Interface conditions for acoustic and elastic wave propagation

The divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media. The physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface. The technique, which can be used with many numerical schemes, is applied to finite differences. A derivation of the acoustic wave equation, which is readily handled by this integration scheme, is outlined. Since this form of the equation is equivalent to the scalar SH wave equation, the scheme can be applied to this equation also. Each component of the elastic P‐SV equation is presented in divergence form to apply this integration scheme, naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces.

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Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

Geophysical Journal International ◽

10.1093/gji/ggz410 ◽

2019 ◽

Vol 219 (3) ◽

pp. 1900-1914 ◽

Cited By ~ 1

Author(s):

T Möller ◽

W Friederich

Keyword(s):

Wave Propagation ◽

Wave Equation ◽

Elastic Wave ◽

Discontinuous Galerkin ◽

Heterogeneous Media ◽

Rheological Behaviour ◽

Displacement Discontinuity ◽

Signal Frequency ◽

Elastic Wave Equation ◽

Numerical Flux

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

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Spectral-element simulation of two-dimensional elastic wave propagation in fully heterogeneous media on a GPU cluster

Journal of Physics Conference Series ◽

10.1088/1742-6596/1011/1/012034 ◽

2018 ◽

Vol 1011 ◽

pp. 012034

Author(s):

Indra Rudianto ◽

Sudarmaji

Keyword(s):

Wave Propagation ◽

Elastic Wave ◽

Heterogeneous Media ◽

Spectral Element ◽

Elastic Wave Propagation ◽

Two Dimensional ◽

Gpu Cluster ◽

Element Simulation

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Elastic Wave Propagation in Heterogeneous Media

The Proceedings of Conference of Kansai Branch ◽

10.1299/jsmekansai.2003.78._5-51_ ◽

2003 ◽

Vol 2003.78 (0) ◽

pp. _5-51_-_5-52_

Author(s):

Masatoshi YAMASHITA ◽

Akihiro NAKATANI ◽

Yoshikazu HIGA ◽

Hiroshi KITAGAWA

Keyword(s):

Wave Propagation ◽

Elastic Wave ◽

Heterogeneous Media ◽

Elastic Wave Propagation

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Weight-adjusted discontinuous Galerkin methods: Matrix-valued weights and elastic wave propagation in heterogeneous media

International Journal for Numerical Methods in Engineering ◽

10.1002/nme.5720 ◽

2017 ◽

Vol 113 (12) ◽

pp. 1779-1809 ◽

Cited By ~ 11

Author(s):

Jesse Chan

Keyword(s):

Wave Propagation ◽

Elastic Wave ◽

Discontinuous Galerkin ◽

Discontinuous Galerkin Methods ◽

Heterogeneous Media ◽

Elastic Wave Propagation ◽

Galerkin Methods

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Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽

10.1190/geo2017-0005.1 ◽

2017 ◽

Vol 82 (5) ◽

pp. T207-T224 ◽

Cited By ~ 15

Author(s):

Zhiming Ren ◽

Zhen Chun Li

Keyword(s):

Wave Propagation ◽

Wave Equation ◽

Finite Difference ◽

Elastic Wave ◽

High Order ◽

Elastic Wave Propagation ◽

Staggered Grid ◽

S Wave ◽

Order Accuracy ◽

Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

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