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Thermodynamic potentials play a substantial role in numerous scientific disciplines and serve as basic constructs for describing the behavior of matter. Despite their significance, comprehensive investigations of their topological characteristics and their connections to molecular interactions have eluded exploration due to experimental inaccessibility issues. This study addresses this gap by analyzing the topology of the Helmholtz energy, Gibbs energy, Grand potential, and Null potential that are associated with different isothermal boundary conditions. By employing Monte Carlo simulations in the NVT, NpT, and μVT ensembles and a molecular-based equation of state, methane, ethane, nitrogen, and methanol are investigated over a broad range of thermodynamic conditions. The predictions from the two independent methods are overall in very good agreement. Although distinct quantitative differences among the fluids are observed, the overall topology of the individual thermodynamic potentials remains unaffected by the molecular architecture, which is in line with the corresponding states principle—as expected. Furthermore, a comparative analysis reveals significant differences between the total potentials and their residual contributions. [ABSTRACT FROM AUTHOR]

Molecular-based equations of state for describing the thermodynamics of chain molecules are often based on mean-field like arguments that reduce the problem of describing the interactions between chains to a simpler one involving only nonbonded monomers. While for dense liquids such arguments are known to work well, at low density they are typically less appropriate due to an incomplete description of the effect of chain connectivity on the local environment of the chains' monomer segments. To address this issue, we develop three semi-empirical approaches that significantly improve the thermodynamic description of chain molecules at low density. The approaches are developed for chain molecules with repulsive intermolecular forces; therefore, they could be used as reference models for developing equations of the state of real fluids based on perturbation theory. All three approaches are extensions of Wertheim's first-order thermodynamic perturbation theory (TPT1) for polymerization. The first model, referred to as TPT1-v, incorporates a second-virial correction that is scaled to zero at liquid-like densities. The second model, referred to as TPT1-y, introduces a Helmholtz-energy contribution to account for correlations between next-nearest-neighbor segments within chain molecules. The third approach, called TPT-E, directly modifies TPT1 without utilizing an additional Helmholtz energy contribution. By employing TPT1 at the core of these approaches, we ensure an accurate description of mixtures and enable a seamless extension from chains of tangentially bonded hard-sphere segments of equal size to hetero-segmented chains, fused chains, and chains of soft repulsive segments (which are influenced by temperature). The low-density corrections implemented in TPT1 are designed to preserve these good characteristics, as confirmed through comparisons with novel molecular simulation results for the pressure of various chain fluids. TPT1-v exhibits excellent transferability across different chain types, but it relies on knowing the second virial coefficient of the chain molecules, which is non-trivial to obtain and determined here using Monte Carlo simulation. The TPT1-y model, on the other hand, achieves comparable accuracy to TPT1-v while being fully predictive, requiring no input besides the geometry of the chain molecules. [ABSTRACT FROM AUTHOR]

In this work, we demonstrate how the ion association constant can be attributed to the difference between the full Poisson–Boltzmann equation and its linearized version in very dilute solutions. We follow a pragmatic approach first by deriving an analytical approximated solution to the Poisson–Boltzmann equation, then calculating its respective Helmholtz free energy and activity coefficient, and then finally comparing it to the contribution from the mass action law principle. The final result is the Ebeling association constant. We conclude that electrostatic ion–ion interaction models miss the ion association contribution naturally introduced in higher-order electrostatic theories. We also demonstrate how the negative deviations from the Debye–Hückel limiting law can be physically attributed to the ion association phenomenon. [ABSTRACT FROM AUTHOR]

We consider propagation of two-dimensional waves on the infinite hexagonal (honeycomb) lattice. Namely, we study the discrete Helmholtz equation in hexagonal lattices without and with a boundary. It is shown that for some configurations these problems can be equivalently reduced to similar problems for the triangular lattice. Based on this fact, new results are obtained for the existence and uniqueness of the solution in the case of the real wave number k ∈ (0 , 6) ∖ { 2 , 3 , 2 } for the non-homogeneous Helmholtz equation in hexagonal lattices with no boundaries and the real wave number k ∈ (0 , 2) ∪ (2 , 6) for the exterior Dirichlet problem. [ABSTRACT FROM AUTHOR]

