Pole condition: A numerical method for Helmholtz-type scattering problems with inhomogeneous exterior domain

This paper presents a new numerical method for the solution of exterior Helmholtz scattering problems, which is applicable to inhomogeneous exterior domains and a wide class of geometries. The algorithm is based on the pole condition, which is a general radiation condition and allows a treatment of exterior Helmholtz problems without an explicit knowledge of Green's functions or a series representation. Our algorithm is based on a numerical approximation of the singularities of a Laplace transform of the exterior solution. Numerical examples illustrate the performance of the method.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

混合边界下开弧外区域声波散射问题的数值解法

关于时间调和声波在一个无限长圆柱形导体上的散射,可以转化为R2中一段光滑开弧上的散射问题.利用单双层位势来逼近散射波,通过单双层位势在开弧两侧的跳跃关系建立了混合边界的积分方程组,然后对此方程组进行参数化和离散化,最终得到离散化后的积分方程组.此边界积分方程组的解是存在唯一的.     

The numerical solution of an inverse scattering problem for acoustic waves

In a previous paper, the authors presented a dual space methodfor the numerical solution of the two-dimensional inverse scatteringproblem for acoustic waves in an inhomogeneous medium. Here,by making major modifications to the dual space method, a dramaticimprovement in the numerical performance of this method is achievedfor solving the inverse scattering problem.     

Inverse acoustic problems: theory and numerical methods

We study the inverse problem for the acoustic equation. The several numerical methods are considered and investigated. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical and numerical investigation of mixed-type functional differential equations

This paper is concerned with the approximate solution of a linear non-autonomous functional differential equation, with both advanced and delayed arguments. We search for a solution x(t), defined for t∈[−1,k], (k∈N), that satisfies this equation almost everywhere on [0,k−1] and assumes specified values on the intervals [−1,0] and (k−1,k]. We provide a discussion of existence and uniqueness theory for the problems under consideration and describe numerical algorithms for their solution, giving an analysis of their convergence.     

Analytical and numerical investigation of the spectra of three-point difference operators

The properties of the spectra of discrete spatially one-dimensional problems of convection — diffusion type with constant coefficients and nonstandard boundary conditions are examined in the framework of stability of explicit algorithms for time-dependent problems of mathematical physics. An analytical method is proposed for finding isolated limit points of the operator spectrum. Limit points are determined for the difference transport equation with different versions of nonreflecting boundary conditions and for an approximation of the heat conduction equation on a grid with condensation near the boundary. Stability and other properties of the spectrum are also established numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 25–45, 2007.     

A point source method for inverse acoustic and electromagnetic obstacle scattering problems

We present the mathematical foundation for a point source methodto solve some inverse acoustic and electromagnetic obstaclescattering problems in three dimensions. We investigate theinverse acoustic scattering problem by a sound-soft and a sound-hardscatterer and the inverse electromagnetic scattering problemby a perfect conductor. Two independent approaches to the methodare presented which reflect its strong relation to basic propertiesof obstacle scattering problems.     

Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems

Analytical and numerical algorithms are proposed for constructing the optimal outcome function and its Lebesgue set for the time-optimal control problem with a circular velocity indicatrix. Our approach to the solution of the time-optimal control problem essentially utilizes the specific dynamic properties of the controlled system. The circular vectogram of possible velocities enables us to interpret the cross sections of the controllability set as wavefronts whose source is uniformly distributed on the boundary of the target set. Procedures have been developed for analytical and numerical construction of the evolution of wavefronts based on prior (given the geometry of the target set boundary) identification of their nonsmoothness sets. An essential feature of the construction is the point-to-set distance function. We investigate the differential properties of this function and identify the manifolds on which it loses its smoothness. The proposed wavefront construction algorithms are of independent interest in so far as they enable us to investigate the geometry of the sets and compute their nonconvexity measure. The results are useful not only when studying the evolution of reachability sets of controlled systems, but also for computing the eikonal in geometrical optics and investigating the solutions of the wave equation. __________ Translated from Prikladnaya Matematika i Informatika, No. 27, pp. 65–79, 2007.     

