First-order Absorbing Boundary Condition Research Articles

Explicit P1 Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space.

Semi-exact local absorbing boundary condition for seismic wave simulation

AbstractAn absorbing boundary condition is necessary in seismic wave simulation for eliminating the unwanted artificial reflections from model boundaries. Existing boundary condition methods often have a trade-off between numerical accuracy and computational efficiency. We proposed a local absorbing boundary condition for frequency-domain finite-difference modelling. The proposed method benefits from exact local plane-wave solution of the acoustic wave equation along predefined directions that effectively reduces the dispersion in other directions. This method has three features: simplicity, accuracy and efficiency. Numerical simulation demonstrated that the proposed method has higher efficiency than the conventional methods such as the second-order absorbing boundary condition and the perfectly matched layer (PML) method. Meanwhile, the proposed method shared the same low-cost feature as the first-order absorbing boundary condition method.

Optimized first-order absorbing boundary conditions for anisotropic elastodynamics

We propose new forms of low-order absorbing boundary conditions (ABC) for time-dependent elastic waves in isotropic and anisotropic media. The configuration considered is that of a two-dimensional elastic waveguide. The ABC shares with the widely-known Lysmer–Kuhlemeyer (LK) boundary condition the ease of implementation, especially in a finite-element (FE) setting, while offering gains in accuracy over the LK-like condition (or the LK condition in the isotropic case). The new conditions are proved to be energy-stable, even in the case when inverse modes are present, for which the normal components of the phase and group velocities have an opposite direction. The developed ABCs are particularly convenient to use in a Finite Element setting as they preserve the symmetry and positivity of the formulation and are simple to implement. The ABC features several parameters, which are optimized for accuracy. The optimization criterion is based on the notion of the energy-rate reflection coefficient. We test the performance of the scheme via a numerical example. While the gain in accuracy is found to be moderate (around 20% relative to the LK condition), the main result of this investigation lies in the development of ABCs for anisotropic elastodynamics that are proved to be stable from the outset, and are amenable for accuracy optimization. We plan to design in a future study analogous stable and optimized high-order ABCs using the methodology developed here.

A Time-Domain Propagator Numerical Method for Computational Electromagnetics

A new time-domain numerical method for full-wave electromagnetic scattering and propagation is presented. The propagator method can be described as a numerical integral operator technique for moving the electromagnetic field through numerical space at successive increments in time. The numerical equations are obtained by discretizing electric and magnetic field propagator integrals. A primary feature of the propagator method is that all six electromagnetic-field components are calculated at each numerical grid point and all components are in time synchronization. Constant spatial and time increments are maintained throughout an inhomogeneous numerical space by employing an extrapolation procedure that ensures causality. A simple and effective first-order absorbing boundary condition, described as the null boundary condition, is introduced. Examples, provided in one-, two-, and three-dimensions, include plane wave reflection from and transmission through a planar boundary, and scattering and radar cross sections for canonical dielectric objects.

A Novel Coupling Algorithm for Perfectly Matched Layer With Wave Equation-Based Discontinuous Galerkin Time-Domain Method

The second-order wave equation-based discontinuous Galerkin time-domain (DGTD) methods typically employ the first-order absorbing boundary condition for modeling open problems. To improve the modeling accuracy, this paper proposes a novel coupling algorithm of the well-posed perfectly matched layer (PML) for wave equation-based DGTD methods. Based on the domain decomposition technique, the proposed coupling algorithm divides the computational domain into two regions, that is, the physical and PML regions, whose meshes can be nonconformal with each other. Instead of introducing time convolution terms, the new coupling scheme is implemented through employing different DGTD frameworks for the two regions. Specifically, the physical region employs the wave equation-based DGTD framework, while the PML region employs the first-order Maxwell’s curl equations-based DGTD framework. To facilitate modeling of electrically small problems, the implicit Newmark-beta time integration is used for the physical region. To conveniently couple with the physical region, the implicit Crank–Nicolson algorithm is used for the PML region. Numerical results are shown to examine the accuracy and efficiency of the proposed coupling algorithm for modeling electrically small problems.