We consider the multi-frequency inverse source problem in the presence of a non-homogeneous medium using passive measurements. Precisely, we derive stability estimates for determining the source from the knowledge of only the imaginary part of the radiated field on the boundary for multiple frequencies. The proof combines a spectral decomposition with a quantification of the unique continuation of the resolvent as a holomorphic function of the frequency. The obtained results show that the inverse problem is well posed when the frequency band is larger than the spatial frequency of the source. [ABSTRACT FROM AUTHOR]

A novel sharp interface method is considered for solving the Helmholtz equation with interface in cylindrical coordinates, and is applied to solve incompressible axisymmetric flows. In order to overcome the singularity at r = 0 and the reduction in precision of numerical solutions near the interface, a modified second-order finite difference scheme is constructed using the immersed interface method and Taylor series expansion, combined with interface relationships on a staggered grid. The projection method is utilized to address the incompressible axisymmetric flow problem with an interface, then the original problem is transformed into several Helmholtz equations with an interface. The sharp interface method is then repeatedly used to numerically determine the physical quantities of fluid velocity and pressure. Numerical experiments verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We present a matrix-free parallel scalable multilevel deflation preconditioned method for heterogeneous time-harmonic wave problems. Building on the higher-order deflation preconditioning proposed by Dwarka and Vuik (SIAM J. Sci. Comput. 42(2):A901-A928, 2020; J. Comput. Phys. 469:111327, 2022) for highly indefinite time-harmonic waves, we adapt these techniques for parallel implementation in the context of solving large-scale heterogeneous problems with minimal pollution error. Our proposed method integrates the Complex Shifted Laplacian preconditioner with deflation approaches. We employ higher-order deflation vectors and re-discretization schemes derived from the Galerkin coarsening approach for a matrix-free parallel implementation. We suggest a robust and efficient configuration of the matrix-free multilevel deflation method, which yields a close to wavenumber-independent convergence and good time efficiency. Numerical experiments demonstrate the effectiveness of our approach for increasingly complex model problems. The matrix-free implementation of the preconditioned Krylov subspace methods reduces memory consumption, and the parallel framework exhibits satisfactory parallel performance and weak parallel scalability. This work represents a significant step towards developing efficient, scalable, and parallel multilevel deflation preconditioning methods for large-scale real-world applications in wave propagation. [ABSTRACT FROM AUTHOR]

We study the Helmholtz equation with periodic coefficients in a closed waveguide. A functional analytic approach is used to formulate and solve the radiation problem in a self‐contained exposition. In this context, we simplify the non‐degeneracy assumption on the frequency. Limiting absorption principles (LAPs) are studied, and the radiation condition corresponding to the chosen LAP is derived; we include an example to show different LAPs lead, in general, to different solutions of the radiation problem. Finally, we characterize the set of all bounded solutions to the homogeneous problem. [ABSTRACT FROM AUTHOR]

The intrigue of waves on periodic lattices and gratings has resonated with physicists and mathematicians alike for decades. In-depth analysis has been devoted to the seemingly simplest array system: a one-dimensionally periodic lattice of two-dimensional scatterers embedded in a dispersionless medium governed by the Helmholtz equation. We investigate such a system and experimentally confirm the existence of a new class of generalised Rayleigh–Bloch waves that have been recently theorised to exist in classical wave regimes, without the need for resonant scatterers. Airborne acoustics serves as such a regime and we experimentally observe the first generalised Rayleigh–Bloch waves above the first cut-off, i.e., in the radiative regime. We consider radiative acoustic lattice resonances along a diffraction grating and connect them to generalised Rayleigh–Bloch waves by considering both short and long arrays of non-resonant 2D cylindrical Neumann scatterers embedded in air. On short arrays, we observe finite lattice resonances under continuous wave excitation, and on long arrays, we observe propagating Rayleigh–Bloch waves under pulsed excitation. We interpret their existence by considering multiple wave scattering theory and, in doing so, unify differing nomenclatures used to describe waves on infinite periodic and finite arrays and the interpretation of their dispersive properties. The interaction of waves with periodic structures is a feature central to many areas of physics from quantum mechanics to acoustics. Here, the authors numerically and experimentally demonstrate the presence of Rayleigh-Bloch waves in the regime above the first cut-off using acoustic gratings. [ABSTRACT FROM AUTHOR]

The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in L 1 − L ∞ operator norm in terms of time t and powers of h. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator L 1 − L ∞ norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem. [ABSTRACT FROM AUTHOR]

