Local Radiation Boundary Conditions Research Articles

An efficient method for transient heat conduction problems with local nonlinearities based on the quasi-superposition principle

PurposeThis paper aims to develop an efficient numerical method for nonlinear transient heat conduction problems with local radiation boundary conditions and nonlinear heat sources.Design/methodology/approachBased on the physical characteristic of the transient heat conduction and the distribution characteristic of the Green’s function, a quasi-superposition principle is presented for the transient heat conduction problems with local nonlinearities. Then, an efficient method is developed, which indicates that the solution of the original nonlinear problem can be derived by solving some nonlinear problems with small structures and a linear problem with the original structure. These problems are independent of each other and can be solved simultaneously by the parallel computing technique.FindingsWithin a small time step, the nonlinear thermal loads can only induce significant temperature responses of the regions near the positions of the nonlinear thermal loads, whereas the temperature responses of the remaining regions are very close to zero. According to the above physical characteristic, the original nonlinear problem can be transformed into some nonlinear problems with small structures and a linear problem with the original structure.Originality/valueAn efficient and accurate numerical method is presented for transient heat conduction problems with local nonlinearities, and some numerical examples demonstrate the high efficiency and accuracy of the proposed 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.

Implementation of High-Order On-Surface Radiation Boundary Conditions

Electromagnetic scattering problems that involve far-field radiation patterns and the calculation of total currents induced in a perfect conductor can be solved using local radiation boundary conditions. These local conditions are often imposed on a domain enclosing the scatterer, and the typical finite-element methods are incorporated so that the well-posedness of the modified problems encompassing the local radiation boundary conditions is preserved. Much effort in recent years has been devoted to attempts to construct higher order far-field conditions, so that the solution accuracy can be improved. In this article, we avoid extensive computations and bring the radiation boundary on the scatter’s surface itself. This procedure is known as the on-surface radiation boundary condition (OSRBC). The limitations in the past have been the implementable order of the OSRBC. Nevertheless, the key feature of the OSRBC to calculate the relevant quantities for engineers is the normal derivative of the solution on the OSRBC. This article introduces a new method for calculating the normal derivative of the electric field on the surface of a scatterer of the known shape. The method is based on a formulation of the boundary conditions through a recursive sequence of differential operators. The numerical implementation of this formulation allows one to extract a relation which is then used to solve for the quantity of interest, such as radar cross section.

Double absorbing boundaries for finite-difference time-domain electromagnetics

We describe the implementation of optimal local radiation boundary condition sequences for second order finite difference approximations to Maxwell's equations and the scalar wave equation using the double absorbing boundary formulation. Numerical experiments are presented which demonstrate that the design accuracy of the boundary conditions is achieved and, for comparable effort, exceeds that of a convolution perfectly matched layer with reasonably chosen parameters. An advantage of the proposed approach is that parameters can be chosen using an accurate a priori error bound.

HIGH-ORDER RADIATION BOUNDARY CONDITIONS FOR STRATIFIED MEDIA AND CURVILINEAR COORDINATES

Optimized local radiation boundary conditions to truncate the computational domain by a rectangular boundary have been constructed for acoustic waves propagating into a homogeneous, isotropic far field. Here we try to achieve comparable efficiencies in stratified media and cylindrical coordinates. We find that conditions constructed for homogeneous media are highly effective in the stratified case. On the circle we derive boundary conditions by optimizing a semidiscretized perfectly matched layer. Though we are unsuccessful in matching the accuracies of the Cartesian case, our experiments show that older sequences based on the progressive wave expansion are surprisingly efficient.

Complete Radiation Boundary Conditions for Convective Waves

AbstractLocal approximate radiation boundary conditions of optimal efficiency for the convective wave equation and the linearized Euler equations in waveguide geometry are formulated, analyzed, and tested. The results extend and improve for the convective case the general formulation of high-order local radiation boundary condition sequences for anisotropic scalar equations developed in [4].

Radiation boundary conditions for time-dependent waves based on complete plane wave expansions

We develop complete plane wave expansions for time-dependent waves in a half-space and use them to construct arbitrary order local radiation boundary conditions for the scalar wave equation and equivalent first order systems. We demonstrate that, unlike other local methods, boundary conditions based on complete plane wave expansions provide nearly uniform accuracy over long time intervals. This is due to their explicit treatment of evanescent modes. Exploiting the close connection between the boundary condition formulations and discretized absorbing layers, corner compatibility conditions are constructed which allow the use of polygonal artificial boundaries. Theoretical arguments and simple numerical experiments are given to establish the accuracy and efficiency of the proposed methods.

