We present the formulation of a finite-element/boundary-integral method for the analysis of three-dimensional doubly periodic structures based on arbitrary nonorthogonal lattice configurations. The method starts from a functional description of the field problem where only a single unit cell of the array is considered. This unit cell is meshed with triangular prismatic volume elements and the electric field intensity is discretized with edge-based expansion functions. On the sidewalls of the unit cell, phase boundary conditions are employed to relate the fields on opposing walls of the unit cell. On the top and/or bottom unit cell planar surfaces, the mesh is terminated using a mixed potential integral equation. The required space-domain periodic Green's function is calculated after applying the Ewald (1921) transformation to convert the slowly converging series representation into two rapidly converging series. The method is validated for simple slot and strip frequency-selective surfaces as well as microstrip dipole arrays. More complex geometries investigated are slot-coupled microstrip patches, photonic bandgap materials, and the so-called "artificial puck plate" frequency-selective surface bandpass structure.

Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation

A parameter study of dual-polarized tapered slot antenna (TSA) arrays shows the key features that affect the wide-band and widescan performance of these arrays. The overall performance can be optimized by judiciously choosing a combination of parameters. In particular, it is found that smaller circular slot cavities terminating the bilateral slotline improve the performance near the low end of the operating band, especially when scanning in the H-plane. The opening rate of the tapered slotline mainly determines the mid-band performance and it is possible to choose an opening rate to obtain balanced overall performance in the mid-band. A longer tapered slotline is shown to increase the bandwidth, especially in the lower end of the operating band. Finally, it is shown that the H-plane anomalies are affected by the array element spacing. A design example demonstrates that the results from the parameter study can be used to design a dual-polarized TSA array with about 4.5:1 bandwidth for a scan volume of not less than /spl theta/=45/spl deg/ from broadside in all planes.

Parameter study and design of wide-band widescan dual-polarized tapered slot antenna arrays

