In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann op...In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann operators corresponding to a potential radial have the same properties for hyperbolic differential equations as for elliptic differential equations. We numerically implement the coefficients of the explicit formulas. Moreover, a Lipschitz type stability is established near the edge of the domain by an estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neumann map in the inverse problem for a hyperbolic differential equation.展开更多
In this paper, a method based on the Dirichlet- to-Neumann map is developed for bandgap calculation of mixed in-plane waves propagating in 2D phononic crystals with square and triangular lattices. The method expresses...In this paper, a method based on the Dirichlet- to-Neumann map is developed for bandgap calculation of mixed in-plane waves propagating in 2D phononic crystals with square and triangular lattices. The method expresses the scattered fields in a unit cell as the cylindrical wave expansions and imposes the Bloch condition on the boundary of the unit cell. The Dirichlet-to-Neumann （DtN） map is applied to obtain a linear eigenvalue equation, from which the Bloch wave vectors along the irreducible Brillouin zone are calculated for a given frequency. Compared with other methods, the present method is memory-saving and time-saving. It can yield accurate results with fast convergence for various material combinations including those with large acoustic mismatch without extra computational cost. The method is also efficient for mixed fluid-solid systems because it considers the different wave modes in the fluid and solid as well as the proper fluid-solid interface condition.展开更多
The present paper focuses on the wave radiation by an oscillating body with six degrees of freedom by using the DtN artifi-cial boundary condition.The artificial boundary is usually selected as a circle or spherical s...The present paper focuses on the wave radiation by an oscillating body with six degrees of freedom by using the DtN artifi-cial boundary condition.The artificial boundary is usually selected as a circle or spherical surface to solve various types of fields,such as sound waves or electromagnetic waves,provided that the considered domain is infinite or unbounded in all directions.However,the substantial wave motion is considered in water of finite depth,that is,the fluid domain is bounded vertically but unbounded horizon-tally.Thus,the DtN boundary condition is given on an artificial cylindrical surface,which divides the water domain into an interior and exterior region.The boundary integral equation is adopted to implement the present model.In the case of a floating cylinder,the results of hydrodynamic coefficients of a chamfer box are discussed.展开更多
Consider the determination of Dirichlet-to-Neumann(D-to-N) map from the far-field pattern in inverse scattering problems,which is the key step in some recently developed inversion schemes such as probe method.Essentia...Consider the determination of Dirichlet-to-Neumann(D-to-N) map from the far-field pattern in inverse scattering problems,which is the key step in some recently developed inversion schemes such as probe method.Essentially,this problem is related to the reconstruction of the scattered wave from its far-field data.We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle.The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map,avoiding the numerical differentiation for the scattered wave.Then combining with the classical potential theory,a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed.Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed.展开更多
In a two-dimensional(2D)photonic crystal(PhC)composed of circular cylinders(dielectric rods or air holes)on a square or triangular lattice,various PhC devices can be created by removing or modifying some cylinders.Mos...In a two-dimensional(2D)photonic crystal(PhC)composed of circular cylinders(dielectric rods or air holes)on a square or triangular lattice,various PhC devices can be created by removing or modifying some cylinders.Most existing numerical methods for PhC devices give rise to large sparse or smaller but dense linear systems,all of which are expensive to solve if the device is large.In a previous work[Z.Hu et al.,Optics Express,16(2008),17383-17399],an efficient Dirichlet-to-Neumann(DtN)map method was developed for general 2D PhC devices with an infinite background PhC to take full advantage of the underlying lattice structure.The DtN map of a unit cell is an operator that maps the wave field to its normal derivative on the cell boundary and it allows one to avoid computing the wave field in the interior of the unit cell.In this paper,we extend the DtN map method to PhC devices with a finite background PhC.Since there is no bandgap effect to confine the light in a finite PhC,a different technique for truncating the domain is needed.We enclose the finite structure with a layer of empty boundary and corner unit cells,and approximate the DtN maps of these cells based on expanding the scattered wave in outgoing plane waves.Our method gives rise to a relatively small and sparse linear systems that are particularly easy to solve.展开更多
