Numerical method for the shape reconstruction of a hard target

IntroductionAninverseproblemofconsiderableimportanceinvariousfieldsofengineeringtechnology ,suchasnondestructivetesting ,medicalimaging ,remotesensingandseismicimaging ,istodeterminetheshapeofascatteringobjectfromitsfar_fieldeffectsontheacousticscatteringwaves.However,thiskindofproblemisparticularlydifficulttosolvesinceitisbothnonlinearandill_posed[1].Fortunately ,therehavebeenseveralmethodsdevelopedforsolvingnumericallytheinverseproblemduringthelastdecade .Ofparticularimportancearenonlinearop…     

Asymptotic approximations for Bloch waves and topological mode steering in a planar array of Neumann scatterers

We study the canonical problem of wave scattering by periodic arrays, either of infinite or finite extent, of Neumann scatterers in the plane; the characteristic lengthscale of the scatterers is considered small relative to the lattice period. We utilise the method of matched asymptotic expansions, together with Fourier series representations, to create an efficient and accurate numerical approach for finding the dispersion curves associated with Floquet–Bloch waves through an infinite array of scatterers. The approach lends itself to direct scattering problems for finite arrays and we illustrate the flexibility of these asymptotic representations on topical examples from topological wave physics.     

Improved algorithm of light scattering by a coated sphere

An efficient numerical algorithm for computing the light scattering by a coated sphere is proposed. The calculation of relevant functions by different recurrence algorithms is discussed. The new algorithm avoids the numerical difficulties, which give rise to significant errors encountered in practice by prior methods. Exemplifying results such as extinction efficiency, scattering efficiency, light scattering intensity as well as calculation speed are provided. The results show that this algorithm is efficient, fast, numerically stable and accurate.     

Space–time focusing of acoustic waves on unknown scatterers

Consider a propagative medium, possibly inhomogeneous, containing some scatterers whose positions are unknown. Using an array of transmit–receive transducers, how can one generate a wave that would focus in space and time near one of the scatterers, that is, a wave whose energy would confine near the scatterer during a short time? The answer proposed in the present paper is based on the so-called DORT method (French acronym for: decomposition of the time reversal operator) which has led to numerous applications owing to the related space-focusing properties in the frequency domain, i.e., for time-harmonic waves. This method essentially consists in a singular value decomposition (SVD) of the scattering operator, that is, the operator which maps the input signals sent to the transducers to the measure of the scattered wave. By introducing a particular SVD related to the symmetry of the scattering operator, we show how to synchronize the time-harmonic signals derived from the DORT method to achieve space–time focusing. We consider the case of the scalar wave equation and we make use of an asymptotic model for small sound-soft scatterers, usually called the Foldy–Lax model. In this context, several mathematical and numerical arguments that support our idea are explored.     

A fast volume integral equation method for the direct/inverse problem in elastic wave scattering phenomena

A fast method for solving the volume integral equation is introduced for the solution of forward and inverse multiple scattering problems in an elastic 3-D full space. For both forward and inverse scattering analysis, the volume integral equation in the wavenumber domain is used. By means of the discrete Fourier transform, the volume integral equation in the wavenumber domain can be dealt with as a Fredholm equation of the 2nd kind with respect to a non-Hermitian operator on a finite dimensional vector space. The Bi-CGSTAB method is employed to construct the Krylov subspace in the wavenumber domain. The current procedure establishes a fast and simplified method without requiring the derivation of a coefficient matrix. Several numerical results validate the accuracy and effectiveness of the current method for both forward and inverse scattering analysis. According to the numerical results, the reconstruction of inhomogeneities of the wave field is successful, even for multiple scattering of several cubes.     

Embedding formulae for wave diffraction by a circular arc

For certain wave diffraction problems, embedding formulae can be derived, which represent the solution (or far-field behaviour of the solution) for all plane wave incident angles in terms of solutions of a (typically small) set of other auxiliary problems. Thus a complete characterisation of the scattering properties of an obstacle can be determined by only determining the solutions of the auxiliary problems, and then implementing the embedding formula. The class of scatterers for which embedding formulae can be derived has previously been limited to obstacles with piecewise linear boundaries; here this class is extended to include a simple curved obstacle, consisting of a thin circular arc. Approximate numerical calculations demonstrate the accuracy of the new embedding formulae.     

Transient gas flow simulation using an Adaptive Method of LinesSimulation d'écoulements dans un gazoduc par la méthode adaptative de lignes

Designing natural gas pipelines to safely and efficiently handle unsteady flows, requires knowledge of pressure drop, flowrate and temperature distribution throughout the system. The accurate prediction of these parameters is essential in order to achieve optimum cumulative deliverability, and safe and reliable operation. An Adaptive Method of Lines algorithm is formulated for the solution of Euler system of equations, which fully simulates slow and fast transients. Two test cases present the improvement of the numerical solution from grid adaptation. Good results are obtained both for slow and fast transients simulations proving that the suggested numerical procedure is appropriate for such predictions. To cite this article: E. Tentis et al., C. R. Mecanique 331 (2003).     

