Analysis and design of cylindrical multilayered MMICs using a unified non-uniform FDTD algorithm

ABSTRACTAs a complementary way to traditional wave-equation-based forward modeling methods, lattice spring model (LSM) is introduced into seismology for wavefield modeling owing to its remarkable stability, high-calculation accuracy, and flexibility in choosing simulation meshes, and so forth. The LSM simulates seismic-wave propagation from a micromechanics perspective, thus enjoying comprehensive characterization of elastic dynamics in complex media. Incorporating an absorbing boundary condition (ABC) is necessary for wavefield modeling to avoid the artificial reflections caused by truncated boundaries. To the best of our knowledge, the perfectly matched layer (PML) method has been a routine ABC in the wave-equation-based numerical modeling of wave physics. However, it has not been used in the nonwave-equation-based LSM simulations. In this work, we want to apply PML to LSM to attenuate the boundary reflections. We divide the whole simulation region into PML region and inner region, PML region surrounds the inner region. To incorporate PML to LSM, we establish elastic-wave equations corresponding to LSM. The simulation in the PML region is conducted using the established wave equations and the simulation in the inner region is conducted using LSM. Three simulation examples show that the PML scheme is effective and outperforms Gaussian ABC.

Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation

The discrete reflection/absorption performance of perfectly matched layer (PML) boundary conditions is a function of the field discretization scheme throughout the computational domain and the form of the complex coordinate extension that characterizes the PML region. Most previous work analyzing PML reflection performance has assumed use of low-order (second- or fourth-), small-stencil (three- or five-point) spatial discretizations whose dispersion is poor outside a narrow low-wavenumber band. Large-stencil differencing schemes, regardless of order, are easily created with physically realistic, wide-band dispersive characteristics, but the effect their adoption has on PML performance has not been analyzed previously. In the context of a semi-discrete approximation to the parabolic/paraxial equation, this work derives the analytic form of the discrete PML reflection coefficient for an arbitrary-order/arbitrary-stencil interior differencing scheme coupled to an arbitrary-order/arbitrary-stencil PML region. This result is then used to show that the absorption performance of the PML regions is largely insensitive to changes in interior differencing. As a consequence, the benefits of superior dispersion characteristics provided by use of large-stencil differencing in the interior physical domain can be had without incurring discrete reflection penalties at the PML.

The absorbing boundary condition (ABC) and the perfectly matched layer (PML) are employed to suppress the numerical reflections in wave propagation methods. The PML has earlier been shown to fail at the grazing angle propagation of beam. The ABC is preferred over the PML where refractive index is not an analytic function. We have explored the performance of the ABC in reflection-less absorption of non-paraxial propagation of beams and have compared it against the PML. We have considered an example of the propagation of Gaussian beams through a homogeneous medium terminated with ABC or PML. We have used the collocation method for propagation. We have found that both the boundary conditions are comparable in performance. The ABC performed relatively better than the PML for beams at smaller angles. However, the performance of the PML is better than that of the ABC at larger tilt angles.

A three-dimensional (3-D) finite difference time domain (FDTD) algorithm with perfectly matched layer (PML) absorbing boundary condition (ABC) is presented for general inhomogeneous, dispersive, conductive media. The modified time-domain Maxwell's equations for dispersive media are expressed in terms of coordinate-stretching variables. We extend the recursive convolution (RC) and piecewise linear recursive convolution (PLRC) approaches to arbitrary dispersive media in a more general form. The algorithm is tested for homogeneous and inhomogeneous media with three typical kinds of dispersive media, i.e., Lorentz medium, unmagnetized plasma, and Debye medium. Excellent agreement between the FDTD results and analytical solutions is obtained for all testing cases with both RC and PLRC approaches. We demonstrate the applications of the algorithm with several examples in subsurface radar detection of mine-like objects, cylinders, and spheres buried in a dispersive half-space and the mapping of a curved interface. Because of their generality, the algorithm and computer program can be used to model biological materials, artificial dielectrics, optical materials, and other dispersive media.

We extend Berenger's perfectly matched layers (PML) to conductive media. A finite-difference-time-domain (FDTD) algorithm with PML as an absorbing boundary condition is developed for solutions of Maxwell's equations in inhomogeneous, conductive media. For a perfectly matched layer in a conductive medium, an additional term involving the time-integrated electric field has to be introduced to account for the coupling between the loss from the PML and the normal conduction loss. This absorbing boundary condition is proven to be highly effective for the absorption of outgoing waves at the computational edge even when a dipping interface intersects the outer boundary. The algorithm is validated by analytical solutions, and is also compared with Liao's absorbing boundary condition. Numerical results for subsurface radar measurements are shown to demonstrate the applications of this method. © 1997 John Wiley & Sons, Inc. Microwave Opt Technol Lett14, 134–137, 1997

For wave propagation problems in infinite-domains, truncation of the far-field by an artificial boundary is essential to define a bounded domain of interest; resultant difficulties are the radiation condition and generation of incident waves. Instead of the radiation condition, an absorbing boundary condition (ABC) is implemented in an inhomogeneous exterior domain, based on the perfectly matched layer (PML) technique. Complex coordinate stretching, a key PML concept, analytically continues a governing equation in the domain of interest into a PML equation as an anisotropic absorbing boundary in the exterior domain; a system of equivalent equations governs waves in the whole computational domain, and thus the PML is reflectionless at the artificial boundary, an interface between the PML and the domain of interest. For generation of incident waves, wave-maker modeling is proposed based on an idea of plain topography and wave-ray tracing. A source function along the interface is a curvilinear integral in a weak form for the governing equation. Finite element examples for wave propagation in one-dimension verify ability and efficiency of techniques with PML ABCs; besides, those for wave scattering by a circular cylinder on a slope show effectiveness of the wave-maker modeling.

