An official website of the United States government
Abstract We have extended the distributional finite difference method (DFDM) to simulate the seismic‐wave propagation in 3D regional earth models. DFDM shares similarities to the discontinuous finite element method on a global scale and to the finite difference method locally. Instead of using linear staggered finite‐difference operators, we employ DFDM operators based on B‐splines and a definition of derivatives in the sense of distributions, to obtain accurate spatial derivatives. The weighted residuals method used in DFDM's locally weak formalism of spatial derivatives accurately and naturally accounts for the free surface, curvilinear meshing, and solid‐fluid coupling, for which it only requires setting the shear modulus and the continuity condition to zero. The computational efficiency of DFDM is comparable to the spectral element method (SEM) due to the more accurate mass matrix and a new band‐diagonal mass factorization. Numerical examples show that the superior accuracy of the band‐diagonal mass and stiffness matrices in DFDM enables fewer points per wavelength, approaching the spectral limit, and compensates for the increased computational burden due to four Lebedev staggered grids. Specifically, DFDM needs 2.5 points per wavelength, compared to the five points per wavelength required in SEM for 0.5% waveform error in a homogeneous model. Notably, while maintaining the same accuracy in the solid domain, DFDM demonstrates superior accuracy in the fluid domain compared to SEM. To validate its accuracy and flexibility, we present various 3D benchmarks involving homogeneous and heterogeneous regional elastic models and solid‐fluid coupling in both Cartesian and spherical settings.
more »
« less
2-D seismic wave propagation using the distributional finite-difference method: further developments and potential for global seismology
SUMMARY We present a time-domain distributional finite-difference scheme based on the Lebedev staggered grid for the numerical simulation of wave propagation in acoustic and elastic media. The central aspect of the proposed method is the representation of the stresses and displacements with different sets of B-splines functions organized according to the staggered grid. The distributional finite-difference approach allows domain-decomposition, heterogeneity of the medium, curvilinear mesh, anisotropy, non-conformal interfaces, discontinuous grid and fluid–solid interfaces. Numerical examples show that the proposed scheme is suitable to model wave propagation through the Earth, where sharp interfaces separate large, relatively homogeneous layers. A few domains or elements are sufficient to represent the Earth’s internal structure without relying on advanced meshing techniques. We compare seismograms obtained with the proposed scheme and the spectral element method, and we show that our approach offers superior accuracy, reduced memory usage, and comparable efficiency.
more »
« less
- Award ID(s):
- 2418419
- PAR ID:
- 10598889
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Geophysical Journal International
- Volume:
- 237
- Issue:
- 1
- ISSN:
- 0956-540X
- Page Range / eLocation ID:
- 339 to 363
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to “upwind” the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.more » « less
-
Abstract Rational approximation recently emerged as an efficient numerical tool for the solution of exterior wave propagation problems. Currently, this technique is limited to wave media which are invariant along the main propagation direction. We propose a new model order reduction-based approach for compressing unbounded waveguides with layered inclusions. It is based on the solution of a nonlinear rational least squares problem using the RKFIT method. We show that approximants can be converted into an accurate finite difference representation within a rational Krylov framework. Numerical experiments indicate that RKFIT computes more accurate grids than previous analytic approaches and even works in the presence of pronounced scattering resonances. Spectral adaptation effects allow for finite difference grids with dimensions near or even below the Nyquist limit.more » « less
-
Abstract The finite‐difference time‐domain (FDTD) method was previously applied to high‐frequency electromagnetic wave propagation through 250 km of theFregion of the ionosphere. That modeling approach was limited to electromagnetic wave propagation above the critical frequency of the ionospheric plasma, and it did not include the lower ionosphere layers or the top of theF‐region. This paper extends the previous modeling methodology to frequencies below the critical frequency of the plasma and to altitudes encompassing the ionosphere. The following changes to the previous work were required to generate this model: (a) theD,Eand top of theFregions of the ionosphere were added; and (b) the perfectly matched layer absorbing boundary on the top side of the grid was replaced with a collisional plasma to prevent reflections. We apply this model to the study of extremely low frequency (ELF) and very low frequency (VLF) electric power line harmonic radiation (PLHR) through the ionosphere. The model is compared against analytical predictions and applied to PLHR propagation in polar, mid‐latitude and equatorial regions. Also, to further demonstrate the advantages of the grid‐based FDTD method, PLHR propagation through a polar cap patch with inhomogeneities is studied. The presented modeling methodology may be applied to additional scenarios in a straightforward manner and can serve as a useful tool for better tracking and studying electromagnetic wave propagation through the ionosphere at any latitude and in the presence of irregularities of any size and shape.more » « less
-
Abstract We develop a generalized interpolation material point method (GIMPM) for the shallow shelf approximation (SSA) of ice flow. The GIMPM, which can be viewed as a particle version of the finite element method, is used here to solve the shallow shelf approximations of the momentum balance and ice thickness evolution equations. We introduce novel numerical schemes for particle splitting and integration at domain boundaries to accurately simulate the spreading of an ice shelf. The advantages of the proposed GIMPM‐SSA framework include efficient advection of history or internal state variables without diffusion errors, automated tracking of the ice front and grounding line at sub‐element scales, and a weak formulation based on well‐established conventions of the finite element method with minimal additional computational cost. We demonstrate the numerical accuracy and stability of the GIMPM using 1‐D and 2‐D benchmark examples. We also compare the accuracy of the GIMPM with the standard material point method (sMPM) and a reweighted form of the sMPM. We find that the grid‐crossing error is very severe for SSA simulations with the sMPM, whereas the GIMPM successfully mitigates this error. While the grid‐crossing error can be reasonably reduced in the sMPM by implementing a simple material point reweighting scheme, this approach it not as accurate as the GIMPM. Thus, we illustrate that the GIMPM‐SSA framework is viable for the simulation of ice sheet‐shelf evolution and enables boundary tracking and error‐free advection of history or state variables, such as ice thickness or damage.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/gji/ggae025
