Helmholtz Equation Research Articles

Holography optimization based on combining iterative Green's function algorithm and deep learning method.

In this Letter, we present a novel, to the best of our knowledge, approach that combines a new numerical iterative algorithm with a physics-informed neural network (PINN) architecture to solve the Helmholtz equation, thereby achieving highly generalized refractive index modulation holography. Firstly, we design a non-uniform refractive index convolutional neural network (NRI-CNN) to modify the refractive index and extract a feature vector. Then we propose an iterative Green's function algorithm (IGFA) to approximately solve the Helmholtz equation. In order to enhance the generalization ability of the solution, the abstracted vector is utilized as a multiplier term in IGFA, obtaining an approximately spatial distribution of the light field. Ultimately, we design a U-net to handle residuals of the Helmholtz equation and phases of optical fields (ERPU-net). We apply this method for holographic reconstructions on random Gaussian beams, beams with image data, and those altered by simulated turbulent phases.

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Finite difference methods for stochastic Helmholtz equation driven by white noise

In this paper, we propose two numerical methods for the stochastic Helmholtz equation driven by white noise. We obtain the approximate stochastic problem by approximating the white noise with piecewise constant process, provide some regularity of its solution and the truncation error between the approximate stochastic problem and the original problem. The limitation on the wave number k of the finite difference method (FDM) is analyzed and a stochastic finite difference (SFD) scheme is presented. The error analysis shows that the stochastic finite difference method is efficient with a certain convergence rate. Numerical experiments are provided to examine our theoretical results.

Deep neural helmholtz operators for 3D elastic wave propagation and inversion

Summary Numerical simulations of seismic wave propagation in heterogeneous 3D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full waveform inversion, which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent works in seismology have shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We employ a Helmholtz neural operator for 2D and 3D elastic wave modeling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly on the surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3D full-waveform inversion, reducing the computation time by a factor of 350.

Study of Dependence of the Energy of the Field in Weakly Conductive Fiberlight-Guide with Gradient Refractive Index Profile Without Regard to Polarization

A round-in-cross-section weakly guiding fiber optic light guide is considered. The system of Maxwellian equations for electrically conductive transparent media without due regard to polarization with an index of refraction in the case under consideration is reduced to the homogeneous Helmholtz equation, a particular solution of which is found using the Green's function. For a unimodalmode a common decision was obtained for the field and energy inside the fiber in the general case of a gradient refractive index profile depending on the radial coordinate. The concept of normalized energy is introduced as the ratio of the energy inside a fiber with gradient refractive index profile to the energy inside a fiber with a step-index profile. Since in a single-mode regime the zero-order Bessel function, which describes the field inside a step-index profile fiber, can be replaced by a Gaussoid, then, when extending this replacement to the case of a single-mode gradient fiber, as a first approximation we shall obtain a finite expression for the normalized energy for a general gradient profile. The dependences of the normalized energy on the waveguide number were obtained for each of the degrees from the first to the fourth inclusive. It is shown that in the considered approximation, the energy increases up to a certain parameter values (for the first degree – a triangle profile) or slowly decreases (other profiles), after this value the energy increases for all profiles. The results obtained will help to select a suitable single-mode fiber for practical application.

The global uniqueness of a dissipative fractional Helmholtz equation

In this paper, we consider the Dirichlet problem for the fractional Helmholtz equation with dissipation, and study the inverse problem of determining the source function, potential function and dissipation of the equation using the Dirichlet-to-Neumann map and Runge approximation. We prove the global uniqueness of these three functions of the equation under low-frequency conditions. As the main result, this implies that one can use the external data to uniquely recover the unknown functions of the equation in the low-frequency case, which will be of great significance in photoacoustic tomography and thermoacoustic tomography.

