Solving Helmholtz Equation Research Articles

Learning based numerical methods for acoustic frequency-domain simulation with high frequency

Acoustic simulation in frequency-domain is related to solving Helmholtz equations, which is still highly challenging at high frequency with complex geometries. In this paper, a learning based numerical method (LbNM) is proposed for general boundary value problems of Helmholtz equation. By using Tikhonov regularization, the solution operator is stably learned from various data solutions especially fundamental solutions. The reconstructed solution operator is reusable and enables fast solution updating with new boundary inputs. The present method is featured by strong interpretability and generalizability. Numerical results demonstrate advantages in precision and efficiency. It can be applied to various engineering problems such as cabin acoustic simulations flexibly.

NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium

Application of Adomian Decomposition Method for Solving Helmholtz Equation

The Adomian decomposition method (ADM) is effective for solving linear and nonlinear ordinary and partial differential equations, integral equations, algebraic equations and integro-differential equations and it leads to an analytical solution in the form of an infinite power series. We will solve the Helmholtz equation in two dimensional by the Adomian decomposition method.
 Key words: Application of Adomian Decomposition Method, Helmholtz Equation, Adomian Decomposition Method. 

Efficient Finite Difference Approaches for Solving Initial Boundary Value Problems in Helmholtz Partial Differential Equations

This study presents numerical solutions for initial boundary value problems of homogeneous and nonhomogeneous Helmholtz equations using first- and second-order difference schemes. The stability of these methods is rigorously analyzed, ensuring their reliability and convergence for a wide range of problem instances. The proposed schemes’ robustness and applicability are demonstrated through several examples, accompanied by an error analysis table and illustrative graphs that visually represent the accuracy of the solutions obtained. The results confirm the effectiveness and efficiency of the proposed schemes, making them valuable tools for solving Helmholtz equations in practical applications.

Meshless generalized finite difference method with a domain-decomposition method for solving Helmholtz equation and its application to caisson resonance problems

This paper aims at presenting a meshless Generalized Finite Difference Method (GFDM) for solving the Helmholtz equation. In particular, in order to tackle the problem where a fluid domain is divided into different parts by a thin-wall structure, a Domain-Decomposition Method (DDM) is introduced to prevent interactions between node pairs coming from different parts of the flow domain. Two rigorous benchmarks are performed to validate the convergence and accuracy of the present GFDM-DDM model. Furthermore, caisson resonance problems are also investigated and discussed based on the novel numerical tool. It is shown that the present GFDM-DDM model can provide satisfactory predictions for resonance modes in caisson resonance problems, and hence can be treated as a powerful numerical tool in ocean engineering design due to its accurate and efficient characteristics.

A quasi Monte Carlo-based model reduction method for solving Helmholtz equation in random media

A quasi Monte Carlo-based model reduction method for solving Helmholtz equation in random media

An efficient neural network method with plane wave activation functions for solving Helmholtz equation

In this paper, we propose a plane wave activation based neural network (PWNN) to solve the Helmholtz equation with constant coefficients and relatively large wave number k efficiently. Since the complex activation function eix→ is introduced to be the activation function of the neural network, PWNN significantly improves the computational speed and accuracy as compared to traditional activation based neural networks (TANN) and the finite difference method (FDM), for relatively large wave number problems. We establish a new upper bound for error estimates for homogeneous Helmholtz solutions in two dimensions by plane waves, which is different from the previous estimates by generalized harmonic polynomials. Based on the new error estimates, the theoretical guidance is given for choosing the number of neural network's neurons and the initial value to accelerate network training. The analyses of computational complexity related to the wave number k are given for PWNN with two layers, TANN, the plane wave partition of unity method (PWPUM) and FDM. Numerical experiments in 2D and 3D are performed to demonstrate the efficiency and accuracy of PWNN. Especially for large wave number problems, like k=500, PWNN can get the solution with relative error less than 10−4 in less than 20 seconds, which is more efficient than other methods.