Preface. Acknowledgments. Chapter 1: Introduction. 1.1 Numerical Simulation of Antennas. 1.2 Finite Element Analysis vs. Other Numerical Methods. 1.3 Frequency- vs. Time-Domain Simulations. 1.4 Brief Review of Past Work. 1.5 Overview of This Book. References. Chapter 2: Finite Element Formulation. 2.1 Finite Element Formulation in the Frequency Domain. 2.2 Finite Element Formulation in the Time Domain. 2.3 Modeling of Complex Materials. 2.3.1 Modeling of Electrically and Magnetically Lossy Materials. 2.3.2 Modeling of Electrically Dispersive Materials. 2.3.3 Modeling of Magnetically Dispersive Materials. 2.3.4 Modeling of Doubly Dispersive Lossy Materials. 2.4 Solution of the Finite Element Equations. 2.5 Higher-Order and Curvilinear Finite Elements. 2.6 Summary. References. Chapter 3: Finite Element Mesh Truncation. 3.1 Absorbing Boundary Conditions. 3.1.1 First-Order Absorbing Boundary Condition. 3.1.2 Second-Order Absorbing Boundary Condition. 3.2 Perfectly Matched Layers. 3.2.1 PML in Terms of Stretched Coordinates. 3.2.2 PML as an Anisotropic Material Absorber. 3.2.3 PML for Truncating Computational Domain. 3.2.4 Finite Element Implementation of PML. 3.2.5 ABC-Backed, Complementary, CFS, and Second-Order PMLs. 3.3 Boundary Integral Equations. 3.3.1 Frequency-Domain Formulations. 3.3.2 Time-Domain Formulations. 3.3.3 Treatment of Infinite Ground Plane. 3.4 Summary. References. Chapter 4: Hybrid FETD-FDTD Technique. 4.1 The FDTD Method. 4.2 PML Implementation in FDTD. 4.2.1 FDTD Stretched-Coordinate PML. 4.2.2 FDTD Anisotropic PML. 4.3 Near-to-Far-Field Transformation in FDTD. 4.4 Alternative FETD Formulation. 4.5 Equivalence between FETD and FDTD. 4.6 Stable FETD-FDTD Interface. 4.6.1 Initial Approaches. 4.6.2 Stable Formulation. 4.7 Building Hybrid Meshes. 4.8 Wave-Equation Stablization. 4.9 Validation Examples. 4.10 Summary. References. Chapter 5: Antenna Source Modeling and Parameter Calculation. 5.1 Antenna Feed Modeling. 5.1.1 Current Probe. 5.1.2 Voltage Gap Generator. 5.1.3 Waveguide Feed Model. 5.2 Plane-Wave Excitation. 5.2.1 Total-Field Formulation. 5.2.2 Scattered-Field Formulation. 5.2.3 Total- and Scattered-Field Decomposition Approach . 5.3 Far-Field Pattern Computation. 5.4 Near-Field Visualization. 5.5 Summary. References. Chapter 6: Modeling of Complex Structures. 6.1 Thin Material Layers and Sheets. 6.1.1 Impedance Boundary Conditions. 6.1.2 Shell Element Formulation. 6.2 Thin Wires and Slots. 6.2.1 Thin Wires. 6.2.2 Thin Slots. 6.3 Lumped Circuit Elements. 6.3.1 Coupled First-Order Equations. 6.3.2 Wave Equation. 6.3.3 Example. 6.4 Distributed Feed Network. 6.5 System-Level Coupling Example. 6.5.1 Internal Dispersive Material Calibration. 6.5.2 External Illumination and Aperture Coupling. 6.6 Summary. References. Chapter 7: Antenna Simulation Examples. 7.1 Narrowband Antennas. 7.1.1 Coaxial-fed Monopole Antenna. 7.1.2 Monopole Antennas on a Plate. 7.1.3 Patch Antennas on a Plate. 7.1.4 Conformal Patch Antenna Array. 7.2 Broadband Antennas. 7.2.1 Ridged Horn Antenna. 7.2.2 Sinuous Antenna. 7.2.3 Logarithmic Spiral Antenna. 7.2.4 Inverted Conical Spiral Antenna. 7.2.5 Antipodal Vivaldi Antenna. 7.2.6 Vlasov Antenna. 7.3 Antenna RCS Simulations. 7.3.1 Microstrip Patch Antenna. 7.3.2 Standard Gain Horn Antenna. 7.4 Summary. References. Chapter 8: Axisymmetric Antenna Modeling. 8.1 Method of Analysis. 8.1.1 Finite Element Formulation. 8.1.2 Mesh Truncation Using Perfectly Matched Layers. 8.1.3 Mesh Truncation Using Boundary Integral Equations. 8.2 Application Examples. 8.2.1 Luneburg Lens. 8.2.2 Corrugated Horn. 8.2.3 Current Loop Inside a Radome. 8.3 Summary. References. Chapter 9: Infinite Phased Array Modeling. 9.1 Frequency-Domain Modeling. 9.1.1 Periodic Boundary Conditions. 9.1.2 Mesh Truncation Techniques. 9.1.3 Extension to Skew Arrays. 9.1.4 Extension to Scattering Analysis. 9.1.5 Application Examples. 9.2 Time-Domain Modeling. 9.2.1 Transformed Field Variable. 9.2.2 Mesh Truncation Techniques. 9.2.3 General Material Modeling. 9.2.4 Application Examples. 9.3 Approximation to Finite Arrays. 9.4 Summary. References. Chapter 10: Finite Phased Array Modeling. 10.1 Frequency-Domain Modeling. 10.1.1 The FETI-DPEM1 Formulation. 10.1.2 The FETI-DPEM2 Formulation. 10.1.3 Nonconforming Domain Decomposition. 10.1.4 Application Examples. 10.2 Time-Domain Modeling. 10.2.1 The Dual-Field Domain Decomposition Method. 10.2.2 Domain Decomposition for Iterative Solutions. 10.2.3 Application Examples. 10.3 Summary. References. Chapter 11: Antenna-Platform Interaction Modeling. 11.1 Coupled Analysis. 11.1.1 FETI-DPEM with Domain Decomposition. 11.1.2 Hybrid FETD-FDTD with Domain Decomposition. 11.1.3 Hybrid FE-BI Method with FMM Acceleration. 11.2 Decoupled Analysis. 11.2.1 Near-Field Calculation. 11.2.2 Far-Field Evaluation by Numerical Methods. 11.2.3 Far-Field Evaluation by Asymptotic Techniques. 11.2.4 Direct and Iterative Improvements. 11.3 Summary. References. Chapter 12: Numerical and Practical Considerations. 12.1 Choice of Simulation Technologies. 12.2 Frequency- vs. Time-Domain Simulation Tools. 12.3 Fast Frequency Sweep. 12.4 Numerical Convergence. 12.5 Domain Decomposition and Parallel Computing. 12.6 Verification and Validation of Predictions. 12.7 Summary. References. Index.

