Wave Equation In Frequency Domain Research Articles

Full wave modeling of radio-frequency beams in tokamaks in the electron cyclotron frequency range

Simulation of full wave, without paraxial approximation, high-resolution solution of wave equations in frequency domain in the electron cyclotron resonance (ECR) frequency range for realistic Tokamak plasma parameters became possible by using recently formulated hybrid iterative algorithm [Svidzinski et al., Phys. Plasmas 25, 082509 (2018)] for numerically solving discretized wave equations. This approach combines time evolution and iterative relaxation techniques into iteration cycles. This algorithm is implemented in 2D code FullWave, solving wave equations in Tokamaks in cold and hot plasma models, and it has been tested in 3D full wave iterative RF beams simulation tool, which is presently being developed to model 3D ECRH RF beams in fusion devices using dynamic grid adaptation. The results of 2D full wave modeling, assuming specified toroidal mode number, of ECRH RF beams in DIII-D plasma, performed in the cold and hot plasma models for outboard and top launch scenarios using FullWave are presented. Nonlocal hot plasma response model, based on accurate numerical solution of linearized Vlasov equation, is used to model beam propagation and absorption in the 2nd electron cyclotron harmonic region. Demonstration of capability of the hybrid iterative algorithm to model ECRH RF beams in 3D is made by simulating a substantial part of realistic beam in DIII-D, launched from outboard side of the machine. All relevant physics of RF beam propagation, most of which is not captured in paraxial approximation, such as beam's divergence, interference between the X and O modes in the beam, X-O mode conversion, beam splitting into the X and O mode beams, transformation of beam's cross section, and absorption at the 2nd electron cyclotron harmonic, is captured in the simulations. A numerical technique to find an optimal beam polarization at the launcher to launch a nearly pure X or O mode beam in plasma is developed and tested.

Practical Aspects of Physics-Informed Neural Networks Applied to Solve Frequency-Domain Acoustic Wave Forward Problem

Abstract Physics-informed neural networks (PINNs) have been used by researchers to solve partial differential equation (PDE)-constrained problems. We evaluate PINNs to solve for frequency-domain acoustic wavefields. PINNs can solely use PDEs to define the loss function for optimization without the need for labels. Partial derivatives of PDEs are calculated by mesh-free automatic differentiations. Thus, PINNs are free of numerical dispersion artifacts. It has been applied to the scattered acoustic wave equation, which relied on boundary conditions (BCs) provided by the background analytical wavefield. For a more direct implementation, we solve the nonscattered acoustic wave equation, avoiding limitations related to relying on the background homogeneous medium for BCs. Experiments support our following insights. Although solving time-domain wave equations using PINNs does not require absorbing boundary conditions (ABCs), ABCs are required to ensure a unique solution for PINNs that solve frequency-domain wave equations, because the single-frequency wavefield is not localized and contains wavefield information over the full domain. However, it is not trivial to include the ABC in the PINN implementation, so we develop an adaptive amplitude-scaled and phase-shifted sine activation function, which performs better than the previous implementations. Because there are only two outputs for the fully connected neural network (FCNN), we validate a linearly shrinking FCNN that can achieve a comparable and even better accuracy with a cheaper computational cost. However, there is a spectral bias problem, that is, PINNs learn low-frequency wavefields far more easily than higher frequencies, and the accuracy of higher frequency wavefields is often poor. Because the shapes of multifrequency wavefields are similar, we initialize the FCNN for higher frequency wavefields by that of the lower frequencies, partly mitigating the spectral bias problem. We further incorporate multiscale positional encoding to alleviate the spectral bias problem. We share our codes, data, and results via a public repository.

Simulating seismic multifrequency wavefields with the Fourier feature physics-informed neural network

SUMMARY To simulate seismic wavefields with a frequency-domain wave equation, conventional numerical methods must solve the equation sequentially to obtain the wavefields for different frequencies. The monofrequency equation has the form of a Helmholtz equation. When solving the Helmholtz equation for seismic wavefields with multiple frequencies, a physics-informed neural network (PINN) can be used. However, the PINN suffers from the problem of spectral bias when approximating high-frequency components. We propose to simulate seismic multifrequency wavefields using a PINN with an embedded Fourier feature. The input to the Fourier feature PINN for simulating multifrequency wavefields is 4-D, namely the horizontal and vertical spatial coordinates of the model, the horizontal position of the source, and the frequency, and the output is multifrequency wavefields at arbitrary source positions. While an effective Fourier feature initialization strategy can lead to optimal convergence in training this network, the Fourier feature PINN simulates multifrequency wavefields with reasonable efficiency and accuracy.