Based on the numerical evidences,an analytical expression of the Dirichletto-Neumann mapping in the form of infinite product was first conjectured for the onedimensional characteristic Schrodinger equation with a sinu...Based on the numerical evidences,an analytical expression of the Dirichletto-Neumann mapping in the form of infinite product was first conjectured for the onedimensional characteristic Schrodinger equation with a sinusoidal potential in[Commun.Comput.Phys.,3(3):641-658,2008].It was later extended for the general secondorder characteristic elliptic equations with symmetric periodic coefficients in[J.Comp.Phys.,227:6877-6894,2008].In this paper,we present a proof for this Dirichlet-toNeumann mapping.展开更多
We study the localisation inverse problem corresponding to Laplacian transport of absorbing cell. Our main goal is to find sufficient Dirichelet-to-Neumann conditions insuring that this inverse problem is uniquely sol...We study the localisation inverse problem corresponding to Laplacian transport of absorbing cell. Our main goal is to find sufficient Dirichelet-to-Neumann conditions insuring that this inverse problem is uniquely soluble. In this paper, we show that the conformal mapping technique is adopted to this type of problem in the two dimensional case.展开更多
In this paper, we are concerned with the coupling of finite element methods and bound- ary integral equation methods solving the classical fluid-solid interaction problem in two dimensions. The original transmission p...In this paper, we are concerned with the coupling of finite element methods and bound- ary integral equation methods solving the classical fluid-solid interaction problem in two dimensions. The original transmission problem is reduced to an equivalent nonlocal bound- ary value problem via introducing a Dirichlet-to-Neumann mapping by the direct boundary integral equation method. We show the existence and uniqueness of the solution for the corresponding variational equation. Numerical results based on the finite element method coupled with the standard Galerkin boundary element method, the fast multipole method and the NystrSm method for approximating the DtN mapping are provided to illustrate the efficiency and accuracy of the numerical schemes.展开更多
A fourth-order operator marching method for the Helmholtz equation in a waveguide is developed in this paper. It is derived from a new fourth-order exponential integrator for linear evolution equations. The method imp...A fourth-order operator marching method for the Helmholtz equation in a waveguide is developed in this paper. It is derived from a new fourth-order exponential integrator for linear evolution equations. The method improves the second-order accuracy associated with the widely used step-wise coupled mode method where the waveguide is approximated by segments that are uniform in the propagation direction. The Helmholtz equation is solved using a one-way reformulation based on the Dirichlet-to-Neumann map. An alternative version closely related to the coupled mode method is also given. Numerical results clearly indicate that the method is more accurate than the coupled mode method while the required computing effort is nearly the same.展开更多
Flattening of the interfaces is necessary in computing wave propagation along strati?ed waveguides in large range step sizes while using marching methods. When the supposition that there exists one horizontal straight...Flattening of the interfaces is necessary in computing wave propagation along strati?ed waveguides in large range step sizes while using marching methods. When the supposition that there exists one horizontal straight line in two adjacent interfaces does not hold, the previously suggested local orthogonal transform method with an analytical formulation is not feasible. This paper presents a numerical coordinate transform and an equation transform to perform the transforms numerically for waveguides without satisfying the supposition. The boundary value problem is then reduced to an initial value problem by one-way reformulation based on the Dirichlet-to-Neumann (DtN) map. This method is applicable in solving long-range wave propagation problems in slowly varying waveguides with a multilayered medium structure.展开更多
In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[J.Lightwave Technology 25(2007),3649-3656]for solving periodic array problems.It turns out that when the perio...In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[J.Lightwave Technology 25(2007),3649-3656]for solving periodic array problems.It turns out that when the periodic array contains an infinite number of periodic cells,our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex,or real but in the stop bands.This technique is also used to solve the time-dependent Schr¨odinger equation in both one and two dimensions,when the periodic potential functions have some local defects.展开更多
文摘In this paper, we provide an explicit expression for the full Dirichlet-to-Neumann map corresponding to a radial potential for a hyperbolic differential equation in 3-dimensional. We show that the Dirichlet-Neumann operators corresponding to a potential radial have the same properties for hyperbolic differential equations as for elliptic differential equations. We numerically implement the coefficients of the explicit formulas. Moreover, a Lipschitz type stability is established near the edge of the domain by an estimation constant. That is necessary for the reconstruction of the potential from Dirichlet-to-Neumann map in the inverse problem for a hyperbolic differential equation.
