In some applications, it is reasonable to assume that geodesics (rays) have a consistent orientation so that a time-harmonic elastic wave equation may be viewed as an evolution equation in one of the spatial directions. With such applications in mind, motivated by our recent work [Hadamard-
Babich ansatz for point-source elastic wave equations in variable media at high frequencies, Multiscale Model Simul. 19/1 (2021) 46–86], we propose a new truncated Hadamard-Babich ansatz based globally valid asymptotic method, dubbed the fast Huygens sweeping method, for computing Green’s functions of frequency-domain point-source elastic wave equations in inhomogeneous
media in the high-frequency asymptotic regime and in the presence of caustics. The first novelty of the fast Huygens sweeping method is that the Huygens-Kirchhoff secondary-source principle is used to integrate many locally valid asymptotic solutions to yield a globally valid asymptotic solution so that caustics can be treated automatically. This yields uniformly accurate solutions
both near the source and away from it. The second novelty is that a butterfly algorithm is adapted to accelerate matrix-vector products induced by the Huygens-Kirchhoff integral. The new method enjoys the following desired features: (1) it treats caustics automatically; (2) precomputed asymptotic
ingredients can be used to construct Green’s functions of elastic wave equations for many different point sources and for arbitrary frequencies; (3) given a specified number of points per wavelength, it constructs Green’s functions in nearly optimal complexity O(N logN) in terms of the total number of mesh points N, where the prefactor of the complexity depends only on the
specified accuracy and is independent of the frequency parameter. Three-dimensional numerical examples are presented to demonstrate the performance and accuracy of the new method.
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Green's function for anisotropic dispersive poroelastic media based on the Radon transform and eigenvector diagonalization
A compact Green's function for general dispersive anisotropic poroelastic media in a full-frequency regime is presented for the first time. First, starting in a frequency domain, the anisotropic dispersion is exactly incorporated into the constitutive relationship, thus avoiding fractional derivatives in a time domain. Then, based on the Radon transform, the original three-dimensional differential equation is effectively reduced to a one-dimensional system in space. Furthermore, inspired by the strategy adopted in the characteristic analysis of hyperbolic equations, the eigenvector diagonalization method is applied to decouple the one-dimensional vector problem into several independent scalar equations. Consequently, the fundamental solutions are easily obtained. A further derivation shows that Green's function can be decomposed into circumferential and spherical integrals, corresponding to static and transient responses, respectively. The procedures shown in this study are also compatible with other pertinent multi-physics coupling problems, such as piezoelectric, magneto-electro-elastic and thermo-elastic materials. Finally, the verifications and validations with existing analytical solutions and numerical solvers corroborate the correctness of the proposed Green's function.
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- Award ID(s):
- 1812573
- NSF-PAR ID:
- 10199183
- Date Published:
- Journal Name:
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- Volume:
- 475
- Issue:
- 2221
- ISSN:
- 1364-5021
- Page Range / eLocation ID:
- 20180610
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1098/rspa.2018.0610
