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Mrinal K Sen - One of the best experts on this subject based on the ideXlab platform.

  • Elastic Wave Propagation in fractured media using the discontinuous galerkin method
    Geophysics, 2016
    Co-Authors: Jonas D De Basabe, Mrinal K Sen, Mary F Wheeler
    Abstract:

    ABSTRACTWe have formulated and implemented a discontinuous Galerkin method (DGM) for Elastic Wave Propagation that allows for discontinuities in the displacement field to simulate fractures or faults. The approach is based on the interior-penalty formulation of DGM, and the fractures are simulated using the linear-slip model, which is incorporated into the weak formulation by including an additional term that is similar to the penalty term but uses the fracture compliance instead of an arbitrary penalty parameter. We have calibrated our results against an analytic solution of fracture-induced anisotropy for a set of elongated horizontal fractures, and we have evaluated numerical examples that simulate the reflection and transmission of Waves at a fracture and at fracture interface Waves. This method can further be used with models containing intersecting fractures and multiple fracture sets in 2D or 3D domains.

  • stability of the high order finite elements for acoustic or Elastic Wave Propagation with high order time stepping
    Geophysical Journal International, 2010
    Co-Authors: Jonas D De Basabe, Mrinal K Sen
    Abstract:

    SUMMARY We investigate the stability of some high-order finite element methods, namely the spectral element method and the interior-penalty discontinuous Galerkin method (IP-DGM), for acoustic or Elastic Wave Propagation that have become increasingly popular in the recent past. We consider the Lax-Wendroff method (LWM) for time stepping and show that it allows for a larger time step than the classical leap-frog finite difference method, with higher-order accuracy. In particular the fourth-order LWM allows for a time step 73 per cent larger than that of the leap-frog method; the computational cost is approximately double per time step, but the larger time step partially compensates for this additional cost. Necessary, but not sufficient, stability conditions are given for the mentioned methods for orders up to 10 in space and time. The stability conditions for IP-DGM are approximately 20 and 60 per cent more restrictive than those for SEM in the acoustic and Elastic cases, respectively.

  • the interior penalty discontinuous galerkin method for Elastic Wave Propagation grid dispersion
    Geophysical Journal International, 2008
    Co-Authors: Jonas D De Basabe, Mrinal K Sen, Mary F Wheeler
    Abstract:

    SUMMARY Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to Wave Propagation. In this work, we investigate the applicability of the interior penalty DGM to Elastic Wave Propagation by analysing it’s grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical Wave Propagation.

Jonas D De Basabe - One of the best experts on this subject based on the ideXlab platform.

  • Elastic Wave Propagation in fractured media using the discontinuous galerkin method
    Geophysics, 2016
    Co-Authors: Jonas D De Basabe, Mrinal K Sen, Mary F Wheeler
    Abstract:

    ABSTRACTWe have formulated and implemented a discontinuous Galerkin method (DGM) for Elastic Wave Propagation that allows for discontinuities in the displacement field to simulate fractures or faults. The approach is based on the interior-penalty formulation of DGM, and the fractures are simulated using the linear-slip model, which is incorporated into the weak formulation by including an additional term that is similar to the penalty term but uses the fracture compliance instead of an arbitrary penalty parameter. We have calibrated our results against an analytic solution of fracture-induced anisotropy for a set of elongated horizontal fractures, and we have evaluated numerical examples that simulate the reflection and transmission of Waves at a fracture and at fracture interface Waves. This method can further be used with models containing intersecting fractures and multiple fracture sets in 2D or 3D domains.