Nonrigorous symmetric second-order ABC applied to large-domain finite element modeling of electromagnetic scatterers

Nonrigorous symmetric second-order absorbing boundary condition (ABC) is presented as a feasible local mesh truncation in the higher-order large-domain finite element method (FEM) for electromagnetic analysis of scatterers in the frequency domain. The ABC is implemented on large generalized curvilinear hexahedral finite elements without imposing normal field continuity and without introducing new variables. As the extension of our previous work, the method is comprehensively evaluated by analyzing several benchmark targets, i.e., a metallic sphere, a dielectric cube, and NASA almond. Numerical examples show that radar cross section (RCS) of analyzed scatterers can be accurately predicted when the divergence term is included in computations nonrigorously. An influence of specific terms in the second-order ABC, which absorb transverse electric (TE) and transverse magnetic (TM) spherical modes, is also investigated. Examples show significant improvements in accuracy of the nonrigorous second-order ABC over the first-order ABC.

Laplace-domain wave-equation modeling and full waveform inversion in 3D isotropic elastic media

Abstract The 3D elastic problem has not been widely studied because of the computational burden. Over the past few years, 3D elastic full waveform inversion (FWI) techniques in the time and frequency domains have been proposed by some researchers based on developments in computer science. However, these techniques still have the non-uniqueness and high nonlinearity problems. In this paper, we propose a 3D elastic FWI algorithm in the Laplace domain that can mitigate these problems. To efficiently solve the impedance matrix, we adopt a first-order absorbing boundary condition that results in a symmetric system. A conjugate gradient (CG) solver can be used because the Laplace-domain wave equation is naturally positive definite. We apply the Jacobi preconditioner to increase the convergence speed. We identify the permissible range of Laplace damping constants through dispersion analysis and accuracy tests. We perform the Laplace-domain FWI based on a logarithmic objective function, and the inversion examples are designed for a land setting, which means that the source is vertically excited and multi-component data are considered. The inversion results indicate that the inversion that uses only the vertical component performs slightly better than the multi-component inversion. This unexpected result is obtained partly because we use a vertically polarized source. We analyze the residuals and Frechet derivatives for each component to examine the characteristics of the Laplace-domain multi-component FWI. The results indicate that the residuals and Frechet derivatives for the horizontal component have a singularity problem. The numerical examples demonstrate that the singularity problem is related to the directivity of the displacement and to taking the logarithm of Laplace-domain wave fields. To avoid this singularity problem, we use a simple method that excludes the data near the singular region. Although we can use either simultaneous or sequential strategies to invert the Laplace-domain data, we apply a simultaneous inversion strategy in this paper. Nonetheless, the numerical examples demonstrate that our inversion algorithm yields reasonable long-wavelength velocity structures.

Modeling of Power/Ground Planes Using Triangular Elements

As the operating frequency of electronic devices increases and system miniaturization continues, designing power delivery networks (PDNs) that supply quality power throughout the system becomes very challenging. To guarantee successful performance of the PDN, an accurate and time-efficient simulation method that computes the PDN impedance is required. In this paper, we propose a computationally efficient PDN modeling method that analyzes multilayer power/ground planes. The proposed method solves the differential form of Maxwell's equation applied on the circuit representation of a plane pair. To use the analogy between the equations of the electromagnetic field and the equivalent circuit, a dual mesh is created on the metal surface. The use of a nonuniform triangular mesh enables effective discretization of multidimensional and irregular geometries. Moreover, the differential equation generates a sparse system matrix, which requires small computer resources. The proposed model is extended to multiple plane pairs, and the modeling of apertures located on any layer is also included. Simulation results show that the proposed method can solve complex structures with less computational effort than other modeling methods, while maintaining accuracy. In addition, the application of the absorbing boundary condition (ABC) to the proposed method is presented. Applying ABC to the plane boundaries prevents outgoing waves from reflecting back into the plane pair, hence removing plane resonances. The use of the first-order ABC results in a simple implementation and the results show good correlation with the results from a full-wave solver using higher order ABC.

