共查询到20条相似文献，搜索用时 31 毫秒
1.
平面波的传播问题通常可以归结为一维波动方程的定解问题。在非均匀介质中,即使简单的一维波动方程也需要借助于数值方法获得近似解。3层5点古典差分格式是计算偏微分方程一种常用算法,作为一种显式迭代格式,需要满足稳定性条件 ,其中 为波速, 为空间采样间隔, 为时间采样间隔。当 时, ,古典差分格式达到临界稳定状态。在这种情况下,平面波在 时间内的传播距离恰好等于空间采样间隔,差分格式真实地反映了平面波的传播原理,因而可以得到一维波动方程的精确解。但是,由于在非均匀介质中存在不连续的波阻抗界面,此方法不适于计算非均匀介质的波场。为了将临界稳定情况下的古典差分格式推广应用至非均匀层状介质,提出了一种能够处理波阻抗界面的有限差分格式,并应用傅里叶分析法得到其稳定性条件。模型算例验证了此算法的正确性。 相似文献
2.
Biot's equations of wave propagation through fluid-saturated porous elastic media are discretized spatially using the finite element method in conjunction with Galerkin's procedure. Laplace transformation of the discretized equations is used to suppress the time variable. Introducing Laplace transforms of constituent velocities at nodal points as additional variables, the quadratic set of equations in the Laplace transform parameter is reduced to a linear form. The solution in the Laplace transform space is inverted, term by term, to get the complete time history of the solid and fluid displacements and velocities. Since the solution is exact in the time domain, the error in the calculated response is entirely due to the spatial approximation. The procedure is applied to one-dimensional wave propagation in a linear elastic material and in a fluid-saturated elastic soil layer with ‘weak’, ‘strong’ as well as ‘moderate’ coupling. With refinement of the spatial mesh, convergence to the exact solution is established. The procedure can provide a useful benchmark for validation of approximate temporal discretization schemes and estimation of errors due to spatial discretization. 相似文献
3.
本文用有限差分法模拟研究二维空间中波的传播、散射及其与介质间断的相互作用。文章第一部分运用所模拟的纵波波源讨论了半无限空间表面台阶地形或槽形裂隙所造成的波的散射。这些间断可模拟具有波长量级的悬崖或狭谷。第二部分用差分法与扰动法结合而成的混合法处理半无限空间表面有另一种物质构成的表面浅障碍时波的传播,以模拟山脊、堑沟等地形的影响。此混合法也被用于讨论半无限空间内含有水平夹层或孔隙的问题。所得研究结果对波的传播及其与介质间断的相互作用的理论研究和地震波破坏效应的分析和隔离等应用研究有一定的意义。 相似文献
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This paper presents a time integration scheme capable of simulating blast loading of relatively high frequency on porous media, using coarse meshes. The scheme is based on the partition of unity finite element method. The discontinuity is imposed on the velocity field, while the displacement field is kept continuous. The velocity discontinuity is postulated to occur in the time domain. The developed time integration scheme is unconditionally stable and has controllable numerical dissipation in the high frequency range. An important feature of the time scheme is that it allows for controlling the numerical damping in a consistent way. The time scheme has been implemented in combination with Biot’s theory of wave propagation in saturated porous media. Numerical examples have demonstrated that the proposed time scheme is, in addition to being accurate and stable, highly effective for coarse meshes. This makes the developed scheme suitable for large scale finite element analysis. 相似文献
6.
Dali Zhang Michael P. Lamoureux Gary F. Margrave Elena Cherkaev 《Computational Geosciences》2011,15(1):117-133
The article presents a numerical inversion method for estimation of quality Q factor and phase velocity in linear, viscoelastic, isotropic media using reconstruction of relaxation spectrum from measured
or computed complex velocity or modulus of the medium. Mathematically, the problem is formulated as an inverse problem for
reconstruction of relaxation spectrum in the analytic Stieltjes representation of the complex modulus using rational approximation.
A rational (Padé) approximation to the relaxation spec trum is derived from a constrained least squares minimization problem
with regularization. The recovered stress-strain relaxation spectrum is applied to numerical calculation of frequency-dependent
Q factor and frequency-dependent phase velocity for known analytical models of a standard linear viscoelastic solid (Zener)
model as well as a nearly constant-Q model which has a continuous spectrum. Numerical results for these analytic models show good agreement between theoretical
and predicted values and demonstrate the validity of the algorithm. The proposed method can be used for evaluating relaxation
mechanisms in seismic wavefield simulation of viscoelastic media. The constructed lower order Padé approximation can be used
for determination of the internal memory variables in time-domain finite difference numerical simulation of viscoelastic wave
propagation. 相似文献
7.
