The integral equation (IE) method is commonly utilized to model time-harmonic electromagnetic (EM) problems. One of the greatest challenges in its applications arises in the solution of the resulting ill-conditioned matrix equation. We introduce a new domain decomposition method (DDM) for the IE solution of EM wave scattering from non-penetrable objects. The proposed method is a non-overlapping/non-conformal DDM and it provides a computationally efficient and effective preconditioner for the IE matrix equations. Moreover, the proposed approach is very suitable for dealing with multi-scale electromagnetic problems since each sub-domain has its own characteristics length and will be meshed independently. Furthermore, for each sub-domain, we are free to choose the most effective IE sub-domain solver based on its local geometrical features and electromagnetic characteristics. Additionally, the multilevel fast multi-pole algorithm (MLFMA) is utilized to accelerate the computations of couplings between sub-domains. Numerical results demonstrate that the proposed method yields rapid convergence in the outer Krylov iterative solution process. Finally, simulations of several large-scale examples testify to the effectiveness and robustness of the proposed IE based DDM.

Integral Equation Based Domain Decomposition Method for Solving Electromagnetic Wave Scattering From Non-Penetrable Objects

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

A novel dual-primal non-overlapping domain decomposition method (DDM) for the finite element solution of three-dimensional (3D) large-scale electromagnetic problems is proposed. This method reduces the computational complexity from solving the original 3D problem to an equivalent interface problem by utilizing the idea of the dual-primal finite element tearing and interconnecting (FETI-DPEM) method. The new method, which is referred to as the FETI-DPEM2, combines the dual-primal idea with two Lagrange multipliers and implements Robin-type transmission condition at the subdomain interfaces to significantly improve the convergence of the interface solution in the high-frequency region. Similar to the original version, a global coarse problem related to the degrees of freedom at the subdomain corner edges is designed to propagate the residual error to the whole computational domain at each iteration, which further increases the convergence. Numerical examples are presented to demonstrate the validity and the capability of this method. The results show that the proposed method produces fast solutions to large-scale problems in any frequency band.

A New Dual-Primal Domain Decomposition Approach for Finite Element Simulation of 3-D Large-Scale Electromagnetic Problems

We design an extremely compact photonic crystal waveguide spatial mode converter which converts the fundamental even mode to the higher order odd mode with nearly 100% efficiency. We adapt a previously developed design and optimization process that allows these types of devices to be designed in a matter of minutes. We also present an extremely compact optical diode device and clarify its general properties and its relation to spatial mode converters. Finally, we connect the results here to a general theory on the complexity of optical designs.

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Ultra-compact photonic crystal waveguide spatial mode converter and its connection to the optical diode effect.

We present a domain decomposition method as a preconditioner for Krylov-type solvers to model complex electromagnetic problems containing periodicities. The method reduces memory requirements by decomposing the original problem into several nonoverlapping sub-domains. The 1st order transmission condition is employed on interfaces between adjacent sub-domains to enforce continuity of electromagnetic fields and to ensure the sub-domain problems are well-posed. By following the spirit of duality paring a symmetric system is obtained. To reduce the computational burdens of the present method, the finite element tearing and interconnecting like algorithm is adopted. This algorithm results in the computation of the so-called "numerical" Green's function, which can be compressed efficiently via a rank-revealing matrix factorization algorithm. The final system matrix is solved by Krylov solvers instead of classical stationary solvers. To improve the convergence of iterative solvers, several robust implementation details are discussed and the choice of some popular Krylov-subspace solvers is studied through numerical examples.

A Domain Decomposition Method With Nonconformal Meshes for Finite Periodic and Semi-Periodic Structures

A Lagrange multiplier based non-overlapping domain decomposition method, referred to as the dual-primal finite element tearing and interconnecting (FETI-DP), is formulated for the finite element simulation of large, three-dimensional (3-D) electromagnetic problems. This formulation extends the FETI-DP for solving the scalar Helmholtz equation to the solution of the vector curl-curl wave equation using edge-based finite elements. It enforces the field continuity explicitly along the edges shared by more than two subdomains and implicitly at the interfaces between two subdomains through the use of Lagrange multipliers. With the aid of a direct sparse solver for each subdomain system, the large global problem is reduced to a much smaller interface problem, from which a Neumann boundary condition is obtained at the interfaces between all the subdomains. This Neumann boundary condition is then used to calculate the field within each subdomain. It is shown that the resulting FETI-DPEM method is scalable with respect to the size of finite elements and the number of subdomains. It is also scalable with respect to the size of the subdomains when the subdomains, with its surfaces enclosed by perfect magnetic conductors, cannot support any resonant modes. The FETI-DPEM method is applied to the electromagnetic simulation of array-type structures where the geometrical redundancy is fully exploited to speedup the simulation and reduce the memory requirement. Numerical results for the simulation of finite antenna arrays and photonic bandgap devices are presented to demonstrate the application, accuracy, efficiency, and capability of the FETI-DPEM method

A Vector Dual-Primal Finite Element Tearing and Interconnecting Method for Solving 3-D Large-Scale Electromagnetic Problems

Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations

Various technical issues are addressed in the frequency- and time-domain finite element analysis of complex antennas and arrays that may consist of fine structures and composite materials. The paper starts with the finite element formulation of a generic antenna problem. Formulation for the modeling of general electrically and magnetically dispersive, lossy, and anisotropic materials in the time-domain finite element analysis is presented for the first time. This is followed by discussion of three typical techniques for truncation of the finite element computational domain via the use of absorbing boundary conditions, perfectly matched layers, and boundary integral equations. It then describes the modeling of antenna feeds using a current probe (including an improved model), a voltage gap, and a waveguide port boundary condition. After that, numerical approaches to modeling infinite phased arrays, large finite arrays, and antenna-platform interaction are presented, including two novel domain decomposition algorithms in both the frequency and time domains. Numerical examples are given to demonstrate the capability and application of the finite element analysis of a variety of complex antennas and arrays.

Finite Element Analysis of Complex Antennas and Arrays

This paper attempts to apply the idea of the FETI-DPH method together with the first-order absorbing boundary condition (ABC) to solve relatively large-scale problems using edge elements. The algorithm for nodal elements is adapted to edge elements to represent the tangential electric and magnetic fields without changing the main idea of the method. The resulting method is denoted as the FETI-DPEM method in this paper

Yu-Jia Li

Papers

A New Dual-Primal Domain Decomposition Approach for Finite Element Simulation of 3-D Large-Scale Electromagnetic Problems

A Vector Dual-Primal Finite Element Tearing and Interconnecting Method for Solving 3-D Large-Scale Electromagnetic Problems

Parallel implementation of the FETI-DPEM algorithm for general 3D EM simulations

Finite Element Analysis of Complex Antennas and Arrays

A Vector Dual-Primal Finite Element Tearing and Interconnecting Method for Solving 3-D Large-Scale Electromagnetic Problems