A Feedforward Neural Network for Modeling of Average Pressure Frequency Response

The Helmholtz equation has been used for modeling the sound pressure field under a harmonic load. Computing harmonic sound pressure fields by means of solving Helmholtz equation can quickly become unfeasible if one wants to study many different geometries for ranges of frequencies. We propose a machine learning approach, namely a feedforward dense neural network, for computing the average sound pressure over a frequency range. The data are generated with finite elements, by numerically computing the response of the average sound pressure, by an eigenmode decomposition of the pressure. We analyze the accuracy of the approximation and determine how much training data is needed in order to reach a certain accuracy in the predictions of the average pressure response.

Trailing-Edge Noise Prediction by Solving Helmholtz Equation with Stochastic Source Term

A numerical approach to predict broadband trailing-edge noise for low Mach number flows is presented. It is based on the combination of a Helmholtz solver to propagate sound waves with a sound source term computed stochastically. The sound propagation is performed in the frequency domain using a high-order finite element solver. The stochastic approach is based on a random particle-mesh method. The performance of this numerical approach is first examined for a Gaussian source. The numerical approach is then applied to a NACA0012 airfoil for flow Mach numbers of 0.11 and 0.16 and to a controlled-diffusion airfoil for a Mach number of 0.047. The predicted sound levels are compared with experimental data and acoustic results obtained from the acoustic perturbations equations. The mean flow does not significantly modify the acoustic propagation. Using a no-flow propagation model like the Helmholtz equation is therefore a valid approach for low Mach numbers. The reduced computational cost of a Helmholtz calculation, together with the speed of the random-particle mesh turbulence synthesis, allows for fast predictions. While this approach provides relative assessments between different configurations (in particular the spectrum shape), it often requires the inclusion of calibration factors determined from reference measurement or numerical data.

Finite difference method based on polynomial interpolation for solving Helmholtz equation

The generalized finite difference formula plays an important role in the meshless method for solving differential equations. The main purpose of this paper is to find the numerical solution of Helmholtz equation, an elliptic partial differential equation describing electromagnetic waves in physics, by using the generalized finite difference formula. Firstly, this paper introduces a simple and practical nodal distribution, which not only guarantees the uniqueness of multivariate polynomial interpolation, but also makes the matrix triangular, so that the constructed basic polynomials can be transformed into Lagrange basis polynomials. Secondly, the generalized finite difference formula is created by polynomial interpolation. Finally, a numerical example of Helmholtz equation under general boundary conditions is given.

Dual system least squares finite element method for the Helmholtz equation

This paper uses LL∗ method to solve Helmholtz equation in case when the frequency parameter is large. The Helmholtz equation is reformulated to the first order system by introducing a new variable and then the corresponding dual problem is considered. The unique existence to the weak solution of the dual problem is proven and the explicit forms of continuity and coercivity constants depending on frequency parameter are developed. We describe numerical experiments with various scalings that result in different performances based on the a priori error estimation. The results are presented and compared with the ones that are obtained by the standard least-squares finite element approach.

The calculation of the wavelengths of natural electromagnetic oscillations of a cell

The equations for calculations of wavelengths of natural electromagnetic oscillations of a spherical cell were obtained. The cell is considered as a spherical resonator filled with an isotropic dielectric. New way to solve Helmholtz equation was obtained. The results of calculations of resonant wavelengths of a cell depending on their size and dielectric permittivity are presented. The calculation shows that the value of the maximum critical wavelengths lie within the submillimeter range.

