 1998, 36(3), 363-378
[58] Wang JS, Ni HG, He YS. Finite-difference TVD scheme for computation of dam-break problems[J]. J. Hydr. Engrg-ASCE. 2000,126(4), 253-262
[59] 胡四一,谭维炎. 用 TVD 格式预测溃坝洪水波的演进[J]. 水利学报,1989(7):1-11
[60] 王嘉松, 倪汉根. 用 TVD 显隐格式摸拟一维溃坝洪水波的演进与反射[J]. 水利学报, 1998(5):7-11
[61] 王嘉松,倪汉根. 二维溃坝波传播和绕流特性的高精度数值模拟[J]. 水利学报,1998(10):1-6
[62] 王嘉松,倪汉根. 二维溃坝问题的高分辨率数值模拟[J]. 上海交通大学学报,1999(10):1213-1216
[63] 王嘉松,倪汉根. 瞬间全溃溃坝波的传播、反射和绕射的数值模拟[J]. 水动力学研究与进展,2000, 15(3):1-7
[64] 潘存鸿,林炳尧. 一维浅水流动方程的 Godunov 格式求解[J]. 水科学进展,2003, 14(4):430-436
[65] 王志峰,李宏. 基于模板选择方法的 Godunov 型差分格式[J]. 中国科学技术大学学报,2001, 31(8):429-435
[66] Savic L, Holly FM. Dambreak flood waves computed by modifed Godunov method[J]. Journal of Hydraulics Research, 1993,31(2):187-204
[67] Garcia-Navarro P, Priestley A. A conservative and shape-preserving semi-Lagrangian method for the solution of the shallow water equations[J]. International Journal for Numerical Methods in Fluids, 1994, 18:273-294
[68] Moin SMA, Lam DCL, Smith AA. Eularian–Lagrangian linked algorithm for simulating discontinuous open channel flows[R]. Proceedings of the VII International Conference on Computer Methods in Water Resources, MIT, USA, 1988, 2:363–368
[69] 张修忠,王光谦. 浅水流动有限元分析及其高分辨率格式[J]. 长江科学院院报,2001, (2):13-15
[70] 汪继文,刘儒勋. 间断解问题的有限体积法[J]. 计算物理,2001, (3):97-105
[71] Hicks FE, SteVer PM. Characteristic dissipative Galerkin scheme for open-channel flow[J]. Journal of Hydraulics Engineering-ASCE, 1992, 118(2):337-352
[72] Hicks FE, SteVer PM, Yasmin N. One-dimensional dam-break solutions for variable width channels[J]. Journal of Hydraulics Engineering-ASCE, 1997, 123(5):464-468
[73] Katopodes ND. A dissipative Galerkin scheme for open-channel flow[J]. Journal of Hydraulics Engineering-ASCE, 1984, 110(4):450-466
[74] Berger RC, Stockstill RL. Finite-element model for high-velocity channels[J]. Journal of Hydraulics Engineering-ASCE, 1995, 121(10):710-716
[75] Katopodes ND, Wu CT. Explicit computation of discontinuous channel flow[J]. Journal of Hydraulics Engineering-ASCE, 1986, 112(6):456-475
[76] Alcrudo F, Garcia NP. A high resolution Godunov type scheme in finite volumes for the 2D shallow water equations[J]. Int. J. Numer. Meth. Fluids., 1993, 16: 489-505
[77] Rao VS, Latha G. A slope modification method for shallow water equations[J]. Int J Numer Meth Fluids, 1992, 14:189-96
[78] Nujic M. Efficient implementation of non-oscillatory schemes for the computation of free-surface flows[J]. J Hydr Res, 1995,3:101-111
[79] Jiasong Wang, Yousheng He, Hangen, Ni. Two-dimensional free surface flow in branch channels by a finite-volum TVD scheme[J]. Advances in Water Resource, 2003,26:623-633
[80] Gwo-Fong Lin, Jihn-Sung Lai, Wen-Dar Guo. Finite-volume component-wise TVD schemes for 2D shallow water equations[J]. Advances in Water Resource, 2003,26:861-873
[81] Ji-Wen Wang, Ru-Xun Liu. A comparative study of finite volume methods on unstructured meshes for simulation of 2D shallow water wave problems[J]. Mathematics and Computers in Simulation, 2000, 53:171-184
[82] Louaked M, Hanich L. TVD scheme for the shallow water equations[J]. J Hydraulic Res, 1998, 36(3):363-378
[83] Van Leer B. Towards the ultimate conservative difference scheme IV: a new approach to numerical convection[J]. Journal of Computational Physics, 1977, 23:276
[84] Van Leer B. Towards the ultimate conservative difference scheme V: a second order sequel to Godunov's method[J]. Journal of Computational Physics, 1979, 32:101-111
[85] DH Zhao, HW Shen et al. Finite-volume two-dimensional unsteady flow model for river basins[J]. Journal of Hydraulics Engineering-ASCE, 1986, 112(6): 456-475
[86] Anastasiou K, Chan CT. Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes[J]. International Journal for Numerical Methods in Fluids, 1997,24:1225 –1245