UNIFORM RADIATION CONDITIONS FOR A SOUND PROPAGATION MODEL

In this work, a new set of uniform radiation boundary conditions for a half-space model are derived and applied to a fundamental problem in outdoor sound propagation. This original approach derived here relies upon a high-order family of local radiation boundary conditions related to plane wave reflection coefficients. Validity of the approximation is carried out by examining sound propagation above an impedance ground using a spectral finite element method. This is followed by computational results verified against an analytic solution for sound propagation over hard ground. Finally, the case of sound propagation above a grass-strip surrounded by rigid ground and a rigid-strip in a grassland environment with atmospheric profiles are studied.

Complete Radiation Boundary Conditions: Minimizing the Long Time Error Growth of Local Methods

We construct and analyze new local radiation boundary condition sequences for first-order, isotropic, hyperbolic systems. The new conditions are based on representations of solutions of the scalar wave equation in terms of modes which both propagate and decay. Employing radiation boundary conditions which are exact on discretizations of the complete wave expansions essentially eliminates the long time nonuniformities encountered when using the standard local methods (perfectly matched layers or Higdon sequences). Specifically, we prove that the cost in terms of degrees-of-freedom per boundary point scales with $\ln\frac{1}{\epsilon}\cdot\ln\frac{cT}{\delta}$, where $\epsilon$ is the error tolerance, T is the simulation time, and $\delta$ is the separation between the source-containing region and the artificial boundary. Choosing $\delta\sim\lambda$, where $\lambda$ is the wavelength, leads to the same estimate which has been obtained for optimal nonlocal approximations. Numerical experiments confirm that the efficiencies predicted by the theory are attained in practice.

Numerical solution of the nonlinear Helmholtz equation using nonorthogonal expansions

In [J. Comput. Phys. 171 (2001) 632–677] we developed a fourth-order numerical method for solving the nonlinear Helmholtz equation which governs the propagation of time-harmonic laser beams in media with a Kerr-type nonlinearity. A key element of the algorithm was a new nonlocal two-way artificial boundary condition (ABC), set in the direction of beam propagation. This two-way ABC provided for reflectionless propagation of the outgoing waves while also fully transmitting the given incoming beam at the boundaries of the computational domain. Altogether, the algorithm of [J. Comput. Phys. 171 (2001) 632–677] has allowed for a direct simulation of nonlinear self-focusing without neglecting nonparaxial effects and backscattering. To the best of our knowledge, this capacity has never been achieved previously in nonlinear optics. In the current paper, we propose an improved version of the algorithm. The principal innovation is that instead of using the Dirichlet boundary conditions in the direction orthogonal to that of the laser beam propagation, we now introduce Sommerfeld-type local radiation boundary conditions, which are constructed directly in the discrete framework. Numerically, implementation of the Sommerfeld conditions requires evaluation of eigenvalues and eigenvectors for a non-Hermitian matrix. Subsequently, the separation of variables, which is a key building block of the aforementioned nonlocal ABC, is implemented through an expansion with respect to the nonorthogonal basis of the eigenvectors. Numerical simulations show that the new algorithm offers a considerable improvement in its numerical performance, as well as in the range of physical phenomena that it is capable of simulating.

A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems

We present a new auxiliary variable formulation of high-order radiation boundary conditions for the numerical simulation of waves on unbounded domains. Retaining the flexibility of Higdon’s wave-product conditions, our approach allows arbitrary-order implementations. When applied to the scalar wave equation, the proposed method leads to balanced, symmetrizable systems of wave equations on the boundary. It can also be extended to first-order systems. Corner compatibility conditions are derived for the auxiliary variable equations. They are shown experimentally to lead to stable, accurate results.

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time-dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first-order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non-reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high-order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.

Accurate radiation boundary conditions for the time-dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time-dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first-order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non-reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high-order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.

Cauchy–Characteristic Evolution and Waveforms

We investigate a new methodology for the computation of waves generated by isolated sources. This approach consists of a global spacetime evolution algorithm based on a Cauchy initial-value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior; we match the two schemes at their common interface. The characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval, so that our numerical solution extends to infinity and accurately models the free-space problem. The matching interface need not be situated far from the sources, the wavefronts may have arbitrary nonspherical geometry, and strong nonlinearity may be present in both the interior and the exterior regions. Stability and second-order convergence of the algorithms (to the exact solution of the infinite-domain problem) are established numerically in three space dimensions. The matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature. For linear problems, matching outperformed the local radiation conditions chosen for testing, and was about as accurate (for the same grid resolution) as the exact nonlocal conditions. However, since the computational cost of the nonlocal conditions is many times that of matching, this algorithm may be used with higher grid resolutions, yielding a significantly higher final accuracy. For strongly nonlinear problems, matching was significantly more accurate than all other methods tested. This seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field, while matching consistently takes nonlinearity into account in both interior and exterior regions.

