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Starting from Kirchhoff-Huygens representation and Duhamel's principle of time-domain wave equations, we propose novel butterfly-compressed Hadamard integrators for self-adjoint wave equations in both time and frequency domain in an inhomogeneous medium. First, we incorporate the leading term of Hadamard's ansatz into the Kirchhoff-Huygens representation to develop a short-time valid propagator. Second, using Fourier transform in time, we derive the corresponding Eulerian short-time propagator in the frequency domain; on top of this propagator, we further develop a time-frequency-time (TFT) method for the Cauchy problem of time-domain wave equations. Third, we further propose a time-frequency-time-frequency (TFTF) method for the corresponding point-source Helmholtz equation, which provides Green's functions of the Helmholtz equation for all angular frequencies within a given frequency band. Fourth, to implement the TFT and TFTF methods efficiently, we introduce butterfly algorithms to compress oscillatory integral kernels at different frequencies. As a result, the proposed methods can construct wave field beyond caustics implicitly and advance spatially overturning waves in time naturally with quasi-optimal computational complexity and memory usage. Furthermore, once constructed the Hadamard integrators can be employed to solve both time-domain wave equations with various initial conditions and frequency-domain wave equations with different point sources. Numerical examples for two-dimensional wave equations illustrate the accuracy and efficiency of the proposed methods.
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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
- 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
