Simulation of electromagnetic scattering by random discrete particles using a hybrid FE-BI-CBFM technique

A hybrid finite element–boundary integral–characteristic basis function method (FE-BI-CBFM) is proposed for an efficient simulation of electromagnetic scattering by random discrete particles. Specifically, the finite element method (FEM) is used to obtain the solution of the vector wave equation inside each particle and the boundary integral equation (BIE) using Green's functions is applied on the surfaces of all the particles as a global boundary condition. The coupling system of equations is solved by employing the characteristic basis function method (CBFM) based on the use of macro-basis functions constructed according to the Foldy–Lax multiple scattering equations. Due to the flexibility of FEM, the proposed hybrid technique can easily deal with the problems of multiple scattering by randomly distributed inhomogeneous particles that are often beyond the scope of traditional numerical methods. Some numerical examples are presented to demonstrate the validity and capability of the proposed method.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

Unified boundary integral equation for the scattering of elastic and acoustic waves: solution by the method of moments

A unified boundary integral equation (BIE) is developed for the scattering of elastic and acoustic waves. Traditionally, the elastic and acoustic wave problems are solved separately with different BIEs. The elastic wave case is represented in a vector BIE with the traction and displacement vectors as unknowns whereas the acoustic wave case is governed by a scalar BIE with velocity potential or pressure as unknowns. Although these two waves can be unified in the form of a partial differential equation, the unified form in its BIE counterpart has not been reported. In this work, we derive the unified BIE for these two waves and then show that the acoustic wave case can be derived from this BIE by introducing a shielding loss for small shear modulus approximation; hence only one code needs to be maintained for both elastic and acoustic wave scattering. We also derive the asymptotic Green's tensor for zero shear modulus and solve the corresponding vector equation. We employ the method of moments, which has been widely used in electromagnetics, as a numerical tool to solve the BIEs involved. Our numerical experiments show that it can also be used robustly in elastodynamics and acoustics.     

An iterative method to solve acoustic scattering problems using a boundary integral equation

In this work, a simple iterative method to solve the acoustic scattering/radiation problems using the boundary integral equation (BIE) formulation is presented. The operator equation obtained in the BIE formulation is converted into a matrix equation using the well-known method of moments solution procedure. The present method requires much fewer mathematical operations per iteration when compared to other available iterative methods. Further, the present iterative method can easily handle multiple incident fields, a highly desirable feature not available in any other iterative method, much the same way as direct solution techniques. Several numerical examples are presented to illustrate the efficiency and accuracy of the method.     

Employment of the scattering amplitude in solving the diffraction problems for waves in a halfspace

The problem on the diffraction of an acoustic wave by a finite-size scatterer (inclusion) located in a halfspace is considered. The method of solving this problem is based on the use of the scattering amplitude of the inclusion. A formula analogous to the Green formula is presented. It allows one to determine the scattering amplitude of the inclusion for an arbitrary incident wave (determined by the directional pattern of the source of primary waves) from the scattering amplitude corresponding to plane incident waves. The algorithm is presented for solving the problem on the operation of an acoustically opaque radiator in a halfspace whose boundary is characterized by an arbitrary reflection coefficient. As an example, the problem is solved on the generation of low-frequency oscillations by a sphere with an acoustically soft boundary near an acoustically hard or soft boundary of the halfspace.     

热应力边界元法中几乎超奇异积分的计算

通过分部积分变换将热弹性力学应力边界积分方程中的超奇异积分转化为强奇异积分,然后与另一个强奇异积分求和,得到仅含几乎强奇异的热应力自然边界积分方程.再对其中的几乎强奇异积分施以正则化,消除了热弹性力学边界元法中的几乎奇异积分,可以准确计算出热弹性力学问题中近边界内点的热应力.算例证明了方法的有效性.     