The generalization of Physics-Informed Neural Networks (PINNs) used to solve the inhomogeneous Helmholtz equation in a simplified three-dimensional room is investigated. PINNs are appealing since they can efficiently integrate a partial differential equation and experimental data by minimizing a loss function. However, a previous study experienced limitations in acoustics regarding the source term. A challenging but realistic excitation case is a confined (e.g., single-point) excitation area, yielding a smooth spatial wave field periodically with the wavelength. Compared to studies using smooth (unrealistic) sound excitation, the network's generalization capabilities regarding a realistic sound excitation are addressed. Different methods like hyperparameter optimization, adaptive refinement, Fourier feature engineering, and locally adaptive activation functions with slope recovery are tested to tailor the PINN's accuracy to an experimentally validated finite element analysis reference solution computed with openCFS. The hyperparameter study and optimization are conducted regarding the network depth and width, the learning rate, the used activation functions, and the deep learning backends (PyTorch 2.5.1, TensorFlow 2.18.0 1, TensorFlow 2.18.0 2, JAX 0.4.39). A modified (feature-engineered) PINN architecture was designed using input feature engineering to include the dispersion relation of the wave in the neural network. For smoothly (unrealistic) distributed sources, it was shown that the standard PINNs and the feature-engineered PINN converge to the analytic solution, with a relative error of 0.28% and 2 × 10 − 4 %, respectively. The locally adaptive activation functions with the slope lead to a relative error of 0.086% with a source sharpness of s = 1 m. Similar relative errors were obtained for the case s = 0.2 m using adaptive refinement. The feature-engineered PINN significantly outperformed the results of previous studies regarding accuracy. Furthermore, the trainable parameters were reduced to a fraction by Bayesian hyperparameter optimization (around 5%), and likewise, the training time (around 3%) was reduced compared to the standard PINN formulation. By narrowing this excitation towards a single point, the convergence rate and minimum errors obtained of all presented network architectures increased. The feature-engineered architecture yielded a one order of magnitude lower accuracy of 0.20% compared to 0.019% of the standard PINN formulation with a source sharpness of s = 1 m. It outperformed the finite element analysis and the standard PINN in terms time needed to obtain the solution, needing 15 min and 30 s on an AMD Ryzen 7 Pro 8840HS CPU (AMD, Santa Clara, CA, USA) for the FEM, compared to about 20 min (standard PINN) and just under a minute of the feature-engineered PINN, both trained on a Tesla T4 GPU (NVIDIA, Santa Clara, CA, USA). [ABSTRACT FROM AUTHOR]

To measure the electroacoustic parameters of transducers in the continuous sound field in a limited water area, a reciprocity calibration method of hydrophones using a spatial sampling average method in a non-anechoic tank was developed. The sound propagation in the non-anechoic tank under the impedance boundary condition, with a sound source producing continuous sound, is introduced based on the Helmholtz equation and Green's function. The reciprocity constant is given using the spatial sampling average sound pressure, and the three-transducer reciprocity calibration procedure was established. The number of spatial sampling points, which must be not less than 6000, and the sampling configuration in the non-anechoic tank are determined. In the non-anechoic tank of 15 m × 9 m × 6 m, the sensitivity of the hydrophone in the frequency range of 250 Hz to 10 kHz was calibrated, and the largest discrepancy between the measurement results from the reciprocity calibration method in the non-anechoic water tank and the standard facility was 0.7 dB. [ABSTRACT FROM AUTHOR]

We present a wavenumber-explicit convergence analysis of the hp Finite Element Method applied to a class of heterogeneous Helmholtz problems with piecewise analytic coefficients at large wavenumber k. Our analysis covers the heterogeneous Helmholtz equation with Robin, exact Dirichlet-to-Neumann, and second order absorbing boundary conditions, as well as perfectly matched layers. [ABSTRACT FROM AUTHOR]

