Absorbing Boundary Research Articles

Magnetization dynamics in nanocontact current controlled oscillators

This work focuses on the magnetization dynamics driven by spin-polarized currents in nonconfined systems when a point-contact geometry is used. We demonstrate that a generalized formulation of the spin-torque efficiency function, which includes space and angular dependences, combined with the usage of abrupt absorbing boundary conditions, which has been found to be suitable to eliminate spin-wave reflection at the computational boundaries, leads to a quantitative agreement between simulations and experimental observations. A study of the properties of spin waves excited close to the threshold clearly indicates the robustness of Slonczewski's prediction. Nevertheless, full-scale investigations reveal the necessity of introducing additional nonlinearities in the equation of motion under the large-angle oscillation regime.

Numerical solution to a linearized KdV equation on unbounded domain

AbstractExact absorbing boundary conditions for a linearized KdV equation are derived in this paper. Applying these boundary conditions at artificial boundary points yields an initial‐boundary value problem defined only on a finite interval. A dual‐Petrov‐Galerkin scheme is proposed for numerical approximation. Fast evaluation method is developed to deal with convolutions involved in the exact absorbing boundary conditions. In the end, some numerical tests are presented to demonstrate the effectiveness and efficiency of the proposed method.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Finite-element time-domain modelling of plane wave scattering by a periodic structure using an exact boundary condition

A two-dimensional singly-periodic structure, at plane wave oblique incidence, is modelled using the finite-element time-domain method. Over the non-periodic boundaries of the computational domain, an exact Floquet modal absorbing boundary condition is developed, in detail, for the finite element method in the time domain. The proposed formulation is validated by comparing its results with that of the finite-element frequency-domain method and with independently obtained published results. Very accurate numerical results, over a wide frequency range and incident angle range, are obtained for both transverse electric and transverse magnetic polarised plane wave illuminations.

Complex frequency shifted convolution PML for FDTD modelling of elastic waves

The perfectly matched layer (PML) is nowadays considered as the best optimum absorbing boundary condition available. However, the PML with the classical stretching tensor has certain limitations. Strangely, these limitations have rarely been addressed in elastic wave modelling. For example, substantial reflections occur when strong evanescent waves are propagating parallel to the interface. To circumvent problems like this, the complex frequency shifted stretching tensor has been introduced in electromagnetic modelling. In this paper we show that the convolution PML with this stretching tensor as used in electromagnetic modelling can be adapted for elastic wave modelling. Numerical results of a model where the presence of evanescent waves is predominant show that the PML based on the complex frequency shifted stretching tensor can improve the performance of the absorbing boundary layer considerably.

Fractional diffusion equation with an absorbent term and a linear external force: Exact solution

We investigate the solutions of a generalized diffusion equation which contains space and time fractional derivatives by taking an absorbent (or source) term and a linear external force into account. They are expressed in terms of the Fox H functions and, for some special cases, they are directly related to the Levy distributions. We also analyze the first passage time distribution for the case characterized by the absence of absorbent (or source) term and external force for a semi-infinite interval with absorbent boundary conditions. We also discuss the connection of the results presented here with the anomalous diffusion.

Adaptive absorbing boundary conditions for Schrödinger-type equations: Application to nonlinear and multi-dimensional problems

We propose an adaptive approach in picking the wave-number parameter of absorbing boundary conditions for Schrödinger-type equations. Based on the Gabor transform which captures local frequency information in the vicinity of artificial boundaries, the parameter is determined by an energy-weighted method and yields a quasi-optimal absorbing boundary conditions. It is shown that this approach can minimize reflected waves even when the wave function is composed of waves with different group velocities. We also extend the split local absorbing boundary (SLAB) method [Z. Xu, H. Han, Phys. Rev. E 74 (2006) 037704] to problems in multi-dimensional nonlinear cases by coupling the adaptive approach. Numerical examples of nonlinear Schrödinger equations in one and two dimensions are presented to demonstrate the properties of the discussed absorbing boundary conditions.

New formulation of the absorbing boundary conditions for the wave equation

New approximate absorbing boundary conditions for the wave equation are derived using the method of multiple scales.

Linear One-Step Processes with Artificial Boundaries

An artificial absorbing boundary is introduced in a linear birth and death stochastic process in order to understand the long time behavior of an ecological community. The solution is obtained by means of a spectral resolution of the probability distribution. A more general linear process with a coefficient of arbitrary strength near the boundary both with absorbing and with reflecting boundary conditions is also studied.