Comparison of Two-Dimensional Conformal Local Adiation Boundary Conditions

ABSTRACT Numerical solutions for openndash;region electromagnetic problems based on differential equations require some means of truncating the computational domain. A number of local Radiation Boundary Conditions (RBCs) for general boundary shapes have been proposed during the past decade. Many are generalizations of the Baylissndash;Turkel RBC for circular truncation boundaries. Tbis paper reviews several twondash;dimensional RBCs for general truncation boundaries. The RBCs are evaluated on the basis of their performance on two separate numerical tests: the annihilation of terms in the Hankel series and the comparison of nearndash;field and radar cross sections for finite element solutions to scattering problems. These tests suggest that the simpler RBCs can be very competitive with RBCs based on more sophisticated derivations.

Comparison of local radiation boundary conditions for the scalar Helmholtz equation with general boundary shapes

The relative accuracy of several local radiation boundary conditions based on the second-order Bayliss-Turkel (1980) condition are evaluated. These boundary conditions permit the approximate solution of the scalar Helmholtz equation in an infinite domain using traditional finite element and finite difference techniques. Unlike the standard Bayliss-Turkel condition, the generalizations considered here are applicable to noncircular solution domains. The accuracy of these conditions are investigated for elliptical and linear/circular boundaries.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Vector finite element formulation for scattering from two-dimensional heterogeneous bodies

A formulation is proposed for electromagnetic scattering from two-dimensional heterogeneous structures that illustrates the combination of the curl-curl form of the vector Helmholtz equation with a local radiation boundary condition (RBC). To eliminate spurious nonzero eigenvalues in the spectrum of the matrix operator, vector basis functions incorporating the Nedelec constraints are employed. Basis functions of linear and quadratic order are presented, and approximations made necessary by the use of the local RBC are discussed. Results obtained with linear-tangential/quadratic normal vector basis functions exhibit excellent agreement with exact solutions for layered circular cylinder geometries, and demonstrate that abrupt jump discontinuities in the normal field components at material interfaces can be accurately modeled. The vector 2D formulation illustrates the features necessary for a general three-dimensional implementation. >

Local high-order radiation boundary conditions for the two-dimensional time-dependent structural acoustics problem

The time-dependent structural acoustics problem involving solution of the coupled wave equation over an infinite fluid domain is posed as a coupled problem over a finite fluid domain with local time-dependent radiation boundary conditions applied to the fluid truncation boundary. The proposed radiation boundary conditions are based on an asymptotic approximation to the exact solution in the frequency domain expressed in negative powers of a nondimensional wave number. A sequence of differential operators that match the leading terms of the asymptotic expansion provide boundary conditions that are of progressively higher order and increasing accuracy. Time-dependent boundary conditions are obtained through an inverse Fourier transform. The relationship of these approximate local operators to the exact nonlocal Dirichlet-to-Neumann map is examined. To illustrate their effectiveness, the boundary conditions are employed in a finite element formulation for the time-dependent structural acoustics problem. In contrast to nonlocal boundary conditions based on the Dirichlet-to-Neumann map or retarded potential integral formulations, the proposed local boundary conditions preserve the data structure of the standard finite element method and do not require storage of a large pool of historical data during the solution process. Numerical results illustrate the accuracy of the proposed boundary conditions as effected by the operator order, acoustic wave number, radiation directionality, and distance from the acoustic source.

Two mixed variational principles for exterior fluid-structure interaction problems

Two mixed variational principles for transient and harmonic analysis of non-conservative coupled exterior fluid-structure systems are proposed. The variational principles accommodate the dissipation associated with viscoelastic constitutive properties of the structure as well as the radiation boundary condition on the fluid. The principles are written for the physical system and its adjoint using the formalism introduced by Morse and Feshbach for dissipative systems. The mixed formulations use pressure and displacement potential scalar variables for the fluid and can incorporate a general class of local radiation boundary conditions. The finite element approximations based on these variational principles display a symmetric form for transient and harmonic problems.

The solution of exterior interface problems using a variational method with Lagrange multipliers

A numerical method is proposed and analyzed for approximately solving a class of exterior interface problems in three dimensions. The solution satisfies the Helmholtz equation near infinity as well as an outgoing or incoming radiation condition. The interface conditions involve jumps in both the function and the normal derivative of the function across a two-dimensional surface Γ. The former condition is an essential interface condition and is treated using Lagrange multipliers. The unboundedness of the exterior domain is treated by introducing an artificial sphere Γ R of sufficiently large radius R and an approximate local radiation boundary condition on this sphere. A variational formulation is obtained for this approximate problem that takes into account the approximate boundary condition on Γ R as well as the natural interface condition (i.e., the jump in the normal derivative on Γ). The resulting variational problem is discretized using the finite element method. It is proved that the discretization error is optimal in the sense of the approximation properties of the finite element subspaces. The discrete Lagrange multiplier formulation is generalized to take into account various perturbations of the method that often occur in its implementation. The error due to the perturbation is analyzed. It is also proved that the error due to the artificial boundary and approximate boundary condition is of order O ( R −2 ) for large R .