  • stability of the high order finite elements for acoustic or Elastic Wave Propagation with high order time stepping
    Geophysical Journal International, 2010
    Co-Authors: Jonas D De Basabe, Mrinal K Sen
    Abstract:

    SUMMARY We investigate the stability of some high-order finite element methods, namely the spectral element method and the interior-penalty discontinuous Galerkin method (IP-DGM), for acoustic or Elastic Wave Propagation that have become increasingly popular in the recent past. We consider the Lax-Wendroff method (LWM) for time stepping and show that it allows for a larger time step than the classical leap-frog finite difference method, with higher-order accuracy. In particular the fourth-order LWM allows for a time step 73 per cent larger than that of the leap-frog method; the computational cost is approximately double per time step, but the larger time step partially compensates for this additional cost. Necessary, but not sufficient, stability conditions are given for the mentioned methods for orders up to 10 in space and time. The stability conditions for IP-DGM are approximately 20 and 60 per cent more restrictive than those for SEM in the acoustic and Elastic cases, respectively.

  • the interior penalty discontinuous galerkin method for Elastic Wave Propagation grid dispersion
    Geophysical Journal International, 2008
    Co-Authors: Jonas D De Basabe, Mrinal K Sen, Mary F Wheeler
    Abstract:

    SUMMARY Recently, there has been an increased interest in applying the discontinuous Galerkin method (DGM) to Wave Propagation. In this work, we investigate the applicability of the interior penalty DGM to Elastic Wave Propagation by analysing it’s grid dispersion properties, with particular attention to the effect that different basis functions have on the numerical dispersion. We consider different types of basis functions that naturally yield a diagonal mass matrix. This is relevant to seismology because a diagonal mass matrix is tantamount to an explicit and efficient time marching scheme. We find that the Legendre basis functions that are traditionally used in the DGM introduce numerical dispersion and anisotropy. Furthermore, we find that using Lagrange basis functions along with the Gauss nodes has attractive advantages for numerical Wave Propagation.

Yangkang Chen - One of the best experts on this subject based on the ideXlab platform.

  • modeling Elastic Wave Propagation using k space operator based temporal high order staggered grid finite difference method
    IEEE Transactions on Geoscience and Remote Sensing, 2017
    Co-Authors: Hanming Chen, Hui Zhou, Qingchen Zhang, Yangkang Chen
    Abstract:

    The traditional high-order staggered-grid finite-difference (SGFD) method has high-order accuracy in space, but only the second-order accuracy in time, which makes the traditional SGFD method suffer from a large temporal dispersion error during long-distance Wave Propagation. This paper develops temporal fourth- and sixth-order and spatial arbitrary evenorder SGFD schemes to model isotropic Elastic Wave Propagation. The temporal high-order SGFD schemes have smaller temporal dispersion than the traditional temporal second-order scheme, and thus allow larger time steps to attain a similar accuracy. The developed temporal high-order SGFD schemes are applied to simulate a quasi-stress–velocity Wave equation (QWE) that is derived in the framework of a $k$ -space approach. A split QWE (SQWE) is further developed, and numerical simulation of SQWE results in separated P (compressional)-Wave and S (shear)-Wave. Theoretical computational cost analysis verifies that the numerical simulation of QWE using the temporal fourthand sixth-order SGFD schemes is more efficient than the numerical simulation of the traditional stress–velocity Wave equation using the traditional temporal second-order SGFD scheme in 2-D. In 3-D, the temporal fourth-order SGFD scheme still runs faster than the traditional temporal second-order scheme; however, the temporal sixth-order scheme is more efficient only when a longer stencil length than 12 is adopted. Numerical examples confirm the correctness of the developed Elastic Wave modeling schemes.

Jesse Chan - One of the best experts on this subject based on the ideXlab platform.

  • weight adjusted discontinuous galerkin methods matrix valued weights and Elastic Wave Propagation in heterogeneous media
    International Journal for Numerical Methods in Engineering, 2018
    Co-Authors: Jesse Chan
    Abstract:

    Summary Weight-adjusted inner products [1,2] are easily invertible approximations to weighted L2 inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time-domain method for Wave Propagation which is low storage, energy stable, and high order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight-adjusted DG (WADG) methods to the case of matrix-valued weights, with the linear Elastic Wave equation as an application. We present a DG formulation of the symmetric form of the linear Elastic Wave equation, with upwind-like dissipation incorporated through simple penalty fluxes. A semi-discrete convergence analysis is given, and numerical results confirm the stability and high order accuracy of WADG for several problems in Elastic Wave Propagation. This article is protected by copyright. All rights reserved.