SUMMARY: This study presents a novel algorithm that combines the Lorenz gauge equations with the Fourier domain technique to simulate magnetotelluric responses in three‐dimensional conductivity structures with general anisotropy. The method initially converts the Helmholtz equations governing vector potentials into one‐dimensional differential equations in the wave number domain via the horizontal two‐dimensional Fourier transform. Subsequently, a one‐dimensional finite element method employing quadratic interpolation is applied to obtain three five‐diagonal linear equation systems. Upon solving these equations, the spatial domain fields are obtained via the inverse Fourier transform. This process guarantees the computational efficiency, memory efficiency and high parallelization of the algorithm. Moreover, an anisotropic medium iteration operator guarantees stable convergence of the method. The correctness, competence and applicability of the algorithm are verified using some synthetic models. The results demonstrate that the new method is efficient and performs well in anisotropic undulating terrain and complex structures. Compared to other Fourier domain methods and the latest edge‐based finite element algorithm, the proposed method exhibits superior computing performance. Finally, the impact of the Euler angles on the magnetotelluric responses is analysed. [ABSTRACT FROM AUTHOR]

Multiple scattering methods are widely used to reduce the computational complexity of acoustic or electromagnetic scattering problems when waves propagate through media containing many identical inclusions. Historically, this numerical technique has been limited to situations in which the inclusions (particles) can be covered by nonoverlapping disks in two dimensions or spheres in three dimensions. This allows for the use of separation of variables in cylindrical or spherical coordinates to represent the solution to the governing partial differential equation. Here, we provide a more flexible approach, applicable to a much larger class of geometries. We use a Green's representation formula and the associated layer potentials to construct incoming and outgoing solutions on rectangular enclosures. The performance and flexibility of the resulting scattering operator formulation in two-dimensions is demonstrated via several numerical examples for multi-particle scattering in free space as well as in layered media. The mathematical formalism extends directly to the three dimensional case as well, and can easily be coupled with several commercial numerical PDE software packages. [ABSTRACT FROM AUTHOR]

ABSTRACT We present a new direct (or quasi‐direct) strategy for solving the three‐dimensional Poisson and Helmholtz problems posed on a Cartesian block subject to Dirichlet boundary conditions. Our approach starts with a spectral approximation of either problem involving modal Chebyshev integration matrices. With N+1$$ N+1 $$ the number of Chebyshev modes associated with each of the coordinate directions, the total number of modes is n=(N+1)3$$ n={\left(N+1\right)}^3 $$. The relevant complexities for our base methods are then similar to certain classical methods; in particular, a set‐up cost scaling like 94n2$$ \frac{9}{4}{n}^2 $$ and, thereafter, an O(n4/3)$$ O\left({n}^{4/3}\right) $$ solve cost. The memory storage for our approach is O(n4/3)$$ O\left({n}^{4/3}\right) $$ and involves no hierarchical data formats. Our approaches exhibit spectral accuracy and are empirically well‐conditioned. We describe acceleration via the introduction of an iterative element. This acceleration yields a method with an O(n4/3)$$ O\left({n}^{4/3}\right) $$ set‐up cost, followed by a sub‐quadratic solve complexity seen empirically to also be O(n4/3)$$ O\left({n}^{4/3}\right) $$. The concluding section remarks on possible further acceleration (not established), targeting O(n4/3)$$ O\left({n}^{4/3}\right) $$ set‐up and O(nlogn)$$ O\left(n\log n\right) $$ solve costs. [ABSTRACT FROM AUTHOR]

The article in the Journal of Fluid Mechanics delves into the impact of fluid viscoelasticity on ultrasound-induced particle transport in microfluidics. Through a combination of theoretical, numerical, and experimental approaches, the study examines how viscoelastic parameters affect acoustic energy density (AED) and particle migration dynamics. The research indicates that increasing fluid elasticity speeds up particle migration, while higher viscosity slows it down. This investigation aims to enhance our comprehension of particle migration in viscoelastic fluids under ultrasound, potentially influencing future studies on particle/cell movement in bio-fluids. [Extracted from the article]

A number of previous papers have used the coupled-mode method to assess three-dimensional scattering by cylindrically symmetric anomalies (seamounts, hills) or anomalies which are invariant in a horizontal direction (wedges, ridges). This paper makes an extension to combinations of these two anomaly types. The upper and lower depth boundaries of the anomalies may be flat or irregular, and the sound source may be anywhere in the medium. After a discretization of the anomalies of the two types with laterally homogeneous rings and strips, respectively, an adaptation of the coupled-mode method yields the solution of the pertinent Helmholtz equation. The adaptation involves a combination of Fourier-series summations to handle the ring anomalies and adaptive wavenumber integrations to handle the strip structure. For each anomaly, recursively computed reflection matrices relate the expansion coefficients for incoming and outgoing normal modes. Iterative solution of a linear equation system for the amplitudes of the scattered cylindrical waves from the ring anomalies, involving formulas for transformation between plane and cylindrical waves, yields an expansion of the field. Related expansions allow isolation of partial waves, multiply scattered among the anomalies. The paper includes examples from underwater acoustics and atmospheric acoustics. [ABSTRACT FROM AUTHOR]

