涡流检测中识别缺陷形状的一种新方法

涡流检测反演技术是一种非常重要的反演缺陷形状尺寸的无损检测方法．运用Dirichlet边界条件下涡流检测反演的远场区域导数,构造了反演缺陷形状的一种新算法,并且给出了二维及三维的算例,数值反演的结果与实际缺陷吻合得较好．从而说明了:对较小的波数,即使用较少的入射和观测方向的远场测量信息,亦可得到未知缺陷形状的一个合理的重构,算法是可行的、正确的．     

Optimal Design of Periodic Diffractive Structures

The problem of designing a periodic interface between two different materials, which gives rise to a specified far-field diffraction pattern for a given incoming plane wave, is considered. The time harmonic waves are assumed to be TM (transverse magnetic) polarized. The diffraction problem is modeled by a generalized Helmholtz equation with transparent boundary conditions. In this paper the design problem is relaxed to include highly oscillatory profiles. Existence of an optimal design is established. The principal method is based on the theory of homogenization for the model equation. Accepted 31 May 2000. Online publication 26 February 2001.     

On the far-field operator for electromagnetic scattering in chiral media

We consider time-harmonic electromagnetic waves propagating in a homogeneous three-dimensional unbounded chiral medium where a perfect conductor has been immersed. Assuming that the incident electric field is a superposition of plane incident electric waves, the corresponding scattered field and the far-field pattern are expressed as the superposition of the scattered fields and the far-field patterns respectively. It is also proved that the sets of far-field patterns are complete if and only if there does not exist an eigenfunction to the interior perfect conductor problem that vanishes on the boundary of the scatterer which is an electric Herglotz field. The Left-Circularly Polarized and the Right-Circularly Polarized far-field operators are defined and studied and using them the electric far-field operator is defined too. The properties of the above operators and Herglotz functions are related to the solution of the interior perfect conductor boundary value problem.     

用正则化方法求解声波散射反问题

研究了从声波散射场的远场模式的信息来再现散射物边界形状的反问题.首先构造表达散射物特征的指示函数,然后利用该函数之特性,建立求解该类反问题的基本方程,从而确定散射物的边界形状.在这个算法中,不需预先知道散射物的边界类型和形状等知识,从T ikhonov正则化方法进行的数值计算结果表明了该方法是有效的和实用的.     

A Regularized Newton Method in Inverse Scattering Problem from Cavities

We consider the inverse problem to determine the shape of a open cavity embedded in the infinite ground plane from knowledge of the far-field pattern of the scattering of TM polarization.For its approximate solution we propose a regularized Newton iteration scheme.For a foundation of Newton type methods we establish the Fréchet differentiability of solution to the scattering problem with respect to the boundary of the cavity.Some numerical examples of the feasibility of the method are presented.     

解洞穴逆散射问题的一种正则化Newton方法

We consider the inverse problem to determine the shape of a open cavity embedded in the infinite ground plane from knowledge of the far-field pattern of the scattering of TM polarization.For its approximate solution we propose a regularized Newton iteration scheme.For a foundation of Newton type methods we establish the Fréchet differentiability of solution to the scattering problem with respect to the boundary of the cavity.Some numerical examples of the feasibility of the method are presented.     

On the reconstruction of Dirichlet-to-Neumann map in inverse scattering problems with stability estimates

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.     

An exact far-field inversion for the Born approximation and plane wave decompositions for Hankel functions

We consider the Born approximation (representative for first-order approximations) of the scattering problem for the scalar Helroholtz equation with a fixed real-valued free-space wavenumber and a complex-valued compactly supported potential. The boundary condition is the Sommerfeld radiation condition. We derive an exact series-integral representation of the potential from the Fourier coefficients of its far-field pattern, suitable for discussion of the connected stability problem. Furthermore we stress the connection between this representation and some plane wave decompositions for Hankel functions. Without loss of generality we restrict ourselves to the case of two space dimensions.     

The inverse electromagnetic scattering problem by a penetrable cylinder at oblique incidence

In this work, we consider the method of non-linear boundary integral equation for solving numerically the inverse scattering problem of obliquely incident electromagnetic waves by a penetrable homogeneous cylinder in three dimensions. We consider the indirect method and simple representations for the electric and the magnetic fields in order to derive a system of five integral equations, four on the boundary of the cylinder and one on the unit circle where we measure the far-field pattern of the scattered wave. We solve the system iteratively by linearizing only the far-field equation. Numerical results illustrate the feasibility of the proposed scheme.     