基金supported by the National Natural Science Foundation of China(51178037,10632020)the 973 State Key Development Program for Basic Research of China(2010CB732104)
文摘In this paper, a method based on the Dirichlet- to-Neumann map is developed for bandgap calculation of mixed in-plane waves propagating in 2D phononic crystals with square and triangular lattices. The method expresses the scattered fields in a unit cell as the cylindrical wave expansions and imposes the Bloch condition on the boundary of the unit cell. The Dirichlet-to-Neumann （DtN） map is applied to obtain a linear eigenvalue equation, from which the Bloch wave vectors along the irreducible Brillouin zone are calculated for a given frequency. Compared with other methods, the present method is memory-saving and time-saving. It can yield accurate results with fast convergence for various material combinations including those with large acoustic mismatch without extra computational cost. The method is also efficient for mixed fluid-solid systems because it considers the different wave modes in the fluid and solid as well as the proper fluid-solid interface condition.
文摘The present paper focuses on the wave radiation by an oscillating body with six degrees of freedom by using the DtN artifi-cial boundary condition.The artificial boundary is usually selected as a circle or spherical surface to solve various types of fields,such as sound waves or electromagnetic waves,provided that the considered domain is infinite or unbounded in all directions.However,the substantial wave motion is considered in water of finite depth,that is,the fluid domain is bounded vertically but unbounded horizon-tally.Thus,the DtN boundary condition is given on an artificial cylindrical surface,which divides the water domain into an interior and exterior region.The boundary integral equation is adopted to implement the present model.In the case of a floating cylinder,the results of hydrodynamic coefficients of a chamfer box are discussed.
基金supported by National Natural Science Foundation of China (Grant No.10771033)
文摘Consider the determination of Dirichlet-to-Neumann(D-to-N) map from the far-field pattern in inverse scattering problems,which is the key step in some recently developed inversion schemes such as probe method.Essentially,this problem is related to the reconstruction of the scattered wave from its far-field data.We firstly prove the well-known uniqueness result of the D-to-N map from the far-field pattern using a new scheme based on the mixed reciprocity principle.The advantage of this new proof scheme is that it provides an efficient algorithm for computing the D-to-N map,avoiding the numerical differentiation for the scattered wave.Then combining with the classical potential theory,a simple and feasible regularizing reconstruction scheme for the D-to-N map is proposed.Finally the stability estimate for the reconstruction with noisy input data is rigorously analyzed.
基金the National Science Foundation of China(Project No.10701016)the open fund of key laboratory of information photonics and optical communications(Beijing University of Posts and Telecommunications)of Ministry of Education+1 种基金the fundamental research funds for the central universities(BUPT2009RC0706)a grant from the Research Grants Council of Hong Kong Special Administrative Region,China(Project No.CityU 102207)。
文摘In a two-dimensional(2D)photonic crystal(PhC)composed of circular cylinders(dielectric rods or air holes)on a square or triangular lattice,various PhC devices can be created by removing or modifying some cylinders.Most existing numerical methods for PhC devices give rise to large sparse or smaller but dense linear systems,all of which are expensive to solve if the device is large.In a previous work[Z.Hu et al.,Optics Express,16(2008),17383-17399],an efficient Dirichlet-to-Neumann(DtN)map method was developed for general 2D PhC devices with an infinite background PhC to take full advantage of the underlying lattice structure.The DtN map of a unit cell is an operator that maps the wave field to its normal derivative on the cell boundary and it allows one to avoid computing the wave field in the interior of the unit cell.In this paper,we extend the DtN map method to PhC devices with a finite background PhC.Since there is no bandgap effect to confine the light in a finite PhC,a different technique for truncating the domain is needed.We enclose the finite structure with a layer of empty boundary and corner unit cells,and approximate the DtN maps of these cells based on expanding the scattered wave in outgoing plane waves.Our method gives rise to a relatively small and sparse linear systems that are particularly easy to solve.
