Estimation of wave responses from subvertical macrofracture systems using a grid characteristic method

This work investigates the formation and propagation of scattered waves forming a response of crack structures on the seismologic record. The initial impulse is a flat wave front propagating into the depth of the medium. The paper considers the periodic structure of the scattered wave response from a system (cluster) of subvertical cracks. The procedures of estimation of geometrical parameters of such crack structures are based on numerical experiments. We use a grid-characteristic method to calculate the triangular grids with the formulation of boundary conditions at the interface of the medium and the cracks, as well as on the borders of the region of integration with the characteristic properties of the governing equations of the hyperbolic type. This numerical method makes it possible to build the most correct numerical algorithms on the boundaries of the integration domain and on the surfaces of the media division (interfaces) and to take into account the domain of the solution’s dependence and the physics of the problem (propagation of disturbances on the characteristic directions). This is why this method is perhaps the most suitable one for the numerical solution of dynamic problems with a distinct wave character in geologically substantially inhomogeneous media, in particular, for the considered problem of seismic waves interaction with fissured structures.     

High Order Accurate Finite Difference Modeling of Seismo-Acoustic Wave Propagation in a Moving Atmosphere and a Heterogeneous Earth Model Coupled Across a Realistic Topography

We develop a numerical method for simultaneously simulating acoustic waves in a realistic moving atmosphere and seismic waves in a heterogeneous earth model, where the motions are coupled across a realistic topography. We model acoustic wave propagation by solving the linearized Euler equations of compressible fluid mechanics. The seismic waves are modeled by the elastic wave equation in a heterogeneous anisotropic material. The motion is coupled by imposing continuity of normal velocity and normal stresses across the topographic interface. Realistic topography is resolved on a curvilinear grid that follows the interface. The governing equations are discretized using high order accurate finite difference methods that satisfy the principle of summation by parts. We apply the energy method to derive the discrete interface conditions and to show that the coupled discretization is stable. The implementation is verified by numerical experiments, and we demonstrate a simulation of coupled wave propagation in a windy atmosphere and a realistic earth model with non-planar topography.     

Numerical study of the anisotropy of wave responses from a fractured reservoir using the grid-characteristic method

This work is aimed at studying the problem of subsurface disturbance propagation in a massive rock containing various inhomogeneities: empty or filled cracks. Numerical solutions to the problems of wave propagation in such substantially inhomogeneous media have been obtained. The dependence that the parameters of the response from a fractured reservoir have on the problem parameters is studied. The latter parameters are the density of the crack location, the fractured reservoir extent, amount of cracks, the initial disturbance location, crack inclination, and disturbance frequency. The concept of response anisotropy is introduced and the dependence that the anisotropy has on the abovementioned parameters is studied.     

Numerical simulation of seismic responses in multilayer geologic media by the grid-characteristic method

The aim of the present work is to simulate and study the propagation of seismic signal through multilayer rocks. Test computations were made in order to check that the problem of contact discontinuity is correctly formulated, and numerical simulation of dynamic processes in multilayer rocks was carried out. The work adopts the grid-characteristic method that permits devising computation algorithms that would perform properly both at the boundaries of a domain of integration and at contact interfaces between layers in a multilayer medium. The suggested approach makes it possible to determine the number of layers and their thicknesses from the response. The possibility to apply such an approach in the calculation of responses from curved interfaces between rock strata is demonstrated.     

全波场地震波数值模拟中的边界处理实践研究

在用有限差分法或有限元法模拟无界区域中的波动时,需要对计算区域的边界做特殊处理,以消除由于把地震波的传播设定在有限区域而产生的边界反射。为了这一目的,人们研究出了多种人工边界处理方法,完全匹配层（PML）吸收边界条件就是理想的方法之一,现已被广泛应用。本文将PML吸收边界条件应用于全波场地震波的数值模拟,数值计算实验表明,对qP波,匹配层的厚度为5个网格间距即可达到要求,而对qSV波与qSH波,为达到理想的吸收效果,匹配层的厚度应当增大,当厚度为13个网格间距时达到了理想的吸收效果。     

Parallel implementation of a velocity-stress staggered-grid finite-difference method for 2-D poroelastic wave propagation

Numerical simulation of wave propagation in poroelastic media demands significantly more computational capability compared to elastic media simulation. Use of serial codes in a single scientific workstation limits the size of problem. To overcome this difficulty, a parallel velocity-stress staggered-grid finite-difference method is developed for efficient simulation of wave propagation in 2-D poroelastic media. The finite difference formulation of Biot's theory has the properties of fourth order accuracy in space and second order accuracy in time. The model is decomposed into small subdomains for each processor. After each processor updates wavefields within its domain, the processors exchange the wavefields via message passing interface (MPI). The parallel implementation reduces the computational time and also allows one to study larger problems. From our numerical experiment, consistent with other 1-D experiments, it is found that the presence of heterogeneity of porous medium can produce significant P-wave attenuation in the seismic frequency range.     

