A Smoothing Scheme for Seismic Wave Propagation Simulation with a Mixed FEM of Diffusionized Wave Equation

2021 ◽  
pp. 2150061
Author(s):  
Ryuta Imai ◽  
Naoki Kasui ◽  
Masayuki Yamada ◽  
Koji Hada ◽  
Hiroyuki Fujiwara
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Seismic Wave ◽  
Time Integration ◽  
Coupled System ◽  
Seismic Wave Propagation ◽  
Implicit Time ◽  
Integration Scheme ◽  
Weak Formulation ◽  
Mixed Fem

In this paper, we propose a smoothing scheme for seismic wave propagation simulation. The proposed scheme is based on a diffusionized wave equation with the fourth-order spatial derivative term. So, the solution requires higher regularity in the usual weak formulation. Reducing the diffusionized wave equation to a coupled system of diffusion equations yields a mixed FEM to ease the regularity. We mathematically explain how our scheme works for smoothing. We construct a semi-implicit time integration scheme and apply it to the wave equation. This numerical experiment reveals that our scheme is effective for filtering short wavelength components in seismic wave propagation simulation.

Optimal Third-Order Symplectic Integration Modeling of Seismic Acoustic Wave Propagation

2020 ◽  
Vol 110 (2) ◽  
pp. 754-762 ◽  
Author(s):  
Chuan Li ◽  
Jianxin Liu ◽  
Bo Chen ◽  
Ya Sun
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Seismic Wave ◽  
Truncation Error ◽  
Seismic Wave Propagation ◽  
Symplectic Integrator ◽  
Third Order ◽  
The Third ◽  
2D And 3D

ABSTRACT Seismic wavefield modeling based on the wave equation is widely used in understanding and predicting the dynamic and kinematic characteristics of seismic wave propagation through media. This article presents an optimal numerical solution for the seismic acoustic wave equation in a Hamiltonian system based on the third-order symplectic integrator method. The least absolute truncation error analysis method is used to determine the optimal coefficients. The analysis of the third-order symplectic integrator shows that the proposed scheme exhibits high stability and minimal truncation error. To illustrate the accuracy of the algorithm, we compare the numerical solutions generated by the proposed method with the theoretical analysis solution for 2D and 3D seismic wave propagation tests. The results show that the proposed method reduced the phase error to the eighth-order magnitude accuracy relative to the exact solution. These simulations also demonstrated that the proposed third-order symplectic method can minimize numerical dispersion and preserve the waveforms during the simulation. In addition, comparing different central frequencies of the source and grid spaces (90, 60, and 20 m) for simulation of seismic wave propagation in 2D and 3D models using symplectic and nearly analytic discretization methods, we deduce that the suitable grid spaces are roughly equivalent to between one-fourth and one-fifth of the wavelength, which can provide a good compromise between accuracy and computational cost.

Finite Element Simulation of Axisymmetric Elastic and Electroelastic Wave Propagation Using Local-Domain Wave Packet Enrichment

2021 ◽  
Vol 144 (2) ◽  
Author(s):  
Amit Kumar ◽  
Santosh Kapuria
Keyword(s):  
Finite Element ◽  
Wave Propagation ◽  
Wave Packet ◽  
Equations Of Motion ◽  
Time Integration ◽  
Coupled System ◽  
Integration Scheme ◽  
Local Domain ◽  
Complex Wave ◽  
Wave Modes

Abstract A local-domain wave packet enriched multiphysics finite element (FE) formulation is employed for accurately solving axisymmetric wave propagation problems in elastic and piezoelastic media, involving complex wave modes and sharp jumps at the wavefronts, which pose challenges to the conventional FE solutions. The conventional Lagrangian interpolations for the displacement and electric potential fields are enriched with the element-domain sinusoidal functions that satisfy the partition of unity condition. The extended Hamilton’s principle is employed to derive the coupled system of equations of motion which is solved using the simple Newmark-β direct time integration scheme without resorting to any remeshing near the wavefronts or post-processing. The performance of the enrichment is assessed for the axisymmetric problems of impact waves in elastic and piezoelectric cylinders and elastic half-space, bulk and Rayleigh waves in the semi-infinite elastic domain and ultrasonic Lamb wave actuation and propagation in plate-piezoelectric transducer system. The element shows significant improvement in the computational efficiency and accuracy over the conventional FE for all problems, including those involving multiple complex wave modes and sharp discontinuities in the fields at the wavefronts.