Elastic wave scattering by a three-dimensional inhomogeneity in an elastic half space

In this paper we will consider scattering of elastic waves in a half space. The half space is an isotropic, linear and homogeneous medium except for a finite inhomogeneity. The T-matrix method (also called the “extended boundary condition method” or “null field approach”) is extended to derive expressions for the elastic field inside the half space and the surface field on the interface. The assumptions on the source that excites the half space are fairly weak. In the numerical applications found in this paper we assume a Rayleigh surface wave to be the incoming field, and we only compute the surface displacements. We make illustrations on some simple types of scatterers (spheres and spheroids; the latter ones can be arbitrarily oriented).     

Subspace projection filters for imaging in random media

We consider the problem of selective imaging extended reflectors in cluttered media. We propose a random travel time model for simulating the array response matrix in clutter and we compare it with the full wave solution. Our simplified model captures very well the full wave random medium behavior as this is illustrated by our numerical results. The algorithm for selective array imaging uses coherent interferometry on a filtered version of the data. The filter, which is based on the singular value decomposition of the response matrix, enhances the signal reflected by the edges of the reflector. We illustrate the performance of the imaging algorithm with numerical simulations in the regime of ultrasonic non-destructive testing in concrete.     

A finite element thermally coupled flow formulation for phase‐change problems

A finite element, thermally coupled incompressible flow formulation considering phase‐change effects is presented. This formulation accounts for natural convection, temperature‐dependent material properties and isothermal and non‐isothermal phase‐change models. In this context, the full Navier–Stokes equations are solved using a generalized streamline operator (GSO) technique. The highly non‐linear phase‐change effects are treated with a temperature‐based algorithm, which provides stability and convergence of the numerical solution. The Boussinesq approximation is used in order to consider the temperature‐dependent density variation. Furthermore, the numerical solution of the coupled problem is approached with a staggered incremental‐iterative solution scheme, such that the convergence criteria are written in terms of the residual vectors. Finally, this formulation is used for the solutions of solidification and melting problems validating some numerical results with other existing solutions obtained with different methodologies. Copyright © 2000 John Wiley & Sons, Ltd.     

Coherent wave propagation in viscoelastic media with mode conversions and pair-correlated scatterers

The influence of correlation between scatterers on coherent waves propagation is studied in the case of a viscoelastic medium hosting a random configuration of either spherical or cylindrical scatterers. A distinction is made between the hole correction and the additional disturbances to the pair correlation function beyond the excluded volume via a radial and concentration dependent Ursell function. The effect of the Ursell function on the effective wavenumber is shown to be of order 3 in concentration and order 2 in scattering, and the corresponding formulas generalize those of Caleap et al. (2012) for an ideal fluid host medium. The whole order 3 in concentration is calculated; its other part is of order 3 in scattering. Both parts of the order 3 in concentration are the sum of two terms, one related to mode conversions, the other not. The numerical study is performed mostly for aluminum spheres in epoxy, which is a rather illustrative situation of the different phenomena that participate to the coherent propagation. The Ursell function effect is enhanced at low frequency, while counteracted partly at higher frequency, by the other term of order 3 in concentration. The most visible effects of both terms are on the attenuation. The Ursell term related to mode conversions is larger than the one with no mode conversions included in the low frequency regime.     

Reflection from a semi-infinite stack of layers using homogenization

A canonical scattering problem is that of a plane wave incident upon a periodic layered medium. Our aim here is to replace the periodic medium by a homogenized counterpart and then to investigate whether this captures the reflection and transmission behaviour accurately at potentially high frequencies.We develop a model based upon high frequency homogenization and compare the reflection coefficients and full fields with the exact solution. For some material properties it is shown that the asymptotic behaviour of the dispersion curves are locally linear near critical frequencies and that low frequency behaviour is replicated at these critical, high, frequencies. The homogenization approach accurately replaces the periodic medium and the precise manner in which this is achieved then opens the way to future numerical implementation of this technique to scattering problems.     

Semi-implicit finite difference methods for three-dimensional shallow water flow

A semi-implicit finite difference method for the numerical solution of three-dimensional shallow water flows is presented and discussed. The governing equations are the primitive three-dimensional turbulent mean flow equations where the pressure distribution in the vertical has been assumed to be hydrostatic. In the method of solution a minimal degree of implicitness has been adopted in such a fashion that the resulting algorithm is stable and gives a maximal computational efficiency at a minimal computational cost. At each time step the numerical method requires the solution of one large linear system which can be formally decomposed into a set of small three-diagonal systems coupled with one five-diagonal system. All these linear systems are symmetric and positive definite. Thus the existence and uniquencess of the numerical solution are assured. When only one vertical layer is specified, this method reduces as a special case to a semi-implicit scheme for solving the corresponding two-dimensional shallow water equations. The resulting two- and three-dimensional algorithm has been shown to be fast, accurate and mass-conservative and can also be applied to simulate flooding and drying of tidal mud-flats in conjunction with three-dimensional flows. Furthermore, the resulting algorithm is fully vectorizable for an efficient implementation on modern vector computers.     