Parallel 3-D simulation of seismic wave propagation in heterogeneous anisotropic media: a grid method approach

SUMMARY This paper presents a parallel numerical technique for modelling wave propagation in 3-D heterogeneous anisotropic media. The scheme is developed by following a so-called 3-D grid method of the elastic-isotropic case. The proposed parallel algorithm needs small data exchanges between subdomains in contrast to that developed based on other numerical techniques; therefore, it is more suitable for a PC-Cluster. The algorithm is implemented on a mesh of mixed tetrahedrons and parallelepipedons, thus providing an accurate description of arbitrary 3-D surface and interface topographies and an easy generation of a non-uniform, unstructured mesh. The unstructured mesh means that the proposed algorithm can reduce the memory requirement by flexibly assigning small grid spacing in regions with low velocities and larger grid spacing in regions with higher velocities. Like the 3-D grid method, the resulting anisotropic scheme naturally satisfies the free-surface boundary conditions of arbitrary surface topography. As a result, the near-surface scattering effects can be more accurately modelled. The proposed scheme can handle a general anisotropy without any interpolations. In this paper, the transversely isotropic medium with a tilted symmetry axis, as typically caused by a system of parallel cracks or fine layers, is discussed in detail. A paraxial absorbing boundary condition in a 3-D general anisotropic case is also proposed. Comparisons with analytical solutions demonstrate the accuracy of the parallel algorithm. Computed 3-D radiation patterns illustrate shear-wave splitting, as predicted by the theory. We show the generality and flexibility of the algorithm by modelling wave propagation in an anisotropic half-space with a hemispherical crater on the surface and in mixed isotropic/anisotropic models with horizontal and inclined interfaces.

High-order accurate modeling of electromagnetic wave propagation across media – Grid conforming bodies

The Maxwell equations contain a dielectric permittivity ε that describes the particular media. For homogeneous materials at low temperatures this coefficient is constant within a material. However, it jumps at the interface between different media. This discontinuity can significantly reduce the order of accuracy of the numerical scheme. We solve the Maxwell equations, with an interface between two media, using a fourth-order accurate algorithm. We regularize the discontinuous dielectric permittivity by a continuous function either locally, near the interface, or globally, in the entire domain. We study the effect of this regularization on the order of accuracy for a one-dimensional time-dependent problem. We then implement this for the three-dimensional Maxwell equations in spherical coordinates with appropriate physical and artificial absorbing boundary conditions. We use Fourier filtering of the high frequency modes near the poles to increase the time-step.

Implementation of the Perfect Matched Layer to the CIP Method

The perfect matched layer (PML) is formulated for the use in the constrained interpolation profile (CIP) method. Numerical results are presented to examine the performance of the proposed formulation of the PML in the case of two-dimensional TM wave. The results show that the proposed methods suppress the reflection effectively in comparison with the natural absorbing boundary condition of the CIP method. We have two methods to formulate the PML, and it is shown that the both methods have equal characteristics.

Role of boundary conditions in dynamic studies of nuclear giant resonances and collisions

Absorbing boundary conditions are often employed in time-dependent mean-field calculations to cope with the problem of emitted particles which would otherwise return back onto the system and falsify the dynamical evolution. We scrutinize two widely used methods, imaginary potentials and gradual attenuation by a mask function. To that end, we consider breathing oscillations of a nucleus computed on a radial one-dimensional grid in coordinate space. The most critical test case is the computation of resonance spectra in the (linear) domain of small amplitude motion. Absorbing bounds turn out to provide a reliable alternative to fully fledged continuum random phase approximation (RPA) calculations, although rather large absorbing bounds are required to simulate reliably well continuum conditions. We also investigate the computation of observables in the nonlinear domain. This regime turns out to be less demanding. Smaller absorbing margin suffice to achieve the wanted absorption effect.

Mean first passage time calculation for a one-dimensional random walk with random absorbing boundary

In this letter, we exactly solve the MFPT problem for a one-dimensional random walk with random step size where there are two fixed reflecting boundaries and a fluctuating absorbing boundary in between. We discuss the consequences of these results in the context of site-specific DNA–protein and DNA–probe interactions.