Bayesian inversion of a fractional elliptic system derived from seismic exploration

In this paper, we concentrate on the Bayesian inversion of a dispersion‐dominated fractional Helmholtz (DDFH) equation, which has been introduced in studies concerning seismic exploration. To establish the inversion theory, we meticulously examine the DDFH equation. We transform it into a system comprising both fractional‐ and integer‐order elliptic equations, extending the conventional definition of the spectral fractional Laplace operator to accommodate non‐homogeneous boundary conditions. Subsequently, we establish the well‐posedness theory for scenarios involving both small and large wavenumbers. Our proof hinges upon the regularity attributes of select fractional elliptic equations and capitalizes fully on the structural peculiarities of the elliptic system, which distinguish it from classical cases. Thereafter, we focus on the inverse medium scattering problem pertinent to the DDFH equation, framed within the Bayesian statistical framework. We address two scenarios: one devoid of model reduction errors and another characterized by such errors—arising from the implementation of certain absorbing boundary conditions. More precisely, based on the properties of the forward operator, well‐posedness of the posterior measures have been proved in both cases, which provide an opportunity to quantify the uncertainties of this problem.

Transverse Spin-Orbit Interaction of Light.

Light carries both longitudinal and transverse spin angular momentum. The spin can couple with its orbital counterpart, known as the spin-orbit interaction (SOI) of light. Complementary to the longitudinal SOI known previously, here we show that transverse SOI of light is inherent in the Helmholtz equation when transverse spinning light propagates in curved paths. It lifts the degeneracy of dispersion relations of light for opposite transverse spin states, analogous to the Dresselhaus effect. Transverse SOI is ubiquitous in nanophotonic systems where transverse spin and optical path bending are inevitable. It can explain anomalous effects like the dispersion relation of surface plasmon polaritons on curved paths and the energy level of whispering gallery modes. Our results reveal the analogies of spin photonics and spintronics and offer a new degree of freedom for integrated photonics, spin photonics, and astrophysics.

Electromagnetic wave scattering in plasma beam-driven waveguides under strong magnetic fields

This study analyzes the scattering of electromagnetic waves in a cold and uniform plasma-filled waveguide driven by an intense relativistic plasma beam under a strong magnetic field. The strong interaction of plasma with electromagnetic waves enables its potential use in different types of waveguides. The Helmholtz equation governs the boundary value problem, which is solved by incorporating the mode matching technique. Invoking the boundary and matching conditions and the derived orthogonality and dispersion relations in this scheme gives an exact solution to the scattering problem. The numerical results shed light on the occurrence of reflection and transmission and flow of power. The power flux is plotted against angular frequency and various duct configurations. The solution is completely validated through the benefit of analytical and numerical results. The investigation of this structure reveals not only its mathematical, but also its physical features.

Theory of the cubic autoproduct and its utility for noisy direction of arrival estimation

Autoproducts are quadratic or higher products of frequency-domain acoustic fields that can mimic genuine fields at frequencies within or outside the original field's bandwidth. Past studies have found a variety of interesting autoproduct properties but have been limited to quadratic autoproducts. This paper presents cubic autoproduct theory and documents how noise suppression may be attained with the cubic frequency-difference autoproduct, a product of three frequency-domain acoustic fields. The cubic autoproduct's field equations, developed from the inhomogeneous Helmholtz equation, and analytical results in single- and two-path environments justify interpretating the cubic autoproduct as a pseudofield and underscore its similarities to the quadratic autoproducts. For nonzero field bandwidth, many frequency triplets satisfy the relationship for a single cubic autoproduct frequency. Thus, bandwidth averaging can lead to serendipitous noise suppression and is shown herein to facilitate field-phase-structure recovery from ideal free space fields corrupted by Gaussian noise. Cubic-autoproduct-based direction of arrival (DOA) estimation using signal and noise recordings collected in the ocean are found to be more accurate than conventional DOA estimates from the same data. In particular, cubic autoproduct results showed a 3–5 dB noise suppression advantage for 4- and 6-kHz direct- and reflected-path sounds broadcast 200 m to a four-element receiving array.

Optimal design of broadband, low-directivity graded index acoustic lenses for underwater communication.

Manipulating underwater pressure waves is crucial for marine exploration, as electromagnetic signals are strongly absorbed in water. However, the multi-path phenomenon complicates the accurate capture of acoustic waves by receivers. Although graded index lenses, based on metamaterials with smoothly varying properties, successfully focus pressure waves, they tend to have high directivity, which hinders practical application. This work introduces three 2D acoustic lenses made from a metamaterial composed of solid inclusions in water. We propose an optimization scheme where the pressure dynamics is governed by Helmholtz's equation, with control parameters affecting each lens cell's density and bulk modulus. Through an appropriate cost function, the optimization encourages a broadband, low-directivity lens. The large-scale optimization is solved using the Lagrangian approach, which provides an analytical expression for the cost gradient. This scheme avoids the need for a separate discretization step, allowing the design to transition directly from the desired smooth refractive index to a practical lattice structure. As a result, the optimized lens closely aligns with real-world behavior. The homogenized numerical model is validated against finite elements, which considers acoustic-elastic coupling at the microstructure level. When homogenization holds, this approach proves to be an effective design tool for achieving broadband, low-directivity acoustic lenses.