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Finite Element Analysis of Antennas and Arrays

This series lecture is an introduction to the finite element method with applications in electromagnetics. The finite element method is a numerical method that is used to solve boundary-value problems characterized by a partial differential equation and a set of boundary conditions. The geometrical domain of a boundary-value problem is discretized using sub-domain elements, called the finite elements, and the differential equation is applied to a single element after it is brought to a “weak” integro-differential form. A set of shape functions is used to represent the primary unknown variable in the element domain. A set of linear equations is obtained for each element in the discretized domain. A global matrix system is formed after the assembly of all elements. This lecture is divided into two chapters. Chapter 1 describes one-dimensional boundary-value problems with applications to electrostatic problems described by the Poisson's equation. The accuracy of the finite element method is evaluated for linear and higher order elements by computing the numerical error based on two different definitions. Chapter 2 describes two-dimensional boundary-value problems in the areas of electrostatics and electrodynamics (time-harmonic problems). For the second category, an absorbing boundary condition was imposed at the exterior boundary to simulate undisturbed wave propagation toward infinity. Computations of the numerical error were performed in order to evaluate the accuracy and effectiveness of the method in solving electromagnetic problems. Both chapters are accompanied by a number of Matlab codes which can be used by the reader to solve one- and two-dimensional boundary-value problems. These codes can be downloaded from the publisher's URL: www.morganclaypool.com/page/polycarpou This lecture is written primarily for the nonexpert engineer or the undergraduate or graduate student who wants to learn, for the first time, the finite element method with applications to electromagnetics. It is also targeted for research engineers who have knowledge of other numerical techniques and want to familiarize themselves with the finite element method. The lecture begins with the basics of the method, including formulating a boundary-value problem using a weighted-residual method and the Galerkin approach, and continues with imposing all three types of boundary conditions including absorbing boundary conditions. Another important topic of emphasis is the development of shape functions including those of higher order. In simple words, this series lecture provides the reader with all information necessary for someone to apply successfully the finite element method to one- and two-dimensional boundary-value problems in electromagnetics. It is suitable for newcomers in the field of finite elements in electromagnetics.

Introduction to the finite element method in electromagnetics

SUMMARY In this paper, classical and enriched flnite element formulations to impose Bloch-periodic boundary conditions are proposed. Bloch-periodic boundary conditions arise in the description of wave-like phenomena in periodic media. We consider the quantum-mechanical problem in a crystalline solid, and derive the weak formulation and matrix equations for the Schrodinger and Poisson equations in a parallelepiped unit cell under Bloch-periodic and periodic boundary conditions, respectively. For such second-order problems, these conditions consist of value- and derivative-periodic parts. The value-periodic part is enforced as an essential boundary condition by construction of a value-periodic basis, whereas the derivative-periodic part is enforced as a natural boundary condition in the weak formulation. We show that the resulting matrix equations can be obtained by suitably specifying the connectivity of element matrices in the assembly of the global matrices or by modifying the Neumann

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Classical and enriched finite element formulations for Bloch-periodic boundary conditions

A new analysis technique for infinite phased array antennas was developed and demonstrated. It consists of the finite element method (FEM) in combination with integral equation radiation conditions and a novel periodic boundary condition for 3-D FEM grids. Accurate modeling of rectangular, circular and circular-coaxial feeds is accomplished by enforcing continuity between the FEM solution and several waveguide modes across an aperture in the array's ground plane. The radiation condition above the array is enforced by a periodic integral equation in the form of a Floquet mode summation, thus reducing the solution to that of a single array unit cell. The periodic boundary condition at unit cell side walls is enforced through a matrix transformation. That mathematically "folds" opposing side walls onto each other with a phase shift appropriate to the array lattice and scan angle. The unit cell electric field is expanded in vector finite elements. Galerkin's method is used to cast the problem as a matrix equation, which is solved by the conjugate gradient method. A general-purpose computer code was developed and validated for cases of open-ended waveguides, microstrip patches, clad monopoles and printed flared notches, showing that the analysis method is accurate and versatile. >

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D.T. McGrath

Papers

Phased array antenna analysis with the hybrid finite element method