各向异性介质中弹性波的数值模拟

提出了一种非均匀各向异性介质中弹性波传播的数值模拟算法。该方法可以灵活地运用于具有任意地表形状、内部孔洞、固液边界和不规则内部交界面的介质情况,另外,该方法自然满足复杂几何边界的自由表面条件。这种基于三角形和四边形离散网格的算法使用的是围绕每个节点的积分平衡方程,而不是其它有限差分法中使用的各个节点满足的弹性动力学的微分方程。该文工作是非均匀各向同性介质中弹性波传播格子法研究的继续。除了研究各向异性介质中波的传播以外,还给出了一种能够省时的四边形网格的格子法。     

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。     

各向异性介质中弹性波传问题的数值解法

通过运用速度－应力有限差分法研究方位各向异性介质中的弹性波传播问题，在计算实施过程中，使用了交错网格技术，为了减少计算量，首次引入了适用于各向异性体的吸收边界条件，并对角点处的吸收做特殊的处理，算例表明，该算法不仅具有较高的精度；与传统方法相比，计算时间也大为缩短，从而可望在实际中获得良好的应用前景。     

基于不同时间步长时域非结构有限体积法模拟声-弹性耦合问题

介绍一种改进的时域非结构有限体积法(FVM),并将其应用于声-弹性耦合问题。在流体与固体介质中分别求解声波动方程与弹性波方程,根据交界面上的力平衡与质点振速连续条件考虑二者的相互作用。同时考虑双线性四边形单元的线性变化项及常数项,并结合常应变三角形单元处理混合网格问题。分别对三角形单元和四边形单元进行色散分析,给出声波动方程的稳定性条件。在不同介质中采用不同时间步长,提高计算效率。求解弹性波问题、声-弹性耦合问题,结果表明,改进后的方法求解声-弹性耦合问题是有效和准确的,并且具有良好的数值稳定性。     

各向异性层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，解析求得各层介质位移位，应力传播矩阵，给出了直角坐标系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。     

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．     

双相各向异性介质弹性波场有限差分正演模拟

从双相各向异性介质模型出发,以Boit理论为基础,推导了斜方晶系各向异性介质-阶弹性波动方程,引入固、流体密度比和孔隙几何参数,将Biot方程系数简化为测量简单、物理意义明确的物理量,采用交错网格技术建立了各向异性孔隙介质波动方程的高精度差分格式,并首次对这类差分格式的频散特性和稳定性作了详细分析讨论,解决了计算稳定性和边界反射问题,与解析解的对比以及理论模型的数值模拟都表明,该方法不仅大大降低了计算量,提高了正演速度,并且具有良好的稳定性和精确性。     

非均匀介质中流动和传热问题的有限分析法

数值求解非均匀介质中的输运问题广泛应用于科学计算和工程领域．介质的强非均匀性给相关问题的准确求解带来极大的困难．近年来，本课题组将有限分析法拓展到该领域，建立了非均匀介质中输运问题的有限分析法．该算法基于网格奇点邻域内类拉普拉斯方程局部解析解构建，算法具有很高的精度，且不依赖于介质的非均匀性强度．不管相邻网格传导率差异如何，仅需对原始网格进行很少地细分就可以获得非常准确的计算结果，因此与其他传统数值算法相比，可以大幅提高计算精度和效率．该算法可广泛应用于求解非均匀多孔介质中的渗流、复合材料中的热传导及电场分布等问题．     

非均匀损伤介质中波传播的数值解

对弹性波在非均匀损伤介质中的传播理论进行了研究。通过将非均匀损伤区域离散成分层均匀的区域，结合相邻区域交界面处的连续条件，推导出了以右行波、左行波为状态向量的波动方程和传递矩阵。对几种非均匀损伤介质中波的传播进行了实例数值计算，并和其解析解的结果进行了比较，讨论了弹性波在非均匀损伤介质中传播的一般性质。     

复杂无粘流场数值模拟的矩形/三角形混合网格技术

建立了一套模拟复杂无粘流场的矩形／三角形混合网格技术，其中三角形仅限于物面附近，发挥非结构网格的几何灵活性，以少量的网格模拟复杂外型；同时在以外的区域采用矩形结构网格，发挥矩形网格计算简单快速的优势，有效地克服全非结构网格计算方法需要较大内存量和较长ＣＰＵ时间的不足．混合网格系统由修正的四分树方法生成．将ＮＮＤ有限差分与ＮＮＤ有限体积格式有机地融合于混合网格计算，消除了全矩形网格模拟曲边界的台阶效应，同时保证了网格间的通量守恒．数值实验表明本方法在模拟复杂无粘流场方面的灵活性和高效性．     

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.     