Uniqueness in Determining a Ball with a Single Incoming Wave

We prove that a ball with the impedance boundary condition is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction. In the uniqueness proof, the impedance parameter in the impedance boundary condition is unknown.     

刚性目标形状反演的一种非线性最优化方法

发展了从声散射场的远场分布的信息来再现声刚性目标形状反问题的一种非线性最优化方法,它是通过独立地求解一个不适定的线性系统和一个适定的非线性最小化问题来实现的。对反问题的非线性和不适定性的这种分离式数值处理,使所建立方法的数值实现是非常容易和快速的,因为在确定声刚性障碍物形状的非线性最优化步中,只需求解一个只有一个未知函数的小规模的最小平方问题。该方法的另一个特别的性质是,只需要远场分布的一个Fourier系数,即可对未知的刚性目标作物形设别。进而提出了数值实现该方法的一种两步调整迭代算法。对具有各种形状的二维刚性障碍物的数值试验保证了本算法是有效和实用的。     

Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

Domain sensitivity analysis of the elastic far-field patterns in scattering from nonsmooth obstacles

We consider elastic scattering problems described by the Dirichlet or the Neumann boundary value problem for the elastodynamic equation in the exterior of a 2D bounded domain or in the exterior of a crack. The boundary of the domain is assumed to have a finite set of corner points where the scattered wave may have singular behaviour. The paper is concerned with the sensitivity of the far scattered field with respect to small perturbations of the shape of the scatterer. Using a modification of the method of adjoint problems (K. Dems, Z. Mróz, Internat. J. Solids Structures 20 (1984) 527-552) we obtain a representation for the shape derivative which is well suited for a numerical realization with boundary element methods and which shows in some cases directly the influence of the singularities of the solution on the sensitivity of the far-field patterns.     

Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern

The problem considered is that of determining the shape of aplane acoustically sound-soft obstacle from the knowledge ofthe far-field pattern for one time-harmonic incident field.An iterative procedure is proposed based on two boundary integralsrepresenting the incident field and the far-field pattern, respectively.Numerical examples are included which show that the proceduregives accurate numerical approximations in relatively few iterations.     

Computation of the scattering amplitude for a scattering wave produced by a disc — Approach by a fundamental solution method

Our fundamental solution method gives an analytic representation of the approximate solution for the reduced wave problem in the exterior region of a disc. The asymptotic behavior of this representation yields an approximate formula for the scattering amplitude. An error estimate for this formula is given. We add two numerical tests: the numerical estimate of errors; and profiles of scattering cross sections and the far-field coefficient. Both tests include cases of high wave numbers.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

Reconstruction of Scattered Field from Far-Field by Regularization

In this paper, we consider an inverse scattering problem for an obstacle D R^2 with Robin boundary condition. By applying the point source, we give a regularizing method to recover the scattered field from the far-field pattern. Numerical implementations are also presented.     

Note on a Uniqueness Theorem of Schiffer

AMS (MOS): 35R30, 76Q05

The uniqueness of the inverse scattering problem is examined. It is shown that the boundary conditions must be suitably restricted to secure uniqueness. For one class the far-field pattern for one incident field is sufficient for uniqueness and, for a wider class, information from a finite number of distinct incident waves suffices. In either case, the far-field also dictates the boundary condition on the scatterer.     

The Factorization Method for an Open Arc

We consider the inverse scattering problem of determining the shape of a thin dielectric infinite cylinder having an open arc as cross section. Assuming that the electric field is polarized in the TM mode, this leads to a mixed boundary value problem for the Helmholtz equation defined in the exterior of an open arc in $R^2$. We suppose that the arc has mixed Dirichlet-impedance boundary condition, and try to recover the shape of the arc through the far field pattern by using the factorization method. However, we are not able to apply the basic theorem introduced by Kirsch to treat the far field operator $F$, and some auxiliary operators have to be considered. The theoretical validation of the factorization method to our problem is given in this paper, and some numerical results are presented to show the viability of our method.     

Inflow problem for the one-dimensional compressible Navier–Stokes equations under large initial perturbation

This paper is concerned with the inflow problem for the one-dimensional compressible Navier–Stokes equations. For such a problem, Matsumura and Nishihara showed in [10] that there exists boundary layer solution to the inflow problem, and that both the boundary layer solution, the rarefaction wave, and the superposition of boundary layer solution and rarefaction wave are nonlinear stable under small initial perturbation. The main purpose of this paper is to show that similar stability results for the boundary layer solution and the supersonic rarefaction wave still hold for a class of large initial perturbation which can allow the initial density to have large oscillation. The proofs are given by an elementary energy method and the key point is to deduce the desired lower and upper bounds on the density function.