This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY 1 A Leap-Frog Discontinuous Galerkin Time-Domain Method for HIRF Assessment Jesus Alvarez, Student Member, IEEE, Luis Diaz Angulo, Amelia Rubio Bretones, Senior Member, IEEE, Miguel Ruiz Cabello, and Salvador G. Garcia, Member, IEEE Abstract—In this paper, we demonstrate the computational af- fordability and accuracy of a leap-frog discontinuous Galerkin (LFDG) time-domain method for high intensity radiated fields as- sessment in electromagnetic compatibility for aerospace. The con- formal truncation of the computational domain is discussed and formulated in the LFDG context. Numerical validations are per- formed on challenging test cases, in comparison to measurements and to other numerical methods, demonstrating the accuracy, effi- ciency, and scalability of the algorithm. Index Terms—Discontinuous Galerkin, finite-element methods (FEMs), high intensity radiated fields (HIRFs), time-domain (TD) analysis. I. INTRODUCTION T HE adverse effects caused by high intensity radiated fields (HIRF) in any electronic device or in a very complex sys- tem, such as an aircraft, is a challenging topic from the stand- point of computational electromagnetics. The typical approach to tackle this electromagnetic compatibility (EMC) problem is based mainly on testing. The development of efficient algo- rithms, able to deal with electrically large structures, and accu- rate methods, capable of estimating transfer functions between incident EM fields and internal fields, or induced currents in bundles, has recently been attracting a great deal of interest in the aerospace industry [1], [2] Typical frequency-domain methods, such as the method of moments or the finite-element method (FEM), are able to cope with electrically large structures having electrically small de- tails. However, the analysis of HIRF requires the computation of wideband frequency responses. In this context, frequency- domain (FD) methods may become computationally ineffi- cient, since each frequency needs one complete simulation requiring the resolution of a linear system of equations. Time- domain (TD) methods are an attractive alternative for these purposes. Some well-known TD methods have been used tradi- Manuscript received August 1, 2012; revised February 26, 2013; accepted May 16, 2013. This work was supported by the European Community’s Sev- enth Framework Programme FP7/2007-2013, under Grant Agreement 205294 (HIRF SE project), by the Spanish National Projects TEC2010-20841-C04-04, CSD2008-00068, by the Junta de Andalucia Project P09-TIC-5327, and by the Granada Excellence Network of Innovation Laboratories project. J. Alvarez is with Cassidian, EADS–CASA, 28906 Getafe, Spain (e-mail: jesus@ieee.org). L. D. Angulo, A. R. Bretones, M. R. Cabello, and S. G. Garcia are with the Department of Electromagnetism, University of Granada, 18071 Granada, Spain (e-mail: lmdiazangulo@gmail.com; arubio@ugr.es; mcabello@ugr.es; salva@ugr.es). 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/TEMC.2013.2265045 tionally in EMC: finite-difference in time-domain (FDTD) [3], transmission-line-matrix method [4], and finite-integral tech- nique [5]. All of these are based on a cubic space partitioning, which impose a significant constraint on the geometrical dis- cretization of complex objects, having arbitrary curvatures and intricate details. To overcome this limitation, advances in fi- nite elements in the time-domain (FEMTD) methods have been made [6], to solve Maxwell’s equations in complex geometries by using an unstructured mesh based, for instance, on tetrahe- dral tessellation. However, classical FEMTD methods are still computationally unaffordable for electrically large problems. Among all FEMTD-based methods in the literature, discon- tinuous Galerkin time-domain (DGTD) approaches are expe- riencing a fast development. On one hand, DGTD have most of the advantages of FDTD; spatial explicit algorithm, memory and computational cost only growing linearly with the number of elements, simplicity, and easy parallelization [7]. Further- more, perfectly matched layer (PML) truncation techniques [8] can also be straightforwardly integrated into DGTD. Several formulations of PML exist; in this paper, we employ an auxil- iary differential equation (ADE) implementation of the uniaxial PML (UPML) technique [9]–[12], in a conformal formulation to achieve an optimum reduction of the computational domain. This conformal capability, with no counterpart in the FDTD context, has been successfully employed in finite-volume TD methods [13]–[15] (which is equivalent to a low-order DGTD), and is highly appropriate for DGTD [16]–[19]. Regarding the time-integration scheme, two ones are com- monly found in the DGTD literature: Runge–Kutta [7], [18] and leap frog (LF) [19], [20]. In this paper, we have chosen a second- order LF for providing a computationally efficient algorithm for which PML can be efficiently formulated. In this paper, an LF algorithm (hereafter LFDG), including the conformal UPML, is described in some detail. We prove that this method is able to simulate very complex electromagnetic prob- lems in an accurate manner, and validate it with a medium-sized 3-D object, compared to measurement, and with an electrically large problem, compared to the well-known FDTD method. We have chosen those two benchmark problems for being available under the HIRF-SE [21] 7PM EU project for the validation of the numerical codes involved in that project [2]. II. FORMULATIONS A. DG Formulation Let us assume Maxwell’s symmetric curl equations for linear isotropic homogeneous media. Now, let us divide the space in M nonoverlapping elements V m , each bounded by ∂V m 0018-9375/$31.00 © 2013 IEEE