从具有水平对称轴的横向各向同性（HTI）介质中的弹性波动方程出发,在交错网格空间中采用高阶差分算子对弹性波动方程进行差分离散,得到了HTI介质中地震波正演的高阶有限差分格式,研究并实现了PML吸收边界条件。在此基础上实现了HTI介质中弹性波方程的多波正演。数值算例表明,该方法能够精确模拟弹性波在复杂各向异性介质中的传播过程,得到高精度的正演记录。 相似文献
8.
TTI介质弹性波频率-空间域有限差分数值模拟 总被引：3，自引：0，他引：3
由周期性薄互层引起的VTI介质是研究比较广泛的一类各向异性介质。当VTI介质对称轴偏离垂向,本构坐标系与观测坐标系不重合时,会形成观测坐标系下的TTI介质。引入25点优化差分算子,推导出二维TTI介质频率域弹性波动方程;为压制边界反射,采用完全匹配层法吸收边界条件,并计算出优化差分系数;最后采用集中力源,模拟了弹性波在TTI介质中的传播过程。从波场快照和地面共炮记录可以看出,笔者采用的数值模拟算法能有效压制数值频散。TTI介质中的波场传播比较复杂,纵波传播相对稳定,横波波前的三分叉现象比较明显,并存在振幅奇异性。当VTI介质的对称轴偏转后,还会增加地面地震记录的复杂性。 相似文献
9.
We present a numerical scheme for reactive contaminant transport with nonequilibrium sorption in porous media. The mass conservative scheme is based on Euler implicit, mixed finite elements, and Newton method. We consider the case of a Freundlich-type sorption. In this case, the sorption isotherm is not Lipschitz but just Hölder continuous. To deal with this, we perform a regularization step. The convergence of the scheme is analyzed. An explicit order of convergence depending only on the regularization parameter, the time step, and the mesh size is derived. We give also a sufficient condition for the quadratic convergence of the Newton method. Finally, relevant numerical results are presented. 相似文献
10.
The numerical error associated with finite-difference simulation of wave propagation in discontinuous media consists of two
components. The first component is a higher-order error that leads to grid dispersion; it can be controlled by higher-order
methods. The second component results from misalignment between numerical grids and material interfaces. We provide an explicit
estimate of the interface misalignment error for the second order in time and space staggered finite-difference scheme applied
to the acoustic wave equation. Our analysis, confirmed by numerical experiments, demonstrates that the interface error results
in a first-order time shift proportional to the distance between the interface and computational grids. A 2D experiment shows
that the interface error cannot be suppressed by higher-order methods and indicates that our 1D analysis gives a good prediction
about the behavior of the numerical solution in higher dimensions.
相似文献
11.
二阶弹性波动方程高精度交错网格波场分离数值模拟 总被引：4，自引：0，他引：4
给出了一种等价的二阶弹性波动方程,以解决弹性波场中完全弹性波动方程不能完全分离耦合的纵、横波波场问题.应用高阶交错网格有限差分法求解该波动方程,并使用通量校正技术(FCT)进一步压制频散,采用均匀介质模型和层状介质模型进行波场分离数值试验,精确得到了混合波场、完全分离的纯纵波及纯横波波场.数值结果分析表明,本文方法在均匀介质情况下准确可靠,在分离后的纯纵、横波波场中可观察到较为丰富的能量转换信息,这对认识复杂弹性波的传播规律及弹性波理论具有重要意义. 相似文献
12.
矿井地质雷达超前探测正演模拟 总被引：5，自引：0，他引：5
用时间域有限差分(FDTD)法建立矿井地质雷达的模型。采用理想频散关系和超吸收边界条件,对几个典型矿井地电模型进行了正演模拟,研究了矿井地质雷达超前探测的剖面特征。巷道对各道直达波能量分布及波列形状有明显影响;煤岩分界面主要影响反射波能量分布。 相似文献
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14.
In this paper we consider the numerical solution of a coupled geomechanics and a stress-sensitive porous media reservoir flow
model. We combine mixed finite elements for Darcy flow and Galerkin finite elements for elasticity. This work focuses on deriving
convergence results for the numerical solution of this nonlinear partial differential system. We establish convergence with
respect to the L
2-norm for the pressure and for the average fluid velocity and with respect to the H
1-norm for the deformation. Estimates with respect to the L
2-norm for mean stress, which is of special importance since it is used in the computation of permeability for poro-elasticity,
can be derived using the estimates in the H
1-norm for the deformation. We start by deriving error estimates in a continuous-in-time setting. A cut-off operator is introduced
in the numerical scheme in order to derive convergence. The spatial grids for the discrete approximations of the pressure
and deformation do not need be the same. Theoretical convergence error estimates in a discrete-in-time setting are also derived
in the scope of this investigation. A numerical example supports the convergence results. 相似文献
15.
