非均匀介质弹性波动方程的不规则网格有限差分方法

从弹性波动方程出发，提出了一种新的空间不规则网格有限差分方法，并用于求解非均匀各向异性介质中的弹性波正演问题。这种方法简单易行，对于复杂几何结构，例如低速层、套管井和非平面界面等，在较细的不规则网格上进行离散，计算时间和占用内存更少。与多重网格差分方法相比，该方法不需要粗、细网格之间的插值，所有网格差分计算在同一次空间迭代中完成。具有复杂几何交界面的模型计算，包括地下透镜体、套管井眼等，在确定弹性常数和密度后，用不规则网格的差分方法更易实现。该方法使用了Higdon吸收边界条件解决人工边界反射问题，引入了新的稳定性条件和网格频散条件，很好地消除了非物理散射波。理论模型的效值计算表明，该方法具有良好的稳定性和计算精度，在模拟非均匀介质弹性波传播时，比相同精度的规则网格有限差分方法计算速度更快。该方法易于推广到非结构网格和三维问题中。     

一维Burgers方程的各种差分格式研究

本文对Burgers方程,一维流体力学动量方程的模型方程,进行了数值实验研究。本文以两类精确解(一类表示定常拟的“激波”解,一类表示非定常的粘性耗散解)为基准。在给定的完全精确的边界条件下,专门探讨各种常用和重要差分格式的优劣。(包括精度、稳定性、计算时间等)。 本文共采用了八种格式(包括Cheng—Allen、修正Cheng格式、MacCormack、分裂格式等)对两类初、边值问题进行了计算和比较。主要的结果是: ①对非定常的粘性耗散问题,采用隐式分裂格式在精度和稳定性方面是各类格式中最好的。 ②对于具有拟“激波”样的定常问题,Cheng的修正格式2具有振幅很小、振动衰减很快的优点,它是各种格式中最佳的。 ③对Mac Cormack格式31,它可以用来算定常和非定常问题,但稳定性很差,要得到好的精度对Re数范围有一定的限制。 ④对于计算含有“激波”的流动,看来差分格式的守恒性是很重要的。 ⑤对像Burgers方程那样的非线性方程,看来Von Neumann的线化稳定性分析仍然是适用的。     

对流扩散方程的迎风变换及相应有限差分方法

本文提出所谓迎风变换,将对流扩散方程分解为对流迎风函数和扩散方程,并构造相应的有限差分格式。对流迎风函数以简明的指数解析形式反映对流扩散现象的迎风效应,原则上消除了源于不对称对流算子的困难,能够便利对流扩散方程的数值求解。有限差分格式具有二阶精度和无条件稳定性,算例表明其准确性、收敛速度及对边界层效应的适应能力均明显优于中心差分格式和迎风差分格式。     

基于MUSTA格式的全非线性Boussinesq波浪传播数值模型

建立了求解二维全非线性布氏（Boussinesq）水波方程的有限差分/有限体积混合数值格式. 针对守恒形式的控制方程,采用有限体积方法并结合 MUSTA格式计算数值通量, 剩余项则采用有限差分方法求解, 采用具有总变差减小（totalvariation diminishing, TVD）性质的三阶龙格-库塔法进行时间积分.该格式具备间断捕捉、程序实现简单、数值稳定性强、海岸动边界以及波浪破碎处理方便和可调参数少等优点.利用典型算例对数值模型进行了验证,计算结果与实验数据吻合较好.      

浅水方程组合型超紧致差分格式

提出一族组合型超紧致差分格式(CSCD),对CSCD的数值特性作了分析,并同其他中心型差分格式进行比较。从定性角度,得出同阶中心差分格式中,CSCD格式的截断误差系数最小的结论。从定量角度,利用Fou-rier分析方法分析了CSCD格式的分辨率,并同其他中心型差分格式比较,得出CSCD格式有较高的分辨率的结论。把10阶CSCD格式应用于KdV-Burgers方程和浅水方程的数值模拟,给出两个应用算例。数值实验表明CSCD格式不仅有理论上的高精度,而且有良好的稳定性和收敛性。     

高精度迎风紧致差分格式与热流计算研究的注记

利用高精度差分格式求解了可压缩 N-S方程球头热流问题。分析了不同差分格式在对球头粘性绕流热流计算中存在的问题 ,并分析了相应的网格雷诺数。在利用高精度迎风紧致 [1 ] 格式求解粘性绕流热流问题时 ,采用 Steger-Warming[2 ]的通量分裂技术将守恒型方程中的流通向量分裂成两部分 ,在此基础上据风向构造逼近于无粘项的高精度迎风格式。对方程中的粘性部分采用中心差分格式。数值结果表明 :高精度差分格式能在较大的网格雷诺数下较好地计算球头驻点热流     

A FULLY NONLINEAR BOUSSINESQ MODELFOR WAVE PROPAGATION BASED ON MUSTA SCHEME

建立了求解二维全非线性布氏（Boussinesq）水波方程的有限差分/有限体积混合数值格式. 针对守恒形式的控制方程,采用有限体积方法并结合 MUSTA格式计算数值通量, 剩余项则采用有限差分方法求解, 采用具有总变差减小（totalvariation diminishing, TVD）性质的三阶龙格-库塔法进行时间积分.该格式具备间断捕捉、程序实现简单、数值稳定性强、海岸动边界以及波浪破碎处理方便和可调参数少等优点.利用典型算例对数值模型进行了验证,计算结果与实验数据吻合较好.     

