IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 57, NO. 8, AUGUST 2009 2365 Time Domain Calderón Identities and Their Application to the Integral Equation Analysis of Scattering by PEC Objects Part II: Stability Francesco P. Andriulli, Member, IEEE, Kristof Cools, Femke Olyslager, Fellow, IEEE, and Eric Michielssen, Fellow, IEEE Abstract—Novel time domain integral equations for analyzing scattering from perfect electrically conducting objects are pre- sented. They are free from DC and resonant instabilities plaguing standard electric field integral equation. The new equations are obtained using operator manipulations originating from the Calderón identities. Theoretical motivations leading to the con- struction of the new equations are explored and numerical results confirming their theoretically predicted behavior are presented. Index Terms—Electric field integral equation (EFIE), stability, time domain analysis. I. INTRODUCTION T HIS IS THE second part of a paper devoted to the ap- plication of time domain Calderón identities to the con- struction of marching-on-in-time (MOT) time domain electric field integral equation (TDEFIE) solvers for analyzing transient scattering from perfect electrically conducting (PEC) surfaces. Part I, (henceforth referenced as [I]), elucidated the construc- tion of Calderón-preconditioned MOT-TDEFIE solvers that re- sult in well-conditioned and rapidly convergent MOT-TDEFIE systems irrespective of the surface mesh density. This paper demonstrates the usefulness of time domain Calderón identities in stabilizing MOT-TDEFIE solvers. The reader is assumed fa- miliar with the ideas and notation of [I], especially those of [I, Section II]. MOT-TDEFIE solvers often are plagued by instabilities, some of which are rooted in the continuous TDEFIE used while others stem from its (improper) discretization. The latter insta- bility type, which typically results in rapidly oscillating and exponentially growing solutions, has been studied extensively Manuscript received February 04, 2008; revised December 22, 2008. First published June 05, 2009; current version published August 05, 2009. This work was supported in part by AFOSR MURI Grant F014432-051936 aimed at mod- eling installed antennas and their feeds, by NSF Grant DMS 0713771, and in part by DARPA-DSO/AFOSR Grant F015123-052587 aimed at constructing compressed scattering matrices and direct solvers. F. P. Andriulli is with the Politecnico di Torino, Turin 10129, Italy. K. Cools is with the Electromagnetics Group, Department of Information Technology (INTEC), Ghent University, 9000 B-Gent, Belgium . F. Olyslager (deceased) was with the Electromagnetics Group, Department of Information Technology, Ghent University, 9000 B-Gent, Belgium. E. Michielssen is with the Radiation Laboratory, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2024464 and remedied by using filters [1]–[3], implicit time stepping [4], [5], and accurate integration schemes [6], [7]. These insta- bilities are no longer present in modern MOT-TDEFIE solvers. This study therefore concerns instabilities stemming from the spectral properties of the continuous TDEFIE operator being discretized. These instabilities are either “DC” or “resonant” in nature. DC instabilities are constant and linear-in-time solutions of MOT-TDEFIE systems that approximately reside in the null space of the TDEFIE and differentiated TDEFIE operator respectively; they arise because divergence-free static currents produce zero electric fields. In the past, these instabilities have been partially cured by using loop-tree decompositions [8] and by enforcing boundary conditions on normal mag- netic field components [9]. Unfortunately, neither technique completely annihilates the static null space of the TDEFIE operator, nor guarantees that MOT-TDEFIE solutions are free of DC remnants. This paper presents a modified TDEFIE that resolves static and linear-in-time currents; upon discretization the resulting MOT-TDEFIE system therefore is immune to DC instabilities. The new equation is obtained by leveraging the time domain Calderón identities in conjunction with the “dot-trick”, viz. a careful rearrangement of temporal derivative operators appearing in sequences of TDEFIE operators. Resonant instabilities are harmonically oscillating solutions of MOT-TDEFIE systems that approximately reside in the null space of the TDEFIE operator; they arise because PEC cavi- ties support discrete interior resonances. In the past, these in- stabilities have been subdued by using time domain combined field integral equations (TDCFIEs) [10]. Unfortunately, MOT- TDCFIE systems, while giving rise to stable solutions, often are slowly convergent due to the presence of the hypersingular TDEFIE operator in the TDCFIE. This paper presents a mod- ified TDCFIE that resolves interior resonances and is devoid of a hypersingular component; upon discretization the resulting MOT-TDCFIE system is immune to resonant instabilities and rapidly convergent, irrespective of the surface mesh density. The new equation is obtained by leveraging the time domain Calderón identities in conjunction with a simple space-time field localization procedure. This paper is organized as follows, Section II elucidates the origins, and provides a classification of MOT-TDEFIE instabil- ities. Sections III and IV introduce a new Calderón-enhanced TDEFIE and TDCFIE immune to DC and resonant instabili- ties, respectively. Section V demonstrates the stability of the 0018-926X/$25.00 © 2009 IEEE