Loop-Star and Loop-Tree Decompositions: Analysis and Efficient Algorithms

A new analysis of the spectral properties of Loop-Star and Loop-Tree decompositions is presented in this work. The analysis will shed light on the behavior of these decompositions when used with regular operators such as the magnetic field and the Calderón preconditioned electric field integral operators. This work will explain the ill-conditioning problems reported in literature and will provide a family of efficient algorithms to solve the ill-conditioning and regularizing several Loop-Star/Tree decomposed equations of interest for applications. The theory will be corroborated by numerical results that will show the practical impact of the theoretical developments.

[1]  G. Vecchi,et al.  Hierarchical Bases for Nonhierarchic 3-D Triangular Meshes , 2008, IEEE Transactions on Antennas and Propagation.

[2]  Weng Cho Chew,et al.  Enhanced A-EFIE With Perturbation Method , 2010, IEEE Transactions on Antennas and Propagation.

[3]  Jin-Fa Lee,et al.  Removal of spurious DC modes in edge element solutions for modeling three-dimensional resonators , 2006, IEEE Transactions on Microwave Theory and Techniques.

[4]  Jin-Fa Lee,et al.  Preconditioned Electric Field Integral Equation Using Calderon Identities and Dual Loop/Star Basis Functions , 2009, IEEE Transactions on Antennas and Propagation.

[5]  Snorre H. Christiansen,et al.  A dual finite element complex on the barycentric refinement , 2005, Math. Comput..

[6]  O. Axelsson.  Iterative solution methods , 1995 .

[7]  Giuseppe Vecchi,et al.  Loop-star decomposition of basis functions in the discretization of the EFIE , 1999 .

[8]  A. Buffa,et al.  A multiplicative Calderón preconditioner for the electric field integral equation , 2008, 2008 IEEE Antennas and Propagation Society International Symposium.

[9]  Ru-Shan Chen,et al.  A Multiresolution Curvilinear Rao–Wilton–Glisson Basis Function for Fast Analysis of Electromagnetic Scattering , 2009, IEEE Transactions on Antennas and Propagation.

[10]  Jin-Fa Lee,et al.  Hierarchical vector finite elements for analyzing waveguiding structures , 2003 .

[11]  Krzysztof A. Michalski,et al.  On the existence of branch points in the eigenvalues of the electric field integral equation operator in the complex frequency plane , 1983 .

[12]  T. Eibert.  Iterative-solver convergence for loop-star and loop-tree decompositions in method-of-moments solutions of the electric-field integral equation , 2004 .

[13]  W. Chew,et al.  Integral equation solution of Maxwell's equations from zero frequency to microwave frequencies , 2000 .

[14]  F. Villone,et al.  A surface integral formulation of Maxwell equations for topologically complex conducting domains , 2005, IEEE Transactions on Antennas and Propagation.

[15]  Weng Cho Chew,et al.  A quantitative study on the low frequency breakdown of EFIE , 2008 .

[16]  A new fast and rapidly converging method for the solution of the electric field integral equation , 2009, 2009 International Conference on Electromagnetics in Advanced Applications.

[17]  Gene H. Golub,et al.  Matrix computations , 1983 .

[18]  Jin-Fa Lee,et al.  Loop star basis functions and a robust preconditioner for EFIE scattering problems , 2003 .

[19]  Israel Gohberg,et al.  Basic Classes of Linear Operators , 2004 .

[20]  Jian-Ming Jin,et al.  EFIE Analysis of Low-Frequency Problems With Loop-Star Decomposition and Calderón Multiplicative Preconditioner , 2010, IEEE Transactions on Antennas and Propagation.

[21]  Weng Cho Chew,et al.  An augmented electric field integral equation for high‐speed interconnect analysis , 2008 .

[22]  G. Vecchi,et al.  A Multiresolution System of Rao–Wilton–Glisson Functions , 2007, IEEE Transactions on Antennas and Propagation.

[23]  Giuseppe Vecchi,et al.  Solving the EFIE at Low Frequencies With a Conditioning That Grows Only Logarithmically With the Number of Unknowns , 2010, IEEE Transactions on Antennas and Propagation.

[24]  A. Quarteroni,et al.  Numerical Approximation of Partial Differential Equations , 2008 .

[25]  E. Michielssen,et al.  A Multiplicative Calderon Preconditioner for the Electric Field Integral Equation , 2008, IEEE Transactions on Antennas and Propagation.