We perform a convergence analysis of the fixed stress split iterative scheme for the Biot system modeling coupled flow and deformation in anisotropic poroelastic media with tensor Biot parameter. The fixed stress split iterative scheme solves the flow subproblem with all components of the stress tensor frozen using a multipoint flux mixed finite element method, followed by the poromechanics subproblem using a conforming Galerkin method in every coupling iteration at each time step. The coupling iterations are repeated until convergence and Backward Euler is employed for time marching. The convergence analysis is based on studying the equations satisfied by the difference of iterates to show that the fixed stress split iterative scheme for anisotropic poroelasticity with Biot tensor is contractive. We also demonstrate that the scheme is numerically convergent using the classical Mandel’s problem solution for transverse isotropy. 相似文献
16.
This paper presents a new approach to a local time-space grid refinement for a staggered-grid finite-difference simulation
of waves. The approach is based on approximation of a wave equation at the interface where two grids are coupled. As no interpolation
or projection techniques are used, the finite-difference scheme preserves second order of convergence. We have proved that
this approach is low-reflecting, the artificial reflections are about 10 − 4 of an incident wave. We have also shown that if a successive refinement is applied, i.e. temporal and spatial steps are refined
at different interfaces, this approach is stable. 相似文献
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18.
Florian Frank Chen Liu Faruk O. Alpak Beatrice Riviere 《Computational Geosciences》2018,22(2):543-563
A numerical method is formulated for the solution of the advective Cahn–Hilliard (CH) equation with constant and degenerate mobility in three-dimensional porous media with non-vanishing velocity on the exterior boundary. The CH equation describes phase separation of an immiscible binary mixture at constant temperature in the presence of a conservation constraint and dissipation of free energy. Porous media / pore-scale problems specifically entail images of rocks in which the solid matrix and pore spaces are fully resolved. The interior penalty discontinuous Galerkin method is used for the spatial discretization of the CH equation in mixed form, while a semi-implicit convex–concave splitting is utilized for temporal discretization. The spatial approximation order is arbitrary, while it reduces to a finite volume scheme for the choice of element-wise constants. The resulting nonlinear systems of equations are reduced using the Schur complement and solved via inexact Newton’s method. The numerical scheme is first validated using numerical convergence tests and then applied to a number of fundamental problems for validation and numerical experimentation purposes including the case of degenerate mobility. First-order physical applicability and robustness of the numerical method are shown in a breakthrough scenario on a voxel set obtained from a micro-CT scan of a real sandstone rock sample. 相似文献
19.
A procedure for numerical approximation to two-dimensional, hydraulically-driven fracture propagation in a poroelastic material is described. The method uses a partitioned solution procedurè to solve a finite element approximation to problems described by the theory of poroelasticity, in conjunction with a finite difference approximation for modelling fluid flow along the fracture. An equilibrium fracture model based on a generalized, Dugdale–Barenblatt concept is used to determine the fracture dimensions. An important feature is that the fracture length is a natural product of the solution algorithm. Two example problems verify the accuracy of the numerical procedure and a third example illustrates a fully-coupled simulation of fracture propagation. Photographs taken from a high-performance engineering workstation provide insight into the nature of the coupling among the physical phenomena. 相似文献
20.
The analysis of the wave propagation in layered rocks masses with periodic fractures is tackled via a two-scale approach in order to consider shape and size of the rock inhomogeneities. To match the displacement fields at the two scales, an approximation of the micro-displacement field is assumed that depends on the first and second gradients of the macro-displacement through micro-fluctuation displacement functions obtained by the finite element solution of cell problems derived by the classical asymptotic homogenization. The resulting equations of motion of the equivalent continuum at the macro-scale result to be not local in space, thus a dispersive wave propagation is obtained from the model. The simplifying hypotheses assumed in the multi-scale kinematics limit the validity of the model to the first dispersive branch in the frequency spectrum corresponding to the lowest modes.Although the homogenization procedure is developed to study the macro-scale wave propagation in rock masses with bounded domain, the reliability of the proposed method has been evaluated in the examples by considering unbounded rock masses and by comparing the dispersion curves provided by the rigorous process of Floquet–Bloch with those obtained by the method presented. The accuracy of the method is analyzed for compressional and shear waves propagating in the intact-layered rocks along the orthotropic axes. Therefore, the influence of crack density in the layered rock mass has been analyzed. Vertical cracks have been considered, periodically located in the stiffer layer, and two different crack densities have been analyzed, which are differentiated in the crack spacing. A good agreement is obtained in case of compressional waves travelling along the layering direction and in case of both shear and compressional waves normal to the layering. The comparison between two crack systems with different spacing has shown this aspect to have a remarkable effect on waves travelling along the direction of layering, and limited in the case of waves propagating normal to the layers.The equivalent continuous model obtained through the dynamic homogenization technique here presented may be applied to the computational analysis of non-stationary wave propagation in rock masses of finite size, also consisting of sub-domains with different macro-mechanical characteristics. This avoids the use of computational models represented at the scale of the heterogeneities, which may be too burdensome or even unfeasible. 相似文献