An improved On-Surface Radiation Condition for acoustic scattering problems in the high-frequency spectrum

This Note addresses the derivation of an improved On-Surface Radiation Condition for the numerical solution of the exterior Helmholtz equation at high-frequencies. This condition is built as an approximation of the Neumann-to-Dirichlet map by using a local regularization of its principal classical symbol in the gliding zone for modelling the creeping waves. The numerical simulation of this pseudodifferential operator is efficiently realized with a linear cost according to the dimension of the boundary element approximation space using suitable complex Padé approximants. A numerical example is provided. To cite this article: X. Antoine et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Analytical and numerical investigation of two families of Lorenz-like dynamical systems

We investigate two families of Lorenz-like three-dimensional nonlinear dynamical systems (i) the generalized Lorenz system and (ii) the Burke–Shaw system. Analytical investigation of the former system is possible under the assumption (I) which in fact concerns four different systems corresponding to  = ±1, m = 0, 1.

(I)

The fixed points and stability characteristics of the Lorenz system under the assumption (I) are also classified. Parametric and temporal (t → ∞) asymptotes are also studied in connection to the memory of both the systems. We calculate the Lyapunov exponents and Lyapunov dimension for the chaotic attractors in order to study the influence of the parameters of the Lorenz system on the attractors obtained not only when the assumption (I) is satisfied but also for other values of the parameters σ, r, b, ω and m.     

Adapted kussmaul formulation of acoustic scattering problems: an overview

Problems of exterior acoustic scattering may be conveniently formulated by means of boundary integral equations. The problem seeks to find a wave function which gives velocity potential profile, pressure density profile, etc. of the acoustic wave at points in space. At the background of the formulations are two theories viz. (Helmholtz) Potential theory and the Green's representation formula. Potential theory gives rise to the so-called indirect formulation and the Green's representation formula to the direct formulations. Classical boundary integral formulations fail at the eigenfrequencies of the interior domain. That is, if a solution is sought of the exterior problem by first solving a homogeneous boundary integral equation, one is inevitably led to the conclusion that these homogeneous boundary equations have nontrivial solutions at certain wave-numbers which are the eigenvalues of the corresponding interior problem. At lower wave-numbers, these eigenfrequencies are thinly distributed but the higher the wave-number, the denser it becomes. This is a well-known drawback for both time-harmonic acoustics and elastodynamics. This is not a physical difficulty but arises entirely as a result of a deficiency in the integral equation is representation. Why then use It? The use has many advantages notably in that the meshing region is reduced from the infinite domain exterior to the body to its finite surface. This created the need for some robust formulations. A proof of the Kussmaul [1] formulation is presented. The formulation has a hypersingular kernel in the integral operator, which creates a havoc in computation (e.g., ill conditioning). The hyper-singularity can be avoided [2], as a result a new formulation is proposed. This paper presents a broad overview of the Adapted Kussmaul Formulation (AKF).     

Analysis of a finite PML approximation for the three dimensional time-harmonic Maxwell and acoustic scattering problems

We consider the approximation of the frequency domain three-dimensional Maxwell scattering problem using a truncated domain perfectly matched layer (PML). We also treat the time-harmonic PML approximation to the acoustic scattering problem. Following work of Lassas and Somersalo in 1998, a transitional layer based on spherical geometry is defined, which results in a constant coefficient problem outside the transition. A truncated (computational) domain is then defined, which covers the transition region. The truncated domain need only have a minimally smooth outer boundary (e.g., Lipschitz continuous). We consider the truncated PML problem which results when a perfectly conducting boundary condition is imposed on the outer boundary of the truncated domain. The existence and uniqueness of solutions to the truncated PML problem will be shown provided that the truncated domain is sufficiently large, e.g., contains a sphere of radius . We also show exponential (in the parameter ) convergence of the truncated PML solution to the solution of the original scattering problem inside the transition layer.

Our results are important in that they are the first to show that the truncated PML problem can be posed on a domain with nonsmooth outer boundary. This allows the use of approximation based on polygonal meshes. In addition, even though the transition coefficients depend on spherical geometry, they can be made arbitrarily smooth and hence the resulting problems are amenable to numerical quadrature. Approximation schemes based on our analysis are the focus of future research.

     

Analytical and numerical results for the Swift-Hohenberg equation

The problem of the Swift-Hohenberg equation is considered in this paper. Using homotopy analysis method (HAM) the series solution is developed and its convergence is discussed and documented here for the first time. In particular, we focus on the roles of the eigenvalue parameter α and the length parameter l on the large time behaviour of the solution. For a given time t, we obtain analytical expressions for eigenvalue parameter α and length l which show how different values of these parameters may lead to qualitatively different large time profiles. Also, the results are presented graphically. The results obtained reveal many interesting behaviors that warrant further study of the equations related to non-Newtonian fluid phenomena, especially the shear-thinning phenomena. Shear thinning reduces the wall shear stress.     

