Fast High Order Algorithm for Three-Dimensional Helmholtz Equation Involving Impedance Boundary Condition with Large Wave Numbers

Acoustic fields with impedance boundary conditions have high engineering applications, such as noise control and evaluation of sound insulation materials, and can be approximated by three-dimensional Helmholtz boundary value problems. Finite difference method is widely applied to solving these problems due to its ease of use. However, when the wave number is large, the pollution effects are still a major difficulty in obtaining accurate numerical solutions. We develop a fast algorithm for solving three-dimensional Helmholtz boundary problems with large wave numbers. The boundary of computational domain is discrete based on high-order compact difference scheme. Using the properties of the tensor product and the discrete Fourier sine transform method, the original problem is solved by splitting it into independent small tridiagonal subsystems. Numerical examples with impedance boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the algorithm has a fourth- order convergence in and -norms, and costs less CPU calculation time and random access memory.

THE BOUNDARY INTEGRAL METHOD FOR THE HELMHOLTZ EQUATION WITH CRACKS INSIDE A BOUNDED DOMAIN

We consider a kind of scattering problem by a crack F that is buried in a bounded domain D, and we put a point source inside the domain D. This leads to a mixed boundary value problem to the Helmholtz equation in the domain D with a crack Г. Both sides of the crack F are given Dirichlet-impedance boundary conditions, and different boundary condition （Dirichlet, Neumann or Impedance boundary condition） is set on the boundary of D. Applying potential theory, the problem can be reformulated as a system of boundary integral equations. We establish the existence and uniqueness of the solution to the system by using the Fredholm theory.

A theorem for quantum operator correspondence to the solution of the Helmholtz equation

We propose a theorem for the quantum operator that corresponds to the solution of the Helmholtz equation, i.e., ∫∫∫V （x1 ,x2,x3）〈X1 ,x2,x3〉〈x1 ,x2,x3〉d3x = V （X1 ,X2,X3） = e-λ2/4 ：V （X1 ,X2,X3）：,where V （X1 ,X2,X3） is the solution to the Helmholtz equation △2V ＋λ2V = 0, the symbol ： ： denotes normal ordering, and X1, X2, X3 are three-dimensional coordinate operators. This helps to derive the normally ordered expansion of Dirac＇s radius operator functions. We also discuss the normally ordered expansion of Bessel operator functions.

A Meshless Collocation Method with Barycentric Lagrange Interpolation for Solving the Helmholtz Equation

In this paper,Chebyshev interpolation nodes and barycentric Lagrange interpolation basis function are used to deduce the scheme for solving the Helmholtz equation.First of all,the interpolation basis function is applied to treat the spatial variables and their partial derivatives,and the collocation method for solving the second order differential equations is established.Secondly,the differential matrix is used to simplify the given differential equations on a given test node.Finally,based on three kinds of test nodes,numerical experiments show that the present scheme can not only calculate the high wave numbers problems,but also calculate the variable wave numbers problems.In addition,the algorithm has the advantages of high calculation accuracy,good numerical stability and less time consuming.

Fourier Moment Method with Regularization for the Cauchy Problem of Helmholtz Equation

In this paper, we consider the reconstruction of the wave field in a bounded domain. By choosing a special family of functions, the Cauchy problem can be transformed into a Fourier moment problem. This problem is ill-posed. We propose a regularization method for obtaining an approximate solution to the wave field on the unspecified boundary. We also give the convergence analysis and error estimate of the numerical algorithm. Finally, we present some numerical examples to show the effectiveness of this method.

The Investigation of the Fractional-View Dynamics of Helmholtz Equations Within Caputo Operator

It is eminent that partial differential equations are extensively meaningful in physics,mathematics and engineering.Natural phenomena are formulated with partial differential equations and are solved analytically or numerically to interrogate the system’s dynamical behavior.In the present research,mathematical modeling is extended and the modeling solutions Helmholtz equations are discussed in the fractional view of derivatives.First,the Helmholtz equations are presented in Caputo’s fractional derivative.Then Natural transformation,along with the decomposition method,is used to attain the series form solutions of the suggested problems.For justification of the proposed technique,it is applied to several numerical examples.The graphical representation of the solutions shows that the suggested technique is an accurate and effective technique with a high convergence rate than other methods.The less calculation and higher rate of convergence have confirmed the present technique’s reliability and applicability to solve partial differential equations and their systems in a fractional framework.

