Second-order Boundary Condition Research Articles

A three-dimensional parabolic equation applied to VHF/UHF propagation over irregular terrain

The two-dimensional (2-D) parabolic equation (PE) is widely used for making radiowave propagation predictions in the troposphere. The effects of transverse terrain gradients, propagation around the sides of obstacles, and scattering from large obstacles to the side of the great circle path are not modeled, leading to prediction errors in many situations. In this paper, these errors are addressed by extending the 2-D PE to three dimensions. This changes the matrix form of the PE making it difficult to solve. A novel iterative solver technique, which is highly efficient and guaranteed to converge, is presented. In order to confine the domain of computation, a three-dimensional (3-D) rectangular box is placed around the region of interest. A new second-order nonreflecting boundary condition is imposed on the surface of this box and its angular validity is established. The boundary condition is shown to keep unwanted fictitious reflections to an acceptable level in the domain of interest. The terrain boundary conditions for this 3-D PE method are developed and an original technique for incorporating them into the matrix form of the iterative solver is described. This is done using the concept of virtual field points below the ground. The prediction accuracy of the 3-D PE in comparison to the 2-D PE is tested both against indoor scaled frequency measurements and very high frequency (VHF) field trials.

SDRA Approach for Higher-Order Impedance Boundary Conditions for Complex Multi-Layer Coatings On Curved Conducting Bodies - Abstract *

A spectral-domain-rational-approximation approach is presented for systematic derivation of higher-order impedance boundary conditions for complex stratified coatings, consisting of stacked homogeneous-(bi)anisotropic and (radially) inhomogeneous-isotropic layers, laid on (locally) cylindrical conducting surfaces. Second-order impedance boundary conditions are shown to provide accurate results even when the local curvature radius is a fraction of a wavelength.

Scattering of plane waves by an anisotropic dielectric half-plane

Scattering of plane waves by a semi-infinite anisotropic thin dielectric layer is investigated, which can be considered as an example for electromagnetic energy absorbing materials. A pair of second-order boundary conditions is used to simulate an anisotropic thin dielectric layer as an infinitesimally thin sheet. Formulation is based on the Fourier integral transform technique, which reduces the scattering problem to two decoupled scalar Wiener-Hopf equations. Diffracted, reflected, and transmitted field terms are evaluated by using the Wiener-Hopf solutions that is obtained by the standard method. The uniqueness of the solution is satisfied by imposing an edge constraint in addition to the classical edge condition.

Fictitious Domain Methods for the Numerical Solution of Two-Dimensional Scattering Problems

Fictitious domain methods for the numerical solution of two-dimensional scattering problems are considered. The original exterior boundary value problem is approximated by truncating the unbounded domain and by imposing a nonreflecting boundary condition on the artificial boundary. First-order, second-order, and exact nonreflecting boundary conditions are tested on rectangular and circular boundaries. The finite element discretizations of the corresponding approximate boundary value problems are performed using locally fitted meshes, and the discrete equations are solved with fictitious domain methods. A special finite element method using nonmatching meshes is considered. This method uses the macro-hybrid formulation based on domain decomposition to couple polar and cartesian coordinate systems. A special preconditioner based on fictitious domains is introduced for the arising algebraic saddle-point system such that the subspace of constraints becomes invariant with respect to the preconditioned iterative procedure. The performance of the new method is compared to the fictitious domain methods both with respect to accuracy and computational cost.

Effective impedance boundary conditions for an inhomogeneous thin layer on a curved metallic surface

Effective impedance boundary conditions for an inhomogeneous thin layer coated on a perfectly conducting object are considered. The permittivity of the thin layer is inhomogeneous along both the normal and tangential directions. Explicit forms of the first- and second-order approximate impedance boundary conditions are derived first for a two-dimensional (2D) thin layer for the TE and TM cases. Numerical results are presented. The case of Maxwell's equations for a three-dimensional inhomogeneous thin layer is also considered.

Generalized impedance boundary conditions for isotropic multilayers

Second-order approximate boundary conditions for multilayered absorbing covering are developed. For dual-layer structures, the coefficients are given in explicit form. The results are applicable to all practically interesting cases, as is demonstrated by numerical examples. These results can be used in analytical and numerical studies of scattering from complex-shaped bodies covered with microwave absorbers. © 1998 John Wiley & Sons, Inc. Microwave Opt Technol Lett 17: 262–265, 1998.

Reflection and transmission coefficients for thin bianisotropic layers

The paper presents approximate analytical solutions for reflection and transmission coefficients of general uniaxial bianisotropic media slabs. The analysis is based on generalised second-order impedance boundary conditions (GIBC) simulating bianisotropic slabs. The GIBC are obtained using the averaging procedure. Layers much thinner than the wavelength are considered. The results are tested by comparing the approximate reflection coefficients from an uniaxial bianisotropic layer with the exact solution.