双重孔隙介质中底水锥进问题的数值解

本文根据首先提出的双重孔隙介质模型及其基本方程,采用线性半隐式差分格式,提供了模拟该介质中轴对称底水锥进问题的数值解法。方法考虑了重力、毛管力、岩块渗吸、流体和介质的压缩性等因素,并将裂缝系统和岩块系统均看作连续介质。作者编制了计算程序,对某油田一口实际油井的生产历史进行了拟合和“动态预测”,结果是相当令人满意的。本文和Kazcmi等人的工作相比,主要有以下几点差别:(1)我们处理的是轴对称单井底水锥进问题;(2)渗吸系数考虑了裂缝的影响;(3)基本方程中没有忽略岩块系统的流动项,即对基本方程未做任何简化和删舍 本文采用的方法可以推广到多维多井的两项流动问题中去     

各向异性展状介质中弹性波的传播矩阵解法

基于广义胡克定律及混和变量弹性波方程，解析求得各层介质内位移、应力传递矩阵，给出了直角坐标系下各向异性层状介质中弹性波的传播矩阵解法．该方法适用于非轴对称各向异性和点源作用，较好地解决了数值计算中有效数字精度损失问题．数值结果表明，计算效率、准确性及稳定性均较好．     

多介质流体非守恒律欧拉方程组的数值计算方法

对多介质流体在界面处满足的Euler方程进行了探讨,方程组中增加了描述材料参数间断性质的对流形式非守恒律方程组 .以波传播算法为基础,通过Roe方程近似求解Riemann问题,同时采用相同的数值差分格式求解流体动力学Euler方程组和界面方程组.该方法可以有效消除多介质流体在界面处压力、速度可能出现的非物理振荡.给出了部分典型一维和二维数值计算结果.     

Seismic propagation simulation in complex media with non-rectangular irregular-grid finite-difference

This paper presents a finite-difference (FD) method with spatially non-rectangular irregular grids to simulate the elastic wave propagation. Staggered irregular grid finite difference operators with a second-order time and spatial accuracy are used to approximate the velocity-stress elastic wave equations. This method is very simple and the cost of computing time is not much. Complicated geometries like curved thin layers, cased borehole and nonplanar interfaces may be treated with nonrectangular irregular grids in a more flexible way. Unlike the multi-grid scheme, this method requires no interpolation between the fine and coarse grids and all grids are computed at the same spatial iteration. Compared with the rectangular irregular grid FD, the spurious diffractions from “staircase” interfaces can easily be eliminated without using finer grids. Dispersion and stability conditions of the proposed method can be established in a similar form as for the rectangular irregular grid scheme. The Higdon‘s absorbing boundary condition is adopted to eliminate boundary reflections. Numerical simulations show that this method has satisfactory stability and accuracy in simulating wave propagation near rough solid-fluid interfaces. The computation costs are less than those using a regular grid and rectangular grid FD method.     

流固耦合地震波动问题的显式谱元模拟方法

流固耦合地震波动问题主要研究由流体和固体构成的复杂系统中地震波传播特性及其规律. 传统模拟方法中一般以声波方程、弹性波方程的数值解分别描述理想流体和弹性固体中的波动, 并实时地处理两种不同性质介质之间的相互耦合作用, 数值格式复杂且限制数值模拟精度与计算效率. 本文采用谱元法结合多次透射公式人工边界条件实现了一种流固耦合地震波动问题的高阶显式数值计算方法. 该方法利用了流固耦合问题统一计算框架,可将饱和多孔介质的Biot波动方程分别退化为理想流体的声波方程和弹性固体的弹性波方程. 通过P波垂直入射的水平成层理想流体-饱和多孔介质-弹性固体场地模型、P波斜入射的不规则层状界面以及任意形状界面的理想流体-饱和多孔介质-弹性固体场地模型等三个算例, 与传递函数法解析解以及集中质量有限元法计算结果进行对比分析, 证明了本文方法的正确性与有效性. 数值模拟结果表明, 本文方法相较传统有限元法可以少得多的节点数量获得更高的数值精度, 并且在较宽的频率范围内都能可靠地模拟出流固耦合系统的动力响应, 充分体现出本文方法兼顾高精度、计算效率和复杂场地建模灵活的特点.      