An analytical method for broadband acoustic analysis of 2D cavities containing or bounded by porous materials

Real-life vibro-acoustic systems often involve porous treatments, resulting in complex-valued and frequency-dependent models that are challenging to solve. Traditional prediction techniques like the finite element (FE) method requires huge computational cost, especially in the mid to high frequency ranges. This paper develops a novel spectral dynamic stiffness (SDS) formulation, using very few number of degrees of freedom but describing the broadband acoustic behaviour of acoustic cavities with porous materials highly accurately. The method employs frequency-dependent shape function that satisfies exactly the (damped) Helmholtz equation to describe the (equivalent) acoustic pressure field, and also features an innovative approach to use the fast-convergent Modified Fourier series to describe any arbitrary acoustic BCs. Finally, the SDS matrices for cavities containing or bounded by porous materials are formulated in an analytical manner. It is demonstrated that the method exhibits a much higher computational efficiency over the FE package COMSOL, at least 6 times faster than the COMSOL with a similar level of accuracy, and benchmark solutions are provided. This promising method can provide a powerful tool for systems with porous materials that have frequency-dependent characteristics, paving the way for efficient and accurate acoustic analysis in complex engineering applications.

Plane-wave and cylindrical-wave acoustic reflection from a marine sediment with layering representative of the New England Mud Patch.

An analysis is presented of reflection from a marine sediment consisting of a homogeneous mud layer overlying a sand-mud basement, the latter with an upward-refracting, inverse-square sound speed profile. Such layering is representative of the sediment at the New England Mud Patch (NEMP). By applying appropriate integral transforms and their inverses to the Helmholtz equations for the ocean and the two sediment layers, along with the boundary conditions, a Sommerfeld-Weyl type of wavenumber integral is obtained for the cylindrical-wave reflection coefficient of the sediment, R. A stationary phase evaluation of this integral yields a closed-form expression for the plane-wave reflection coefficient, R0. In the absence of attenuation, the plane-wave solution exhibits total reflection up to a critical grazing angle, ac, but when attenuation in the sediment is introduced, the region of total reflection in |R0| is replaced by a sequence of contiguous peaks. With realistic levels of sediment attenuation, the cylindrical-wave solution, |R|, exhibits a quasi-critical grazing angle, less than ac, which is strongly dependent on the source-plus-receiver height above the seabed, which is mildly dependent on the depth of the mud layer but is essentially independent of frequency. Such behavior is consistent with independent experimental observations at the NEMP.

Free-Form Deformation as a non-invasive, discrete unfitted domain method: Application to the time-harmonic acoustic response of a saxophone

The Finite Element method, widely used for solving Partial Differential Equations, may result in suboptimal computational costs when computing smooth fields within complex geometries. In such situations, IsoGeometric Analysis often offers improved per degree-of-freedom accuracy but building analysis-suitable representation of complex shapes is generally not obvious. This paper introduces a non-invasive, spline-based fictitious domain method using Free-Form Deformation to efficiently solve the Helmholtz equation in complex domains, such as in musical instruments. By immersing a fine FE mesh into a simple B-spline box, the approximation subspace size is significantly reduced without compromising accuracy. Accompanied by specific conditioning treatment, the method not only proves to be efficient, but also robust and easy to implement in existing FE software. Applied to an alto saxophone, the method reduces the number of degrees of freedom by over two orders of magnitude and the computation time by more than one compared to standard FE methods with comparable accuracy when compared to experimental tests.

