Exact Radiation Boundary Condition Research Articles

Exact radiation boundary conditions to determine the complex wavenumber of an underwater acoustic leaky wave antenna.

Underwater elastic leaky wave antennas (LWAs) steer acoustic energy as a function of frequency by exploiting fluid-solid coupling. LWAs present a modeling challenge due to complex radiation impedance on the waveguide surface that leads to changes in dynamic response. This work presents an approach to model underwater LWAs that considers an elastic unit cell surrounded by a fluid domain and includes a radiation boundary condition to simulate an open boundary. The model solves an eigenvalue problem for the complex-valued wavenumber given a specified frequency, forming an accurate representation for the free response of an elastic LWA in an underwater environment.

Spectral-Element Spectral-Integral (SESI) Method for the 1-D Bloch (Floquet) Periodic Problems With Scatterers Embedded in Multiple Regions of 2-D Layered Media

The hybrid spectral-element spectral-integral (SESI) method is developed for the electromagnetic (EM) scattering in 1-D Bloch (Floquet) periodic problems with scatterers embedded in multiple regions of 2-D layered media. The periodic layered medium Green’s function (PLMGF) is derived for the SESI method so that it can simulate EM scattering by horizontally periodic scatterers placed in an arbitrary number of layers. The SESI method is a combination of the spectral-element method (SEM) and the spectral-integral method (SIM). The SEM is applied only in regions with scatterers and has the merits of exponential convergence at a low spatial sampling density, while the SIM serves as an exact radiation boundary condition for the bottom and top boundaries of each SEM region. Therefore, there is no need to discretize the regions with pure layered media; furthermore, the inverses of these SIM matrices are given exactly and explicitly with only <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N\log N)$ </tex-math></inline-formula> CPU time. Three numerical experiments are performed to show the accuracy and efficiency of the SESI method.

Complete Radiation Boundary Conditions for Maxwell’s Equations

We describe the construction, analysis, and implementation of arbitrary-order local radiation boundary condition sequences for Maxwell’s equations. In particular we use the complete radiation boundary conditions which implicitly apply uniformly accurate exponentially convergent rational approximants to the exact radiation boundary conditions. Numerical experiments for waveguide and free space problems using high- order discontinuous Galerkin spatial discretizations are presented.

Distributional analysis of radiation conditions for the $3+1$ wave equation

Consider the Cauchy problem for the ordinary 3+1 wave equation. Reduction of the spatial domain to a half-space involves an exact radiation boundary condition enforced on a planar boundary. This boundary condition is most easily formulated in terms of the tangential-Fourier and time-Laplace transform of the solution. Using the Schwartz theory of distributions, we examine two other formulations: (i) the nonlocal spacetime form and (ii) its three-dimensional (tangential/time) Fourier transform. The spacetime form features a convolution between two tempered distributions.

A Hybrid FEM/MoM Method for 3-D Electromagnetic Scattering in Layered Medium

Accurate and efficient prediction of electromagnetic scattering from inhomogeneous objects in layered medium is one of the most challenging issues in engineering applications. This paper presents the first 3-D higher order hybrid finite-element method (FEM) and method of moments (MoM) for the accurate modeling of inhomogeneous dielectric objects in multilayered medium. The main challenges of this paper include: 1) the integration of these algorithms for layered medium and 2) the higher order computational approach involved in layered medium for high efficiency. In the proposed method, the MoM with the layered medium dyadic Green’s function is used as the exact radiation boundary condition in an inhomogeneous background, and the FEM is applied to model the inhomogeneous objects. Furthermore, the higher order maximally orthogonal basis functions with curl-conforming and divergence-conforming properties are used in the FEM and MoM, respectively to improve the modeling capability of this algorithm. For 3-D inhomogeneous objects scattering in multilayered medium, this new method requires a much more tightly truncated simulation domain than the traditional FEM, and provides much higher flexibility than the pure surface integral equation method. Finally, some numerical results are provided to validate the accuracy, efficiency, and flexibility of this method.

A Hybrid SIM-SEM Method for 3-D Electromagnetic Scattering Problems

A new method combining the spectral integral method and spectral element method (SIM-SEM) is proposed to simulate 3-D electromagnetic scattering from inhomogeneous objects. In this hybrid technique (a special case of the finite element boundary integral (FEM-BI) combination), the SEM with the mixed-order curl conforming vector Gauss-Lobatto-Legendre (GLL) basis functions are used to represent the interior electric field with high accuracy, while the SIM on a cuboid surface is used as an exact radiation boundary condition. The Toeplitz property of the SIM matrix is utilized to reduce the memory and CPU time costs in an iterative solver by using the fast Fourier transform algorithm. Unlike the traditional FEM-BI combination where the BI portion usually dominates the computational complexity, the computational costs are much lower in the SIM-SEM method. Numerical results verify the accuracy and capability of this method, confirming that the SIM-SEM method is a good alternative for solving scattering problems from inhomogeneous objects.

Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction

Exact radiation boundary conditions on the surface of a sphere are presented for the single-particle time-dependent Schrödinger equation with a localized interaction. With these boundary conditions, numerical computations of spatially unbounded outgoing wave solutions can be restricted to the finite volume of a sphere. The boundary conditions are expressed in terms of the free-particle Green's function for the outside region. The Green's function is analytically calculated by an expansion in spherical harmonics and by the method of Laplace transformation. For each harmonic number a discrete boundary condition between the function values at adjacent radial grid points is obtained. The numerical method is applied to quantum tunneling through a spherically symmetric potential barrier with different angular-momentum quantum numbers l . Calculations for l=0 are compared to exact theoretical results.

A Spectral Integral Method and Hybrid SIM/FEM for Layered Media

This paper first presents a spectral integral method (SIM) for electromagnetic scattering from homogeneous dielectric and perfectly electric conducting objects straddling several layers of a multilayered medium. It then uses this SIM as an exact radiation boundary condition to truncate the computational domain in the finite-element method (FEM) to form a hybrid SIM/FEM, which is applicable to arbitrary inhomogeneous objects. Due to the high accuracy of the SIM, the sampling density on the radiation boundary requires less than five points per wavelength to achieve 1% accuracy. The efficiency and accuracy of the developed methods have been demonstrated with several numerical experiments for the TMz case. The TEz case can be obtained by duality

Plane-wave time-domain accelerated radiation boundary kernels for FDTD analysis of 3-D electromagnetic phenomena

Truncating finite difference time domain meshes using exact radiation boundary conditions is computationally expensive. The computational bottleneck stems from the global nature of the resulting boundary update scheme, which calls for the evaluation of retarded-time boundary integrals at each time step. The classical evaluation of such integrals requires O(N/sub s//sup 2/) operations per time step where N/sub s/ is the number of spatial field sampling points on the boundary. Here, the plane wave time domain algorithm is used to evaluate retarded-time boundary integrals in O(N/sub s/log/sup 2/N/sub s/) operations per time step and thereby accelerate finite difference time domain solvers that impose exact radiation boundary conditions. The effectiveness of the resulting approach and its computational complexity are demonstrated through several numerical examples.

Exact radiation conditions on spheroidal boundaries with sparse iterative methods for efficient computation of exterior structural acoustics

Exact radiation boundary conditions are derived from harmonic expansions in spheroidal coordinates and formulated in a finite-element method for the Helmholtz equation in unbounded domains. The use of spheroidal boundaries enables the efficient solution of scattering from elongated objects. Extending the ideas of Malhotra for the exact Dirichlet-to-Neumann (DtN) map defined on a sphere, a matrix-free implementation of the nonlocal DtN map for spheroidal boundaries, suitable for iterative solution of the resulting complex–symmetric system, is described. Efficient SSOR preconditioners together with Eisenstat’s trick based on the sparse matrix partition associated with the interior mesh and local part of the radiation boundary operator are used to accelerate convergence of Krylov-subspace iterative solution methods. Numerical examples are computed to demonstrate the efficiency and accuracy of the boundary treatments for high-frequency scattering from elongated structures. [Work supported by NSF PECASE.]

Application of volume discretization methods to oblique scattering from high-contrast penetrable cylinders

Two triangular-cell volume discretization methods for oblique scattering from heterogeneous, penetrable cylinders are shown to maintain their accuracy and ease of applicability when applied to high-contrast scatterers. The first method is an integral equation formulation; the second is a differential equation formulation incorporating an exact radiation boundary condition. For a cylinder aligned parallel to the z-axis, the primary unknowns in both procedures are the z-components of the electric and magnetic fields. Potential applications include electromagnetic penetration into biological tissue and scattering from radar absorbing materials.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Two‐dimensional finite element analysis of microwave heating

AbstractTransient temperature profiles for long rods of lossy dielectric materials with thermally‐dependent dielectric properties exposed to uniform plane waves are obtained. Maxwell's equation and the heat equation are simultaneously solved using the finite element method to predict the power absorbed and the resulting temperature rise in samples of square and circular cross‐section. Following the method introduced recently, we derive an exact radiation boundary condition which is independent of the rod cross‐section. For a cylindrical sample, the boundary condition is imposed on the cylinder itself. For a square rod, the boundary condition is imposed on a cylinder containing the rod. The temperature dependence of dielectric properties and sample dimensions appreciably influence heating patterns. For square samples, the edges focus radiation, causing preferential heating at the edges. This effect is pronounced for larger samples. In addition, the incident wave polarization influences the heating of the rod. For waves where the electric field is polarized along the long axis of the sample (TMz polarization) the power absorbed is higher than when the electric field is perpendicular to the axis (TEz polarization). A case involving runaway heating is also investigated.

An exact radiation boundary condition for the finite-element solution of electromagnetic scattering on an open domain

A major difficulty in applying the finite-element method (FEM) to electromagnetic scattering problems is the lack of a means to truncate the solution domain to lie within a finite boundary on which a condition is enforced, thus ensuring that the fields will obey the Sommerfeld radiation condition at distances asymptotically far from the object. An exact radiation boundary condition (RBC) that embodies the Sommerfeld radiation condition exactly is discussed. The price paid for exactness is that the exact RBC operator is global on the exterior boundary.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>