[87] 李玉兰,陈景秋. 现代医学手段诱发空蚀的形成及其影响[J]. 重庆大学学报, 2002,25(12):114-118
[88] Chaussy C., Brendel W., Schmiedt E. Extracorporesally induced destruction of kidney stones by shock waves[J]. Lancet 1980,8207,2:1265-1268
[89] Chaussy C., and Fuchs G.J., Current state and future developments of noninvasive treatment of human urinary stones with extracorporeal shock wave lithotripsy[J]. J.Urol.,1989,141:782-792
[90] Chaussy C., Extracorporeal Shock Wave Lithotripsy: New Aspect in the Treatment of KidneyStone Disease[M]. 1982
[91] Gavestein D. Extracorporeal shock wave lithotripsy and percutaneous nephrolithotomy[J]. Anesthesiol Clin North America 2000,18(4):953-971
[92] Delius M. Lithotripsy[J]. Ultrasound Med Biol 2000,26 Suppl 1:S55-8
[93] Delius M., Medical applications and bioeffects of extracorporeal shock waves[J]. Shock Waves, 1994,4:55-72
[94] Delius M., Enders G.,Xuan Z., Liebich H., and Brendel W. Biological effects of shock waves: Kindey damage by shock waves in dogs-Dose dependence[J]. Ultrasound in Med. & Biol., 1988,14:117-122
[95] Blomgren P., Connors B.A., Lingeman J.E., Willis L.R., and Evan A.P., Quantitation of shock wave lithotripsy-induced lesion in small and large pig kidneys[J]. Aatomical Rec., 1997,249:341-348
[96] Evan A., Willis L.R., Lingeman J.E. and McAteer J.A., Renal trauma and the risk of long-term complications in shock wave lithotripsy[J]. Nephron, 1998,78:1-8
[97] Teichman J.M.H., Portis A.J., Cecconi P.P., Bub W.L., Endicott R.C., Denes B., Pearle M.S. and Clayman R.V., In vitro comparison of shock wave lithotripsy machines[J]. J. Urol., 2000,164:1259-1264
[98] Mueller M, Stosswellenfokusierung in Wasser[D], Dissertation der RWTH Aachen. Germany. 1988
[99] Bailey M.R., Blackstock D.T., Cleveland R.O., and Crum L.A., Comparison of electrohydraulic lithotripters with rigid and pressure-release ellipsoidal reflectors[J]. II. Acoustic fields, J. Acoust. Soc. Am., 1999,104:2517-2524
[100] 中华人民共和国行业标准 YY0001-90 体外冲击波碎石机通用技术条件.国家医药管理局发布.1990:1-2
[101] Wolfgang Eisenmenger. The mechanisms of stone fragmentation in ESWL[J]. Ultrasound in Medicine & Biology, 2001,27(5): 683-693
[102] Lokhandwall M, Sturtevant B. Fracture mechanics model of stone comminution in ESWL and implications for tissue damage[J]. Phys Med Biol, 2000,45:1923-1940
[103] Andreas Skolarikos, Gerasimos Alivizatos, Jean de la Rosette. Extracorpeal shock wave lithotripsy 25 year later: Complications and their prevention[J]. European Urology, 2006,1207:1-10
[104] Christopher E.Brennen. Cavitation and Bubble Dynamics[M].Oxford University Press,1995:18-25
[105] Jens J.Rassweiler, Geert G.Tailly, Christian Chaussy. Progress in Lithotriptor Technology[J].EAU Update Series, 2005,3: 17-36
[106] Songlin Zhu, Franklin H.Cocks. The role of stress waves and cavitation in stone comminution in shock wave lithotripsy[J]. Ultrasound in Med.& Biol., 2002, 28(5):661-671
[107] Bailey M.R., Blackstock D.T., Cleveland R.O., and Crum L.A., Comparison of electrohydraulic lithotripters with rigid and pressure-release ellipsoidal reflectors. II. Acoustic fields[J]. J. Acoust. Soc. Am., 1999,104:2517-2524
[108] Xi X, Zhong P. Dynamic photoelastic study of transient stress field in solids during shock wave lithotripsy[J]. J Acoust Soc Am 2001;109(3):1226-39
[109] Sapozhnikov OA, Khoklova VA, Bailey MR, et al. Effect of overpressure and pulse repetition frequency on cavitation in shock wave lithotripsy[J]. J Acoust Soc Am 2002, 3:45-82
[110] Huber P, Debus J, Jochle K, et al. Control of cavitation activity by different shockwave pulsing regimes[J]. Phys Med and Biol 1999; 44(6): 1427-37
[111] Sturtevant B, Kulkarny V A. The focusing of weak shock waves[J]. Journal of Fluid mechanics, 1976,73(4):651-671
[112] Sommerfeld M, Mueller M. Experimental and numerical studies of shock wave focusing in water [J]. Experiments in Fluids, 1988,6:209-216
