On Resolving Inverse Nonstationary Scattering Problems in a Two-Dimensional Homogeneous Layered Medium by the τ–<Emphasis Type="Italic">p</Emphasis> Radon Transform

We consider a two-dimensional nonstationary inverse scattering problem in a layered homogeneous acoustic medium. The data consist of a scattered wavefield from a surface point source registered on the boundary of the half-plane. We prove the uniqueness of the recovery of an acoustic impedance and velocity in a medium from the scattering data. An algorithm for solving an inverse twodimensional scattering problem as a one-dimensional problem with the parameter based on the τ–p Radon transformation is constructed. Also, the numerical modeling results for the direct scattering problem and solutions of a pair of inverse scattering problems in a layered homogeneous acoustic medium are presented. The proposed algorithm is applicable to data processing in geophysical prospecting both for surface seismics and vertical seismic profiling.     

A finite element scheme with a high order absorbing boundary condition for elastodynamics

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom–Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.     

Multiple circular hole problem for an elastic half-plane

The multiple circular hole problem for an elastic half-plane is considered in this paper. The linear combination of some particular solutions which satisfy the traction free condition along the boundary of the half-plane is called the eigenfunction expansion form (EE) in the analysis. The EE for the problem investigated has been derived. The undetermined coefficients in the EE are determined by the use of the variational principle in elasticity. The whole process for solving the boundary value problem is named the eigenfunction expansion variational method (EEVM) in this paper. Finally, several numerical examples are given.     

The far field operator for a multilayered scatterer

In this paper, we consider the inverse problem of the scattering of a plane acoustic wave by a multilayered scatterer; especially we study the properties of the corresponding far field operator. This problem models time-harmonic acoustic or electromagnetic scattering by a penetrable homogeneous medium with an impenetrable core. The discussion is essential for numerical approximation of the corresponding inverse problem, i.e., from the knowledge of the far field patterns to model the shape of the scatterer.     

Boundary element solution for elastic orthotropic half-plane problems

This paper presents the boundary element solution for orthotropic half-plane problems. The complete fundamental solutions due to unit point loads within the half-plane are given. The boundary integral formulation using these fundamental solutions is presented. Expressions for stresses and displacements at internal points are also given. This formulation is applied to some classical problems. This solution procedure is highly accurate and computationally more efficient than the boundary element formulation using the Kelvin fundamental solution for orthotropic half-plane problems.     

Third Order Explicit Runge-Kutta Discontinuous Galerkin Method for Linear Conservation Law with Inflow Boundary Condition

In this paper we will present the stability in L²-norm and the optimal a priori error estimate for the Runge-Kutta discontinuous Galerkin method to solve linear conservation law with inflow boundary condition. Semi-discrete version and fully-discrete version of this method are considered respectively, where time is advanced by the explicit third order total variation diminishing Runge-Kutta algorithm. To avoid the reduction of accuracy, two correction techniques are given for the intermediate boundary condition. Numerical experiments are also given to verify the above results.     

A Complete Factorization Method for Scattering by Periodic Surfaces

The Factorization Method, a well established method in inverse scattering problems for bounded obstacles, is extended to the case of scattering by a periodic surface. The method is rigorously proved to provide accurate reconstructions for the cases of the total field satisfying a Dirichlet or an impedance boundary condition on the scattering surface. A number of computational examples are given with an emphasis on exploring the number of evanescent modes for which data has to be reliably measured to obtain satisfactory reconstructions.     

A Time Domain Point Source Method for Inverse Scattering by Rough Surfaces

In this paper we propose a new method to determine the location and shape of an unbounded rough surface from measurements of scattered electromagnetic waves. The proposed method is based on the point source method of Potthast (IMA J. Appl. Math. 61, 119–140, 1998) for inverse scattering by bounded obstacles. We propose a version for inverse rough surface scattering which can reconstruct the total field when the incident field is not necessarily time harmonic. We present numerical results for the case of a perfectly conducting surface in TE polarization, in which case a homogeneous Dirichlet condition applies on the boundary. The results show great accuracy of reconstruction of the total field and of the prediction of the surface location.     