Inverse Solution of Thermoacoustic Wave Equation for Cylindrical Layered Media

Thermoacoustic imaging is a crossbred approach taking advantages of electromagnetic and ultrasound disciplines, together. A significant number of current medical imaging strategies are based on reconstruction of source distribution from information collected by sensors over a surface covering the region to be imaged. Reconstruction in thermoacoustic imaging depends on the inverse solution of thermoacoustic wave equation. Homogeneous assumption of tissue to be imaged results in degradation of image quality. In our paper, inverse solution of the thermoacoustic wave equation using layered tissue model consisting of concentric annular layers on a cylindrical cross-section is investigated for cross-sectional thermoacustic imaging of breast and brain. By using Green’s functions and surface integral methods we derive an exact analytic inverse solution of thermoacoustic wave equation in frequency domain. Our inverse solution is an extension of conventional solution to layered cylindrical structures. By carrying out simulations, using numerical test phantoms consisting of thermoacoustic sources distributed in the layered model, our layered medium assumption solution was tested and benchmarked with conventional solutions based on homogeneous medium assumption in frequency domain. In thermoacoustic image reconstruction, where the medium is assumed as homogeneous medium, the solution of nonhomogeneous thermoacoustic wave equation results in geometrical distortions, artifacts and reduced image resolution due to inconvenient medium assumptions.

On the convective wave equation for the investigation of combustor stability using FEM-methods

Solving the Helmholtz equation with spatially resolved finite element method (FEM) approaches is a well-established and cost-efficient methodology to numerically predict thermoacoustic instabilities. With the implied zero Mach number assumption all interaction mechanisms between acoustics and the mean flow velocity including the advection of acoustic waves are neglected. Incorporating these mechanisms requires higher-order approaches that come at massively increased computational cost. A tradeoff might be the convective wave equation in frequency domain, which covers the advection of waves and comes at equivalent cost as the Helmholtz equation. However, with this equation only being valid for uniform mean flow velocities it is normally not applicable to combustion processes. The present paper strives for analyzing the introduced errors when applying the convective wave equation to thermoacoustic stability analyses. Therefore, an acoustically consistent, inhomogeneous convective wave equation is derived first. Similar to Lighthill’s analogy, terms describing the interaction between acoustics and non-uniform mean flows are considered as sources. For the use with FEM approaches, a complete framework of the equation in weak formulation is provided. This includes suitable impedance boundary conditions and a transfer matrix coupling procedure. In a modal stability analysis of an industrial gas turbine combustion chamber, the homogeneous wave equation in frequency domain is subsequently compared to the Helmholtz equation and the consistent Acoustic Perturbation Equations (APE). The impact of selected source terms on the solution is investigated. Finally, a methodology using the convective wave equation in frequency domain with vanishing source terms in arbitrary mean flow fields is presented.

A versatile framework to solve the Helmholtz equation using physics-informed neural networks

SUMMARYSolving the wave equation to obtain wavefield solutions is an essential step in illuminating the subsurface using seismic imaging and waveform inversion methods. Here, we utilize a recently introduced machine-learning based framework called physics-informed neural networks (PINNs) to solve the frequency-domain wave equation, which is also referred to as the Helmholtz equation, for isotropic and anisotropic media. Like functions, PINNs are formed by using a fully connected neural network (NN) to provide the wavefield solution at spatial points in the domain of interest, in which the coordinates of the point form the input to the network. We train such a network by backpropagating the misfit in the wave equation for the output wavefield values and their derivatives for many points in the model space. Generally, a hyperbolic tangent activation is used with PINNs, however, we use an adaptive sinusoidal activation function to optimize the training process. Numerical results show that PINNs with adaptive sinusoidal activation functions are able to generate frequency-domain wavefield solutions that satisfy wave equations. We also show the flexibility and versatility of the proposed method for various media, including anisotropy, and for models with strong irregular topography.

