Analysis and Application of Perfectly Matched Layer Absorbing Boundary Conditions for Computational Aeroacoustics

The Perfectly Matched Layer (PML) was originally proposed by Berenger as an absorbing boundary condition for Maxwell\u27s equations in 1994 and is still used extensively in the field of electromagnetics. The idea was extended to Computational Aeroacoustics in 1996, when Hu applied the method to Euler\u27s equations. Since that time much of the work done on PML in the field of acoustics has been specific to the case where mean flow is perpendicular to a boundary, with an emphasis on Cartesian coordinates. The goal of this work is to further extend the PML methodology in a two-fold manner: First, to handle the more general case of an oblique mean flow, where mean velocities strike the boundary at an arbitrary angle, and second, to adapt the equations for use in a cylindrical coordinate system. These extensions to the PML methodology are effectively carried out in this dissertation. Perfectly Matched Layer absorbing boundary conditions are presented for the linearized and nonlinear Euler equations in two dimensions. Such boundary conditions are presented in both Cartesian and cylindrical coordinates for the case of an oblique mean flow. In Cartesian coordinates, the PML equations for the side layers and corner layers of a rectangular domain will be derived independently. The approach used in the formation of side layer equations guarantees that the side layers will be perfectly matched at the interface between the interior and PML regions. Because of the perfect matching of the side layers, the equations are guaranteed to be stable. However, a somewhat different approach is used in the formation of the corner layer equations. Therefore, the stability of linear waves in the corner layer is analyzed. The results of the analysis indicate that the proposed corner equations are indeed stable. For the PML equations in cylindrical coordinates, there is no need for separate derivations of side and corner layers, and in this case, the stability of the equations is achieved through an appropriate space-time transformation. As is shown, such a transformation is needed for correcting the inconsistencies in phase and group velocities which can negatively affect the stability of the equations. After this correction has been made, the cylindrical PML can be implemented without risk of instability. In both Cartesian and cylindrical coordinates, the PML for the linearized Euler equations are presented in primitive variables, while conservation form is used for the nonlinear Euler equations. Numerical examples are also included to support the validity of the proposed equations. Specifically, the equations are tested for a combination of acoustic, vorticity and entropy waves. In each example, high-accuracy solutions are obtained, indicating that the PML conditions are effective in minimizing boundary reflections

Advances in Time-Domain Electromagnetic Simulation Capabilities Through the Use of Overset Grids and Massively Parallel Computing

A new methodology is presented for conducting numerical simulations of electromagnetic scattering and wave propagation phenomena. Technologies from several scientific disciplines, including computational fluid dynamics, computational electromagnetics, and parallel computing, are uniquely combined to form a simulation capability that is both versatile and practical. In the process of creating this capability, work is accomplished to conduct the first study designed to quantify the effects of domain decomposition on the performance of a class of explicit hyperbolic partial differential equations solvers; to develop a new method of partitioning computational domains comprised of overset grids; and to provide the first detailed assessment of the applicability of overset grids to the field of computational electromagnetics. Furthermore, the first Finite Volume Time Domain (FVTD) algorithm capable of utilizing overset grids on massively parallel computing platforms is developed and implemented. Results are presented for a number of scattering and wave propagation simulations conducted using this algorithm, including two spheres in close proximity and a finned missile

Development and Application of Compatible Discretizations of Maxwell's Equations

We present the development and application of compatible finite element discretizations of electromagnetics problems derived from the time dependent, full wave Maxwell equations. We review the H(curl)-conforming finite element method, using the concepts and notations of differential forms as a theoretical framework. We chose this approach because it can handle complex geometries, it is free of spurious modes, it is numerically stable without the need for filtering or artificial diffusion, it correctly models the discontinuity of fields across material boundaries, and it can be very high order. Higher-order H(curl) and H(div) conforming basis functions are not unique and we have designed an extensible C++ framework that supports a variety of specific instantiations of these such as standard interpolatory bases, spectral bases, hierarchical bases, and semi-orthogonal bases. Virtually any electromagnetics problem that can be cast in the language of differential forms can be solved using our framework. For time dependent problems a method-of-lines scheme is used where the Galerkin method reduces the PDE to a semi-discrete system of ODE's, which are then integrated in time using finite difference methods. For time integration of wave equations we employ the unconditionally stable implicit Newmark-Beta method, as well as the high order energy conserving explicit Maxwell Symplectic method; for diffusion equations, we employ a generalized Crank-Nicholson method. We conclude with computational examples from resonant cavity problems, time-dependent wave propagation problems, and transient eddy current problems, all obtained using the authors massively parallel computational electromagnetics code EMSolve

Modeling EMI Resulting from a Signal Via Transition Through Power/Ground Layers

Signal transitioning through layers on vias are very common in multi-layer printed circuit board (PCB) design. For a signal via transitioning through the internal power and ground planes, the return current must switch from one reference plane to another reference plane. The discontinuity of the return current at the via excites the power and ground planes, and results in noise on the power bus that can lead to signal integrity, as well as EMI problems. Numerical methods, such as the finite-difference time-domain (FDTD), Moment of Methods (MoM), and partial element equivalent circuit (PEEC) method, were employed herein to study this problem. The modeled results are supported by measurements. In addition, a common EMI mitigation approach of adding a decoupling capacitor was investigated with the FDTD method