Xenon fluid state model and evaluation of working medium in satellite electric propulsion system

Xenon, the working medium of satellite electric propulsion system, is easy to enter the supercritical state, and does not follow the ideal gas equation of state in filling and in-orbit application. The error of fluid state and parameter evaluation of xenon is large. In view of this difficulty, the working fluid state test system and test method of xenon were established. The saturation vapor pressure, gas phase PVT and supercritical PVT characteristics of xenon were tested. A new saturation vapor pressure equation of xenon was established. The relative deviation between the saturated vapor pressure test results and the NIST equation is less than 0.2 %. The Helmholtz equation established on the basis of xenon experimental data has a high accuracy, and the deviation from the experimental data is less than 0.5 % within 273 K–358 K. In 358 K–373 K, the deviation from the test data is better than 0.05 %, and the second dimension coefficient is used to prove the accuracy and reliability of the model, which provides a state model for satellite xenon filling and in-orbit application.

Inverse source problem for discrete Helmholtz equation

Abstract We consider multi-frequency inverse source problem for the discrete Helmholtz operator on the square lattice Z d , d ⩾ 1 . We consider this problem for the cases with and without phase information. We prove uniqueness results and present examples of non-uniqueness for this problem for the case of compactly supported source function, and a Lipshitz stability estimate for the phased case is established. Relations with inverse scattering problem for the discrete Schrödinger operators in the Born approximation are also provided.

Increasing stability of the acoustic and elastic inverse source problems in multi-layered media

Abstract This paper investigates inverse source problems for the Helmholtz and Navier equations in multi-layered media, considering both two and three-dimensional cases respectively. The study reveals a consistent increase in stability for each scenario, characterized by two main terms: a Hölder-type term associated with data discrepancy, and a logarithmic-type term that diminishes as more frequencies are considered. In the two-dimensional case, measurements on interfaces and far-field data are essential. By employing the fundamental solution in free-space as the test function and utilizing the asymptotic behavior of the solution and continuation principle, stability results are obtained. In the three-dimensional case, measurements on interfaces and artificial boundaries are taken, and the stability result can be derived by applying the arguments for inverse source problems in homogeneous media.

On the Optimum Distribution of Multiple Helmholtz Resonators for Annular Combustors

Abstract The combustors of many modern land-based gas turbines and aero-engines are annular. This kind of combustors often suffer from thermoacoustic oscillations, with the occurrence of mostly the circumferential first-order, second-order and third-order oscillation modes in the annular combustion chamber. To solve this problem, passive control methods are widely used adding passive acoustic dampers such as Helmholtz resonators (HRs). Depending on the type and number of HRs, and the possible positions over the circumference to install them, there could easily have millions, or even billions, of possible arrangment patterns. Finding a good design is the key to solve this problem. In this paper, we perform a theoretical and numerical study of an annular combustor installed with multiple types of HRs over the circumference. Firstly, a simple annular duct with arbitrary distributions of these HRs is studied analytically by solving the nonlinear eigenvalue problem of a 1D network model. Based on the results of the analytical method, the impact of the HRs on the acoustic modes of the combustion chamber is studied and the optimum arrangement of multiple resonators is obtained. This arrangement usually gives a null (or small) mode splitting strength and a good damping effect. Finally, we apply the optimum arrangement to damp the thermoacoustic modes that have been captured by numerical simulation for a real annular combustor. We use numerical simulations based on solving the 3D Helmholtz equation in COMSOL to verify the feasibility of the optimum arrangement.

Numerical Solution of the Cauchy Problem for the Helmholtz Equation Using Nesterov’s Accelerated Method

In this paper, the Cauchy problem for the Helmholtz equation, also known as the continuation problem, is considered. The continuation problem is reduced to a boundary inverse problem for a well-posed direct problem. A generalized solution to the direct problem is obtained and an estimate of its stability is given. The inverse problem is reduced to an optimization problem solved using the gradient method. The convergence of the Landweber method with respect to the functionals is compared with the convergence of the Nesterov method. The calculation of the gradient in discrete form, which is often used in the numerical solutions of the inverse problem, is described. The formulation of the conjugate problem in discrete form is presented. After calculating the gradient, an algorithm for solving the inverse problem using the Nesterov method is constructed. A computational experiment for the boundary inverse problem is carried out, and the results of the comparative analysis of the Landweber and Nesterov methods in a graphical form are presented.

Approximations of the Helmholtz equation with variable wave number in one dimension

Approximations of the Helmholtz equation with variable wave number in one dimension