THE BEM FOR SOLVING THE NONHOMOGFNEOUS HELMHOLTZ EQUATION WITH VARIABLE COEF

Considering the fundamental solution of the Laplace equation as the weight function,we give the iterative format for solving the nonhomogeneous Helmholtz equation with variable coefficients. Furthermore, the iteration method of BEM for solving the equation mentioned above is obtained. The numerical example is given in this paper. Finally, the iteration method of BUM mentioned above is compared with the coupled method of BEM that was presented before then by authors.

A Kind of Boundary Element Methods for Boundary Value Problem of Helmholtz Equation

1.Problems for electromagnetic scattering are of significant importance in many areas oftechnology.In this paper we discuss the scattering problem of electromagnetic wave incidentby using boundary element method associated with splines.The problem is modelled by aboundary value problem for the Helmholtz equation

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

In this paper,the authors propose Neumann series neural operator(NSNO)to learn the solution operator of Helmholtz equation from inhomogeneity coefficients and source terms to solutions.Helmholtz equation is a crucial partial differential equation(PDE)with applications in various scientific and engineering fields.However,efficient solver of Helmholtz equation is still a big challenge especially in the case of high wavenumber.Recently,deep learning has shown great potential in solving PDEs especially in learning solution operators.Inspired by Neumann series in Helmholtz equation,the authors design a novel network architecture in which U-Net is embedded inside to capture the multiscale feature.Extensive experiments show that the proposed NSNO significantly outperforms the state-of-the-art FNO with at least 60%lower relative L^(2)-error,especially in the large wavenumber case,and has 50%lower computational cost and less data requirement.Moreover,NSNO can be used as the surrogate model in inverse scattering problems.Numerical tests show that NSNO is able to give comparable results with traditional finite difference forward solver while the computational cost is reduced tremendously.

New Sixth-Order Compact Schemes for Poisson/Helmholtz Equations

Some new sixth-order compact finite difference schemes for Poisson/Helmholtz equations on rectangular domains in both two-and three-dimensions are developed and analyzed.Different from a few sixth-order compact finite difference schemes in the literature,the finite difference and weight coefficients of the new methods have analytic simple expressions.One of the new ideas is to use a weighted combination of the source term at staggered grid points which is important for grid points near the boundary and avoids partial derivatives of the source term.Furthermore,the new compact schemes are exact for 2D and 3D Poisson equations if the solution is a polynomial less than or equal to 6.The coefficient matrices of the new schemes are M-matrices for Helmholtz equations with wave number K≤0,which guarantee the discrete maximum principle and lead to the convergence of the new sixth-order compact schemes.Numerical examples in both 2D and 3D are presented to verify the effectiveness of the proposed schemes.

Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D

An adapted-bubbles approach which is a modification of the residualfree bubbles(RFB)method,is proposed for the Helmholtz problem in 2D.A new two-level finite element method is introduced for the approximations of the bubble functions.Unlike the other equations such as the advection-diffusion equation,RFB method when applied to the Helmholtz equation,does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems.Adapted-bubbles(AB)are obtained by a simple modification of the sub-problems.This modification increases the accuracy of the numerical solution impressively.We provide numerical experiments with the AB method up to ch=5 where c is the wavenumber and h is the mesh size.Numerical tests show that the AB method is better by far than higher order methods available in the literature.

Deep Domain Decomposition Methods:Helmholtz Equation

This paper proposes a deep-learning-based Robin-Robin domain decomposition method(DeepDDM)for Helmholtz equations.We first present the plane wave activation-based neural network(PWNN),which is more efficient for solving Helmholtz equations with constant coefficients and wavenumber k than finite difference methods(FDM).On this basis,we use PWNN to discretize the subproblems divided by domain decomposition methods(DDM),which is the main idea of DeepDDM.This paper will investigate the number of iterations of using DeepDDM for continuous and discontinuous Helmholtz equations.The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domain decomposition method(FDM-DDM)under the same Robin parameters,i.e.,the number of iterations by DeepDDM is almost the same as that of FDM-DDM.By choosing suitable Robin parameters on different subdomains,the convergence rate is almost constant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights for improving the PDE solver by deep learning.

