A generalized multiscale finite element method for elastic wave propagation in fractured media

Chung, Eric T.; Efendiev, Yalchin; Gibson, Richard L.; Vasilyeva, Maria

doi:10.1007/s13137-016-0081-4

Cited by 34 publications

(16 citation statements)

References 56 publications

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“…Brief introduction to the GMsFEM. The GMsFEM [9,10,12] follows the framework of the Multiscale Finite Element Method (MsFEM) and is introduced to systematically add new degrees of freedom in each coarse block. The new basis functions are computed by constructing the snapshots and performing local spectral decomposition in the snapshot space.…”

Section: Introductionmentioning

confidence: 99%

Coupling of multiscale and multi-continuum approaches

et al. 2017

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Simulating complex processes in fractured media requires some type of model reduction. Well-known approaches include multi-continuum techniques, which have been commonly used in approximating subgrid effects for flow and transport in fractured media. Our goal in this paper is to (1) show a relation between multi-continuum approaches and Generalized Multiscale Finite Element Method (GMsFEM) and (2) to discuss coupling these approaches for solving problems in complex multiscale fractured media. The GMsFEM, a systematic approach, constructs multiscale basis functions via local spectral decomposition in pre-computed snapshot spaces. We show that GMsFEM can automatically identify separate fracture networks via local spectral problems. We discuss the relation between these basis functions and continuums in multi-continuum methods. The GMsFEM can automatically detect each continuum and represent the interaction between the continuum and its surrounding (matrix). For problems with simplified fracture networks, we propose a simplified basis construction with the GMs-FEM. This simplified approach is effective when the fracture networks are known and have simplified geometries. We show that this approach can achieve a similar result compared to the results using the GMsFEM with spectral basis functions. Further, we discuss the coupling between the GMsFEM and multi-continuum approaches. In this case, many fractures are resolved while for unresolved fractures, we use a multicontinuum approach with local Representative Volume Element (RVE) information. As a result, the method deals with a system of equations on a coarse grid, where each equation represents one of the continua on the fine grid. We present various basis construction mechanisms and numerical results. The GMsFEM framework, in addition, can provide adaptive and online basis functions to improve the accuracy of coarse-grid

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Section: Introductionmentioning

confidence: 99%

Coupling of multiscale and multi-continuum approaches

et al. 2017

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“…In Schoenberg and Sayers (1995); Schubnel and Gueguen (2003), plane wave propagation was studied in elastic media weakened by open fractures; this included problems on effects of anisotropy of open crack distribution at wave propagation speeds and decay characteristics. Numerical studies were performed by Zhang (2005);Chung et al (2016) and other. In this research, the authors applied a linear-slip displacementdiscontinuity model where fracture is assumed to have a vanishing width across which the tractions are taken to be continuous; however, displacements can be discontinuous.…”

Section: Introductionmentioning

confidence: 99%

Elastic waves in fractured rocks under periodic compression

Kossovich

Talonov

Savatorova

2017

Int J Mech Mater Eng

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Background: One of the current problems in studying the mechanical properties and behavior of structurally inhomogeneous media with cracks is the characterization of acoustic wave propagation. This is especially important in Geomechanics and prognosis of earthquakes. Methods: In this work, the authors propose an approach that could simplify characterization of wave propagation in medium with cracks. It is based on homogenization procedure performed at a set of equations characterizing acoustic wave propagation in media weakened by fractures under condition of external distributed loading. Such kind of loading in most cases is close to the real one in case of consideration of Geomechanics problems. Results: On the basis of the proposed homogenization technique, we performed characterization of elastic properties and plane acoustic waves propagation in a pre-loaded linear elastic medium weakened by a large amount of cracks. We have investigated two special cases of loading: uniaxial compression and complex compression. We have also studied how the wavespeeds depend on averaged concentration and distribution of craks. Conclusions: Effective elastic properties were theoretically characterized for fractured media under external loading. The results revealed high dependency of the longitudinal wave propagation speed on the relation between stresses reasoned by an external loading.

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“…For Φ ∈ L 1 (Ω × Y ) we define 25) where [x/ε] denotes the "integer" part of x/ε with respect to the unit cube Y and {x/ε} = x/ε − [x/ε]. The operator U ε satisfies: 26) for all Φ ∈ L 1 (D × Y ), where D ε is the 2ε neighbourhood of D; Φ is regarded as 0 when x is outside D. The proof of this proposition is quite straightforward; we refer to [29] for details. We then have the following corrector result.…”

Section: Numerical Correctorsmentioning

confidence: 99%

“…The advantage of the multiscale basis is that it can be reused for different forcing functions in the right hand side. Recently the method has been extended to "Generalized Multiscale FE method" (see [37], [26] and [42]). The Heterogeneous Multiscale Method ( [35] and [1]) solves the cell problem for each macroscopic degree of freedom to account for the microscopic information.…”

Section: Introductionmentioning

confidence: 99%

Sparse tensor product finite element method for some linear and nonlinear multiscale problems

Tan¹

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W 1,4 0 (D) versus the mesh size h for 1 dimensional problem at t = 1 for Crank-Nicholson method.. . 105 3.8 The error u 1 −û 1 L 4 (D,W 1,4 # (Y)) versus the mesh size h for 1 dimensional problem at t = 1 for Crank-Nicholson method.. 105 3.9 The error u 0 −û L 0 W 1,4 0 (D) versus the mesh size h for two dimensional problem at t = 1 for backward Euler method.. 107 3.10 The error u 1 −û 1 L 4 (D,W 1,4 # (Y)) versus the mesh size h for two dimensional problem at t = 1 for backward Euler method.107 3.11 The error u 0 −û L 0 W 1,4 0 (D) versus the mesh size h for two dimensional problem at t = 1 for Crank-Nicholson method.. 108 LIST OF FIGURES xiii 3.12 The error u 1 −û 1 L 4 (D,W 1,4 # (Y)) versus the mesh size h for two dimensional problem at t = for Crank-Nicholson method. .

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A generalized multiscale finite element method for elastic wave propagation in fractured media

Cited by 34 publications

References 56 publications

Coupling of multiscale and multi-continuum approaches

Coupling of multiscale and multi-continuum approaches

Elastic waves in fractured rocks under periodic compression

Sparse tensor product finite element method for some linear and nonlinear multiscale problems

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