The effect of media parameters on wave propagation in a chiral metamaterials slab using FDTD

SUMMARY Because the permittivity, permeability, and chirality parameters of chiral metamaterials (CMMs) are frequency dependent, the wave equations that describe the characters of electromagnetic wave propagation in CMMs are presented and discretized on the basis of auxiliary differential equation technique in finite-difference time-domain method. The total-field/scattered-field, Mur's first-order absorbing and dielectric boundary conditions for CMMs slab are discussed in the paper. Numerical results show that the cross-polarized reflected coefficient of the CMMs slab is zero. Negative index of refraction phenomenon and optical property of giant optical activity in CMMs slabs are illustrated with 1D auxiliary differential equation–finite-difference time-domain method. The effects to positive or negative phase velocity caused by media parameters of CMMs are studied. Copyright © 2013 John Wiley & Sons, Ltd.

A Novel Absorbing Boundary Condition for the 3-D Split-Step FDTD Method

In this letter, a novel absorbing boundary condition (ABC) is proposed for the 3-D split-step finite-difference time-domain (FDTD) method. Compared with the traditional Mur first-order ABC, the proposed ABC is simpler and requires less additional computation. Numerical simulation results show that the reflection error of the proposed ABC is much lower than that of the traditional Mur first-order ABC and its modifications. Further investigation shows that the accuracy of the proposed ABC is almost equivalent to the Mur second-order ABC.

Micro-Differential Boundary Conditions Modelling the Absorption of Acoustic Waves by 2D Arbitrarily-Shaped Convex Surfaces

AbstractWe propose a new Absorbing Boundary Condition (ABC) for the acoustic wave equation which is derived from a micro-local diagonalization process formerly defined by M.E. Taylor and which does not depend on the geometry of the surface bearing the ABC. By considering the principal symbol of the wave equation both in the hyperbolic and the elliptic regions, we show that a second-order ABC can be constructed as the combination of an existing first-order ABC and a Fourier-Robin condition. We compare the new ABC with other ABCs and we show that it performs well in simple configurations and that it improves the accuracy of the numerical solution without increasing the computational burden.

The closed-form solution of vector finite element integral equations for three dimensional electromagnetic analysis

In this paper, a set of closed-form formulas for vector Finite Element Method (FEM) to analyze three dimensional electromagnetic problems is presented on the basis of Gaussian quadrature integration scheme. By analyzing the open region problems, the first-order Absorbing Boundary Condition (ABC) is considered as the truncation boundary condition and the equation is carried out in a closed-form. Based on the formulas, the hybrid Expanded Cholesky Method (ECM) and MultiFrontal algorithm (MF) is applied to solve finite element equations. Using the closed-form solution, the electromagnetic field of three dimensional targets can be studied sententiously and accurately. Simulation results show that the presented formulas are successfully and concise, which can be easily used to analyze three dimensional electromagnetic problems.

Mur Absorbing Boundary Condition for Three-Step 3-D LOD-FDTD Method

A consistent implementation of Mur first-order absorbing boundary condition (ABC) for three-step 3-D locally 1-D finite-difference time- domain (LOD-FDTD) method is presented. Numerical examples are included to validate the proposed scheme in comparison with Mur ABC for the standard FDTD and two other Mur ABC schemes for the LOD-FDTD. The numerical results illustrate that the proposed scheme can achieve good wave-absorbing capability.

Mur Absorbing Boundary Condition for Efficient Fundamental 3-D LOD-FDTD

This letter presents the implementations of Mur first-order absorbing boundary condition (ABC) for efficient fundamental scheme of 3-D locally one-dimensional finite-difference time-domain (LOD-FDTD) method. The Mur ABC is incorporated into the efficient fundamental scheme using consistent implementation and a novel implementation with lower reflection coefficient. Both implementations in the efficient fundamental scheme are compared and validated with the conventional LOD-FDTD scheme. By comparing the CPU time of both conventional and efficient fundamental schemes, it is ascertained that substantial gain in the overall efficiency may be achieved for the latter even with Mur ABC incorporated.

Implementation of the Second-Order ABC in the FETI-DPEM Method for 3D EM Problems

The electromagnetic dual-primal finite element tearing and interconnecting (FETI-DPEM) method, originally equipped with the first-order absorbing boundary condition (ABC), is combined with the second-order ABC implemented using auxiliary variables to enhance the accuracy of electromagnetic scattering and radiation simulations. When the second-order ABC surface is partitioned by domain decomposition, it is necessary to enforce an extra continuity condition on the auxiliary variables representing a continuous surface divergence for the electric field at the junctions of the subdomain interfaces and the truncation surface. A simple approach to enforce such a continuity is proposed without altering the overall framework of the FETI-DPEM method. Numerical examples are provided to demonstrate the accuracy improvement due to the use of the second-order ABC.