Electromagnetic scattering from finite conductivity wind-roughened water surfaces

Numerical moment-method calculations of the electromagnetic backscattering from experimentally measured wind-roughened water surfaces that were previously made assuming perfectly conducting surfaces have been repeated taking into account the finite conductivity of sea water. The finite conductivity of the scattering medium was treated using impedance boundary conditions. Comparison with the earlier calculations shows that the backscattering drops slightly at horizontal polarization and much more dramatically at vertical polarization when the finite conductivity is considered. At small and moderate incidence angles, the magnitudes of the scattering drops are consistent with that predicted by the two-scale scattering model. The asymmetry in the upwind and downwind looking scattering that results from the non-uniform distribution of the Braggresonant electromagnetically small-scale waves across the larger scale waves is unaffected by the reduced conductivity at horizontal polarization and reduced very slightly at vertical polarization. The limitations of the two-scale model are essentially the same whether the surface is treatedwith perfect or sea-water conductivity.     

Inverse scattering by a multilayered obstacle

In this paper, we consider the inverse scattering problem of a plane acoustic wave by a multilayered obstacle which is important in various areas of imaging and nondestructive testing. When the core is penetrable (with transmission boundary conditions), we obtain that the core is determined uniquely by the corresponding far field pattern.     

Application of the FEM with adaptivity for electromagnetic inverse medium scattering problems

The paper presents a study of solution of inverse medium scattering problems for 3D time-harmonic electromagnetics. The solution is based on minimization of the misfit function between the observed scattered fields and trial solutions of direct problems for the sought approximate distribution of electric permittivity. The sources of illuminating waves and the observation points of the scattered fields are located in the nearfield the scatterer. The simulations of the direct problems are performed with the Finite Element Method. Application of adaptive mesh refinement to improve accuracy of solutions is illustrated. Tests with real- and complex-valued distributions of electric permittivity, and with complete and incomplete measurement data are presented.     

Momentum-time flux conservation method for one-dimensional wave equations

We present a conservation element and solution element method in time and momentum space. Several paradigmatic wave problems including simple wave equation, convection-diffusion equation, driven harmonic oscillating charge and nonlinear Korteweg-de Vries (KdV) equation are solved with this method and calibrated with known solutions to demonstrate its use. With this method, time marching scheme is explicit, and the nonreflecting boundary condition is automatically fulfilled. Compared to other solution methods in coordinate space, this method preserves the complete information of the wave during time evolution which is an useful feature especially for scattering problems.     

An investigation of boundary element methods for the exterior acoustic problem

This paper is concerned with the efficient determination of acoustic fields around arbitrary-shaped finite structures in an infinite three-dimensional acoustic medium. The boundary integral formulation due to Burton and Miller is used to overcome the non-existence and non-uniqueness problems associated with classical integral equation formulations of this problem. A class of numerical approximation schemes is developed and the results of applying these schemes to a number of test problems are discussed and compared. The choice of the parameters of the method is critically considered. In particular, the gains to be made, in terms of solution accuracy, by using higher-order approximations to the unknown boundary function, rather than the commonly employed piecewise constant representation, are examined in view of their additional computational costs.     

Boundary Element Simulation of Thermal Waves

In this paper we survey computational techniques based on boundary integral formulations for the simulation of thermal waves. Time-harmonic solutions to diffusion problems appear in many physical situations of interest and give rise to many interesting problems related to material characterization, parameter assessment or detection of defects. We review the main direct, indirect and mixed integral numerical methods for a model of scattering of thermal waves by many obstacles and discuss how multiple scattering techniques and other physical tools can be understood as iterative methods or used as preconditioners. We also deal with some transient problems that can be solved with boundary element methods using the Laplace transform and with coupled finite and boundary element schemes for non-homogeneous obstacles.     

On Solving an Acoustic Wave Problem Via Frequency-Domain Approach and Tensorial Spline Galerkin Method

In this paper, we introduce a numerical method for solving the dynamical acoustic wave propagation problem with Robin boundary conditions. The method used here is divided into two stages. In the first stage, the equations are transformed, via the Fourier Transform, into an equivalent problem for the frequency variables. This allow us to avoid a discretization of the time variable in the considered system. Existence and uniqueness for the equation in frequency-domain are given. An approximation of the acoustic density in frequency-domain approach is also proposed by using a tensorial spline finite element Galerkin method. In the second stage, a Gauss–Hermite quadrature method is used for the computation of inverse Fourier transform of the frequency acoustic density to obtain the time-dependent solution of the acoustic wave problem. Error estimates in Sobolev spaces and convergence behavior of the presented methods are studied. Several numerical test examples are given to illustrate the performance of the proposed method, effectiveness and good resolution properties for smooth and discontinuous heterogeneous solutions.     