Solving the frequency-domain acoustic VTI wave equation using physics-informed neural networks

SUMMARY Frequency-domain wavefield solutions corresponding to the anisotropic acoustic wave equation can be used to describe the anisotropic nature of the Earth. To solve a frequency-domain wave equation, we often need to invert the impedance matrix. This results in a dramatic increase in computational cost as the model size increases. It is even a bigger challenge for anisotropic media, where the impedance matrix is far more complex. In addition, the conventional finite-difference method produces numerical dispersion artefacts in solving acoustic wave equations for anisotropic media. To address these issues, we use the emerging paradigm of physics-informed neural networks (PINNs) to obtain wavefield solutions for an acoustic wave equation for transversely isotropic (TI) media with a vertical axis of symmetry (VTI). PINNs utilize the concept of automatic differentiation to calculate their partial derivatives, which are free of numerical dispersion artefacts. Thus, we use the wave equation as a loss function to train a neural network to provide functional solutions to the acoustic VTI form of the wave equation. Instead of predicting the pressure wavefields directly, we solve for the scattered pressure wavefields to avoid dealing with the point-source singularity. We use the spatial coordinates as input data to the network, which outputs the real and imaginary parts of the scattered wavefields and auxiliary function. After training a deep neural network, we can evaluate the wavefield at any point in space almost instantly using this trained neural network without calculating the impedance matrix inverse. We demonstrate these features on a simple 2-D anomaly model and a 2-D layered model. Additional tests on a modified 3-D Overthrust model and a 2-D model with irregular topography further validate the effectiveness of the proposed method.

A high-precision low-dispersive nearly analytic difference method with its application in frequency-domain seismic waveform inversion

ABSTRACTTo improve the efficiency of seismic waveform inversion, we construct a novel high-order (6th-order) frequency-domain nearly analytic discrete method for forward modelling in inversion processes. Compared with some existing numerical schemes, this method is more powerful in suppressing numerical dispersion and enhancing the accuracy of computational results based on coarse discrete grids. We describe the discretisation of the frequency-domain wave equation, investigate the efficiency of wave-field simulation and perform numerical dispersion analysis. The inversion efficiency can also be improved when the inexact rotated block triangular preconditioned Krylov iteration solvers are incorporated into the corresponding linear system. Moreover, the inversion behaviour of frequency-domain NAD methods is also systematically discussed as one of its important applications. We take three representative media models as examples and the reliable inversion results are obtained reflecting the effectiveness of the proposed methods.

Generalized stiffness reduction method to remove the artificial edge-effects for seismic wave modelling in elastic anisotropic media

SUMMARYIn seismic wave modelling, the boundary reflections caused by the computational grid edges should be reduced to produce accurate simulation results. The perfectly matched layer (PML) method is one of the popular techniques to suppress such artificial reflections, because it can be easily applied to the first-order wave equation in many numerical methods. However, one issue of the PML method is that the stability condition might be violated in complex elastic anisotropic media. In these cases, the PML method will not attenuate the boundary reflections but rather introduce strong artefacts in the simulation results. To tackle this problem, we propose a generalized stiffness reduction method (GSRM) as a substitute for the PML method. We first derive the stability conditions of the PML method and analyse the suitable conditions for their application to time- and frequency-domain seismic wave modelling. Then, we develop a simple and effective numerical implementation of the GSRM to attenuate the boundary reflections and apply it to seismic wave modelling in elastic anisotropic media. We give some numerical experiments to demonstrate the feasibility and advantages of the GSRM compared to the PML method. Numerical examples show the GSRM is conceptually simpler, more computationally efficient and more straightforward in terms of numerical implementation than the PML method for seismic modelling using either first- or second-order time- and frequency-domain wave equations.

Optimized compact finite difference scheme for frequency-domain acoustic wave equation

Frequency-domain numerical simulation is the most important foundation of frequency-domain full-waveform inversion and reverse time migration. The accuracy of numerical simulation seriously affects the results of the seismic inversion and image. In this article, we develop an optimized compact finite difference scheme for acoustic wave equation in frequency domain to improve numerical simulation accuracy. For the sake of avoiding the extra memory and computational costs caused by solving the inverse of a pentadiagonal band matrix, we calculate the optimized compact finite difference discrete operator for the Laplace operator in the numerical simulation. Although the optimized compact finite difference scheme has only second-order formal accuracy, it has a spectral-like resolution feature. This method can significantly reduce the numerical dispersion and the numerical anisotropy. We find that the results of the optimized compact finite difference scheme agree well with the analytic solution according to accuracy analysis. Two numerical simulations are done to verify the theoretical analysis of the optimized compact finite difference scheme.

