# Milestones

## 2005 Annual Report

#### ANNUAL PROGRESS SUMMARY

### Reporting Period: 1 September 2004 to 31 August 2005

#### OBJECTIVES

The objective of this project is to develop novel and accurate numerical techniques to model complex antenna feeds, feed networks, and active antenna elements for analyzing/ designing/optimizing multifunctional, conformal antennas/arrays involving complex/ nonlinear materials/devices.

Robust schemes for modeling feeds and distribution networks are critical to synthesizing advanced antenna/array systems. Indeed, while simplified feed models permit the calculation of reasonably accurate radiation patterns, they fail for all but the simplest systems when input impedances and mutual coupling (S-parameters) are to be accurately predicted. In practice, most antennas are fed by coaxial lines, waveguides, or microstrip elements and their analysis is complicated by the presence of baluns and tuning circuits. Simplified models present in most antenna analysis packages often entirely fail to accurately account for the geometric details of the feed and use nonvariational equations to estimate input impedances.

Direct integration of nonlinear circuitry into antennas/arrays often permits feed/ distribution networks and packages to be simplified and reduced in size and weight. However, one of the principal challenges in designing active antenna arrays involves the prediction of, and compensation for, electromagnetic interference and signal distortion caused by the nonlinear behavior of the active devices, which may result in spurious, unpredictable, and undesirable antenna responses. Current computational tools capable of modeling nonlinear packages are often low-order finite difference based and fail to capture nonlinear effects with accuracies that the antenna engineer has become accustomed to when using integral equation methods.

To overcome the problems described above, we will:

- Subtask 1: Develop accurate antenna feed models for the time- and frequency-domain finite element and integral equation based simulation techniques. Develop stable and variational formulations for the calculation of the input and mutual impedances or S-parameters that fully characterize the mutual coupling in conformal arrays.
- Subtask 2: Implement models for passive and active circuit elements in time-domain finite element field solvers to enable active antenna element design optimization;
- Subtask 3: Develop broadband macromodels for the portions of a feed network that do not interact with the active elements and their stable incorporation in the nonlinear field solvers used for active array modeling and simulation;
- Subtask 4: Develop efficient simulation techniques that are compatible with fast finite element and integral equation field solvers and thus can be used to enable efficient multi time-scale transient simulation of active antenna arrays with uncompromised electromagnetic accuracy;
- Subtask 5: Inherent and central to each subtask is a validation and verification effort involving the application of the developed technology to existing antenna/array models and government-provided antennas and arrays.

#### STATUS OF EFFORT/ACCOMPLISHMENTS/NEW FINDINGS

**1. Antenna Feed and Phased Array Modeling in the Frequency and Time Domains**

An important component in the numerical simulation of antennas is the accurate modeling of the antenna feed. A robust scheme for modeling antenna feeds is critical to the design of advanced antenna systems such as multi-functional antenna arrays. While simplified feed models permit the calculation of reasonably accurate radiation patterns, they fail for all but the simplest systems when input impedances and mutual coupling (S-parameters) are to be accurately predicted. In practice, most antennas are fed by coaxial lines, waveguides, or microstrip elements. Simplified models present in most antenna analysis packages often entirely fail to accurately account for the geometric details of the feed.

During the previous performance period of the contract (1 May 04 - 31 Aug 04), we developed an accurate method to model a variety of antenna feeds. This model is directly based on the full-wave analysis of electric and magnetic fields in the feed structures, instead of using the voltage and current concepts. This model involves a reference surface, often chosen to be close to the coaxial or waveguide opening, and then represents the total field as the superposition of the incident and reflected waves (including higher-order modes excited by the structures in the vicinity of the feed). An exact boundary condition can then be derived, which can be incorporated into the FEM or MoM solution of the antenna problem. As a result, the input for the numerical analysis is the incident field in the feed line and the output is the reflection coefficient (S11), from which the input impedance can be readily calculated. In the case for multiple antennas, a full S-parameter (or mutual impedance) matrix can be computed by exciting each antenna. This model is applicable to most antenna feeds. Except for numerical discretization, the model is exact and its numerical solution is error controllable. More important, since this model calculates the reflected waves in the feed lines, one can incorporate the effect of feed structures, such as corporate feed, into the antenna array analysis. Since this model, termed as the waveguide port boundary condition (WPBC), includes all higher-order modes, its implementation in the time domain is nontrivial.

During this performance period, we have implemented this new antenna modeling technique into a higher-order time-domain finite element method for analysis of broadband antennas. In this higher-order time-domain finite element method, we implemented perfectly matched layers (PML) for mesh truncation and the WPBC for antenna feed modeling. We demonstrated its application and capability through several numerical examples. These include a two-arm logarithmic spiral antenna, an antipodal Vivaldi antenna, and a Vlasov antenna, and in all the cases, we obtained excellent results. In particular, the Vlasov antenna was a benchmark problem proposed by Electromagnetic Code Consortium (EMCC) and was considered a very challenging problem because it contains electrically very large and very small structures. We also used our higher-order time-domain finite element method and participated this year's EMCC benchmark testing. To be more specific, we computed the input impedances, the mutual impedances, and the radiation patterns of a monopole antenna, a monopole array, a microstrip antenna, and a microstrip array, all with a detailed modeling of the feeds.