NUMERICAL SIMULATION OF ELASTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

A new numerical simulation algorithm is presented for the elastic wave propagation in heterogeneous anisotropic media. We make discretization of the computational domain by using triangular and quadrangular grids. The scheme is based on integral equilibrium at each node to simulate the elastic wave propagation in heterogeneous anisotropic media. The scheme is very flexible in dealing with arbitrary surface topography, inner openings, liquid-solid boundaries and irregular interfaces. Moreover, the free-surface condition of complex geometrical boundaries can be satisfied naturally. This work is an extension of the grid method for the elastic wave propagation in heterogeneous isotropic media, and a quadrangular grid with low computational cost is also introduced. Project supported by the National Natural Science Foundation of China(19672016).     

Simulation seismic wave propagation in topographic structures using asymmetric staggered grids

A new 3 D finite- difference ( FD ) method of spatially asymmetric staggered grids was presented to simulate elastic wave propagation in topographic structures. The method approximated the first-order elastic wave equations by irregular grids finite difference operator with second-order time precise and fourth-order spatial precise. Additional introduced finite difference formula solved the asymmetric problem arisen in non-uniform staggered grid scheme, The method had no interpolation between the fine and coarse grids. All grids were computed at the same spatial iteration. Complicated geometrical structures like rough submarine interface, fault and nonplanar interfaces were treated with fine irregular grids. Theoretical analysis and numerical simulations show that this method saves considerable memory and computing time, at the same time, has satisfactory stability and accuracy.     

Discontinuous wave fronts propagation in anisotropic layered media

The problem of dynamic interaction of wave phase fronts with anisotropic elastic media interfaces is considered. A technique based on joint use of the ray theory, locally plane approach and theory of stereomechanical impact is elaborated. It is employed for the investigation of discontinuous waves propagation in anisotropic tectonic structures. The cases of interaction of quasi-longitudinal and quasi-shear discontinuous waves with the interfaces separating different anisotropic elastic media are treated. The issues are considered which are associated with the wave front surfaces bifurcations, generation of their singularities and caustics, as well as with stress concentration and formation of zones where the stresses tend to infinity.     

A NUMERICAL METHOD OF MODELING ELASTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

The velocity-stress finite-difference method is adopted to simulate the elastic wave propa-gation in azimuthal anisotropic media.The difference grids are completely staggered in the numerical im-plementation.To reduce the computational work,the absorbin8 boundary conditions for anisotropic mediaare introduced first and the corner points are specially treated.Examples show that more accurate resultscan be obtained from the modeling algorithm,which cost much less computational time than the conven-tional methods.Therefore,the algorithm has broad application prospects in engineering.     

On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles

This paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles. A point model is used in the vertical, with a no-slip condition at the sea bed. In the finite difference approach the high-shear bottom layer is resolved using either a regular grid on a logarithmic or log-linear transformed co-ordinate or an irregular grid, varying in such a manner as to retain second-order accuracy. The accuracy of these various grid schemes is considered in detail. The relative merits of using either the Crank-Nicolson or Dufort-Frankel time integration methods are considered; in the case of a fine grid in a high-viscosity region, some numerical problems are found with the Dufort-Frankel method. An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high-shear bottom boundary layer. Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers. The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near-bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.     

ON FREE WAVE PROPAGATION IN ANISOTROPIC LAYERED MEDIA

The method of reverberation-ray matrix （MRRM） is extended and modified for the analysis of free wave propagation in anisotropic layered elastic media. A general, numerically stable formulation is established within the state space framework. The compatibility of physical variables in local dual coordinates gives the phase relation, from which exponentially growing functions are excluded. The interface and boundary conditions lead to the scattering relation, which avoids matrix inversion operation. Numerical examples are given to show the high accuracy of the present MRRM.