Two-dimensional shear waves scattering by a large rigidity strip with interface crack

A two-dimensional problem of shear horizontal (SH) waves scattering by a finite width planar elastic (piezoelectric) inclusion partially debonded from its surrounding elastic matrix is investigated using the effective boundary conditions and singular integral equations technique. The case of large rigidity inclusions with blunted tips is considered, in which the upper face of the inclusion is perfectly bonded to the matrix. The debonding region is modeled as interface crack with non-contacting faces. Using the Green theorem the mixed boundary value problem is reduced to a system of the hypersingular integral equations. Numerical results of the scattering fields characteristics are presented. The effects of incidence direction, various material parameters of the strip on the scattering field are discussed and phenomenon of the non-specular reflection of SH waves is considered. The accuracy of the numerical results is confirmed by the use of analytical approximate problem solution of high-frequency SH waves scattering on a finite hard/soft inclusion.     

Efficient wave scattering analysis for damaged cylindrical waveguides

This paper addresses scattering effects at arbitrarily shaped defects in waveguides with finite cross-sections. The main subject is to predict scattered wavefields induced by an incident wavefield in order to localize and characterize damages. The numerical approach involves the solution of a boundary value problem in combination with a decomposition method of scattered wavefields. In order to investigate mode conversion phenomena at defects, a boundary element model of the damaged waveguide section is built up. Implementation of the Boundary Element Method based on the elastodynamic boundary integral equation and the Waveguide Finite Element Method allows for a numerically efficient calculation of scattering coefficients. For defect characterization and mode sensitivity analysis, various types of surface opening defects as well as shear fractures are considered. Numerical results are presented for cylindrical waveguides and are verified experimentally.     

On multidimensional ultrasonic scattering in an inhomogeneous elastic background.

This work is concerned with the modeling of elastic wave scattering by solid or fluid-filled objects embedded in an inhomogeneous elastic background. The medium is probed by a monochromatic force and the scattered field is computed (forward problem) or observed (inverse problem) at some known receiver locations. Based on vector integral equations for elastic scattering, a general framework is developed, independent of both the problem geometry and the transmitter-receiver characteristics. This framework encompasses both forward and inverse modeling. In the forward model, a Born approximation for an inhomogeneous background is applied to obtain a closed form expression for the scattered field. In the inverse model, this approximation is also invoked to linearize for the multiparameter characteristic of the object. Finally, an iterative inversion scheme alternating forward and inverse modeling is proposed to improve the resolution and accuracy of the reconstruction algorithm.     

Effective dynamic properties of 3D composite materials containing rigid penny-shaped inclusions

The propagation of time-harmonic plane elastic waves in infinite elastic composite materials consisting of linear elastic matrix and rigid penny-shaped inclusions is investigated in this paper. The inclusions are allowed to translate and rotate in the matrix. First, the three-dimensional (3D) wave scattering problem by a single inclusion is reduced to a system of boundary integral equations for the stress jumps across the inclusion surfaces. A boundary element method (BEM) is developed for solving the boundary integral equations numerically. Far-field scattering amplitudes and complex wavenumbers are computed by using the stress jumps. Then the solution of the single scattering problem is applied to estimate the effective dynamic parameters of the composite materials containing randomly distributed inclusions of dilute concentration. Numerical results for the attenuation coefficient and the effective velocity of longitudinal and transverse waves in infinite elastic composites containing parallel and randomly oriented rigid penny-shaped inclusions of equal size and equal mass are presented and discussed. The effects of the wave frequency, the inclusion mass, the inclusion density, and the inclusion orientation or the direction of the wave incidence on the attenuation coefficient and the effective wave velocities are analysed. The results presented in this paper are compared with the available analytical results in the low-frequency range.     