[113] Hamilton, M. Transient axial solution for the reflection of a spherical wave from a concave ellipsoidal mirror[J]. Journal of the Acoustical Society of America, 1993, 93(3):1256-1266
[114] Christopher T. Modeling the Dornier HM3 lithotripter[J]. Journal of the Acoustical Society of America, 1994,96(5):3088-3095
[115] Coleman A M, Choi, J Saunders. Theoretical predictions of the acoustic pressure generated by a shock wave lithotripter[J]. Ultrasound in Medicine and Biology, 1991,17:245-255
[116] Averkiou M, R Cleveland. Moldeling of an electrohydraulic lithotripter with the KZK equation[J]. Journal of the Acoustical Society of America, 1999, 106(1):102-112
[117] Cates J. B Sturtevant. Shock wave focusing using geometrical shock dynamics[J]. Physics of Fluids, 1997,9(10), 3058-3068
[118] Sokolov DL, Bailey MR, Crum LA. Use of a dual-pulse lithotripter to generate a localized and intensified cavitation field[J]. J Acoust Soc Am, 2001,110: 1685-95
[119] Zhong P., Cioanta I., Cocks F.H., Preminger G.M. Inertial cavitation and associated acoustic emission produced during electrohydraulic shock wave lithotripsy[J]. J Acoust. Soc. Am., 1997,2940-2950
[120] A J Coleman. M J Choi, J E Sauders . and T G Leighton. Acoustic emission and sonoluminescence due to cavitation at the beam focus of an electrohydraulic shock wave lithotripter[J]. Ultrasound Med. Biol., 1992,18:267-281
[121] S.C.Chang. The method of space-time conservation element and solution element-a new approach for solving the Navier-Stokes and Euler equations[J]. J Comput Phys 1995; 119:295-324
[122] ZC Zhang, ST John Yu, SC Chang. A space-time conservation element and solution element method for solving the two- and three-dimensional unsteady Euler equations using quadrilateral and hexahedral meshes[J]. J Comput Phys 2002; 175:168-199
[123] XY Wang, SC Chang. A 2D non-splitting unstructured triangular mesh Euler solver based on the space-time conservation element and solution element metho[J]d, Comput. Fluid Dyn. J 1999; 8:309-315
[124] S Jerez, JV Romero. A Semi-Implicit space-time CE-SE method to improve mass conservation through tapered ducts in internal combustion engines[J]. Mathematical and computer modeling 2004; 40:941-951
[125] Y Guo, AT Hsu, J Wu. Extension of CE/SE method to 2D viscous flows[J]. computers and fluids 2004; 33:1349-1361
[126] XY Wang, SC Chang. Accuracy Study of the Space-Time CE/SE Method for Computational Aeroacoustics Problems Involving Shock Waves[J]. AIAA, 2000,0474:1-4
[127] YH.Guo, Andrew T. Hsu. Extension of CE/SE method to 2D viscous flows[J]. Computers and Fliods, 2004, 33:1349-1361
[128] Zeng-chan Zhang, S.T. John Yu. Direct calculations of two and three- dimensional detonations by an extended CE/SE method[J]. AIAA, 2001-0476
[129] S.C. Chang, X.Y.Wang, W.M. To. Application of the space-time conservation element and solution element method to one-dimensional convection-diffusion problems[J]. J. Com. Phy. 2000, 165:189-215
[130] Ching Y. Loh, Cleveland OH. Noise computation of a shock-containing supersonic axisymmetric jet by the CE/SE method[J]. AIAA, 2000-0475
[131] Ching Y. Loh, Lennart S. Hultgren. Near field screech noise computation for an underexpanded supersonic jet by the CE/SE method[J]. AIAA, 2001-2252
[132] Philip C.E. Jorgenson, Ching Y.Loh. computing axsymmetric jet screech tones using unstructured grids[J]. AIAA, 2000-3889
[133] 张增产,沈孟育.改进德时空守恒元和解元方法[J].清华大学学报,1997,37(8):65-68
[134] 张增产,沈孟育.用时空守恒方法求带源项及刚性源项德守恒律方程[J].清华大学学报,1998,38(11):87-90
[135] 张增产,沈孟育.一种严格保证时-空守恒律德数值方法[J].计算物理,1997,14(6):835-840
[136] 张增产,沈孟育.求解二维 Euler 方程的时-空守恒格式[J].力学学报,1999,31(2):152-158
[137] 张增产,沈孟育.一般坐标系下的二维 Euler 方程时-空守恒格式[J].应用力学学报,1999,16(1):10-14
[138] 张增产.一般形式的二维时-空守恒格式[J].计算力学学报,1997,14(4):377-381
[139] 张增产.二维时-空守恒格式及其在激波传播计算中的应用.爆炸与冲击,1999,19(1):7-12