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 an...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.展开更多
The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and ...The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.展开更多
A regularized recursive linearization method is developed for a two-dimensional in-verse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium deposite...A regularized recursive linearization method is developed for a two-dimensional in-verse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium deposited on a homogeneous substrate from data accessible through photon scanning tunneling microscopy experiments. In addition to the ill-posedness of the inverse scattering problems, two difficulties arise from the layered back-ground medium and limited aperture data. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to the weak scattering at a low frequency, each update is obtained via recursive linearization with respect to the wavenumber by solving one forward problem and one adjoint problem of the Helmholtz equation. Numerical experiments are included to illustrate the feasibility of the proposed method.展开更多
In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced...In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced. Taking advantage of the relationship between Euler-Maclaurin formula and trapezoidal quadrature rule,and the relationship between trapezoidal and square quadrature rule,sharp computable bound with analytical form on the error of numerical integration of Bessel integral identity by square quadrature rule is derived in this paper. Numerical experiments are presented at the end to demonstrate the accuracy of the sharp computable bound on the numerical integration error.展开更多
Noise reduction for aircraft engine has attracted great concern due to the strict noise control regulation nowadays. Conventional perforated sheet-over-honeycomb acoustic liners have been widely used to attenuate turb...Noise reduction for aircraft engine has attracted great concern due to the strict noise control regulation nowadays. Conventional perforated sheet-over-honeycomb acoustic liners have been widely used to attenuate turbofan engine noise. To dampen the broadband noise and resist the harsh service conditions with high temperature and pressure in modern turbofan engine,new acoustic liner concepts are proposed and evaluated in the latest decade. In this review,available studies regarding the recent development of liners are gathered. The paper starts with the introduction of acoustic absorption mechanism of local-reacting and extended-reacting liners. The progress of novel passive liners(e.g.,mesh-cap liner,variable-depth liner,metal foam liner,hybrid liner,etc.) is summarized. Furthermore,adaptive liners with tunable geometry dimension or bias flow are illustrated in details.Metamaterial is also mentioned as a hot candidate in the next generation of acoustic liners. Finally,this review identifies benefits and some technical challenges with the goal of unveiling the potential of novel acoustic liner technologies in aero engine.展开更多
In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transp...In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.展开更多
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 ...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.展开更多
Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave veloc...Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave velocity is assumed to have power-law variation in the radial direction only, and the shear modulus is constant. The Helmholtz equation with a variable coefficient is equivalently transformed into a standard Helmholtz equation with a general conformal transformation method（GCTM）. The displacements and stress fields are proposed. Numerical results show that the wave number and the inhomogeneity parameter of the medium have significant effects on the dynamic stress concentration around the circular cavity. The dynamic stress concentration factor（DSCF） becomes singular when the inhomogeneity parameter of medium is close to zero.展开更多
This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x...This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.展开更多
In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave func...In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.展开更多
The aim of this lab was to determine an experimental value for the charge-to-mass ratio e/m<sub>e</sub> of the electron. In order to do this, an assembly consisting of Helmholtz coils and a helium-filled f...The aim of this lab was to determine an experimental value for the charge-to-mass ratio e/m<sub>e</sub> of the electron. In order to do this, an assembly consisting of Helmholtz coils and a helium-filled fine beam tube containing an electron gun was used. Electrons were accelerated from rest by the electron gun at a voltage of 201.3 V kept constant across trials. When the accelerated electrons collided with the helium atoms in the fine beam tube, the helium atoms entered an excited state and released energy as light. Since the Helmholtz coils put the electrons into centripetal motion, this resulted in a circular beam of light, the radius of which was measured by taking a picture and using photo analysis. This procedure was used to test currents through the Helmholtz coils ranging from 1.3 A to 1.7 A in increments of 0.1 A. Using a linearization of these data, the experimental value for the charge-to-mass ratio of the electron was found to be 1.850 × 10<sup>11</sup> C/kg, bounded between 1.440 × 10<sup>11</sup> C/kg and 2.465 × 10<sup>11</sup> C/kg. This range of values includes the accepted value of 1.759 × 10<sup>11</sup> C/kg, and yields a percent error of 5.17%. The rather low percent error is a testament to the accuracy of this procedure. During this experiment, the orientation of the ambient magnetic field due to the Earth at the center of the apparatus was not considered. In the future, it would be worthwhile to repeat this procedure, taking care to position the Helmholtz coils in such a way to negate the effects of the Earth’s magnetic field on the centripetal motion of electrons.展开更多
We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered fie...We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem.Some numerical examples to show the feasibility of the method are presented.展开更多
In this article, we study numerically a Helmholtz decomposition methodology, based on a formulation of the mathematical model as a saddle-point problem. We use a preconditioned conjugate gradient algorithm, applied to...In this article, we study numerically a Helmholtz decomposition methodology, based on a formulation of the mathematical model as a saddle-point problem. We use a preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To solve the elliptic partial differential equations, we use a second order mixed finite element approximation for discretization. We show, using 2-D synthetic vector fields, that this approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation.展开更多
基金supported by Natural Science Foundation of China under grant number 10471047
文摘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.