For h-FEM discretisations of the Helmholtz equation with wavenumber k, we obtain k-explicit analogues of the classic local FEM error bounds of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9), Demlow et al.(Math. Comput. 80(273), 1–9 2011), showing that these bounds hold with constants independent of k, provided one works in Sobolev norms weighted with k in the natural way. We prove two main results: (i) a bound on the local H 1 error by the best approximation error plus the L 2 error, both on a slightly larger set, and (ii) the bound in (i) but now with the L 2 error replaced by the error in a negative Sobolev norm. The result (i) is valid for shape-regular triangulations, and is the k-explicit analogue of the main result of Demlow et al. (Math. Comput. 80(273), 1–9 2011). The result (ii) is valid when the mesh is locally quasi-uniform on the scale of the wavelength (i.e., on the scale of k - 1 ) and is the k-explicit analogue of the results of Nitsche and Schatz (Math. Comput. 28(128), 937–958 1974), Wahlbin (1991, §9). Since our Sobolev spaces are weighted with k in the natural way, the result (ii) indicates that the Helmholtz FEM solution is locally quasi-optimal modulo low frequencies (i.e., frequencies ≲ k ). Numerical experiments confirm this property, and also highlight interesting propagation phenomena in the Helmholtz FEM error. [ABSTRACT FROM AUTHOR]

In this study, we explore the effectiveness of elliptic partial differential equations (PDEs) in two and three dimensional space based on Helmholtz equation for simulations of acoustic sound in a very complex environment of propagation. For this purpose, we use an advance and robust numerical technique by utilizing the properties of shifted Chebyshev spectral collocation method. This technique is an extension of the traditional Chebyshev polynomials, incorporating a shift in their argument to enhance flexibility across a wider domain, while retaining an extraordinary numerical characteristic such as orthogonality and spectral convergence making them exceptionally effective in finding the approximate solutions. The exponential order of convergence of the proposed approach is shown both through theoretical and numerical approaches. We provide a number of numerical experiments to verify the theoretical results. The spectral convergence has been substantially enhanced by these numerical examples. The exponential order is further validated by numerical error behaviour in both $ L_2 $ L 2 and $ L_\infty $ L ∞ norms. [ABSTRACT FROM AUTHOR]

Numerical simulations of seismic wave propagation in heterogeneous 3-D 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 (FWI), 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 work in seismology has shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3-D 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 3-D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We use a Helmholtz neural operator for 2-D and 3-D elastic wave modelling, 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 at the free 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 3-D FWI, reducing the computation time by a factor of 350. [ABSTRACT FROM AUTHOR]

Directional interpolation is a fast and efficient compression technique for high‐frequency Helmholtz boundary integral equations, but requires a very large amount of storage in its original form. Algebraic recompression can significantly reduce the storage requirements and speed up the solution process accordingly. During the recompression process, weight matrices are required to correctly measure the influence of different basis vectors on the final result, and for highly accurate approximations, these weight matrices require more storage than the final compressed matrix. We present a compression method for the weight matrices and demonstrate that it introduces only a controllable error to the overall approximation. Numerical experiments show that the new method leads to a significant reduction in storage requirements. [ABSTRACT FROM AUTHOR]

We show that a tempered distribution u on ℝd is a solution of (-Δ)su = u if and only if -Δu = u. This result holds for any s ∈ (0,∞) and any dimension d ≥ 1. Our proof uses Fourier analysis. [ABSTRACT FROM AUTHOR]

For a two-block splitting iterative scheme to solve the complex linear equations system resulting from the complex Helmholtz equation, the iterative form using descent vector and residual vector is formulated. We propose splitting iterative schemes by considering the perpendicular property of consecutive residual vector. The two-block splitting iterative schemes are proven to have absolute convergence, and the residual is minimized at each iteration step. Single and double parameters in the two-block splitting iterative schemes are derived explicitly utilizing the orthogonality condition or the minimality conditions. Some simulations of complex Helmholtz equations are performed to exhibit the performance of the proposed two-block iterative schemes endowed with optimal values of parameters. The primary novelty and major contribution of this paper lies in using the orthogonality condition of residual vectors to optimize the iterative process. The proposed method might fill a gap in the current literature, where existing iterative methods either lack explicit parameter optimization or struggle with high wave numbers and large damping constants in the complex Helmholtz equation. The two-block splitting iterative scheme provides an efficient and convergent solution, even in challenging cases. [ABSTRACT FROM AUTHOR]