Finite-Volumes Based FDTD Material Dispersion Modeling

The conventional FDTD, based on second-order central difference formula, is useful only so long as the electrical size of the structure is small. Phase error accumulates in the field calculations as the dimensions of the numerical FDTD lattice become larger. The Finite Volumes-Based 3-D second-order in time, fourth-order in space (FV24) modeling is highly capable of controlling such phase errors. Therefore, it is suitable for electrically large problems at coarse grid resolutions. This work models the frequency dependence of material losses using an Auxiliary Differential Equation (ADE), a technique extensively discussed in the literature. (Elsherbeni and Demir, 2016.) To account for material dispersion, the derivation of ADE is extended for FV24 by modifying the electric field update equations. A Multipole Debye model, which provides an auxiliary differential equation in time domain and also produces a causal response, is used in the current analysis. This model, suitable for FDTD simulations, can simulate relative permittivity and conductivity of materials with high degree of accuracy over a wide bandwidth. For the present study, a simple dielectric scatterer and breast tumor model are used as the problem space. The planewave excitation is provided using the total field/scattered field-based leakage free technique. (R. C. Bollimuntha et al., IET Microwaves, Antennas and Propagation, 2016.) The FV24 algorithm, being accurate even at coarse discretizations, provides excellent wideband performance. Keeping low number of cells per wavelength provides a substantial decrease in floating-point operations per wavelength, enabling faster computation. This fact allows significant reduction in memory usage. This feature of FV24 renders it relatively less expensive than FDTD to model three-dimensional (3-D) problems that are hundreds of wavelengths large. A comparison of accuracy and performance in terms of memory usage and simulation time of conventional FDTD versus FV24 will be presented

Finite-Volumes Based FDTD Material Dispersion Modeling

The conventional FDTD, based on second-order central difference formula, is useful only so long as the electrical size of the structure is small. Phase error accumulates in the field calculations as the dimensions of the numerical FDTD lattice become larger. The Finite Volumes-Based 3-D second-order in time, fourth-order in space (FV24) modeling is highly capable of controlling such phase errors. Therefore, it is suitable for electrically large problems at coarse grid resolutions. This work models the frequency dependence of material losses using an Auxiliary Differential Equation (ADE), a technique extensively discussed in the literature. (Elsherbeni and Demir, 2016.) To account for material dispersion, the derivation of ADE is extended for FV24 by modifying the electric field update equations. A Multipole Debye model, which provides an auxiliary differential equation in time domain and also produces a causal response, is used in the current analysis. This model, suitable for FDTD simulations, can simulate relative permittivity and conductivity of materials with high degree of accuracy over a wide bandwidth. For the present study, a simple dielectric scatterer and breast tumor model are used as the problem space. The planewave excitation is provided using the total field/scattered field-based leakage free technique. (R. C. Bollimuntha et al., IET Microwaves, Antennas and Propagation, 2016.) The FV24 algorithm, being accurate even at coarse discretizations, provides excellent wideband performance. Keeping low number of cells per wavelength provides a substantial decrease in floating-point operations per wavelength, enabling faster computation. This fact allows significant reduction in memory usage. This feature of FV24 renders it relatively less expensive than FDTD to model three-dimensional (3-D) problems that are hundreds of wavelengths large. A comparison of accuracy and performance in terms of memory usage and simulation time of conventional FDTD versus FV24 will be presented

Numerical methods for electromagnetic wave propagation and scattering in complex media

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.Vita.Includes bibliographical references (p. 227-242).Numerical methods are developed to study various applications in electromagnetic wave propagation and scattering. Analytical methods are used where possible to enhance the efficiency, accuracy, and applicability of the numerical methods. Electromagnetic induction (EMI) sensing is a popular technique to detect and discriminate buried unexploded ordnance (UXO). Time domain EMI sensing uses a transient primary magnetic field to induce currents within the UXO. These currents induce a secondary field that is measured and used to determine characteristics of the UXO. It is shown that the EMI response is difficult to calculate in early time when the skin depth is small. A new numerical method is developed to obtain an accurate and fast solution of the early time EMI response. The method is combined with the finite element method to provide the entire time domain response. The results are compared with analytical solutions and experimental data, and excellent agreement is obtained. A fast Method of Moments is presented to calculate electromagnetic wave scattering from layered one dimensional rough surfaces. To facilitate the solution, the Forward Backward method with Spectral Acceleration is applied. As an example, a dielectric layer on a perfect electric conductor surface is studied. First, the numerical results are compared with the analytical solution for layered flat surfaces to partly validate the formulation. Second, the accuracy, efficiency, and convergence of the method are studied for various rough surfaces and layer permittivities. The Finite Difference Time Domain (FDTD) method is used to study metamaterials exhibiting both negative permittivity and permeability in certain frequency bands.(cont.) The structure under study is the well-known periodic arrangement of rods and split-ring resonators, previously used in experimental setups. For the first time, the numerical results of this work show that fields propagating inside the metamaterial with a forward power direction exhibit a backward phase velocity and negative index of refraction. A new metamaterial design is presented that is less lossy than previous designs. The effects of numerical dispersion in the FDTD method are investigated for layered, anisotropic media. The numerical dispersion relation is derived for diagonally anisotropic media. The analysis is applied to minimize the numerical dispersion error of Huygens' plane wave sources in layered, uniaxial media. For usual discretization sizes, a typical reduction of the scattered field error on the order of 30 dB is demonstrated. The new FDTD method is then used to study the Angular Correlation Function (ACF) of the scattered fields from continuous random media with and without a target object present. The ACF is shown to be as much as 10 dB greater when a target object is present for situations where the target is undetectable by examination of the radar cross section only.by Christopher D. Moss.Ph.D

Annual Review of Progress in Applied Computational Electromagnetics

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