In the course of our research, we have also developed an improvement for a simpilified model of antenna feeds. Although the accurate feed modeling described above is very accurate, it can be quite time consuming since the antenna feeds often consist of very small structures and their accurate modeling often results in an extremely unbalanced mesh density and an excessively large numerical system. For initial fast designs, it is still desirable to use a simplified model that is both efficient and convenient for numerical simulations. In such a simplified model, the original feed structure is replaced with a simplified source, typically an electric current probe, which provides the current flow between two terminals of an antenna. This probe feed model has been extensively used in the finite element simulations for a variety of antenna structures. Although simple and efficient, this model is approximate; it works well for microstrip patch antennas with a very thin substrate. For most other types of antennas, its calculation of the input impedance deviates significantly from the true solution or the actual measured data. Our improved model was found to substantially improve the accuracy of the input impedance calculation.

During this performance period, we have also started to develop a three-dimensional time-domain finite-element method that can be used to analyze infinite periodic phased arrays using one unit cell. This method permits the calculation of the broadband frequency response of an phased array in a single execution. The method solves for a transformed field variable (instead of solving directly for the electric field) in order to easily enable periodic boundary conditions in the time domain. The accuracy and stability of the method has been demonstrated by a series of examples where the new formulation is compared with reference solutions. The method is found to be stable even for angles of incidence close to grazing.

#### Publications:

- Z. Lou and J. M. Jin, "Modeling and simulation of broadband antennas using the time-domain finite element method,"
*IEEE Trans. Antennas Propagat.,*submitted for publication, March 2005. - L. E. R. Petersson and J. M. Jin, "A three-dimensional time-domain finite element formulation for periodic structures,"
*IEEE Trans. Antennas Propagat.,*submitted for publication, Feb. 2005. - Z. Lou and J. M. Jin, "An accurate waveguide port boundary condition for the time-domain finite element method," IEEE Antennas and Propagation Society International Symposium, Washington, DC, July 2005.
- L. E. R. Petersson, Z. Lou, and J. M. Jin, "A 2-D Floquet boundary condition for time-domain finite element analysis of periodic geometries," IEEE Antennas and Propagation Society International Symposium, Washington, DC, July 2005.

2. Development of a Time-Domain Integral Equation (TDIE) Based Antenna Analysis Tool

During the first 6 months of the contract period, a self-consistent hybrid scheme for analyzing coupling of electromagnetic transients into coaxial cables mounted on electrically-large and geometrically-intricate platforms was developed [H. Bagci et al., 2004 *URSI Symp*. *Digest*]. This scheme used a three-dimensional (3-D) TDIE-based field solver, a one-dimensional (1-D) finite-difference time-domain (FDTD)-based trans-mission-line solver, and a modified nodal analysis (MNA)-based circuit solver to compute external fields on platforms, guided fields along coaxial cables, and node voltages on equivalent circuits of cable connectors and loads, respectively. These three solvers were interfaced at coaxial-cable shields (via surface impedance and admittance mechanisms) and at circuit terminals (via a current injection mechanism). The 3-D TDIE-based field solver, which required most of the computational resources, was accelerated by parallel computation and the time-domain adaptive integral method (TD-AIM) [A. E. Yilmaz et al., *IEEE Trans. Antennas Propagat.*, 10, 2692-2708, 2004]. Even though this hybrid scheme was proven useful in the analysis of several antennas including a log periodic wing-trailing monopole array [H. Bagci et al., 2005 *IEEE AP Symp. Digest*], it suffered from important drawbacks relating to (i) accuracy and (ii) modeling versatility stemming from its reliance on a (second-order accurate) FDTD-scheme for tracking guided fields along cables.

During months 6-12 of the contract period, improvements to the above-described solver that alleviate these drawbacks were developed. Specifically, a 1-D implicit TDIE-based transmission-line solver to compute guided fields along cables was implemented [H. Bagci et al., 2005 *URSI Symp*. *Digest*]; this solver replaces the FDTD cable analyzer in the previous version of the code. The new transmission-line solver either uses analytically derived time domain Green functions for low- or constant-loss cables or numerically constructed Green functions (that are specified either by means of frequency- or time-domain samples) for cables with frequency-dependent losses. This *full* TDIE approach has three important features of note: (i) It enforces boundary conditions at cable ends *exactly*, (ii) it does not suffer from the buildup of dispersion errors, and (iii) as it is a fully implicit solver, it does not impose CFL stability restrictions on the (global) time-step size (based on the spatial discretization along the cable). The last property is especially important for accurate modeling of coupling through cable shields, as such requires the computation of external and guided fields at many sample (coupling) points along the cable. The principle disadvantage of the new transmission-line solver, however, is its high computational complexity: it requires operations to compute guided fields along each cable; here is the number of coupling points along all cables. To alleviate this computational bottleneck, a 1-D TD-AIM accelerator for the TDIE-based cable solver was implemented. This accelerator reduces the guided-field computation cost to operations, where is the number of nodes on a 1-D uniform grid defined along the cable.