Features of electromagnetic wave propagation in a longitudinally magnetized gyrotropic elliptic waveguide

In scientific literature the electromagnetic waves propagation in gyrotropic elliptic waveguides has not been studied enough due to: 1) difficulties of deriving Helmholtz equations for gyrotropic elliptic waveguides in arbitrary magnetization directions, 2) difficulties of solving Helmholtz equations and deriving the dispersion equations, 3) difficulties of solving the dispersion equations themselves. In this work, for the first time, dispersion equations for longitudinally magnetized gyrotropic elliptic waveguides were solved. Basing on the solutions results, the dependencies of the constant propagation: 1) on the strength and direction of the magnetizing field, 2) on the waveguide ellipticity at a constant ferrite magnetization were investigated. Studies have revealed previously unknown features inherent only to the electromagnetic waves in such waveguides propagation and showed that: 1) with the ferrite magnetization increase -3,1 times the polarization plane rotation angle increases by «5,4 times; 2) the gyrotropic elliptic waveguide polarization plane rotation angle is greater than that of a similar round waveguide. The results show the main advantage of gyrotropic elliptic guide systems over circular ones consists in the possibilities of varying both external and geometric parameters of gyrotropic elliptic guide systems and obtaining significant phase raids, which in turn allows developing more efficient phase shifters.

Dynamic Response of Piles under Wave Loading Based on Comsol Multiphysics

In order to simulate the action of waves on Submarine Structures, COMSOL Multiphysics Finite Element Analysis Software was used. By solving Helmholtz equation in frequency domain of given frequency and introducing artificially applied background acoustics field, considering incidence and penetration of acoustics pressure at the junction of seabed and water surface, a total acoustics field is established. Porous media was characterized by Biot’s consolidation theory to compute the displacement field and acoustics pressure fluctuation in porous media propagating porous elastic waves, considering the viscous loss of the model,so the deformation of Solid Matrix can be described. Finally, a numerical model of wave-pile-seabed was established by coupling the pressure acoustic physical field with the physical field of porous media. The influence of different wave frequencies and wave incidence angles on the displacement of piles and the influence of different wave incidence angles on the stress of piles were analyzed. Finally, the liquefaction of soil around piles was studied.

Spectral generalized function method for solving homogeneous partial differential equations with constant coefficients

A method based on a generalized function in Fourier space gives analytical solutions to homogeneous partial differential equations with constant coefficients of any order in any number of dimensions. The method exploits well-known properties of the Dirac delta, reducing the differential mathematical problem into the factorization of an algebraic expression that finally has to be integrated. In particular, the method was utilized to solve the most general homogeneous second order partial differential equation in Cartesian coordinates, finding a general solution for non-parabolic partial differential equations, which can be seen as a generalization of d'Alambert solution. We found that the traditional classification, i.e., parabolic, hyperbolic and elliptic, is not necessary reducing the classification to only parabolic and non-parabolic cases. We put special attention for parabolic partial differential equations, analyzing the general 1D homogeneous solution of the Photoacoustic and Photothermal equations in the frequency and time domain. Finally, we also used the method to solve Helmholtz equation in cylindrical coordinates, showing that it can be used in other coordinates systems.

Preconditioning RBF collocation method by adapted finite difference matrix

In this paper, we shall consider an adapted finite difference preconditioning technique for radial basis function (RBF) method. This technique, for the system of equations that arises from solving Helmholtz equation with Dirichlet boundary conditions by RBF collocation method, is considered. We prove that the eigenvalues of preconditioned matrices are bounded for one and two-dimensional cases. We also show that RBF interpolating matrix can be well-conditioned by using appropriate number of polynomial bases in augmented term of RBF approximation. Numerical experiments show the efficiency and robustness of the preconditioning procedure, particularly when there are non-zero boundary conditions.

Solving Helmholtz Equation with Local Fractional Derivative Operators

The paper presents a new analytical method called the local fractional Laplace variational iteration method (LFLVIM), which is a combination of the local fractional Laplace transform (LFLT) and the local fractional variational iteration method (LFVIM), for solving the two-dimensional Helmholtz and coupled Helmholtz equations with local fractional derivative operators (LFDOs). The operators are taken in the local fractional sense. Two test problems are presented to demonstrate the efficiency and the accuracy of the proposed method. The approximate solutions obtained are compared with the results obtained by the local fractional Laplace decomposition method (LFLDM). The results reveal that the LFLVIM is very effective and convenient to solve linear and nonlinear PDEs.