A new point-weighting finite-difference modelling for the frequency-domain wave equation

In this paper, we propose a new 21 point finite-difference scheme for modelling the frequency-domain wave equation in 2-dimensional domain. To discretize the second derivative term, we develop a modified central difference scheme in which each of the gridpoints is replaced with a linear combination of it and its neighbouring girdpoints along one direction. For the discretization of the zeroth-order term, we use the weighted average of all the 21 points. The combination coefficients and the weight parameters are determined by minimizing the numerical dispersion. In comparison with the scheme of a derivative-weighting one, the new scheme has a much better performance in reducing the numerical dispersion when the step sizes are not equal in different directions. Numerical experiments are presented to illustrate the effectiveness of the new scheme.

Modeling the interference between shear and longitudinal waves under high intensity focused ultrasound propagation in bone

Magnetic resonance-guided high intensity focused ultrasound (MR-HIFU) is a noninvasive thermal technique that enables rapid heating of a specific area in the human body. Its clinical relevance has been proven for the treatments of soft tissue tumors, like uterine fibroids, and for the treatments of solid tumors in bone. In MR-HIFU treatment, MR-thermometry is used to monitor the temperature evolution in soft tissue. However, this technique is currently unavailable for bone tissue. Computer models can play a key role in the accurate prediction and monitoring of temperature. Here, we present a computer ray tracing model that calculates the heat production density in the focal region. This model accounts for both the propagation of shear waves and the interference between longitudinal and shear waves. The model was first compared with a finite element approach which solves the Helmholtz equation in soft tissue and the frequency-domain wave equation in bone. To obtain the temperature evolution in the focal region, the heat equation was solved using the heat production density generated by the raytracer as a heat source. Then, we investigated the role of the interaction between shear and longitudinal waves in terms of dissipated power and temperature output. The results of our model were in agreement with the results obtained by solving the Helmholtz equation and the frequency-domain wave equation, both in soft tissue and bone. Our results suggest that it is imperative to include both shear waves and their interference with longitudinal waves in the model when simulating high intensity focused ultrasound propagation in solids. In fact, when modeling HIFU treatments, omitting the interference between shear and longitudinal waves leads to an over-estimation of the temperature increase in the tissues.

A LBIE-RBF solution to the convected wave equation for flow acoustics

In this paper, the meshless local boundary integral equation (LBIE) method is implemented for the solution of convected wave equation in frequency domain. Such a problem plays an important role for the noise prediction in flow acoustics. The case of a uniform flow in x-direction is analyzed. For the resultant convected Helmholtz equation, the free-space Green's function has a more complicated form than the Green's function for the classical Helmholtz equation, though it reduces to the latter when a coordinate (Prandtl–Glauert) transformation is applied. Furthermore, the LBIE method requires the determination of the solution of the corresponding Dirichlet's problem in local subdomains (companion solution). Therefore, the relevant companion solution is derived. Radial basis functions are used for the interpolations of the field variable. Numerical examples are shown for cases up to frequency of 5000 Hz and Mach number of 0.5, and the results are compared with FEM software (COMSOL).

An equivalent fluid model based finite-difference time-domain algorithm for sound propagation in porous material with rigid frame.

An equivalent fluid model based finite-difference time-domain algorithm is proposed to simulate the frequency characteristic of porous material with rigid frame. The rigid-frame porous material is replaced by an equivalent fluid characterized by the effective density and the effective bulk modulus, which reflect the viscous effects and thermal effects in the porous material, respectively. The effective density and the effective bulk modulus are frequency-dependent complex values, which are designed in the form of infinite impulse response filters in the frequency domain. The Z transform theory is used to discretize the frequency-domain wave equations. The accuracy of the proposed algorithm is preliminarily validated by good agreement between the calculated and measured normal-incidence sound absorption coefficients of two samples composed of different multiple-layered materials in a wide frequency range.