Application of the boundary integral equation (BIE) method to transient response analysis of inclusions in a half space

Transient scattering of elastic waves by inclusions in a half space is investigated by the boundary integral equation (BIE) method. The formulation of BIE presented here is based on the Fourier transform method, and involves the analysis of transformed problems and the reconstitution of transient solutions by Fourier inversion. After the BIE has been solved numerically in the transformed domain, the transient wave fields are obtained with the help of the fast Fourier transform (FFT) algorithm. After confirmation of the accuracy of the present method, some numerical examples are shown for various inclusions in a half space, such as a cavity, an elastic inclusion, and a fluid inclusion.     

An indirect shooting method based on the POD/DEIM technique for distributed optimal control of the wave equation

This paper presents a fast numerical method, based on the indirect shooting method and Proper Orthogonal Decomposition (POD) technique, for solving distributed optimal control of the wave equation. To solve this problem, we consider the first‐order optimality conditions and then by using finite element spatial discretization and shooting strategy, the solution of the optimality conditions is reduced to the solution of a series of initial value problems (IVPs). Generally, these IVPs are high‐order and thus their solution is time‐consuming. To overcome this drawback, we present a POD indirect shooting method, which uses the POD technique to approximate IVPs with smaller ones and faster run times. Moreover, in the presence of the nonlinear term, to reduce the order of the nonlinear calculations, a discrete empirical interpolation method (DEIM) is applied and a POD/DEIM indirect shooting method is developed. We investigate the performance and accuracy of the proposed methods by means of 4 numerical experiments. We show that the presented POD and POD/DEIM indirect shooting methods dramatically reduce the CPU time compared to the full indirect shooting method, whereas there is no significant difference between the accuracy of the reduced and full indirect shooting methods.     

A fast algorithm for soil dynamics calculations by wavelet decomposition

Summary The article presents a fast numerical algorithm for calculating the response of a halfspace under any surface loads. Under certain conditions there exists an analytical solution to the problem in the Fourier domain. To get the desired response, a numerical inverse Fourier transform of this analytic solution has to be made. By using a wavelet decomposition, the proposed algorithm can reduce the calculation time significantly, thus allowing the computation of complex problems. As an example, the response of the beam-halfspace coupled system under moving load is presented. Received 6 March 1997; accepted for publication 20 May 1997     

Calculation of three‐dimensional turbulent flow with a finite volume multigrid method

A numerical method for the efficient calculation of three‐dimensional incompressible turbulent flow in curvilinear co‐ordinates is presented. The mathematical model consists of the Reynolds averaged Navier–Stokes equations and the k–ε turbulence model. The numerical method is based on the SIMPLE pressure‐correction algorithm with finite volume discretization in curvilinear co‐ordinates. To accelerate the convergence of the solution method a full approximation scheme‐full multigrid (FAS‐FMG) method is utilized. The solution of the k–ε transport equations is embedded in the multigrid iteration. The improved convergence characteristic of the multigrid method is demonstrated by means of several calculations of three‐dimensional flow cases. Copyright © 1999 John Wiley & Sons, Ltd.     

复数矢径虚拟边界谱方法在二维空穴声辐射和散射问题中的应用

基于复数矢径虚拟边界积分法,通过将虚拟积分曲线上的未知源强密度函数用Fourier级数展开,同时借助快速数值Fourier变换计算程序,提出了一种求解二维任意形状空穴声辐射和散射问题的复数矢径虚拟边界谱方法．该方法具有以下特点：(1)不存在奇异积分处理;(2)采用复数矢径虚拟边界积分方法,不仅保证了解的唯一性,而且由于虚拟源强密度函数采用Fourier级数展开,克服了用单元离散方法不能用于较高频率范围的缺点;(3)采用快速数值Fourier变换技术使计算效率大幅度提高．文中给出的计算结果表明：在求解任意形状二维空穴声辐射和散射问题上较通常采用的FEM、BEM和VBEM更为有效．     

Solution of acoustic scattering problems by means of second kind integral equations

In the present paper, the problem of scattering of acoustic waves from a fluid inclusion in two dimensions is solved numerically. The boundary conditions are those of continuous pressure and normal displacement. First, the problem in the frequency domain is reduced to a pair of second kind Fredholm integral equations on the boundary of the scatterer. Then, the integral equations are discretized by means of the Nyström algorithm, and the resulting linear systems are solved iteratively. Finally, the time domain solution is obtained from a sequence of frequency domain values. The integral equations of the present paper possess a very simple physical interpretation which guarantees the stability of their numerical solution and rapid convergence of the iterative solver. The resulting algorithm is an efficient tool for solving relatively large scale two-dimensional scattering problems.     

含非线性摩擦系数的单自由度颤振系统全局动力学行为数值研究

摩擦系数模型取更具普适性的Stribeck非线性模型,基于事件驱动理论,利用C与Matlab联合仿真的方法开发了干摩擦颤振问题的快速求解程序。给出改进的胞映射算法,对含非线性摩擦的单自由度摩擦颤振系统的演化过程及其全局性态进行数值分析和研究,得到系统在任意的初始状态下的响应特性、系统收敛域的数值计算分析结果。分析结果表...