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 ...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.展开更多
We consider the numerical solution for the Helmholtz equation in R^2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based ...We consider the numerical solution for the Helmholtz equation in R^2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.展开更多
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
In this paper, we give a hybrid method to numerically solve the inverse open cavity scattering problem for cavity shape, given the scattered solution on the opening of the cavity. This method is a hybrid between an it...In this paper, we give a hybrid method to numerically solve the inverse open cavity scattering problem for cavity shape, given the scattered solution on the opening of the cavity. This method is a hybrid between an iterative method and an integral equations method for solving the Cauchy problem. The idea of this hybrid method is simple, the operation is easy, and the computation cost is small. Numerical experiments show the feasibility of this method, even for cases with noise.展开更多
This paper is concerned with the problem of scattering of time-harmonic electromag- netic waves from penetrable diffraction gratings in the 2D polarization case. We propose a new, weakly singular, integral equation fo...This paper is concerned with the problem of scattering of time-harmonic electromag- netic waves from penetrable diffraction gratings in the 2D polarization case. We propose a new, weakly singular, integral equation formulation for the scattering problem which is proved to be uniquely solvable. A main feature of the new integral equation formula- tion is that it avoids the computation of the normal derivative of double-layer potentials which is difficult and time consuming. A fast numerical algorithm is also developed for the scattering problem, based on the NystrSm method for the new integral equation. Nu- merical examples are also shown to illustrate the applicability of the new integral equation formulation.展开更多
Consider the inverse scattering problem in terms of Helmholtz equation.We study a simply connected domain with oblique derivative boundary condition.In the case of constant l,given a finite number of incident wave,the...Consider the inverse scattering problem in terms of Helmholtz equation.We study a simply connected domain with oblique derivative boundary condition.In the case of constant l,given a finite number of incident wave,the shape of the scatterer is reconstructed from the measured far-field data.We propose a Newton method which is based on the nonlinear boundary integral equation.After computing the Fr´echet derivatives with respect to the unknown boundary,the nonlinear equation is transformed to its linear form,then we show the iteration scheme for the inverse problem.We conclude our paper by presenting several numerical examples for shape reconstruction to show the validity of the method we presented.展开更多
基金supported by the grant from the National Natural Science Foundation of China(11301405)supported by the grants from the National Natural Science Foundation of China(11171127 and 10871080)
文摘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.
文摘We consider the numerical solution for the Helmholtz equation in R^2 with mixed boundary conditions.The solvability of this mixed boundary value problem is estab- lished by the boundary integral equation method.Based on the Green formula,we express the solution in terms of the boundary data.The key to the numerical real- ization of this method is the computation of weakly singular integrals.Numerical performances show the validity and feasibility of our method.The numerical schemes proposed in this paper have been applied in the realization of probe method for inverse scattering problems.
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Supported by Major State Research Development Program of China(2005CB321701)National Natural Science Foundation of China(10971083)
文摘In this paper, we give a hybrid method to numerically solve the inverse open cavity scattering problem for cavity shape, given the scattered solution on the opening of the cavity. This method is a hybrid between an iterative method and an integral equations method for solving the Cauchy problem. The idea of this hybrid method is simple, the operation is easy, and the computation cost is small. Numerical experiments show the feasibility of this method, even for cases with noise.
文摘This paper is concerned with the problem of scattering of time-harmonic electromag- netic waves from penetrable diffraction gratings in the 2D polarization case. We propose a new, weakly singular, integral equation formulation for the scattering problem which is proved to be uniquely solvable. A main feature of the new integral equation formula- tion is that it avoids the computation of the normal derivative of double-layer potentials which is difficult and time consuming. A fast numerical algorithm is also developed for the scattering problem, based on the NystrSm method for the new integral equation. Nu- merical examples are also shown to illustrate the applicability of the new integral equation formulation.
基金foundation of Jinling Institute of Technology(No.jit-b-201524)the Science Foundation of Jinling Institute of Technology(No.Jit-fhxm-201809).
文摘Consider the inverse scattering problem in terms of Helmholtz equation.We study a simply connected domain with oblique derivative boundary condition.In the case of constant l,given a finite number of incident wave,the shape of the scatterer is reconstructed from the measured far-field data.We propose a Newton method which is based on the nonlinear boundary integral equation.After computing the Fr´echet derivatives with respect to the unknown boundary,the nonlinear equation is transformed to its linear form,then we show the iteration scheme for the inverse problem.We conclude our paper by presenting several numerical examples for shape reconstruction to show the validity of the method we presented.