Herein, analyzed is a symmetric finite element method (FEM) formulation that can be used to calculate time-harmonic acoustic waves in external domains by using finite elements. The dispersion study shows how mesh refining affects the discrete representation of the FEM parameters. In the Helmholtz area, stabilization through coefficient modification is used in conjunction with conventionally stabilized finite elements to enhance FEM performance. Numerical evidence backs up the robust performance of this finite element perfectly matched layer (PML) approach. We suggest and evaluate a quick technique for calculating the answer to the Helmholtz equation in a confined region using a changing wave speed function. Wave splitting is the method's foundation. To solve iteratively for a specified tolerance, the Helmholtz equation is first divided into one-way wave equations. The wave speed function and the previously solved one-way wave equations are both necessary for the source functions to function. Then, using the sum of one-way solutions for each iteration, the Helmholtz equation's solution is roughly determined. to decrease computational expenses. The findings show that each model under consideration has significant variances in density and speed. The findings show the effective application of MATLAB R2021 software and the finite element method to solve both first and second-order Helmholtz equations. [ABSTRACT FROM AUTHOR]

The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations. [ABSTRACT FROM AUTHOR]

Scattering problems have wide applications in the medical and military fields. In this paper, the weighted least-squares (WLS) collocation method based on radial basis functions (RBFs) is developed to solve elastic wave scattering problems, which are governed by the Navier equation and the Helmholtz equations with coupled boundary conditions. The perfectly matched layer (PML) technique is used to truncate the unbounded domain into a bounded domain. The WLS method is constructed by setting the collocation points denser than the trial centers and imposing different weights on different types of boundary conditions. The WLS method can overcome the matrix singularity problem encountered in the Kansa method, and the convergence rate of WLS is m − 2 for Sobolev kernel with kernel smoothness m. Furthermore, compared with the finite element method (FEM) and the Kansa method, WLS can provide higher accuracy and more stable solutions for relatively large angular frequencies. The numerical example with a circular obstacle is used to verify the effectiveness and convergence behavior of the WLS. Besides, the proposed scheme can easily handle irregular obstacles and obtain stable results with high accuracy, which is validated through experiments with ellipse and kite-shaped obstacles. [ABSTRACT FROM AUTHOR]

We present a method to compute the photoionization spectra of atoms and molecules in linear-response, time-dependent density functional theory. The electronic orbital variations corresponding to ionized electrons are expanded on a basis set of delocalized functions, obtained as the solution of the inhomogeneous Helmholtz equation, with gaussian basis set functions as the right-hand side. The resulting scheme is able to reproduce the photoionization spectra without any need for artificial regularization or localization. We demonstrate that this Green's function-based approach is able to produce accurate spectra for semilocal exchange-correlation functionals, even using relatively small standard gaussian basis sets. [ABSTRACT FROM AUTHOR]

We study the validity of the Neumann or Born series approach in solving the Helmholtz equation and coefficient identification in related inverse scattering problems. Precisely, we derive a sufficient and necessary condition under which the series is strongly convergent. We also investigate the rate of convergence of the series. The obtained condition is optimal and it can be much weaker than the traditional requirement for the convergence of the series. Our approach makes use of reduction space techniques proposed by Suzuki [21]. Furthermore we propose an interpolation method that allows the use of the Neumann series in all cases. Finally, we provide several numerical tests with different medium functions and frequency values to validate our theoretical results.

Two new assertions characterizing analytically disks in the Euclidean plane $\mathbb{R}^2$ are proved. Weighted mean value property of positive solutions to the Helmholtz and modified Helmholtz equations are used for this purpose; the weight has a logarithmic singularity. The obtained results are compared with those without weight that were found earlier.

Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.