COMPACT FOURTH-ORDER FINITE DIFFERENCE SCHEMES FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS

In this paper, two fourth-order accurate compact difference schemes are presented for solving the Helmholtz equation in two space dimensions when the corresponding wave numbers are large. The main idea is to derive and to study a fourth-order accurate compact difference scheme whose leading truncation term, namely, the O（h^4） term, is independent of the wave number and the solution of the Helmholtz equation. The convergence property of the compact schemes are analyzed and the implementation of solving the resulting linear algebraic system based on a FFT approach is considered. Numerical results are presented, which support our theoretical predictions.

A Source Transfer Domain Decomposition Method For Helmholtz Equations in Unbounded Domain Part II: Extensions

In this paper we extend the source transfer domain decomposition method(STDDM)introduced by the authors to solve the Helmholtz problems in two-layered media,the Helmholtz scattering problems with bounded scatterer,and Helmholtz problems in 3D unbounded domains.The STDDM is based on the decomposition of the domain into non-overlapping layers and the idea of source transfer which transfers the sources equivalently layer by layer so that the solution in the final layer can be solved using a PML method defined locally outside the last two layers.The details of STDDM is given for each extension.Numerical results are presented to demonstrate the efficiency of STDDM as a preconditioner for solving the discretization problem of the Helmholtz problems considered in the paper.

FOURTH-ORDER COMPACT SCHEMES FOR HELMHOLTZ EQUATIONS WITH PIECEWISE WAVE NUMBERS IN THE POLAR COORDINATES

In this paper, fourth-order compact finite difference schemes are proposed for solving Helmholtz equation with piecewise wave numbers in polar coordinates with axis-symmetric and in some cases that the solution depends both of independent variables. The idea of the immersed interface method is applied to deal with the discontinuities in the wave number and certain derivatives of the solution. Numerical experiments are included to confirm the accuracy and efficiency of the proposed method.

A TAILORED FINITE POINT METHOD FOR THE HELMHOLTZ EQUATION WITH HIGH WAVE NUMBERS IN HETEROGENEOUS MEDIUM

In this paper, we propose a tailored-finite-point method for the numerical simulation of the Helmholtz equation with high wave numbers in heterogeneous medium. Our finite point method has been tailored to some particular properties of the problem, which allows us to obtain approximate solutions with the same behaviors as that of the exact solution very naturally. Especially, when the coefficients are piecewise constant, we can get the exact solution with only one point in each subdomain. Our finite-point method has uniformly convergent rate with respect to wave number k in L^2-norm.

FINITE ELEMENT AND DISCONTINUOUS GALERKIN METHOD FOR STOCHASTIC HELMHOLTZ EQUATION IN TWO-AND THREE-DIMENSIONS

In this paper, we consider the finite element method and discontinuous Galerkin method for the stochastic Helmholtz equation in R^d （d = 2, 3）. Convergence analysis and error estimates are presented for the numerical solutions. The effects of the noises on the accuracy of the approximations are illustrated. Numerical experiments are carried out to verify our theoretical results.

INFINITE ELEMENT METHOD FOR THE EXTERIOR PROBLEMS OF THE HELMHOLTZ EQUATIONS

There are two cases of the exterior problems of the Helmholtz equation. If λ ≥ 0 the bilinear form is coercive, and if λ < 0 it is the scattering problem. We give a new approach of the infinite element method, which enables us to solve these exterior problems as well as corner problems. A numerical example of the scattering problem is given. [ABSTRACT FROM AUTHOR]

TAILORED FINITE CELL METHOD FOR SOLVING HELMHOLTZ EQUATION IN LAYERED HETEROGENEOUS MEDIUM

In this paper, we propose a tailored finite cell method for the computation of two- dimensional Helmholtz equation in layered heterogeneous medium. The idea underlying the method is to construct a numerical scheme based on a local approximation of the solution to Helmholtz equation. This provides a computational tool of achieving high accuracy with coarse mesh even for large wave number （high frequency）. The stability analysis and error estimates of this method are also proved. We present several numerical results to show its efficiency and accuracy.

A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation

A weak Galerkin(WG)method is introduced and numerically tested for the Helmholtz equation.This method is flexible by using discontinuous piecewise polynomials and retains the mass conservation property.At the same time,the WG finite element formulation is symmetric and parameter free.Several test scenarios are designed for a numerical investigation on the accuracy,convergence,and robustness of the WG method in both inhomogeneous and homogeneous media over convex and non-convex domains.Challenging problems with high wave numbers are also examined.Our numerical experiments indicate that the weak Galerkin is a finite element technique that is easy to implement,and provides very accurate and robust numerical solutions for the Helmholtz problem with high wave numbers.