On the formulation of a BEM in the Bézier–Bernstein space for the solution of Helmholtz equation

• Novel boundary element approach to yield a geometry-independent field approximation. • Geometrically based on both computer aid design (CAD). • Field variables are independently approximated from the geometry. • Generalised Lagrange interpolation functions are used as element shape functions . • Applicability for solving Helmholtz equation in longitudinally invariant problems. This paper proposes a novel boundary element approach formulated on the Bézier-Bernstein basis to yield a geometry-independent field approximation. The proposed method is geometrically based on both computer aid design (CAD) and isogeometric analysis (IGA), but field variables are independently approximated from the geometry. This approach allows the appropriate approximation functions for the geometry and variable field to be chosen. We use the Bézier–Bernstein form of a polynomial as an approximation basis to represent both geometry and field variables. The solution of the element interpolation problem in the Bézier–Bernstein space defines generalised Lagrange interpolation functions that are used as element shape functions. The resulting Bernstein–Vandermonde matrix related to the Bézier–Bernstein interpolation problem is inverted using the Newton-Bernstein algorithm. The applicability of the proposed method is demonstrated solving the Helmholtz equation over an unbounded region in a two-and-a-half dimensional (2.5D) domain.

Critical Review of Preconditioners for Helmholtz Equation and their Spectral Analysis

Objectives: The study reviews the iterative solvers for solving Helmholtz equation. It also surveys the preconditioners for Krylov subspace methods for discrete Helmholtz equation. Numerical experiments based conclusions are drawn, after a thorough comparison of various preconditioners. Methods/Statistical Analysis: Five point finite difference scheme for two-dimensional Helmholtz equation is employed to obtain discrete analogue of Helmholtz equation. Krylov subspace method for is considered for iterations. Matrix based preconditioner ILU with variety of fill-in and operator based preconditioner complex shifted Laplace preconditioner (CSLP) are tested. Different shifts are proposed and experimented in CSLP. Optimization of shifts is recommended on basis of results. Spectral analysis is performed to validate results. Findings: For small wave number problem, ILU type preconditioned performs better and is fairly comparable with CSLP. However, increasing wave number makes unaffordable in terms of computational cost. The pure imaginary shift (0,1) and complex shift (1, / 4 π ) costsless time and number of iterations compared to other choices of shift. These choices are recommended when preconditioner is inverted directly. Approximate inversion may affect iterations, but will save solve time for large problems. Extensive spectral analysis, validating results, highlights this work. Application/Improvements: Dependency upon wave number is decreased with this recommended choice of shifts in CSLP. The number of iterations increases at very low rate when wave number is increased at rate of 2. Keywords: Helmholtz Equation, Iterative Methods, Krylov Subspace Methods, Preconditioners, Spectral Analysis

Two-Level Block Decompositions for Solving Helmholtz Equation via Chebyshev Pseudo Spectral Method

In this paper, we consider solving the Helmholtz equation in the Cartesian domain , subject to homogeneous Dirichlet boundary condition, discretized with the Chebyshev pseudo-spectral method. The main purpose of this paper is to present the formulation of a two-level decomposition scheme for decoupling the linear system obtained from the discretization into independent subsystems. This scheme takes advantage of the homogeneity property of the physical problem along one direction to reduce a 2D problem to several 1D problems via a block diagonalization approach and the reflexivity property along the second direction to decompose each of the 1D problems to two independent subproblems using a reflexive decomposition, effectively doubling the number of subproblems. Based on the special structure of the coefficient matrix of the linear system derived from the discretization and a reflexivity property of the second-order Chebyshev differentiation matrix, we show that the decomposed submatrices exhibits a similar property, enabling the system to be decomposed using reflexive decompositions. Explicit forms of the decomposed submatrices are derived. The decomposition not only yields more efficient algorithm but introduces coarse-grain parallelism. Furthermore, it preserves all eigenvalues of the original matrix.