Introduction. Optical fibers are widely used for high-bandwidth transmitting communication signals over long distances. The key feature enabling this performance is signal low attenuation, that is signals experience minimal power loss propagating along the optical fiber. One of the factors influencing power loss during information transmission is the fiber bending. Bending can increase the signal transmission power loss of an optical fiber because of both macrobending and microbending. Studying the dependence of signal power losses when bending on waveguide parameters makes it possible to control the signal power losses of an optical fiber during information transmission. Aim of the Study. The study ia aimed at evaluating the effect of microbending on the field and energy of a weakly guiding optical fiber with a gradient refractive index profile in a single-mode regime. Materials and Methods. From the equations for the fields of straight and curved sections of weakly guiding fiber for an arbitrary gradient profile of the refractive index with the help of the subsequent solution of inhomogeneous Helmholtz equations by the Green’s function method, there were obtained expressions for relative energy: the ratio of the field energy of the fiber curved section to the field energy of the fiber straight section (in the first approximation for a single-mode regime). Results. The obtained expression for the relative energy depends on two parameters: the waveguide parameter and the ratio of the optical fiber radius to the radius of curvature. For the quadratic case of a power-law profile, as the closest to the actually used one, numerical calculations were used to construct the dependences of the relative energy on the parameter characterizing the bending for different values of the waveguide parameter. Discussion and Conclusion. It has been shown that in the case of microbending, the longer the wavelength or the smaller the fiber radius, the lower the losses. The results obtained can be used in calculating optical fiber profiles designed to operate in a bent state and eliminate expensive experimental modeling of light guides and in designing waveguides to solve specific applications, in particular, to increase energy efficiency, reliability and performance of the measuring instruments.

Unique continuation is one of the most important properties for the solution of partial differential equation, which means the local information of the solution can determine the global one. In this paper, we discuss a special unique continuation for Helmholtz equations on a sphere in ℝ3$$ {\mathbb{R}}^3 $$, which is different with the classical unique continuation. The unique continuation holds only on the sphere and may not be extended to the domain in ℝ3$$ {\mathbb{R}}^3 $$. A Hölder type conditional stability of unique continuation is proved by complex extension method. Then a Tikhonov regularized scheme is proposed and the convergence rate is obtained by using the conditional stability estimate. Numerical examples are presented to show the performance of the scheme. It should be remarked here that our results may be applied to the problem of recovering a far field pattern with its local information. [ABSTRACT FROM AUTHOR]

We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry. [ABSTRACT FROM AUTHOR]

The Weyl expansion technique, also known as the angular spectrum expansion, expresses an outgoing spherical wave as a linear combination of plane waves. The scalar spherical waves are the solutions of the homogeneous Helmholtz equation and therefore have direct relation to the scalar multipole fields. This paper gives the Weyl expansion of multipole fields, scalar and vector, of any degree and order for spherical wave functions. The expressions are given in closed form for the scalar, ψℓm(τ)$$ {\psi}_{\mathit{\ell m}}^{\left(\tau \right)} $$, and vector, Mℓm(τ),Nℓm(τ)$$ {\mathbf{M}}_{\mathit{\ell m}}^{\left(\tau \right)},{\mathbf{N}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, Lℓm(τ)$$ {\mathbf{L}}_{\mathit{\ell m}}^{\left(\tau \right)} $$, multipole fields, evaluated across a plane orthogonal to any given direction. In the case of scalar spherical multipoles, the spherical gradient operator has been used, while for the vector spherical multipoles, the vector spherical wave operator has been constructed. [ABSTRACT FROM AUTHOR]

Block matrices with simultaneously diagonalizable blocks arise in diverse application areas, including, e.g., numerical methods for engineering based on partial differential equations as well as network synchronization, cryptography and control theory. In the present paper, we develop a parallel algorithm for the inversion of m × m block matrices with simultaneously-diagonalizable blocks of order n. First, a sequential version of the algorithm is presented and its computational complexity is determined. Then, a parallelization of the algorithm is implemented and analyzed. The complexity of the derived parallel algorithm is expressed as a function of m and n as well as of the number μ of utilized CPU threads. Results of numerical experiments demonstrate the CPU time superiority of the parallel algorithm versus the respective sequential version and a standard inversion method applied to the original block matrix. An efficient parallelizable procedure to compute the determinants of such block matrices is also described. Numerical examples are presented for using the developed serial and parallel inversion algorithms for boundary-value problems involving transmission problems for the Helmholtz partial differential equation in piecewise homogeneous media. [ABSTRACT FROM AUTHOR]