Numerical analyses of operator-splitting algorithms for the two-dimensional advection-diffusion equation

Holly and Preissmann's (HP) scheme has been the basis of a large number of operator splitting algorithms for the solution of the advection-diffusion equation. However, these algorithms, including HP, are first-order accurate in time due to splitting errors. Error analyses of these algorithms, incorporating splitting error and errors resulting from numerical solutions of the split advection and diffusion equations, are lacking. In this paper, error analysis of a second-order accurate adaptation of the HP scheme (AHP) is presented for the two-dimensional advection-diffusion equation. A modified AHP scheme (MAHP) is suggested to remove the ad-hoc nature of boundary conditions during the diffusion step of computations in both HP and AHP. As boundary conditions specified for the advection-diffusion equation are not applicable to the split equations, second-order accurate boundary conditions for the split advection and diffusion equations are derived. An analysis of numerical dispersion and dissipation associated with the numerical procedure for the advection equation is presented. The analysis establishes a criterion so that computational errors are small in two-dimensional advection dominated transport problems. Several numerical examples are presented to verify the numerical analyses presented in the paper. In addition, a review of the current status of operator splitting algorithms for the advection-diffusion equation is presented. The objective of the review is to identify the issues that have not been addressed in the previous studies.

Scattering by a right-angled lossy dielectric wedge

Electromagnetic scattering by a right-angled lossy dielectric wedge at normal incidence is considered in detail. The lossy dielectric material of the wedge is simulated using second-order impedance boundary conditions on its faces. The Maliuzhinets method is employed to obtain the total exterior field. A high-frequency asymptotic approximation of the Sommerfeld integral is carried out to obtain a UTD expression for the total field. The predictions are validated against experimental measurements taken in a controlled laboratory environment. A comparison with the widely used heuristic UTD diffraction coefficient of Luebbers is presented and the validity of the latter coefficient is discussed.

Second-order small-disturbance solution for wing at supersonic speeds

An integral equation of the transonic small-disturbance differential equation is extended to second order for supersonic flows around three-dimensional thin wings, allowing bow as well as embedded shock waves. An explicit solution of the second-order small-disturbance equations satisfying the second-order boundary conditions of a wing is obtained. The particular solution is expressed as the differentiation of a volume integral with specifically chosen boundaries, and the complementary solution satisfies the linearized velocity potential equation with appropriate boundary conditions. A numerical procedure based on this solution for three-dimensional wing calculations is proposed. The flows about a delta wing with a wedge cross section and a supersonic leading edge at zero attack angle are calculated by the numerical methods.

Steady viscous flow in a triangular cavity by efficient numerical techniques

Accurate and efficient calculations of the flow inside a triangular cavity are presented for high Reynolds numbers. The Navier-Stokes equations, expressed in a stream function and vorticity formulation, are solved numerically using finite differences on a transformed geometry. Second-order numerical boundary conditions are derived and Newton's iteration is employed to solve the nonlinear system resulting from the finite difference discretization. Aside from solving the equilateral triangular cavity problem, we have also been able to compute numerical solutions for scalene triangular cavity problems. Our coarse-mesh results for the equilateral triangular cavity problem are compared with finer mesh results in the literature and the agreement is good.

TIME-DOMAIN SIMULATION OF THE GENERATION AND PROPAGATION OF SECOND-ORDER STOKES WAVES IN A TWO-DIMENSIONAL WAVE FLUME. PART I: MONOCHROMATIC WAVEMAKER MOTIONS

A time-domain simulation of the generation and propagation of second-order Stokes waves in a semi-infinite, narrow channel of uniform depth is presented. The wave motion is generated by the small-amplitude, oscillatory motion of a generic planar wavemaker. A novel second-order radiation boundary condition is applied at the far-field boundary. The suitability of this condition is verified by considering the forms of the computed second-order wave profiles for various locations of the radiation boundary. The accuracy of the solution is studied by monitoring the difference between the energy flux at the boundaries of the computational domain and the rate of change of kinetic energy of the fluid. Free-surface elevations predicted by the present approach exhibit good agreement with experimental results in a laboratory wavetank, and with those obtained from a second-order frequency-domain solution. In order to test the solution methodology further, the problem of wave propagation in a channel containing a rectangular step of infinite extent is also considered. The numerical results for this case indicate that the step may have a significant influence on the second-order wave field. It is concluded that the present approach provides an accurate and efficient technique to simulate the generation and propagation of nonlinear Stokes waves in a two-dimensional wavetank.

On the Neumann-Kelvin Problem in Bounded Domains

A theorem on unique solvability for an elliptic problem in a polygonal domain with a second-order boundary condition on part of the boundary is proved. The relation with a well-posed formulation of the plane Neumann-Kelvin problem is discussed.

High-frequency scattering of a whispering gallery mode by a cylindrically curved surface with second-order generalized impedance boundary conditions

A high-frequency solution is derived for the diffraction of electromagnetic waves by a cylindrically curved metallic sheet whose concave and convex faces are coated by dielectric layers of the same physical properties and thicknesses. After simulating the effects of the grounded dielectric layer by second-order generalized impedance boundary conditions, a canonical problem is formulated in an infinitely extended angular space from which explicit asymptotic expressions for the diffraction coefficients are obtained through the asymptotic solution of a Hilbert problem.

