Mingwei Zhuang

Efficient Ordinary Differential Equation-Based Discontinuous Galerkin Method for Viscoelastic Wave Modeling

We present an efficient nonconformal-mesh discontinuous Galerkin (DG) method for elastic wave propagation in viscous media. To include the attenuation and dispersion due to the quality factor in time domain, several sets of auxiliary ordinary differential equations (AODEs) are added. Unlike the conventional auxiliary partial differential equation-based algorithm, this new method is highly parallel with its lossless counterpart, thus requiring much less time and storage consumption. Another superior property of the AODE-based DG method is that a novel exact Riemann solver can be derived, which allows heterogeneous viscoelastic coupling, in addition to accurate coupling with purely elastic media and fluid. Furthermore, thanks to the nonconformal-mesh technique, adaptive hp-refinement and flexible memory allocation for the auxiliary variables are achieved. Numerical results demonstrate the efficiency and accuracy of our method.

Spectral element method for elastic and acoustic waves in frequency domain

Numerical techniques in time domain are widespread in seismic and acoustic modeling. In some applications, however, frequency-domain techniques can be advantageous over the time-domain approach when narrow band results are desired, especially if multiple sources can be handled more conveniently in the frequency domain. Moreover, the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain. In this paper, we present a spectral-element method (SEM) in frequency domain to simulate elastic and acoustic waves in anisotropic, heterogeneous, and lossy media. The SEM is based upon the finite-element framework and has exponential convergence because of the use of GLL basis functions. The anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems. Compared with the conventional finite-element method, the number of unknowns in the SEM is significantly reduced, and higher order accuracy is obtained due to its spectral accuracy. To account for the acoustic-solid interaction, the domain decomposition method (DDM) based upon the discontinuous Galerkin spectral-element method is proposed. Numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain.

The rotated cartesian coordinate method to remove the axial singularity of cylindrical coordinates in finite-difference schemes

SUMMARY When modelling the propagation of 3-D non-axisymmetric viscoelastic waves in cylindrical coordinates using the finite-difference time-domain (FDTD) method, one encounters a mathematical singularity due to the presence of 1/r terms in the viscoelastic wave equations. For many years this issue has been impeding the accurate numerical solution near the axis. In this paper, we propose a simple but effective method for the treatment of this numerical singularity problem. By rotating the Cartesian coordinate (RCC) system around the z-axis in cylindrical coordinates, the numerical singularity problems in both 2-D and 3-D cylindrical coordinates can be removed. This algorithm has three advantages over the conventional treatment techniques: 1) the excitation source can be directly loaded at r = 0; 2) the central difference scheme with second-order accuracy is maintained; 3) the stability condition at the axis is consistent with the FDTD in Cartesian coordinates. This method is verified by several 3-D numerical examples. Results show that the method is accurate and stable at the singularity point. The improved FDTD algorithm is also applied to sonic logging simulations in non-axisymmetric formations and sources. This article is protected by copyright. All rights reserved

The Auxiliary Differential Equations Perfectly Matched Layers Based on the Hybrid SETD and PSTD Algorithms for Acoustic Waves

The perfectly matched layer (PML) absorbing boundary condition has been proven to absorb body waves and surface waves very efficiently at non-grazing incidence. However, the traditional PML would generate large spurious reflections at grazing incidence, for example, when the sources are located near the truncating boundary and the receivers are at a large offset. In this paper, a new PML implementation is presented for the boundary truncation in three-dimensional spectral element time domain (SETD) for solving acoustic wave equations. This method utilizes pseudospectral time-domain (PSTD) method to solve first-order auxiliary differential equations (ADEs), which is more straightforward than that in the classical FEM framework.

Reverse Time Migration of Elastic Waves Using the Pseudospectral Time-Domain Method

Reverse time migration (RTM), especially that for elastic waves, consumes massive computation resources which limit its wide applications in industry. We suggest to use the pseudospectral time-domain (PSTD) method in elastic wave RTM. RTM using PSTD can significantly reduce the computational requirements compared with RTM using the traditional finite difference time domain method (FDTD). In addition to the advantage of low sampling rate with high accuracy, the PSTD method also eliminates the periodicity (or wraparround) limitation caused by fast Fourier transform in the conventional pseudospectral method. To achieve accurate results, the PSTD method needs only about half the spatial sampling rate of the twelfth-order FDTD method. Thus, the PSTD method can save up to 87.5% storage memory and 90% computation time over the twelfth-order FDTD method. We implement RTM using PSTD for elastic wave equations and accelerate it by Open Multi-Processing technology. To keep the computational load balance in parallel ...

Spectral-Element Method With Divergence-Free Constraint for 2.5-D Marine CSEM Hydrocarbon Exploration

Rapid simulations of large-scale low-frequency subsurface electromagnetic measurements are still a challenge because of the low-frequency breakdown phenomenon that makes the system matrix extremely poor-conditioned. Hence, significant attention has been paid to accelerate the numerical algorithms for Maxwell’s equations in both integral and partial differential forms. In this letter, we develop a novel 2.5-D method to overcome the low-frequency breakdown problem by using the mixed spectral element method with the divergence-free constraint and apply it to solve the marine-controlled-source electromagnetic systems. By imposing the divergence-free constraint, the proposed method considers the law of conservation of charges, unlike the conventional governing equation for these problems. Therefore, at low frequencies, the Gauss law guarantees the stability of the solution, and we can obtain a well-conditioned system matrix even as the frequency approaches zero. Several numerical experiments show that the proposed method is well suited for solving low-frequency electromagnetic problems.

New advances in FDTD methods for electromagnetic and elastic waves for probing complex media

For modeling large-scale 3-D problems in electro-magnetic and elastic waves in the probing of complex media, the finite-difference time domain (FDTD) method is widely used. However, there are still challenges in the high-frequency regime and extremely low-frequency regime, as well as in the accurate curved boundary treatment in this method. In this work, we report several new improvements in the FDTD method to address these challenging issues: (a) We have developed a high-order FDTD method for elastic waves to greatly reduce the numerical dispersion errors, and computer memory and CPU time requirements, with a novel treatment of the free ground boundary condition and the fluid-solid interface condition. (b) We have proposed a new numerical method, finite volume method and enlarged cell technique (ECT), to accurately and efficiently implement the free-surface boundary conditions in FDTD elastic wave simulations of an arbitrary ground-surface topography. (c) We have developed an implicit FDTD scheme that allows a time step increment many orders of magnitude beyond the stability condition in the explicit FDTD method for extremely low frequency electromagnetic probing of subsurface, based on the Crank-Nicolson scheme together with the perfectly matched layer. The efficacy of these methods and their large scale applications will be demonstrated in the presentation.