We study a reduced order model (ROM) based waveform inversion method applied to a Helmholtz problem with impedance boundary conditions and variable refractive index. The first goal of this paper is to obtain relations that allow the reconstruction of the Galerkin projection of the continuous problem onto the space spanned by solutions of the Helmholtz equation. The second goal is to study the introduced nonlinear optimization method based on the ROM aimed to estimate the refractive index from reflection and transmission data. Finally we compare numerically our method to the conventional least squares inversion based on minimizing the distance between modelled to measured data. [ABSTRACT FROM AUTHOR]

We consider the inverse interior problem of recovering the surface impedances of the cavity from sources and measurements placed on a curve inside of it. The uniqueness issue is investigated, and a hybrid method is proposed for the numerical solution. The approach takes advantages of both direct and iterative schemes, such as it does not require an initial guess and has an accuracy of a Newton‐type method. Presented numerical experiments demonstrate the feasibility and effectiveness of the approach. [ABSTRACT FROM AUTHOR]

The paper presents analogues of Poisson's formula for solving interior and exterior boundary value problems with Dirichlet or Neumann boundary conditions on a sphere for the Helmholtz equation, with the kernels of the integrals in the form of series. [ABSTRACT FROM AUTHOR]

We present algorithms for computing strongly singular and near-singular surface integrals over curved triangular patches, based on singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy and robustness of our method for quadratic basis functions and quadratic triangles by integrating it into a boundary element code and solving several scattering problems in three dimensions. We also give numerical evidence that the utilization of curved boundary elements enhances computational efficiency compared to conventional planar elements. [ABSTRACT FROM AUTHOR]

This work is devoted to the numerical solution of the Helmholtz equation with variable wave number and including a point source in appropriately truncated infinite domains. Motivated by a two‐dimensional model, we formulate a simplified one‐dimensional model. We study its well posedness via wave number explicit stability estimates and prove convergence of the finite element approximations. As a proof of concept, we present the outcome of some numerical experiments for various wave number configurations. Our experiments indicate that the introduction of the artificial boundary near the source and the associated boundary condition lead to an efficient model that accurately captures the wave propagation features. [ABSTRACT FROM AUTHOR]

A model that approximates the sound field well is useful in various fields, such as acoustic signal processing and numerical simulation. We have proposed an effective model in which the wideband instantaneous sound field is regarded as an element of a spherically band-limited function space, using the reproducing kernel of that space. In this paper, the frequency-domain sound field is regarded as an element of some band-limited function space, and a representation of the field as a linear combination of the reproducing kernel in that space is proposed. This model has the strongest representational capacity of all function systems when we know only the sound pressure information at arbitrary positions. The proposed model can be considered a generalization of the existing three-dimensional sound field model using the reproducing kernel of the solution space of the Helmholtz equation to the spatial dimension. One of the advantages of capturing the frequency-domain sound field in this way is the simplicity achieved for the estimation formula of the wavenumber spectrum. Two numerical simulations were conducted to validate the proposed methods. [ABSTRACT FROM AUTHOR]

This study proposes a two-dimensional, eight-noded automated mesh generator for precise and efficient finite element analysis (FEA) in microwave applications. The suggested method for solving the Helmholtz problem employs an optimal domain discretization procedure. MAPLE-13 software's advanced automatic mesh generator was developed specifically for this work. To demonstrate the effectiveness of the approach, three distinct waveguide structures are analyzed, with the results compared to the best available analytical or numerical solutions. The findings indicate that the proposed method yields highly accurate and efficient finite element results, particularly for waveguide structures containing singularities. In microwave applications, this method can significantly enhance energy transmission efficiency. [ABSTRACT FROM AUTHOR]

This paper addresses the electromagnetic inverse scattering problem of determining the location and shape of anisotropic objects from near-field Cauchy data. We investigate both cases involving the Helmholtz equation and Maxwell's equations for this inverse problem. Our study focuses on developing efficient imaging functionals that enable a fast and stable recovery of the anisotropic object. The implementation of the imaging functionals is simple and avoids the need to solve an ill-posed problem. The resolution analysis of the imaging functionals is conducted using the Green representation formula. Furthermore, we establish stability estimates for these imaging functionals when noise is present in the data. To illustrate the effectiveness of the methods, we present numerical examples showcasing their performance. [ABSTRACT FROM AUTHOR]