基金The authors would like to thank Prof.Matthias Ehrhardt for the inspiring discussion on this work.C.Zheng was supported by the National Natural Science Foundation of China under Grant No.11371218.
文摘Based on the numerical evidences,an analytical expression of the Dirichletto-Neumann mapping in the form of infinite product was first conjectured for the onedimensional characteristic Schrodinger equation with a sinusoidal potential in[Commun.Comput.Phys.,3(3):641-658,2008].It was later extended for the general secondorder characteristic elliptic equations with symmetric periodic coefficients in[J.Comp.Phys.,227:6877-6894,2008].In this paper,we present a proof for this Dirichlet-toNeumann mapping.
文摘We study the localisation inverse problem corresponding to Laplacian transport of absorbing cell. Our main goal is to find sufficient Dirichelet-to-Neumann conditions insuring that this inverse problem is uniquely soluble. In this paper, we show that the conformal mapping technique is adopted to this type of problem in the two dimensional case.
文摘In this paper, we are concerned with the coupling of finite element methods and bound- ary integral equation methods solving the classical fluid-solid interaction problem in two dimensions. The original transmission problem is reduced to an equivalent nonlocal bound- ary value problem via introducing a Dirichlet-to-Neumann mapping by the direct boundary integral equation method. We show the existence and uniqueness of the solution for the corresponding variational equation. Numerical results based on the finite element method coupled with the standard Galerkin boundary element method, the fast multipole method and the NystrSm method for approximating the DtN mapping are provided to illustrate the efficiency and accuracy of the numerical schemes.
文摘A fourth-order operator marching method for the Helmholtz equation in a waveguide is developed in this paper. It is derived from a new fourth-order exponential integrator for linear evolution equations. The method improves the second-order accuracy associated with the widely used step-wise coupled mode method where the waveguide is approximated by segments that are uniform in the propagation direction. The Helmholtz equation is solved using a one-way reformulation based on the Dirichlet-to-Neumann map. An alternative version closely related to the coupled mode method is also given. Numerical results clearly indicate that the method is more accurate than the coupled mode method while the required computing effort is nearly the same.
基金Project supported by the Program for New Century Excel-lent Talents in University (No. NCET-08-0450)the 985 II of Xi’an Jiaotong University, and the High Talented Person Scientific Research Start Project of North China University of Water Resources and Electric Power (No. 003001)
文摘Flattening of the interfaces is necessary in computing wave propagation along strati?ed waveguides in large range step sizes while using marching methods. When the supposition that there exists one horizontal straight line in two adjacent interfaces does not hold, the previously suggested local orthogonal transform method with an analytical formulation is not feasible. This paper presents a numerical coordinate transform and an equation transform to perform the transforms numerically for waveguides without satisfying the supposition. The boundary value problem is then reduced to an initial value problem by one-way reformulation based on the Dirichlet-to-Neumann (DtN) map. This method is applicable in solving long-range wave propagation problems in slowly varying waveguides with a multilayered medium structure.
文摘In this work we improve and extend a technique named recursive doubling procedure developed by Yuan and Lu[J.Lightwave Technology 25(2007),3649-3656]for solving periodic array problems.It turns out that when the periodic array contains an infinite number of periodic cells,our method gives a fast evaluation of the exact boundary Robin-to-Robin mapping if the wave number is complex,or real but in the stop bands.This technique is also used to solve the time-dependent Schr¨odinger equation in both one and two dimensions,when the periodic potential functions have some local defects.