Elastic-wave scattering and Stoneley wave localization by anisotropic imperfect interfaces between solids

SUMMARY In this paper, elastic scattering and localization of guided waves on a thin anisotropic imperfect interfacial layer between two solids are studied. We have proposed a second-order asymptotic boundary condition approach to model such an interfacial layer. Here, using previous results, we derive simple stiffness-matrix representations of stress-displacement relations on the interface for the decomposed symmetric and anti-symmetric elastic motions. The stiffness matrices are given for an off-axis orthotropic layer or, equivalently, for a monoclinic interfacial layer. For the problem of scattering on such a thin anisotropic layer between identical isotropic semi-spaces the scattering matrices are obtained in explicit forms. Analytical dispersion equations for Stoneley-type interfacial waves localized in such a system are also given. Additional results are included for imperfect interfaces, such as fractured interfaces, modelled by spring boundary conditions. The applicability of the stiffness-matrix approach to the layer model is analysed by numerical comparison between the approximate and exact solutions. The numerical examples, which include reflection transmission on the interphase and dispersion curves of the interfacial waves, show that the stiffness-matrix method is a simple and accurate approach to describe wave interaction with a thin anisotropic interfacial layer between two solids.

Two-phase flows - Second-order schemes and boundary conditions

A second-order numerical method suitable to unstructured grids is developed for the solution of two-phase flow equations. A number of assumptions are introduced, justified on physical grounds, when the dispersed phase concentration is low; their effect is to change the mathematical nature of the system of equations. The new numerical method put forward builds on Van Leer's original ideas. The solution of two Riemann problems is involved: one for the gas phase and a second for the dispersed phase. The method is first tested on a one-dimensional case, with excellent results; it is then generalized to two dimensions

Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids. II. Second-order asymptotic boundary conditions

In this paper, using a transfer matrix method, second-order asymptotic boundary conditions (B.C.) are introduced to model an anisotropic interfacial layer when the layer is thin compared to the wavelength. Compared with the first-order asymptotic B.C. which have previously been introduced by Rokhlin and Huang [J. Acoust. Soc. Am. 92, 1729–1742 (1992)], the second-order B.C. greatly improve the accuracy and consistency of approximation and at the same time preserve the same order of simplicity. Other attractive features are that unlike the first-order approximation the wave solutions for the second-order B.C. exactly satisfy energy balance and give zero scattering from a homogeneous substrate/layer/substrate system. Furthermore, for decoupled symmetric and antisymmetric problems, the second-order B.C. can be simplified through lower-order stiffness matrices so that the rank of the system of equations is reduced by half. Here the second-order B.C. are presented for a thin orthotropic layer between two solids which is rotated about one of its axes of symmetry with respect to the incident plane. Such boundary conditions couple the in-plane and out-of-plane stresses and displacements on the interface even for isotropic substrates. The results are illustrated by numerical examples of ultrasonic wave scattering from an imperfect interface between anisotropic solids by comparison with exact solutions.

Second-order wave interaction with two-dimensional floating bodies by a time-domain method

In this paper a time-domain method is used to simulate second-order wave interactions with a large floating body in two dimensions. In the numerical scheme adopted, a boundary integral equation method based on Green's theorem is used to calculate the velocity potential of the resulting flow field at each time step, and the second-order free-surface boundary conditions and the radiation condition of the corresponding initial-boundary value problem are treated by a time-integration scheme to obtain the development of the flow. The equations of motion of the body are solved numerically by the fourth-order Runge—Kutta algorithm. The method is applied to the case of a semi-submerged circular cylinder. Numerical calculations presented include the transient motion of a freely-floating cylinder with a specified initial displacement, and the diffraction—radiation of Stokes second-order waves by a moored floating cylinder. Important second-order wave effects associated with the hydrodynamic forces and motions of the floating structures in regular waves are highlighted. It is found that the present approach is both computationally efficient and stable.

Modifications to a parabolic equation algorithm for the prediction of atmospheric sound fields which extend its range and improve its accuracy

A two-dimensional algorithm based on Gilbert and White’s method [West etal., Appl. Acoust. 36, 1–19 (1992)] coded for a 486 PC, has served as a test bed for the following modifications: (i) a new second-order ground boundary condition, (ii) a ‘‘tuned’’ upper absorbing layer, (iii) a novel upper boundary condition, which replaces the absorbing layer, and (iv) a new procedure for calculation of the starter field. The algorithm with the above modifications has also been adapted to operate with range-dependent sound-speed profiles obtained from an isoturbulent atmospheric model. [Work supported by MOD and DRA.]

A Sommerfeld‐Type Diffraction Problem with Second‐Order Boundary Conditions

AbstractAn operator‐theoretic approach is used to study the problem of diffraction of time‐harmonic electromagnetic waves by a metalbacked dielectric half‐plane. The correspondent boundary value problem for the two dimensional Helmholtz equation is considered in a Sobolev space setting and is reduced to equivalent boundary integral equations of Wiener‐Hopf type in L+2(ℝ). An explicit analytical solution is obtained for the particular case of having the sam dielectric layer on both banks of the half‐plane.