文摘The present work describes the application of the method of fundamental solutions （MFS） along with the analog equation method （AEM） and radial basis function （RBF） approximation for solving the 2D isotropic and anisotropic Helmholtz problems with different wave numbers. The AEM is used to convert the original governing equation into the classical Poisson＇s equation, and the MFS and RBF approximations are used to derive the homogeneous and particular solutions, respectively. Finally, the satisfaction of the solution consisting of the homogeneous and particular parts to the related governing equation and boundary conditions can produce a system of linear equations, which can be solved with the singular value decomposition （SVD） technique. In the computation, such crucial factors related to the MFS-RBF as the location of the virtual boundary, the differential and integrating strategies, and the variation of shape parameters in multi-quadric （MQ） are fully analyzed to provide useful reference.
基金The research was supported in part by the ONR grant N000140210365the NSF grants DMS-0604790 and CCF-0514078the National Science Foundation of China grant 10428105.
文摘A regularized recursive linearization method is developed for a two-dimensional in-verse medium scattering problem that arises in near-field optics, which reconstructs the scatterer of an inhomogeneous medium deposited on a homogeneous substrate from data accessible through photon scanning tunneling microscopy experiments. In addition to the ill-posedness of the inverse scattering problems, two difficulties arise from the layered back-ground medium and limited aperture data. Based on multiple frequency scattering data, the method starts from the Born approximation corresponding to the weak scattering at a low frequency, each update is obtained via recursive linearization with respect to the wavenumber by solving one forward problem and one adjoint problem of the Helmholtz equation. Numerical experiments are included to illustrate the feasibility of the proposed method.
基金the National Natural Science Foundation of China (No. 11074170)the Independent Research Program of State Key Laboratory of Machinery System and Vibration (SKLMSV) (No. MSV-MS-2008-05)the Visiting Scholar Program of SKLMSV (No. MSV-2009-06)
文摘In 2D fast multipole method for scattering problems,square quadrature rule is used to discretize the Bessel integral identity for diagonal expansion of 2D Helmholtz kernel,and numerical integration error is introduced. Taking advantage of the relationship between Euler-Maclaurin formula and trapezoidal quadrature rule,and the relationship between trapezoidal and square quadrature rule,sharp computable bound with analytical form on the error of numerical integration of Bessel integral identity by square quadrature rule is derived in this paper. Numerical experiments are presented at the end to demonstrate the accuracy of the sharp computable bound on the numerical integration error.
文摘Noise reduction for aircraft engine has attracted great concern due to the strict noise control regulation nowadays. Conventional perforated sheet-over-honeycomb acoustic liners have been widely used to attenuate turbofan engine noise. To dampen the broadband noise and resist the harsh service conditions with high temperature and pressure in modern turbofan engine,new acoustic liner concepts are proposed and evaluated in the latest decade. In this review,available studies regarding the recent development of liners are gathered. The paper starts with the introduction of acoustic absorption mechanism of local-reacting and extended-reacting liners. The progress of novel passive liners(e.g.,mesh-cap liner,variable-depth liner,metal foam liner,hybrid liner,etc.) is summarized. Furthermore,adaptive liners with tunable geometry dimension or bias flow are illustrated in details.Metamaterial is also mentioned as a hot candidate in the next generation of acoustic liners. Finally,this review identifies benefits and some technical challenges with the goal of unveiling the potential of novel acoustic liner technologies in aero engine.
基金supported in part by a grant from the Research Grants Council of the Hong Kong Special Administrative Region,China (Project No.CityU 102204).
文摘In this paper,we study the electromagnetic scattering from a two dimen- sional large rectangular open cavity embedded in an infinite ground plane,which is modelled by Helmholtz equations.By introducing nonlocal transparent boundary con- ditions,the problem in the open cavity is reduced to a bounded domain problem.A hypersingular integral operator and a weakly singular integral operator are involved in the TM and TE cases,respectively.A new second-order Toeplitz type approximation and a second-order finite difference scheme are proposed for approximating the hyper- singular integral operator on the aperture and the Helmholtz in the cavity,respectively. The existence and uniqueness of the numerical solution in the TE case are established for arbitrary wavenumbers.A fast algorithm for the second-order approximation is pro- posed for solving the cavity model with layered media.Numerical results show the second-order accuracy and efficiency of the fast algorithm.More important is that the algorithm is easy to implement as a preconditioner for cavity models with more general media.