管道内壁侵蚀形状识别的无网格法研究

本文使用直接配点无网格法结合共轭梯度法对管道内壁面侵蚀状况进行了反演识别。直接配点无网格法使用节点离散求解区域,采用移动最小二乘近似构造试函数,直接配点法构造线性方程组进行导热正问题求解;反演过程采用共轭梯度法使目标函数最小化,Akima三次样条插值将连续的几何边界反演问题转化为离散点几何位置的反演,并最终将这些离散点拟合成为光滑曲线。文中选择两个典型算例对数值方法进行验证,模拟结果表明使用直接配点无网格法结合共轭梯度法进行管道内壁几何边界识别具有较高精度。     

Use of nonsingular boundary integral formulation for reducing errors due to near-field measurements in the boundary element method based near-field acoustic holography

In the conformal near-field acoustic holography (NAH) using the boundary element method (BEM), the transfer matrix relating the vibro-acoustic properties of source and field depends solely on the geometrical condition of the problem. This kind of NAH is known to be very powerful in dealing with the sources having irregular shaped boundaries. When the vibro-acoustic source field is reconstructed by using this conformal NAH, one tends to position the sensors as close as possible to the source surface in order to get rich information on the nonpropagating wave components. The conventional acoustic BEM based on the Kirchhoff-Helmholtz integral equation has the singularity problem in the close near field of the source surface. This problem stems from the singular kernel of the Green function of the boundary integral equation (BIE) and the singularity can influence the reconstruction accuracy greatly. In this paper, the nonsingular BIE is introduced to the NAH calculation and the holographic BIE is reformulated. The effectiveness of nonsingular BEM has been investigated for the reduction of reconstruction error. Through interior and exterior examples, it is shown that the resolution of predicted field pressure could be improved in the close near field by employing the nonsingular BIE. Because the BEM-based NAH inevitably requires the field pressure measured in the close proximity to the source surface, the present approach is recommended for improving the resolution of the reconstructed source field.     

Scattering of Gaussian beam by a spheroidal particle with a spherical inclusion at the center

An analytic solution to electromagnetic scattering by a spheroidal particle having a spherical inclusion at the center, for oblique incidence of a Gaussian beam, is obtained within the framework of the generalized Lorenz–Mie theory (GLMT). By virtue of a transformation between the spheroidal and spherical vector wave functions, a theoretical procedure is developed to deal with the boundary conditions. Numerical results of the normalized differential scattering cross section are evaluated.     

Plane wave interpolation in direct collocation boundary element method for radiation and wave scattering: numerical aspects and applications

The classical boundary element formulation for the Helmholtz equation is rehearsed, and its limitations with respect to the number of variables needed to model a wavelength are explained. A new type of interpolation for the potential is then described in which the usual boundary element shape functions are modified by the inclusion of a set of plane waves, propagating in a range of directions. This is termed the plane wave basis boundary element method. The modifications needed to the classical procedures, in terms of integration of the element matrices, and location of collocation points are described. The well-known Singular Value Decomposition solution technique, which is adopted here for the solution of the system matrix equation in its complex form, is briefly outlined. The conditioning of the system matrix is analysed for a simple radiation problem. The corresponding diffraction problem is also analysed and results are compared with analytical and classical boundary element solutions. The CHIEF method is adopted to enhance the quality of the solution, particularly in the vicinity of irregular frequencies. The plane wave basis boundary element method is then applied to two problems: scattering of plane waves by an elliptical cylinder and the multiple circular cylinder plane wave scattering problem. In both cases results are compared with analytical solutions. The results clearly demonstrate that the new method is considerably more efficient than the classical approach. For a given number of degrees of freedom, the frequency for which accurate results can be obtained, using the new technique, can be up to three or four times higher than that of the classical method. This makes the method a powerful new addition to our tools for tackling high-frequency radiation and scattering problems.     

Sound propagation above an inhomogeneous plane: Boundary integral equation methods

A boundary integral equation method is used to compute the sound pressure emitted by a harmonic source above an inhomogeneous plane. First, the theoretical aspects of the problem (behaviour of the pressure around the discontinuities,…) are studied. Then, a comparison between theoretical levels and experimental levels obtained in an anechoic room is presented. It shows that the boundary integral equation (BIE) method is quite convenient for solving this kind of problem. Two interesting results are pointed out: (i) if only a prediction of maximum sound levels is needed, the attenuation is the same for a cylindrical source, a spherical source and N spherical sources, and so it is possible to transform some three-dimensional problems into two-dimensional ones; (ii) a numerical method of computation of the sound field above an inhomogeneous plane does not provide a correct prediction if each part of the plane is not accurately described by the boundary condition chosen.     