Analytical and numerical results for first escape time in 2D

We consider the problem of a particle subject to Brownian motion in a 2D circular domain with reflecting boundaries except for an absorbing gate. An exact solution for the mean first escape time is given for a gate of any size. Also obtained is the exact probability density of the location of an exiting particle. Numerical simulations of the stochastic process with finite step size are compared with the exact solution to the Brownian motion (the limit of zero step size). The difference between the two appears to decrease with diminishing step size.     

Assessment of the surface radiation condition by reference to acoustic scattering by a hard sphere

Two forms of the surface radiation condition (SRC) introducedby different approaches are considered and their relative meritsare examined via a special problem, namely acoustic scatteringby a hard spherical object. Each of the methods offers a differentialequation on the scatterer to determine the surface field andeventually die calculation of the scattered field is reducedto quadra-tures. These equations are first solved exactly bya series expansion method for an incident plane wave and thefar field is obtained analytically. A comparison between theexact series solutions constructed by the SRC techniques andthe exact answer of the problem shows that both approaches,almost equivalently, provide the scattered field very accuratelyin the low and middle frequency ranges, i.e. for ka 5, but asthe frequency increases, although the phase remains remarkablyaccurate, the relative error in the far-field amplitude growsin the forward region; nevertheless, the results are still qualitativelyquite satisfactory and the accuracy increases with increasingfrequency in the backward direction. Since the series expansionmethod is limited to the geometries where the Helmholtz operatoris separable, some supplementary techniques are needed to applythe SRC concept for arbitrary convex objects. For this purpose,the effects of introducing a high-frequency asymptotic expansionand then an iterative technique for the solution of the SRCequations are investigated. The first-order approximations ofboth techniques also yield sufficiently accurate results forthe far field. Thus they appear to be reliable supplementarymethods for a general obstacle.     

Radiation condition and scattering problem for time-harmonic acoustic waves in a stratified medium with a nonstratified inhomogeneity

In this paper, a generalized Sommerfeld radiation conditionis presented for the scattering waves in a stratified mediumwith a nonstratified inhomogeneity. Using integral equationmethods, the uniqueness and existence of the direct scatteringproblem are proved. Relations between the scattered acousticwaves in the far field and the sound profile of the inhomogeneityare obtained. Using these relations, the author proves threereciprocity relations between the free-wave far-field patternsand the guided-wave far-field pattern vectors correspondingto incident distorted plane waves and normal mode waves. Thenconditions under which a set of far-field patterns is completein a Hilbert space are determined using the reciprocity relation.These properties are important in investigating inverse scatteringproblems.     

The performance of numerical methods for elliptic problems with mixed boundary conditions

We consider solving linear, second order, elliptic partial differential equations with boundary conditions of types Dirichlet (DIR), mixed (MIX), and nearly Neumann (Neu) by using software modules that implement five numerical methods (one finite element and four finite differences). They represent both the new generation of improved methods and the traditional ones; they are: Hermite collocation plus band Gauss elimination (HC), ordinary finite differences plus band Gauss elimination (5P), ordinary finite differences with Dyaknov iteration (DY), DY with Richardson extrapolation to achieve fourth order convergence (D4), and ordinary finite differences with multigrid iteration (MG). We carry out a performance evaluation in which we measure the grid size and the computer time needed to achieve three significant digits of accuracy in the solution. We compute the changes in these two measures as we change boundary condition types from DIR to MIX and MIX to NEU and then test the following hypotheses: (i) the performance of all the modules is degraded by introducing the derivative terms into the boundary conditions; (ii) finite element collocation (HC) is least affected; (iii) the fourth order modules (HC and D4) are less affected than the other second order modules; and (iv) the traditional 5-point finite differences (5P) are most affected. We establish these hypotheses with high levels of confidence by using several sample problems. The most significant conclusion is that a high order collocation method is preferred for problems with general operators and derivatives in the boundary conditions. We also establish with considerable confidence that these modules have the following rankings in absolute comparative time performance: MG (best), HC and D4, DY, and 5P (worst).