3. Domain Decomposition Methodologies for FEM-based Modeling of Planar, Multilayered, Signal Distribution Networks

The primary focus of this task is the development and demonstration of a systematic and computationally-efficient, finite element-based methodology for the robust electromagnetic modeling of the planar multi-layered signal distribution network (SDN) of complex antenna arrays. The proposed methodology is aimed at the following two major obstacles associated with the finite element-based modeling of such networks. The first concerns the meshing of the signal distribution structure, a task that is hindered by the presence of multiple layers of metallization of fairly disparate layouts. We address this issue by means of a layer-by-layer decomposition of the structure that allows for the finite element discretization to be performed one metallization layer at a time, independently of the other layers. The second concerns the convergence of finite element solvers, when dealing with structures exhibiting large variation in feature sizes from sub-wavelength features (e.g., the cross-sectional dimensions of the conductors and the substrate thickness) to multiple wavelengths. For this purpose, Krylov subspace techniques and/or multigrid methods are utilized.

As an example of the implemented domain decomposition methodology, the electromagnetic transmission properties of the signal via penetrating through multiple pairs of power and ground planes in a multi-layered printed circuit board are analyzed. The structure is depicted in Fig. 1. Two ground pins, are placed on either side of the signal via to provide for impedance control in this vertical interconnection. The ground pins make electric contact with the ground planes and go through via holes in the power planes in the printed-circuit-board stack-up. For the electromagnetic modeling of this structure a sub-domain is defined to consist of a pair of ground planes with a power plane sandwiched in between. The associated ports for the definition of the network macro-model for each via are also shown in Fig. 1. The specific structure is formed by cascading two identical sub-domains, each one formed by a pair of ground planes with a power plane sandwiched between them. Clearly, once the two-port network is extracted for the single sub-domain, it can be used repeatedly to model more complicated, multi-layered structures, formed through the concatenation of any number of the generated two-port macro-model.

**Figure 1.** Cross-sectional view of a multi-layered signal via structure.

The scattering parameters for the multi-layered via structure of Fig. 1 were computed for three different cases. Case 1, which is denoted as "GPG" in the plots of Fig. 2, is the single sub-domain case. Case 2, which is denoted as "GPGPGPG," is for the via structure formed through the concatenation of three sub-domains. The final case (GPGPGPGPGPG) is for the structure formed through the concatenation of five sub-domains. From the plots it is immediately evident that as the number of layers increases the transmission bandwidth of the via is reduced. It is noted that for the generation of the 40-GHz bandwidth, two-port macro-model of the single sub-domain, a Krylov subspace-based model order reduction process with a single frequency expansion point (at 20 GHz) was used. Thus, the bulk of the computational cost for the broadband macro-model generation is associated with the finite element solution of the single sub-domain at a single frequency. The accuracy of the layer-by-layer domain decomposition solution was validated through comparisons with the solution obtained from the modeling of the entire structure using HFSS for the cases GPG and GPGPGPG. In both cases, very good agreement was observed between the calculated S-parameters.

**Figure 4.** Magnitude of the scattering parameters for a multi-layered signal via.

#### Publications:

- H. Wu and A. C. Cangellaris, "A domain decomposition methodology for electromagnetic modeling of multi-layered planar waveguides,"
*Proc.**2005**IEEE**International Antennas & Propagation Symposium and URSI Meeting*, Washington, D.C., Jul. 2005. - A. C. Cangellaris and H. Wu, "Domain decomposition and multi-scale finite elements for electromagnetic analysis of integrated electronic systems,"
*Proc. 2005 IEEE International Symposium on Electromagnetic Compatibility*, Vol. 2, pp. 817-822, Chicago, IL, Aug. 2005.

**4. Fast Multipole Algorithm for Low-Frequency and Large Object Problems**

- At the request of Northrop Grumman Corporation, ScaleMe was successfully ported from a Unix environment to a Linux environment. A simple case was tested to validate the success of the porting.
- We studied a low-frequency acceleration technique using mixed-form fast multipole algorithm. In this form, the nondiagonalized factorization of the Green's function was used when the low frequency regime, and the diagonalized form of the factorization of the Green's function was adopted in the mid frequency regime. Then a transformer was used to transform the outgoing wave basis from multipole basis to plane-wave basis and similar operation was done in the reverse direction. Results successfully showed that the smallest box size of the multilevel fast multipole algorithm can be reduced.
- We have developed an alternative Nystrom method where the local correction is done differently from the Nystrom method for electromagnetic Green's function that are in existence in the literature. In this method, the local correction is done without having to invert a local correction matrix. Higher order convergence of this method is demonstrated. Also extensive testing of the method for geometries with sharp edges and corners has been performed.
- We have studied the use of loop basis to reduce the computational load of volume integral equation. This can reduce the number of unknowns in the volume integral equation.
- We have trained a master's student to evaluate the use of Huygens' principle to represent field accurately, as well as training him in the art of geometrical optics and physical optics. We are also currently training a master's level student in the art of layered medium Green's function.