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  1. UIUC MURI Review J.-M. Jin, A. C. Cangellaris, and W. C. Chew Center for Computational Electromagnetics Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois 61801-2991 Program Director: Dr. Arje Nachman, AFOSR June 19, 2006

  2. Time-Domain Finite Element Method for Analysis of Broadband Antennas and Arrays J.-M. Jin Center for Computational Electromagnetics Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois 61801-2991 Acknowledgment: This work is sponsored by AFOSR via a MURI grant (Program Director: Dr. Arje Nachman)

  3. TDFEM Antenna Analysis Year 1: • Truncation of Open Free Space • Absorbing boundary condition (ABC) • Perfectly matched layers (PML) • Feed Modeling • Simplified feed model: electric probe feed • Waveguide port boundary condition (WPBC) Year 2: • Novel, Highly Efficient Domain Decomposition • Large antennas • Finite array antennas • Periodic TDFEM • Infinite periodic phased-array antennas

  4. Perfectly Matched Layers (PML) Spatial FEM discretization: Time-Domain Discretization • {e} and {u} are discretized in time domain according to Newmark-Beta method • Resultant system is stable for time marching Convolution

  5. Time-Domain WPBC Time-Domain Formulation: Assume dominant mode incidence:

  6. Monopole Antenna Measured data: J. Maloney, G. Smith, and W. Scott, “Accurate computation of the radiation from simple antennas using the finite difference time-domain method,” IEEE Trans. A.P., vol. 38, July 1990.

  7. Logarithmic Spiral Antenna Probe Feed G. Deschamps, “Impedance properties of complementary multiterminalplanar structures,” IRE Trans. Antennas Propagat., vol. AP-7, Dec. 1959.

  8. Antipodal Vivaldi Antenna Reflection at the TEM port “The 2000 CAD benchmark unveiled,” Microwave Engineering Online, July 2001

  9. Antipodal Vivaldi Antenna E-plane Radiation patterns at 10 GHz H-plane

  10. Interfaces Domain Decomposition • Traditional Methods: • Schwartz Methods • Schur Complement (Substructuring) Method • Finite Element Tearing and Interconnecting (FETI): • Use Lagrange multiplier to formulate the interface problem, usually solved by an iterative solver • Subdomain problems can be solved independently based on interface solutions • Time-domain FETI is less efficient since the interface problem needs to be solved repeatedly at each time step • Time-Domain Dual-Field Domain Decomposition (DFDD): • Does not require solving the interface problem • Computes both electric and magnetic fields • Employs a leapfrog time-marching scheme similar to the FDTD

  11. Weak-Form Representation: Two-Domain DFDD Second-Order Vector Wave Equation (m=1,2): • Boundary Conditions: • SMmetallic surface • SA impedance surface

  12. Two-Domain DFDD Equivalent Surface Currents:

  13. Temporal Discretization • Leapfrog on subdomain interfaces t t • Newmark-Beta method inside each subdomain

  14. Computational Performance (Serial) • Tested on SGI-Altix 350 system with Intel Itanium II 1.5GHz processor • Subdomain problems are solved in serial on a single processor • Each subdomain system is pre-factorized using direct solver before time marching Peak Memory Total CPU Time: Factorization Time CPU Time per Step

  15. Computational Performance (Parallel) Speedup • Tested on SGI-Altix 350 system with multiple Intel Itanium II 1.5GHz processors • Each subdomain is assigned to a different processor

  16. 10-by-10 Vivaldi Array • 2.8 million unknowns • Distributed on 72 processors • Solving time per step: 0.3 s

  17. 1m 9 m FEM Discretization • Element size: 0.08 ~ 0.25 m • 208,747 mixed-2nd order tets • 1.4 million unknowns • Partitioned into 9 subdomains 9 m Photonic Bandgap: Dipole Radiating in Photonic BandGap Air er = 11.56

  18. Dipole Radiating in Photonic BandGap f = 0.25 c/a f = 0.35 c/a f = 0.50 c/a Reflection at Coaxial Port

  19. A Generic Periodic Phased Array • Technical challenges: • Enforcement of periodic boundary conditions • Mesh truncation in the non-periodic direction

  20. Transformed Field Variable • Periodic boundary conditions in the frequency domain • Introduce a transformed field variable M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-difference time-domain analysis of wave scattering from periodic structures: oblique incidence case,” J. Electromag. Waves Appl., vol. 7, pp. 1595-1607, Dec. 1993.