A developed nearly analytic discrete method for forward modeling in the frequency domain

High-efficiency forward modeling methods play a fundamental role in full waveform inversion (FWI). In this paper, the developed nearly analytic discrete (DNAD) method is proposed to accelerate frequency-domain forward modeling processes. We first derive the discretization of frequency-domain wave equations via numerical schemes based on the nearly analytic discrete (NAD) method to obtain a linear system. The coefficients of numerical stencils are optimized to make the linear system easier to solve and to minimize computing time. Wavefield simulation and numerical dispersion analysis are performed to compare the numerical behavior of DNAD method with that of the conventional NAD method. The results demonstrate the superiority of our proposed method. Finally, the DNAD method is implemented in frequency-domain FWI, and high-resolution inverse results are obtained.

Frequency Domain Wave Equation Inversion and Its Application on the Heterogeneous Reservoir Model Data

Seismic full waveform inversion seeks to make use of the full information based on full wave field modeling to extract quantitative information from seismograms. Its serious nonlinearity and high dependence on initial velocity model often results in unsatisfactory inversion results in paleo-karsts carbonate reservoir characterized by strong heterogeneity. The paper presents an improved strategy of multi-scale inversion to establish velocity field model of waveform tomography. the forward wave equation algorithm was derived in frequency domain, and then the Matrix formalism for the iterative inverse methods is derived by gradient methods to speed up calculation and to avoid convergence to local minimum value. After massive amount of frequencies tests, the appropriate bandwidth are extracted, and the velocity field calculated at low frequency is used as the input of the high frequency. After the iteration, the accurate velocity field is inverted. Finally, frequency domain wave equation full waveform inversion in mathematical and physical models is conducted in order to verify the inverse program. The method of selecting the inverse frequencies is proved to be effective.

Numerical investigation of vibration and dynamic pressure of a vertical axis wind turbine

In the environmental field, the problems of noise reduction have become a major preoccupation, particularly on the noise generated by the acoustic radiation pressure produced by wind turbines. This paper is aimed at presenting the investigation on the application of variational indirect boundary element method for study the acoustic radiation pressure produced by vertical-axis wind turbine. For this initiative, we considered Neumann boundary condition. The formulation has two advantages: the first one is to avoid the meshing of the fluid domain; the second advantage is to treat the singular integral of the Green's function, solution of fundamental solution of the wave equation in frequency domain.

Frequency domain optoacoustic tomography using amplitude and phase

We introduce optoacoustic tomographic imaging using intensity modulated light sources and collecting amplitude and phase information in the frequency domain. Imaging is performed at multiple modulation frequencies. The forward modeling uses the Green's function solution to the pressure wave equation in frequency domain and the resulting inverse problem is solved using regularized least squares minimization. We study the effect of the number of frequencies and of the bandwidth employed on the image quality achieved. The possibility of employing an all-frequency domain optoacoustic imaging for experimental measurements is studied as a function of noise. We conclude that frequency domain optoacoustic tomography may evolve to a practical experimental method using light intensity modulated sources, with advantages over time-domain optoacoustics.

Wave equation tomographic velocity inversion method based on the Born/Rytov approximation

This paper discusses Born/Rytov approximation tomographic velocity inversion methods constrained by the Fresnel zone. Calculations of the sensitivity kernel function and traveltime residuals are critical in tomographic velocity inversion. Based on the Born/Rytov approximation of the frequency-domain wave equation, we derive the traveltime sensitivity kernels of the wave equation on the band-limited wave field and simultaneously obtain the traveltime residuals based on the Rytov approximation. In contrast to single-ray tomography, the modified velocity inversion method improves the inversion stability. Tests of the near-surface velocity model and field data prove that the proposed method has higher accuracy and Computational efficiency than ray theory tomography and full waveform inversion methods.

Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media

We interpret the purely spectral forward Maxwell equation with up to 3${^{\rm rd}}$ order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named nonlinear wave equation in frequency domain, includes both quadratic and cubic nonlinearities, delayed Raman effects and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due to competition between self-steepening, Raman effects and self-steepening-like effects from cascading originating in the group-velocity mismatch between the pump and second harmonic. We analyze the first-order contributions, and show that this balance can be broken to create fast or slow pulses. Through further simulations we demonstrate few-cycle compressed solitons in extremely short crystals, where spectral phenomena such as blue/red shifting, non-stationary radiation in accordance with the non-local phase matching condition and dispersive-wave generation are observed and marked, which help improving the experimental knowledge of cascading nonlinear soliton pulse compression.