Electromagnetic wave propagation based upon spectral-element methodology in dispersive and attenuating media

2019 ◽  
Vol 220 (2) ◽  
pp. 951-966
Author(s):  
Christina Morency
Keyword(s):  
Wave Propagation ◽  
Dielectric Permittivity ◽  
Seismic Wave ◽  
Wave Equations ◽  
Mass Matrix ◽  
Time Integration ◽  
Spectral Element ◽  
Seismic Wave Propagation ◽  
Transverse Magnetic ◽  
Integration Schemes

SUMMARY We build on mathematical equivalences between Maxwell’s wave equations for an electromagnetic medium and elastic seismic wave equations. This allows us to readily model Maxwell’s wave propagation in the spectral-element codes SPECFEM2D and SPECFEM3D, written for acoustic, viscoelastic and poroelastic seismic wave propagation, providing the ability to handle complex geometries, inherent to finite-element methods and retaining the strength of exponential convergence and accuracy due to the use of high-degree polynomials to interpolate field functions on the elements, characteristic to spectral-element methods (SEMs). Attenuation and dispersion processes related to the frequency dependence of dielectric permittivity and conductivity are also included using a Zener model, similar to shear attenuation in viscoelastic media or viscous diffusion in poroelastic media, and a Kelvin–Voigt model, respectively. Ability to account for anisotropic media is also discussed. Here, we limit ourselves to certain dielectric permittivity tensor geometries, in order to conserve a diagonal mass matrix after discretization of the system of equations. Doing so, simulation of Maxwell’s wave equations in the radar frequency range based on SEM can be solved using explicit time integration schemes well suited for parallel computation. We validate our formulation with analytical solutions. In 2-D, our implementation allows for the modelling of both a transverse magnetic (TM) mode, suitable for surface based reflection ground penetration radar type of applications, and a transverse electric (TE) mode more suitable for crosshole and vertical radar profiling setups. Two 2-D examples are designed to demonstrated the use of the TM and TE modes. A 3-D example is also presented, which allows for the full TEM solution, different antenna orientations, and out-of-plane variations in material properties.

scholarly journals Seismic Wave Propagation and Inversion with Neural Operators

2021 ◽  
Vol 1 (3) ◽  
pp. 126-134
Author(s):  
Yan Yang ◽  
Angela F. Gao ◽  
Jorge C. Castellanos ◽  
Zachary E. Ross ◽  
Kamyar Azizzadenesheli ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Seismic Wave ◽  
Automatic Differentiation ◽  
Velocity Structure ◽  
Waveform Inversion ◽  
Seismic Wave Propagation ◽  
Source Location ◽  
Velocity Models ◽  
Reverse Mode

Abstract Seismic wave propagation forms the basis for most aspects of seismological research, yet solving the wave equation is a major computational burden that inhibits the progress of research. This is exacerbated by the fact that new simulations must be performed whenever the velocity structure or source location is perturbed. Here, we explore a prototype framework for learning general solutions using a recently developed machine learning paradigm called neural operator. A trained neural operator can compute a solution in negligible time for any velocity structure or source location. We develop a scheme to train neural operators on an ensemble of simulations performed with random velocity models and source locations. As neural operators are grid free, it is possible to evaluate solutions on higher resolution velocity models than trained on, providing additional computational efficiency. We illustrate the method with the 2D acoustic wave equation and demonstrate the method’s applicability to seismic tomography, using reverse-mode automatic differentiation to compute gradients of the wavefield with respect to the velocity structure. The developed procedure is nearly an order of magnitude faster than using conventional numerical methods for full waveform inversion.

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