  21. Transformed Field Variable • Second-order vector wave equation: where • Solved via a Galerkin method in space using vector testing functions residing in a tetrahedral mesh and time-integration via the Newmark-b method

  22. Higher-Order Floquet ABC • Floquet expansion for the transformed field variable • Time-domain expression • A very accurate truncation condition can be constructed

  23. Reflection from an Array of Spheres qi = 20o with the ABC a small distance from the surface of the sphere TM-polarization TE-polarization M. Inoue, “Enhancement of local field by a two-dimensional array of dielectric spheres placed on a substrate,” Physical Review B, vol. 36, pp. 2852-2862, Aug. 1987.

  24. A Vivaldi Phased-Array Antenna D. T. McGrath and V. P. Pyati, “Phased array antenna analysis with the hybrid finite element method,” IEEE Trans. Antennas Propagat., vol. 42, pp. 1625-1630, Dec. 1994.

  25. Summary Achievements: • A complete TDFEM modeling of broadband antennas and arrays involving complex geometry and material • A highly effective PML formulation to emulate a free-space environment • A highly accurate waveguide port boundary condition for a physical modeling of antenna feeds • A novel, highly efficient dual-field domain decomposition technique to handle large-scale simulations • TDFEM analysis of infinite phased arrays Work in Progress: • Hybridization of TDFEM and ROM to interface antenna feeds and feed network • Hybridization of TDFEM and TDIE (TD-AIM & PWTD) to model antenna/platform interaction

  26. Finite Element Based, Broadband Macro-modeling of Antenna Array Feed Networks H. Wu and A. C. Cangellaris Center for Computational Electromagnetics & EM Lab Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign Urbana, Illinois 61801-2991 cangella@uiuc.edu

  27. Objectives • Generate compact, multi-port macromodels for antenna feed network • Broadband macro-models • Generated directly from FEM model using Krylov subspace model order reduction methods • Compatible with both frequency-domain and time-domain EM solvers • Cast in terms of generalized impedance matrix for the electromagnetic multiport • Both waveguide mode-based ports and lumped-circuit ports supported • Impedance matrix elements in terms of rational function of frequency • Frequency interpolation for use with frequency-domain solvers • Computationally-efficient interfacing with time-domain solvers

  28. Multi-Layered Feed Network • Radiating Elements Customization ofsupporting substrate for improved array performance (patterned substrate, embedded EM band-gap structures, …) • Spacer - Custom patterning for enhanced array performance; layer for integration of active electronics • Slots • Feed network - Extended to multiple layers to support biasing network for any active electronics

  29. Dispersive Attributes of Feed Network • Conductor loss • Dielectric loss • Dispersive (macroscopic) properties of artificially-designed substrates • Network matrix abstraction of a portion of the feed network in terms of frequency-dependent multiport

  30. Macromodeling of FEM Models With Dispersion • Krylov subspace-based model order reduction • Surface impedance boundary conditions • Skin effect in lossy conductors • Generalized surface impedance boundary conditions • Frequency-dependent permittivity and permeability • Debye media, Lorentz media, Drude media,… • Incorporation of frequency-dependent electromagnetic multi-ports in FEM models • Frequency-dependent, multi-port macromodel abstractions of sub-domains

  31. 1 4 2 3 The Finite Element Model

  32. Skin-effect Surface Impedance

  33. Frequency-dependent Permittivity

  34. d Two-Port Macro-model of Metal Plates

  35. Definition of Reduced-Order Model

  36. FEM Model for Dispersive Media

  37. Rational function fit of H(s)

  38. Moments of ZG(s) (1)

  39. Moments of ZG(s) (2)

  40. Moments of ZG(s) (3)

  41. Construction of the Krylov Subspace

  42. Generation of Reduced-order Model

  43. 0.5 0.25 0.6 0.6 0.02 0.2 Example 1: Microstrip On Debye substrate • 6 cm-long line terminated at a 60-Ohm load • Substrate relative permittivity: εr = 2 +10(1+ jω2×10-10)-1 • All dimensions in mm

  44. Example 1: Microstrip on Debye substrate

  45. Example 2: Microstrip band-pass filter Substrate relative permittivity = 9.8

  46. Example 2: Microstrip band-pass filter

  47. Example 3: Coupling through lossy ground

  48. Example 3: Coupling through lossy ground

  49. Summary • Krylov subspace, equation preserving, model order reduction of FEM models that include frequency-dependent features • Hybrid distributed-lumped element models • Dispersive media • Cost of reduction dominated by the solution of E-field finite element equation at the expansion frequency • Generated reduced-order model provides for: • Fast frequency interpolation of system’s electromagnetic response • Rational function matrix macro-modeling of complex, passive multiports for computationally efficient interfacing with time-domain solvers

  50. W. C. ChewCenter for Computational Electromagneticsand Electromagnetics LaboratoryDepartment of Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign, Urbana, IL MURI REVIEW 2006 OSU MURI Review June 19, 2006