The nonlinear Landweber method applied to an inverse scattering problem for sound-soft obstacles in 3D

The inverse scattering problem for sound-soft obstacles is considered for both smooth and piecewise smooth surfaces in 3D. The nonlinear and ill-posed integral equation of the first kind is solved by the nonlinear Landweber method. It is an iterative regularization scheme to obtain approximations for the unknown boundary of the obstacle. It is stable with respect to noise and essentially no extra work is required to incorporate several incident waves. So far, it has only been applied to the two dimensional case. Two different integral equations are presented to obtain far-field data. Furthermore, the domain derivative and its adjoint are characterized. The integral equations of the second kind are approximated by a boundary element collocation method. The two-grid method is used to solve the large and dense linear systems. Numerical examples are illustrated to show that both smooth and piecewise smooth obstacles can be reconstructed with this method, where the latter case has not yet been reported.     

Adaptive multilevel BEM for acoustic scattering

We study the h- and p-versions of the Galerkin boundary element method for integral equations of the first kind in 2D and 3D which result from the scattering of time harmonic acoustic waves at hard or soft scatterers. We derive an abstract a-posteriori error estimate for indefinite problems which is based on stable multilevel decompositions of our test and trial spaces. The Galerkin error is estimated by easily computable local error indicators and an adaptive algorithm for h- or p-adaptivity is formulated. The theoretical results are illustrated by numerical examples for hard and soft scatterers in 2D and 3D.     

Development of a noise prediction software NASEFA and its application in medium-to-high frequency ranges

For the analysis of noise problems in medium-to-high frequency ranges, the energy flow boundary element method (EFBEM) has been studied. EFBEM is numerical analysis method of energy flow analysis (EFA), and solves energy governing equations using a boundary element method in complex structures. Based on EFBEM, a noise prediction software, “noise analysis system by energy flow analysis” (NASEFA), was developed. For effective maintenance, NASEFA is composed of three main modules: the translator, the model converter, and the main solver. The translator changes the FE model to the NASEFA BE model, and the model converter changes the BE model to an EFBE model, including various data, such as structural materials, medium properties, sources, and boundary conditions. NASEFA then solves the acoustic energy density and intensity on boundary and in the field. Moreover, it analyzes interior and exterior noise problems for single and multiple domains in two and three dimensions. Finally, for the validation of the software developed, interior and exterior noise predictions of various structures were performed. The results obtained with NASEFA were compared with those of the commercial SEA program and experiment. From these comparative studies, the usefulness of NASEFA was established.     

Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation

One approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain. A perfectly matched layer (PML) is an unphysical absorbing layer model for linear wave equations that absorbs, almost perfectly, outgoing waves of all non-tangential angles-of-incidence and of all non-zero frequencies. This paper develops the PML concept for time-harmonic elastodynamics in Cartesian coordinates, utilising insights obtained with electromagnetics PMLs, and presents a novel displacement-based, symmetric finite-element implementation of the PML for time-harmonic plane-strain or three-dimensional motion. The PML concept is illustrated through the example of a one-dimensional rod on elastic foundation and through the anti-plane motion of a two-dimensional continuum. The concept is explored in detail through analytical and numerical results from a PML model of the semi-infinite rod on elastic foundation, and through numerical results for the anti-plane motion of a semi-infinite layer on a rigid base. Numerical results are presented for the classical soil–structure interaction problems of a rigid strip-footing on a (i) half-plane, (ii) layer on a half-plane, and (iii) layer on a rigid base. The analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by PML models even with small bounded domains.     

Lattice-Boltzmann finite-difference model with optical phonons for nanoscale thermal conduction

A recently developed multiscale model is used to study thermal conduction in silicon. In this work, the role of optical phonons is included in the nanoscale by introducing phonons with zero velocity in the lattice-Boltzmann domain. In the model, only the optical phonons are heated, and the energy transfer rate from optical to acoustic phonons is described with a relaxation time. As a test case, a nanoscale hot spot is introduced into the system, and thermal conduction to ambient medium is calculated. The results show a temperature step at the spot boundary, while elsewhere the results are identical to thermal diffusion. Optical phonons are seen to increase the spot boundary thermal resistance, which is also heavily dependent on the relaxation time of the optical-acoustic phonon scattering.