Domain decomposition for wave propagation problems

The problem posed by domain decomposition methods is to find the correct modeling of physical phenomena across the interfaces separating the subdomains. The technique described here for wave propagation problems is based on physical grounds since it relies on the fact that the wave equation can be decomposed into incoming and outgoing wave modes at the boundaries of the subdomains. The inward propagating waves depend on the solution exterior to the subdomains and therefore are computed from the appropriate boundary conditions, while the behavior of the outward propagating waves is determined by the solution inside the subdomains. The technique is applied to the anisotropic-viscoelastic wave equation, which practically includes all the possible rheologies of one-phase media.     

Identification of a source of oceanic ring waves near Darwin’s Island based on space photos

A system of ring surface waves in the Pacific Ocean is identified, which was registered near Darwin’s Island in one of the photos taken according to the program of space experiment Uragan. Within the scope the hypothesis that the waves were generated as a result of a harmonic action on the ocean bed, a mathematical model, describing the excitation and propagation of waves in the approximation of the theory of long waves, was employed in image processing. As a result of calculations, the simulated wave system was approached to the one registered in the photo, which made it possible to estimate the physical parameters of the source of waves.     

基于多重反射法对周期结构的频率响应研究

采用多重反射法对受到外扰的二组元周期梁结构的频率响应进行了研究．施加至Ⅱ周期梁结构上的外部扰动被假定为一入射波，传播波入射到不连续处会产生反射波和透射波，进而在周期结构中会产生多重的反射和透射．首先，基于波的多重反射，考虑施加扰动的组元上的波场；其次，由于波的透射，分别考虑两个传播方向上的其他组元的波场，作为初始波场；最后，可先考虑某个组元右侧的所有组元上的向左传播的波在其上的叠加，作为一次迭代波场；再考虑某个组元左侧的所有组元上的向右传播的波在其上的叠加，作为二次迭代波场．依次类推，基于多重反射法，叠加了入射波引起的多重反射和透射，得到了所有组元的波场．给出了周期梁结构中任一点的波幅与入射波幅之间的函数关系，确定了受外扰的周期梁结构的传播常数及相应的波场的迭代次数．     

Parallel adaptive simulations of dynamic fracture events

Finite element simulations of dynamic fracture problems usually require very fine discretizations in the vicinity of the propagating stress waves and advancing crack fronts, while coarser meshes can be used in the remainder of the domain. This need for a constantly evolving discretization poses several challenges, especially when the simulation is performed on a parallel computing platform. To address this issue, we present a parallel computational framework developed specifically for unstructured meshes. This framework allows dynamic adaptive refinement and coarsening of finite element meshes and also performs load balancing between processors. We demonstrate the capability of this framework, called ParFUM, using two-dimensional structural dynamic problems involving the propagation of elastodynamic waves and the spontaneous initiation and propagation of cracks through a domain discretized with triangular finite elements.     

Propagation and transformation of periodic nonlinear shallow-water waves in basins with selected breakwater systems

This paper describes the development of a Boussinesq three-equation model for simulating propagation and transformation of periodic nonlinear waves (cnoidal waves) in an arbitrary shallow-water basin. The Boussinesq equations in terms of depth-averaged horizontal velocities and free-surface elevation are solved numerically in a curvilinear coordinate system. An Euler’s predictor-corrector finite-difference algorithm is applied for numerical computation. The effects of irregular boundary, non-uniform water depth and coastal structures inside a basin are all included in the model simulation. A second-order cnoidal wave solution for the Boussinesq equations is used as an incident wave condition. A set of open boundary conditions is also applied to effectively transmit waves out of the computational domain. Model tests were conducted by simulating waves propagating past an isolated breakwater. The effect of variable depth was examined with modeling waves over an uneven bottom with convex ramp topography. The overall evolution of wave propagation, diffraction and reflection in coupled harbors with various layouts of inner and outer breakwaters was also studied. Data comparisons reveal that the simulated wave heights agree reasonably well with laboratory measurements, especially in the region of inner basin.     