Transmission boundary condition for mixed variable finite-difference method

All kinds of numerical methods based on the discretization of first-order velocity-stress hyperbolic system of elastic wave equations are named mixed variable finite-difference method in this paper, and the transmission boundary condition of artificial boundaries is studied here by the mixed variable finite-difference method. The transmission condition of complex geometrical boundaries of a transversely isotropic medium is presented based on the characteristic variables of wavefields propagating in the normal direction of the boundary. The boundary condition proposed in this paper is a local artificial boundary condition, with which the computation cost is very low. Elastic wave propagations in transversely isotropic medium are modelled by the staggered grid finite-difference method incorporated with the boundary condition presented. Numerical results and analysis of reflection coefficients show that the reflections of incoming waves by artificial boundaries are efficiently reduced.Supported by National Natural Science Foundation and Liaoning Province Science Foundation.     

Numerical simulation of standing wave with 3D predictor-corrector finite difference method for potential flow equations

A three-dimensional （3D） predictor-corrector finite difference method for standing wave is developed. It is applied to solve the 3D nonlinear potential flow equa- tions with a free surface. The 3D irregular tank is mapped onto a fixed cubic tank through the proper coordinate transform schemes. The cubic tank is distributed by the staggered meshgrid, and the staggered meshgrid is used to denote the variables of the flow field. The predictor-corrector finite difference method is given to develop the difference equa- tions of the dynamic boundary equation and kinematic boundary equation. Experimental results show that, using the finite difference method of the predictor-corrector scheme, the numerical solutions agree well with the published results. The wave profiles of the standing wave with different amplitudes and wave lengths are studied. The numerical solutions are also analyzed and presented graphically.     

Boundary element simulation of surface waves on a deformed half-space

Homogeneous and two-layer half-spaces consisting of an anisotropic elastic, isotropic viscoelastic, or poroelastic material are considered. The Kelvin–Voigt model and the model with the Abel kernel are used as models of the viscoelastic material; the poroelastic material is studied within the framework of the model of the compressible Biot material. The case where the half-space contains a cavity is also considered. Propagation of surface waves is studied by the boundary element method. The numerical solution involves the method of collocations for a regularized boundary integral equation.     

三维非均匀介质中弹性波传播的数值模拟

提出了一种三维非均匀介质中弹性波传播数值模拟的方法,文中称为三维格子法。该算法是二维格子法（一种二维非均匀介质中P－SV波传播的数值模拟算法）向三维非均匀介质情况的推广。在空间离散上该文方法与有限元方法类似,容许根据连续体的形状和介质分界面任意剖面网格,且自然满足自由表面边界条件。不同于常规有限差分法在各个节点上满足动力学微分方程,该算法通过满足各节点周围格子的整体平衡（积分平衡方程）来对问题进行求解,三维格子法所需的计算机内存及计算耗时与同阶精度的规则网格有限差分法相当。算例表明,该文提出的三维格子法具有较高的精度且可很好地模拟三维复杂形状地表对弹性波的反射和绕射。     

A boundary element solution to scattering of elastic SH wave by arbitrarily shaped inclusions embedded in an infinite anisotropic medium

Scattering problems for inhomogeneous bodies are investigated by the integral equation method. The boundary integral equation (BIE) for the scattered displacement field associated with finite inhomogeneities in an anisotropic medium are derived with the help of the generalized Green's identity. The discretization of BIE is based upon the constant element, linear element and quadratic element. Several numerical examples for calculating the scattering displacement, stress and scattering cross section from a cylinder, an interface crack, and two elliptic cylinders are given. Results show that the present method can be advantageously applied to a wide range of scattering problems of elastic waves.     

A boundary element method for calculating the SH waves scattering from arbitrarily shaped holes in an anisotropic medium

The scattering problem of elastic wave by arbitrarily shaped cavities in an infinite anisotropic medium is investigated by the boundary integral equation (BIE) method. The formulations of BIE are derived with the help of generalized Green's formula. The discretization of BIE is based upon constant elements. After confirmation of the accuracy of the present method, some numerical examples are given for various cavities in a full space, in which an isotropic body with a circular cylinder hole is used for comparison and good agreement is observed. It has been proved that the method developed in this paper is effective.     

各向异性体内含任意孔洞对反平面波散射的边界元方法

本文借助于广义格林公式导出了用位移表示的各向异性介质中SH波入射时的边界积分方程.根据本文作者在文献[8]给出的基本解,求解了各向异性介质中孔洞对SH波的散射问题.边界积分方程的离散基于常数元模式.文中给出了一个圆柱、一个椭圆柱和两个椭圆柱形式的孔洞周围的位移场和应力场的数值结果.最后,对入射波频率较高时的情形作了说明.