基金supported by the National Science Foundation of China under Grant No.92370125the National Key R&D Program of China under Grant Nos.2019YFA0709600 and 2019YFA0709602.
文摘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.
基金Project supported by the Earthquake Industry Special Science Research Foundation Project(No.201508026-02)the Natural Science Foundation of Heilongjiang Province of China(No.A201310)
文摘Based on the theory of complex function and the principle of homogenization, harmonic dynamics stress of a radially infinite inhomogeneous medium with a circular cavity is investigated. Due to the symmetry, wave velocity is assumed to have power-law variation in the radial direction only, and the shear modulus is constant. The Helmholtz equation with a variable coefficient is equivalently transformed into a standard Helmholtz equation with a general conformal transformation method（GCTM）. The displacements and stress fields are proposed. Numerical results show that the wave number and the inhomogeneity parameter of the medium have significant effects on the dynamic stress concentration around the circular cavity. The dynamic stress concentration factor（DSCF） becomes singular when the inhomogeneity parameter of medium is close to zero.
基金supported by the NSF of China(10571079,10671085)and the program of NCET
文摘This paper is concerned with the Cauchy problem for the modified Helmholtz equation in an infinite strip domain 0<x≤1,y∈R.The Cauchy data at x = 0 is given and the solution is then sought for the interval 0<x≤1.This problem is highly ill-posed and the solution(if it exists) does not depend continuously on the given data. In this paper,we propose a fourth-order modified method to solve the Cauchy problem. Convergence estimates are presented under the suitable choices of regularization parameters and the a priori assumption on the bounds of the exact solution.Numerical implementation is considered and the numerical examples show that our proposed method is effective and stable.
基金The authors would like to thank the reviewers and Dr.Zheng Enxi for many valuable suggcstions. This work is supported by the National Natural Science Foundation of China （Grant No. 11371172, 51178001）, Science and technology research project of the education department of Jilin Province （Grant No. 2014213）.
文摘In this paper, we investigate the use of ultra weak variational formulation to solve a wave scattering problem in near field optics. In order to capture the sub-scale features of waves, we utilize evanescent wave functions together with plane wave functions to approximate the local properties of the field. We analyze the global convergence and give an error estimation of the method. Numerical examples are also presented to demonstrate the effectiveness of the strategy.
文摘The aim of this lab was to determine an experimental value for the charge-to-mass ratio e/m<sub>e</sub> of the electron. In order to do this, an assembly consisting of Helmholtz coils and a helium-filled fine beam tube containing an electron gun was used. Electrons were accelerated from rest by the electron gun at a voltage of 201.3 V kept constant across trials. When the accelerated electrons collided with the helium atoms in the fine beam tube, the helium atoms entered an excited state and released energy as light. Since the Helmholtz coils put the electrons into centripetal motion, this resulted in a circular beam of light, the radius of which was measured by taking a picture and using photo analysis. This procedure was used to test currents through the Helmholtz coils ranging from 1.3 A to 1.7 A in increments of 0.1 A. Using a linearization of these data, the experimental value for the charge-to-mass ratio of the electron was found to be 1.850 × 10<sup>11</sup> C/kg, bounded between 1.440 × 10<sup>11</sup> C/kg and 2.465 × 10<sup>11</sup> C/kg. This range of values includes the accepted value of 1.759 × 10<sup>11</sup> C/kg, and yields a percent error of 5.17%. The rather low percent error is a testament to the accuracy of this procedure. During this experiment, the orientation of the ambient magnetic field due to the Earth at the center of the apparatus was not considered. In the future, it would be worthwhile to repeat this procedure, taking care to position the Helmholtz coils in such a way to negate the effects of the Earth’s magnetic field on the centripetal motion of electrons.
基金the National Natural Science Foundation of China(Grant No.10431030)
文摘We consider a mathematical problem modelling some characteristics of near field optical microscope.We take a monofrequency line source to illuminate a sample with constant index of refraction and use the scattered field data measured near the sample to reconstruct the shape of it. Mixed reciprocity relation and factorization method are applied to solve our problem.Some numerical examples to show the feasibility of the method are presented.
文摘In this article, we study numerically a Helmholtz decomposition methodology, based on a formulation of the mathematical model as a saddle-point problem. We use a preconditioned conjugate gradient algorithm, applied to an associated operator equation of elliptic type, to solve the problem. To solve the elliptic partial differential equations, we use a second order mixed finite element approximation for discretization. We show, using 2-D synthetic vector fields, that this approach, yields very accurate solutions at a low computational cost compared to traditional methods with the same order of approximation.
