Andreas C. Cangellaris

Time-domain finite-element methods

Various time-domain finite-element methods for the simulation of transient electromagnetic wave phenomena are discussed. Detailed descriptions of test/trial spaces, explicit and implicit formulations, nodal and edge/facet element basis functions are given, along with the numerical stability properties of the different methods. The advantages and disadvantages of mass lumping are examined. Finally, the various formulations are compared on the basis of their numerical dispersion performance.

Analysis of the numerical error caused by the stair-stepped approximation of a conducting boundary in FDTD simulations of electromagnetic phenomena

A rigorous analysis of the numerical error associated with the use of stair-stepped (saw-toothed) approximation of a conducting boundary for finite-difference time-domain (FDTD) simulations is presented. First, a dispersion analysis in two dimensions is performed to obtain the numerical reflection coefficient for a plane wave scattered by a perfectly conducting wall, tilted with respect to the axes of the finite-difference grid, under both transverse electric and transverse magnetic polarizations. The characteristic equation for surface waves that can be supported by such saw-tooth conducting surfaces is derived. This equation leads to expressions that show the dependence of the propagation constant along the boundary and the attenuation constant perpendicular to it on cell size and wavelength. Numerical simulations that demonstrate the effects predicted by the dispersion analysis are presented. >

Point-matched time domain finite element methods for electromagnetic radiation and scattering

Direct time domain computation of Maxwells differential equations will soon become a practical technique because of the availability of supercomputers. The principal methods used in time domain computations and the supporting theories are presented. The point-matched finite element method is chosen as the main feature of this presentation, which includes the discretization of equations, conforming mesh generation, dielectric and metallic interfaces, numerical stability and simulation of radiation conditions. Numerical results of scattering of Gaussian pulses are presented in time sequences.

Numerical stability and numerical dispersion of a compact 2-D/FDTD method used for the dispersion analysis of waveguides

The stability condition is derived for the compact two-dimensional finite-difference-time-domain (2-D/FDTD) scheme which was recently proposed for the dispersion analysis of waveguiding structures. It is shown that the upper limit of the Courant number depends on the desirable propagation constant beta and is always smaller than the one for the standard FDTD scheme in two dimensions. The dispersion equation for the numerical scheme is derived also and is used to examine the impact of grid size on the accuracy of the calculated eigenvalues (frequencies) for the dominant and higher-order modes.<<ETX>>

Rigorous electromagnetic modeling of chip-to-package (first-level) interconnections

A methodology is presented for the rigorous electromagnetic analysis of pulse transmission through first-level interconnects. The methodology combines a full-wave, vectorial, time-dependent Maxwells equations solver with SPICE circuit models for the nonlinear drivers, to facilitate the accurate modeling of the electromagnetic phenomena occurring at the chip-to-package interface. Comparisons of the results obtained using this method with others calculated using SPICE simulations are used to validate the method and demonstrate its application in the electromagnetic modeling of high-speed packaging structures. >

A study of discretization error in the finite element approximation of wave solutions

A dispersion analysis is used to study the errors caused by the spatial discretization of the finite-element method for the two-dimensional scalar Helmholtz equation. It is shown that the error can be determined analytically for a uniform mesh of infinite extent. Numerical results are presented to show the effects of several parameters on the error. These parameters are the nodal density, the electrical size of the mesh, the direction of propagation of the incident wave, the type of element, and the type of boundary condition. >

A general approach for the development of unsplit-field time-domain implementations of perfectly matched layers for FDTD grid truncation

It is shown that the anisotropic perfectly matched medium, proposed recently for the construction of reflectionless absorbing boundaries for differential equation-based electromagnetic simulations in unbounded domains, can be made equivalent to the Chew-Weedon perfectly matched medium developed from a modified Maxwells system with coordinate stretching. Consequently, despite the apparently nonphysical coordinate stretching, Chew-Weedons formulation, with an appropriate definition of the involved electric and magnetic fields, is merely an alternative mathematical form of Maxwells system in an anisotropic medium. Finally, a more convenient time-domain implementation of the perfectly matched layer without splitting of the field components is derived.

Efficient transient simulation of lossy packaging interconnects using moment-matching techniques

This paper introduces enhancements to moment-matching techniques for rapid time-domain computer simulation of packaging interconnect structures. New moment calculating methods are proposed to handle dispersive, lossy interconnects and related package discontinuities, transitions and junctions that are modeled in terms of either numerically extracted or measured frequency-dependent, equivalent transmission line parameters, or scattering parameters. Computer simulations of pulse propagation in interconnects with frequency-dependent ohmic losses are used to demonstrate the validity and accuracy of the proposed methods.

On the accuracy of numerical wave simulations based on finite methods

The numerical error associated with the simulation of linear wave phenomena using finite methods in both the frequency and time domain is considered. Both exact and numerically generated solutions of finite difference and finite element approximations to the scalar Helmholtz equation in one and two dimensions are used to demonstrate the dependence of the accuracy of the discrete solution on the number of nodes per wavelength, the electrical size of the computational domain, the order of the discretization, and the type of boundary conditions used. The results from these studies, as well as results from similar studies for finite difference approximations of Maxwells equations in the time domain, are used to generate simple expressions for selecting the nodal density to maintain a desirable accuracy.

Frequency-dependent inductance and resistance calculation for three-dimensional structures in high-speed interconnect systems

Frequency-dependent inductances and resistances of three-dimensional structures commonly encountered in modern electronic interconnections are calculated by a combined finite-element/integral-equation method. the mathematical formulation and the numerical method of solution are discussed. Numerical results for microstrip bends and vias are presented. These results compare well with experiment. The effect of the frequency dependence of the inductance on pulse propagation is discussed. >