  • weight adjusted discontinuous galerkin methods matrix valued weights and Elastic Wave Propagation in heterogeneous media
    arXiv: Numerical Analysis, 2017
    Co-Authors: Jesse Chan
    Abstract:

    Weight-adjusted inner products are easily invertible approximations to weighted $L^2$ inner products. These approximations can be paired with a discontinuous Galerkin (DG) discretization to produce a time-domain method for Wave Propagation which is low storage, energy stable, and high order accurate for arbitrary heterogeneous media and curvilinear meshes. In this work, we extend weight-adjusted DG (WADG) methods to the case of matrix-valued weights, with the linear Elastic Wave equation as an application. We present a DG formulation of the symmetric form of the linear Elastic Wave equation, with upwind-like dissipation incorporated through simple penalty fluxes. A semi-discrete convergence analysis is given, and numerical results confirm the stability and high order accuracy of WADG for several problems in Elastic Wave Propagation.

Yalchin Efendiev - One of the best experts on this subject based on the ideXlab platform.

  • A generalized multiscale finite element method for Elastic Wave Propagation in fractured media
    GEM - International Journal on Geomathematics, 2016
    Co-Authors: Eric T Chung, Richard L Gibson, Yalchin Efendiev, Maria Vasilyeva
    Abstract:

    In this paper, we consider Elastic Wave Propagation in fractured media applying a linear-slip model to represent the effects of fractures on the Wavefield. Fractured media, typically, are highly heterogeneous due to multiple length scales. Direct numerical simulations for Wave Propagation in highly heterogeneous fractured media can be computationally expensive and require some type of model reduction. We develop a multiscale model reduction technique that captures the complex nature of the media (heterogeneities and fractures) in the coarse scale system. The proposed method is based on the generalized multiscale finite element method, where the multiscale basis functions are constructed to capture the fine-scale information of the heterogeneous, fractured media and effectively reduce the degrees of freedom. These multiscale basis functions are coupled via the interior penalty discontinuous Galerkin method, which provides a block-diagonal mass matrix. The latter is needed for fast computation in an explicit time discretization, which is used in our simulations. Numerical results are presented to show the performance of the presented multiscale method for fractured media. We consider several cases where fractured media contain fractures of multiple lengths. Our numerical results show that the proposed reduced-order models can provide accurate approximations for the fine-scale solution.

  • generalized multiscale finite element method gmsfem for Elastic Wave Propagation in heterogeneous anisotropic media
    Journal of Computational Physics, 2015
    Co-Authors: Kai Gao, Richard L Gibson, Eric T Chung, Yalchin Efendiev
    Abstract:

    It is important to develop fast yet accurate numerical methods for seismic Wave Propagation to characterize complex geological structures and oil and gas reservoirs. However, the computational cost of conventional numerical modeling methods, such as finite-difference method and finite-element method, becomes prohibitively expensive when applied to very large models. We propose a Generalized Multiscale Finite-Element Method (GMsFEM) for Elastic Wave Propagation in heterogeneous, anisotropic media, where we construct basis functions from multiple local problems for both the boundaries and interior of a coarse node support or coarse element. The application of multiscale basis functions can capture the fine scale medium property variations, and allows us to greatly reduce the degrees of freedom that are required to implement the modeling compared with conventional finite-element method for Wave equation, while restricting the error to low values. We formulate the continuous Galerkin and discontinuous Galerkin formulation of the multiscale method, both of which have pros and cons. Applications of the multiscale method to three heterogeneous models show that our multiscale method can effectively model the Elastic Wave Propagation in anisotropic media with a significant reduction in the degrees of freedom in the modeling system.