Nonconforming finite element methods for the simulation of waves in viscoelastic solids

The propagation of waves in two- and three-dimensional bounded viscoelastic media is described in the space–frequency domain, leading to a Helmholtz-type boundary value problem, which is noncoercive, non-Hermitian, and complex valued. First-order absorbing boundary conditions are derived and used to minimize spurious reflections from the artificial boundaries. The paper consists of two parts. In Part I we describe the global procedures for the approximate solution of the problem. Simplicial and rectangular nonconforming finite element methods are employed for the spatial discretization. Optimal error estimate in a broken energy and L 2(Ω) norms are derived using a bootstrapping argument of Schatz. Also a hybridization of these procedures is analyzed. In Part II we define and analyze nonoverlapping domain decomposition iterative methods. Convergence results are derived and numerical experiments showing the potential applicability in seismology are presented.

APPLICATION OF A DOMAIN DECOMPOSITION METHOD WITH LAGRANGE MULTIPLIERS TO ACOUSTIC PROBLEMS ARISING FROM THE AUTOMOTIVE INDUSTRY

The Finite Element Tearing and Interconnecting method for the Helmholtz equation is a recent nonoverlapping domain decomposition method for solving linear systems arising from the finite element discretization of Helmholtz problems in bounded domains. This method was validated on two-dimensional external problems with first-order absorbing boundary conditions. The purpose of this paper is to study the robustness and efficiency of iterative methods for the solution of the associated interface problem for three-dimensional interior problems arising from the automotive industry.

A new absorbing boundary condition structure for waveguide analysis

The existence of evanescent waves and waves near cutoff frequencies limits the accuracy of the fields computed in waveguides using the finite-difference time-domain method, and prompted several researchers to design complicated boundary conditions, including combinations of perfectly matched layers and Higdon's higher order absorbing boundary conditions (ABCs). Instead, we employ a terminating structure in which the lateral walls are made absorbing in addition to the longitudinal walls. The undesirable lateral waves at the normal boundary interface are slowed down and effectively attenuated in the lateral walls, while the propagating waves are absorbed in the longitudinal walls. Numerical calculations for pulse excitation of a rectangular waveguide, using the simple Mur's first-order ABC, demonstrate the usefulness of the method.

Generalized Galerkin approximations of elastic waves with absorbing boundary conditions

For the propagation of elastic waves in unbounded domains, absorbing boundary conditions (ABCs) at the fictitious numerical boundaries have been proposed. In this paper we focus on both first- and second-order ABCs in the framework of variational (weak) approximations, like those stemming from Galerkin method (or its variants) for finite element or spectral approximations [1]. In particular, we recover first order conditions as natural (or Neumann) conditions, whereas we propose a penalty residual method for the treatment of second order ABCs. The time discretization is based on implicit backward finite differences, whereas we use spectral Legendre collocation methods set in a variational form for the spatial discretization (treatment of finite element or spectral element approximations is completely similar). Numerical experiments exhibit that the present formulation of second-order ABCs improves the one based on first-order ABCs with regard to both the reduction of the total energy in the computational domain, and the Fourier spectrum of the displacement field at selected points of the elastic medium. A stability analysis is developed for the variational problem in the continuous case both for first- and second-order ABCs. A suitable treatment of ABCs at corners is also proposed.

On the use of rational iterations and domain decomposition methods for the Helmholtz problem

An iterative algorithm for the numerical solution of the Helmholtz problem is considered. It is difficult to solve the problem numerically, in particular, when the imaginary part of the wave number is zero or small. We develop a parallel iterative algorithm based on a rational iteration and a nonoverlapping domain decomposition method for such a non-Hermitian, non-coercive problem. Algorithm parameters (artificial damping and relaxation) are introduced to accelerate the convergence speed of the iteration. Convergence analysis and effective strategies for finding efficient algorithm parameters are presented. Numerical results carried out on an nCUBE2 are given to show the efficiency of the algorithm. To reduce the boundary reflection, we employ a hybrid absorbing boundary condition (ABC) which combines the first-order ABC and the physical $Q$ ABC. Computational results comparing the hybrid ABC with non-hybrid ones are presented.