From the point of view of energy analysis, the cause that the uniqueness of the boundary integral equation induced from the exterior Helmholtz problem does not hold is investigated in this paper. It is proved that the...From the point of view of energy analysis, the cause that the uniqueness of the boundary integral equation induced from the exterior Helmholtz problem does not hold is investigated in this paper. It is proved that the Sommerfeld's condition at the infinity is changed so that it is suitable not only for the radiative wave but also for the absorptive wave when we use the boundary integral equation to describe the exterior Helmholtz problem. There fore, the total energy of the system is conservative. The mathematical dealings to guarantee the uniqueness are discussed based upon this explanation.展开更多
A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Severa...A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.展开更多
This paper develops and analyzes interior penalty discontinuous Galerkin(IPDG)method by patch reconstruction technique for Helmholtz problems.The technique achieves high order approximation by locally solving a discre...This paper develops and analyzes interior penalty discontinuous Galerkin(IPDG)method by patch reconstruction technique for Helmholtz problems.The technique achieves high order approximation by locally solving a discrete least-squares over a neighboring element patch.We prove a prior error estimates in the L 2 norm and energy norm.For each fixed wave number k,the accuracy and efficiency of the method up to order five with high-order polynomials.Numerical examples are carried out to validate the theoretical results.展开更多
In this paper,we investigate the method of fundamental solutions(MFS)for solving exterior Helmholtz problems with high wave-number in axisymmetric domains.Since the coefficientmatrix in the linear system resulting fro...In this paper,we investigate the method of fundamental solutions(MFS)for solving exterior Helmholtz problems with high wave-number in axisymmetric domains.Since the coefficientmatrix in the linear system resulting fromtheMFS approximation has a block circulant structure,it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space.Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.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
There are N domains Dj(j=0,1,...,N-1) of different physical parameters in the whole space and their interfaces S,are non-horizontally smooth curved surfaces. The following boundary problem is called Hclinholiz boundar...There are N domains Dj(j=0,1,...,N-1) of different physical parameters in the whole space and their interfaces S,are non-horizontally smooth curved surfaces. The following boundary problem is called Hclinholiz boundary problem:The analytical solution of the above problem is given in this paper.展开更多
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-...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.展开更多
The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose bounda...The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica? is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.展开更多
Fundamental solution of Dirichlet boundary value problem of axisymmetric Helmholtz equation is constructed via modi?ed Bessel function of the second kind, which uni?ed the formulas of fundamental solution of Helmholtz...Fundamental solution of Dirichlet boundary value problem of axisymmetric Helmholtz equation is constructed via modi?ed Bessel function of the second kind, which uni?ed the formulas of fundamental solution of Helmholtz equation, elliptic type Euler-Poisson-Darboux equation and Laplace equation in any dimensional space.展开更多
In this paper,the boundary knot method is applied to simulate inverse problems of determining physical boundary occurring in an inaccessible interior part.This is fulfilled from measurements of the partially accessibl...In this paper,the boundary knot method is applied to simulate inverse problems of determining physical boundary occurring in an inaccessible interior part.This is fulfilled from measurements of the partially accessible outer boundary.The truncated singular value decomposition under parameter choice of the cross validation method is employed for noisy boundary data cases.Numerical results for two benchmark problems show that the boundary knot method is simple,accurate,stable and computationally efficient for inverse problems under domains with doubly connected domains.展开更多
文摘From the point of view of energy analysis, the cause that the uniqueness of the boundary integral equation induced from the exterior Helmholtz problem does not hold is investigated in this paper. It is proved that the Sommerfeld's condition at the infinity is changed so that it is suitable not only for the radiative wave but also for the absorptive wave when we use the boundary integral equation to describe the exterior Helmholtz problem. There fore, the total energy of the system is conservative. The mathematical dealings to guarantee the uniqueness are discussed based upon this explanation.
基金This work was supported by Natural Science Foundation of Fujian under Grant A0310002the Excellent Young Teachers Program (EYTP) of the Ministry of Education of China.
文摘A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.
基金the National Key Research and Development Program of China(No.2020YFA0714200)the Science and Technology Major Project of Hubei Province under Grant No.2021AAA010+1 种基金the National Science Foundation of China(Nos.12125103 and 12071362)the Natural Science Foundation of Hubei Province(No.2019CFA007).
文摘This paper develops and analyzes interior penalty discontinuous Galerkin(IPDG)method by patch reconstruction technique for Helmholtz problems.The technique achieves high order approximation by locally solving a discrete least-squares over a neighboring element patch.We prove a prior error estimates in the L 2 norm and energy norm.For each fixed wave number k,the accuracy and efficiency of the method up to order five with high-order polynomials.Numerical examples are carried out to validate the theoretical results.
基金The work described in this paper was supported by National Basic Research Program of China(973 Project No.2010CB832702)the R&D Special Fund for Public Welfare Industry(Hydrodynamics,Project No.201101014 and the 111 project under grant B12032)National Science Funds for Distinguished Young Scholars(Grant No.11125208).The third author acknowledges the support of Distinguished Overseas Visiting Scholar Fellowship provided by the Ministry of Education of China.
文摘In this paper,we investigate the method of fundamental solutions(MFS)for solving exterior Helmholtz problems with high wave-number in axisymmetric domains.Since the coefficientmatrix in the linear system resulting fromtheMFS approximation has a block circulant structure,it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space.Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.
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.
文摘There are N domains Dj(j=0,1,...,N-1) of different physical parameters in the whole space and their interfaces S,are non-horizontally smooth curved surfaces. The following boundary problem is called Hclinholiz boundary problem:The analytical solution of the above problem is given in this paper.
文摘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.
文摘The Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical harmonic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are defined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica? is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtained.
基金The NSF(11326152) of Chinathe NSF(BK20130736) of Jiangsu Province of Chinathe NSF(CKJB201709) of Nanjing Institute of Technology
文摘Fundamental solution of Dirichlet boundary value problem of axisymmetric Helmholtz equation is constructed via modi?ed Bessel function of the second kind, which uni?ed the formulas of fundamental solution of Helmholtz equation, elliptic type Euler-Poisson-Darboux equation and Laplace equation in any dimensional space.
基金Supported by the Natural Science Foundation of Anhui Province(1908085QA09)the Natural Science Research Project of Anhui Province(KJ2019A0591&KJ2017B015)Higher Education Department of the Ministry of Education(201802358008)。
文摘In this paper,the boundary knot method is applied to simulate inverse problems of determining physical boundary occurring in an inaccessible interior part.This is fulfilled from measurements of the partially accessible outer boundary.The truncated singular value decomposition under parameter choice of the cross validation method is employed for noisy boundary data cases.Numerical results for two benchmark problems show that the boundary knot method is simple,accurate,stable and computationally efficient for inverse problems under domains with doubly connected domains.