An improved boundary element-free method （IBEFM） for two-dimensional potential problems

The interpolating moving least-squares (IMLS) method is discussed first in this paper. And the formulae of the IMLS method obtained by Lancaster are revised. Then on the basis of the boundary element-free method (BEFM), combining the boundary integral equation (BIE) method with the IMLS method, the improved boundary element-free method (IBEFM) for two-dimensional potential problems is presented, and the corresponding formulae of the IBEFM are obtained. In the BEFM, boundary conditions are applied directly, but the shape function in the MLS does not satisfy the property of the Kronecker δ function. This is a problem of the BEFM, and must be solved theoretically. In the IMLS method, when the shape function satisfies the property of the Kronecker δ function, then the boundary conditions, in the meshless method based on the IMLS method, can be applied directly. Then the IBEFM, based on the IMLS method, is a direct meshless boundary integral equation method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied directly and easily, thus it gives a greater computational precision. Some numerical examples are presented to demonstrate the method.     

An inverse scattering problem from an impedance obstacle

In this paper we consider an inverse scattering problem from an obstacle with impedance boundary condition. Our aim is to recover the unknown scatterer from the far field pattern iteratively assuming the impedance function. Our method, while remaining in the framework of Newton’s method, based on a system of two nonlinear integral equations which is equivalent to the original inverse problem, avoids the need of calculating a direct problem at each iteration. Because of the ill-posedness of this problem, regularization method for example, Tikhonov regularization, is incorporated in our solution scheme. Several numerical examples with only one incident wave are given at the end of the paper to show the feasibility of our method.     

Albedo problem for finite plane-parallel medium

The problem of radiation transfer through a scattering and absorbing finite plane-parallel medium is solved using an efficient and accurate method of analysis which utilizes trial functions based on Case's eigenvalues plus a linear combination of exponential integral functions. The proposed trial functions are used on the integral equation reducing it to a system of algebraic equations to be solved for the expansion coefficients which are used to calculate some interesting physical quantities such as the angular radiation intensity and the reflection and the transmission coefficients. Numerical results are obtained for two different external incidence on the left boundary, x=0. The results are compared with the exact results and with those calculated by the Pomraning-Eddington variational method.     

新的对角形式快速多极边界元法求解声学Helmholtz方程

传统外部声学Helmholtz边界积分方程无法在个人计算机上求解大规模工程问题. 为了有效解决这个问题, 将快速多极方法引入到边界积分方程中, 加速系统矩阵方程组的迭代求解. 由于在边界积分方程中引入基本解的对角形式多极扩展, 新的快速多极边界元法的计算效率与传统边界元相比显著提高, 计算量和存储量减少到O(N)量级(N为问题的自由度数). 包括含有420000个自由度的大型潜艇模型数值算例验证了快速多极边界元法的准确性和高效性, 清楚表明新算法在求解大规模声学问题中的优势,     

On Uniqueness in the General Inverse Transmisson Problem

In this paper we demonstrate uniqueness of a transparent obstacle, of coefficients of rather general boundary transmission condition, and of a potential coefficient inside obstacle from partial Dirichlet-to Neumann map or from complete scattering data at fixed frequency. The proposed transmission problem includes in particular the isotropic elliptic equation with discontinuous conductivity coefficient. Uniqueness results are shown to be optimal. Hence the considered form can be viewed as a canonical form of isotropic elliptic transmission problems. Proofs use singular solutions of elliptic equations and complex geometrical optics. Determining an obstacle and boundary conditions (i.e. reflecting and transmitting properties of its boundary and interior) is of interest for acoustical and electromagnetic inverse scattering, for modeling fluid/structure interaction, and for defects detection.