Analysis of Periodic Structures via a Time-Domain Finite-Element Formulation With a Floquet ABC

An accurate absorbing boundary condition (ABC) is presented for three-dimensional time-domain finite-element (FEM-TD) analysis of infinitely periodic structures. The proposed ABC serves to truncate the computational domain in the nonperiodic directions and it is highly effective to absorb both the fundamental and the higher order Floquet modes. The formulation is applicable both for scattering and radiation analysis. Validation results are presented that demonstrate the ability of the FEM-TD to obtain highly accurate results over a broad frequency band that includes frequencies where a higher order Floquet mode is propagating (nonevanescent)

Algebraic Dirichlet-to-Neumann mapping for linear elasticity problems with extreme contrasts in the coefficients

The convergence of iterative based domain decomposition methods is linked with the absorbing boundary conditions defined on the interface between the sub-domains. For linear elasticity problems, the optimal absorbing boundary conditions are associated with non-local Dirichlet-to-Neumann maps. Most of the methods to approximate these non-local maps are based on a continuous analysis. In this paper, an original algebraic technique based on the computation of local Dirichlet-to-Neumann maps is investigated. Numerical experiments are presented for linear elasticity problems with extreme contrasts in the coefficients.

Benchmarking two simulation models for underwater and atmospheric sound propagation

Computational wave propagation models are widely used in underwater and atmospheric sound propagation simulation. In most realistic cases the physical domains involved are irregular. We have developed finite element techniques, applied to general irregular meshes and coupled with discrete, artificial absorbing boundary conditions of nonlocal type, for the Helmholtz equation and its ‘standard’ parabolic approximation. The physical domain is axially symmetric, with several fluid layers of variable acoustic properties. Boundaries and interfaces of general topography are allowed. The resulting models are referred to as the FENL and CNP1-NL models, respectively. We present results of the FENL model for underwater acoustic applications related to object identification and of the CNP1-NL for atmospheric sound propagation over an irregular terrain.

An accurate waveguide port boundary condition for the time-domain finite-element method

This paper presents an accurate time-domain waveguide port boundary condition (WPBC) that can be used as a mesh truncation boundary condition to terminate a finite-element mesh. The formulation for deriving the boundary condition is based on multimodal expansion, and thus is related to the modal absorbing boundary conditions developed for the finite-difference time-domain method. The boundary condition is formulated for the general three-dimensional analysis of waveguides with an arbitrary cross section. All waveguide modes, including transverse electromagnetic, transverse electric, and transverse magnetic modes, are accounted for in the boundary condition in a unified formulation. Both propagating and evanescent modes can be properly absorbed. The proposed formulation is validated by numerical examples, and the accuracy and stability of the algorithm are investigated.

OPTIMAL CONVERGENCE OF NON-OVERLAPPING SCHWARZ METHODS FOR THE HELMHOLTZ EQUATION

The non-overlapping Schwarz method with absorbing boundary conditions instead of the Dirichlet boundary conditions is an efficient variant of the overlapping Schwarz method for the Helmholtz equation. These absorbing boundary conditions defined on the interface between the subdomains are the key ingredients to obtain a fast convergence of the iterative Schwarz algorithm. In a one-way subdomains splitting, non-local optimal absorbing boundary conditions can be obtained and leads to the convergence of the Schwarz algorithm in a number of iterations equal to the number of subdomains minus one. This paper investigates different local approximations of these optimal absorbing boundary conditions for finite element computations in acoustics. Different approaches are presented both in the continuous and in the discrete analysis, including high-order optimized continuous absorbing boundary conditions, and discrete absorbing boundary conditions based on algebraic approximation. A wide range of new numerical experiments performed on unbounded acoustics problems demonstrate the comparative performance and the robustness of the proposed methods on general unstructured mesh partitioning.

Padded continued fraction absorbing boundary conditions for dispersive waves

Continued fraction absorbing boundary conditions (CFABCs) are new arbitrarily high-order local absorbing boundary conditions that are highly effective in simulating wave propagation in unbounded domains. The current versions of CFABCs are developed for the non-dispersive acoustic wave equation with convex polygonal computational domains. In this paper, the CFABCs are modified through augmentation of special padding elements, and are effective for absorbing evanescent waves occurring in dispersive wave problems while retaining their absorption properties for propagating waves. The padded CFABCs for dispersive wave equations result in fourth order evolution equations, which are solved using an efficient combination of Crank Nicholson method, Newmark time-stepping scheme, and operator splitting ideas. The effectiveness of the padded CFABCs and their implementation is illustrated through numerical examples with varying levels of dispersion.

Algebraic approximation of Dirichlet-to-Neumann maps for the equations of linear elasticity

The absorbing boundary conditions defined on the interface between the sub-domains are of major importance for the convergence of domain decomposition methods. In linear elasticity, optimal absorbing boundary conditions can be derived and are associated with a Dirichlet-to-Neumann map. In this paper, several original algebraic techniques of approximation of this Dirichlet-to-Neumann map are investigated. Asymptotic, spectral and numerical analysis of these techniques are successively presented for linear elasticity problems. Various numerical experiments illustrate the convergence properties of these original techniques.