微结构固体中孤立波的动力学稳定性

描述微结构固体中波传播的一种KdV类方程作为控制方程并利用积分因子方法,对微结构固体中传播的孤立波的动力学稳定性进行了数值模拟研究。主要以高斯波、Ricker子波以及双曲正割波作为初始扰动,考察了不同小扰动下孤立波能否较长时间保持波形结构和传播速度而稳定传播问题。模拟结果表明,不同的小扰动对孤立波的影响不同,孤立波的稳定传播与扰动幅度和宽度都有关系,只有受到幅度和宽度都非常小的扰动下在微结构固体中传播的孤立波才能显现出一定程度的抗干扰性和动力学稳定性,可在微结构固体中较长时间稳定传播。     

Monitoring the state of the moving train by use of high performance systems and modern computation methods

The objective of this work has been to study the propagation of elastic waves in rails. It presents the comparison of calculations obtained by the grid-characteristic and discontinuous Galerkin methods. The propagation of elastic waves in the presence and absence of the karst inclusion in the ground under the embankment, diagnosed in these cases from the rails, are compared. The wave pictures and diagnosed signals for four types of defects of a fractured character: vertical and horizontal head layering, cross fracture in the head and cracks in the rail web are given. The grid-characteristic method on the curvilinear structural meshes and the discontinuous Galerkin method on the nonstructured triangular meshes make it possible to solve efficiently the tasks on monitoring the state of the moving train and rail, including a great number of integrity violations, dynamic interactions in the train-rail system, and obtain the full wave picture.     

基于分层模型的复合材料耦合Lamb波频率—波数域特性研究

针对复合材料层合板中耦合Lamb波的传播问题,基于分层模型提出解析建模与有限元数值模拟相结合的方法对其进行预测和评估。利用Legendre正交多项式展开法推导多层各向异性复合材料层合板中耦合Lamb波的控制方程,并对频率-波数域频散特性曲线实现数值求解。基于平面壳单元构建复合材料层合板的有限元模型,采用波结构加载法生成单一Lamb波基本模态,设计复合材料层合板的不同纤维取向、边界和界面约束条件,并经二维傅里叶变换获得有限元模拟数据的频率-波数域频散特性曲线。通过对比验证,结果表明两种方法均有较好的吻合性。     

Influence of natural disasters on ground facilities

The purpose of this article is to study the problem of the propagation of waves that result in earthquakes in different geological media: homogeneous, multilayer, gradient, with fractured layer, and karst cavern. The authors pose the problem of analyzing the impact of waves on ground structures: buildings and dams. Numerical solutions of problems of wave propagation in heterogeneous media are obtained. On the basis of the analysis of wave patterns, the types of waves propagated from the focus of the earthquake are qualified. The comparison of the impact of elastic waves on the day surface for the cases of different geological media is done. Synthetic seismograms for these media are obtained. The influence of elastic waves on the stability of ground structures is qualitatively examined. The grid-characteristic method for triangle meshes with the formulation of boundary conditions on interfaces of rock-crack, building-rock, rock-water, and dam-water, as well as free surfaces in an explicit form, is used in this paper.     

Recursive algorithm and accurate computation of dyadic Green’s functions for stratified uniaxial anisotropic media

A recursive algorithm is adopted for the computation of dyadic Green＇s functions in three-dimensional stratified uniaxial anisotropic media with arbitrary number of layers. Three linear equation groups for computing the coefficients of the Sommerfeld integrals are obtained according to the continuity condition of electric and magnetic fields across the interface between different layers, which are in correspondence with the TM wave produced by a vertical unit electric dipole and the TE or TM wave produced by a horizontal unit electric dipole, respectively. All the linear equation groups can be solved via the recursive algorithm. The dyadic Green＇s functions with source point and field point being in any layer can be conveniently obtained by merely changing the position of the elements within the source term of the linear equation groups. The problem of singularities occurring in the Sommerfeld integrals is efficiently solved by deforming the integration path in the complex plane. The expression of the dyadic Green＇s functions provided by this paper is terse in form and is easy to be programmed, and it does not overflow. Theoretical analysis and numerical examples show the accuracy and effectivity of the algorithm.     

Mesh domain decomposition in the finite-difference solution of Maxwell’s equations

We discuss the mesh domain decomposition when studying optical diffraction from sub-wavelength structures using the finite-difference solution of Maxwell’s equations. Special consideration is given to the case of decomposition into non-overlapping sub-domains. Theoretical estimates of the algorithms’ and computational procedures’ speedup at various decomposition parameters are compared.     

Study of stress waves in geomedia and effect of a soil cover layer on wave attenuation using a 1-D finite-difference method

The propagation and attenuation of blast-induced stress waves differs between geomedia such as rock or soil mass. This paper numerically studies the propagation and attenuation of blast-induced elastoplastic waves in deep geomedia by using a one-dimensional (1-D) finite-difference code. Firstly, the elastoplastic Cap models for rock and soil masses are introduced into the governing equations of spherical wave motion and a FORTRAN code based on the finite difference method is developed. Secondly, an underground spherical blast is simulated with this code and verified by software, RENEWTO. The propagation of stress-waves in rock and soil masses is numerically investigated, respectively. Finally, the effect of a soil cover layer on the attenuation of stress waves in the rear rock mass is studied. It is determined that large plastic deformation of geomedia can effectively dissipate the energy of stress-waves inward and the developed 1-D finite difference code coupled with elastoplastic Cap models is convenient and effective in the numerical simulations for underground spherical explosion.