In this work, we present a numerical approach for simulating wave propagation in unbounded domains which combines discontinuous Galerkin methods with arbitrary high order time integration (ADER-DG) and a stabilized modification of perfectly matched layers (PML). Here, the ADER-DG method is applied to Bérenger’s formulation of PML. The instabilities caused by the original PML formulation are treated by a fractional step method that allows to monitor whether waves are damped in PML region. In grid cells where waves are amplified by the PML, the contribution of damping terms is neglected and auxiliary variables are reset. Results of 2D simulations in acoustic media with constant and discontinuous material parameters are presented to illustrate the performance of the method.

The finite-difference time-domain (FDTD) method in conjunction with the perfectly matched layers (PML) absorbing boundary condition (ABC) has been a popular choice for the simulation of ground penetrating radars (GPRs) due to the following reasons: FDTD is suitable for modeling inhomogeneities. FDTD is also suitable for the computation of time-dependent radar signals. PML is a very effective ABC that renders the FDTD method quite efficient. In this paper, in addition to the conventional usage of PML to truncate the FDTD computational domain, we report on how the PML can also be used to simulate an absorbing material inside the FDTD computational domain.

In this paper, we present a new unstaggered colocated finite-difference scheme for solving time-domain Maxwell's equations in a curvilinear coordinate system. All components of the electric and magnetic fields are defined at the same spatial point. A combination of one-sided forward- and backward-difference (FD/BD) operators for the spatial derivatives is used to produce the same order of accuracy as a staggered, central differencing scheme. In the temporal variable, the usual leapfrog integration is used. The computational domain is bounded at the far end by a curvilinear perfectly matched layer (PML). The PML region is terminated with a first-order Engquist-Majda-type absorbing boundary condition (ABC). A comparison is shown with results available in the literature for TE/sub z/ scattering by conducting cylinders. Equations are also presented for the three-dimensional (3-D) case.

The method of perfectly matched layers (PML) is based on the elegant idea of introducing attenuation while keeping impedance unaltered, thus resulting in effective absorption of waves without spurious reflections. Due to its simplicity and effectiveness, PML has been widely used over the past 25 years to model wave propagation in unbounded domains. However, an acknowledged challenge is that PML no longer preserves impedance once the domain and the PML region are discretized, leading to spurious reflection at the interface. This issue has been fixed with the help of so-called impedance preserving discretization, leading to the method of perfectly matched discrete layers (PMDL), developed more than a decade ago. The idea is based on the observation that midpoint integration completely eliminates the discretization error in impedance that arises from linear finite element discretization. Beyond eliminating reflections at the interface, impedance preserving discretization is shown to be crucial to achieving stability of absorbing boundary conditions in situations with differing signs of phase and group velocities, e.g., anisotropic media. This talk will be an exposition of the development of the PMDL by the author and his collaborators, staring with fundamental ideas impedance preserving discretization followed by more advanced ABCs for anisotropic and periodic media.

The Perfectly Matched Layer (PML) concept, introduced by Berenger with the aim of synthesizing an absorbing boundary condition (ABC) for the Finite Difference Time Domain (FDTD) method, was recently modified and extended to Finite Element Frequency Domain (FEFD) applications. The modified equations, which neither require the splitting of the field components of interest nor involve negative conductivity parameters, are employed, in this paper, in conjunction with analytic ABC's to obtain a boundary condition (BC) for scattering problems. This BC exhibits an improved performance over a PML medium terminated by a perfect conductor, or an ABC termination alone. >

A method is presented for modeling open periodic structures by using the finite element method truncated by an anisotropic perfectly matched layer (PML). The method can be used to simulate periodic structures with inhomogeneous profile and arbitrary geometries. The PML is shown to be an effective method for mesh termination, even in the presence the grazing and evanescent fields generated by periodic structures. Numerical analysis is conducted for validation and to demonstrate the effectiveness of the proposed method. © 2002 Wiley Periodicals, Inc. Microwave Opt Technol Lett 35: 106–110, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10530

Because of finite computational domain, an absorbing boundary condition (ABC) is significant in the numerical modelling of wave equations. Liu and Sen (2010 and 2012) proposed a hybrid ABC for time-domain acoustic and elastic modelling. Their scheme has the advantages of small memory, small computation time and nearly perfect absorption. Ren and Liu (2012) extended the scheme to the frequency domain and performed visco-acoustic modelling. To validate the scheme, we develop the hybrid ABC for frequency-domain elastic modelling and compare it with perfectly matched layer (PML). Synthetic data tests demonstrate two ABCs are similarly efficient for high-accuracy modelling with nearly the same computation time and memory, and the hybrid ABC is better than PML for low- or mid-accuracy modelling.

The anisotropic perfectly matched layer (PML) is employed in the mode-matching analysis of an open-ended rectangular waveguide radiating into a layered half space. In the analysis, the layered half space is approximated by a large rectangular waveguide coated with anisotropic PMLs along its walls. Closed form expressions of the eigenvalues and the electromagnetic fields for both TE and TM modes in the anisotropic PML-filled rectangular waveguide are first derived and then used in the mode-matching analysis of the resultant waveguide-junction problem. Numerical results for the aperture admittance and reflection coefficient are presented and compared with available data in the literature. Good agreement is achieved.