IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 12, DECEMBER 2011 4643 An Improved Marching-on-in-Degree Method Using a New Temporal Basis Zicong Mei, Yu Zhang, TapanK. Sarkar, Fellow, IEEE, Baek Ho Jung, Alejandro García-Lampérez, Member, IEEE, and Magdalena Salazar-Palma, Senior Member, IEEE Abstract—The marching-on-in-degree (MOD) method has been presented earlier for solving time domain electric field integral equations in a stable fashion. This is accomplished by expanding the transient responses by a complete set of orthogonal entire domain associated Laguerre functions, which helps one to analyt- ically integrate out the time variable from the final computations in a Galerkin methodology. So, the final computations are car- ried out using only the spatial variables. However, the existing MOD method suffers from low computational efficiency over a marching-on-in-time (MOT) method. The two main causes of the computational inefficiency in the previous MOD method are now addressed using a new form of the temporal basis functions and through a different computational arrangement for the Green’s function. In this paper, it is shown that incorporating these two new concepts can speed up the computational process and make it comparable to a MOT algorithm. Sample numerical results are presented to illustrate the validity of these claims in solution of large problems using the MOD method. Index Terms—Laguerre polynomials, marching-on-in-degree (MOD) method, marching-on-in-time (MOT) method, method of moments (MoM), time domain electric field integral equation (TD-EFIE). I. INTRODUCTION T IME DOMAIN electric field integral equation (TD-EFIE) has been of interest for solving transient problems for a long time [1]–[12]. The conventional methodology to solve the TD-EFIE takes the form of a marching-on-in-time (MOT) algo- rithm [9]–[12]. In an alternate formulation, using the marching-on-in-degree (MOD) method, the transient response is approximated by a set of associated Laguerre functions, which are a set of causal or- Manuscript received August 12, 2010; revised May 25, 2011; accepted May 30, 2011. Date of publication August 18, 2011; date of current version De- cember 02, 2011. This work was partially supported by the Fundamental Re- search Funds for the Central Universities of China (JY10000902002) and NFSC (61072019). Z. Mei and T. K. Sarkar are with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244-1240 USA (e-mail: zmei@syr.edu; tksarkar@syr.edu). Y. Zhang is with the School of Electronic and Engineering, Xidian University, Xi’an 710071, China (e-mail: yuzhang@mail.xidian.edu.cn). B. H. Jung is with the Department of Information and Communication Engi- neering, Hoseo University, Chungnam 336-795, Korea (e-mail: bhjung@hoseo. edu). A. García-Lampérez and M. Salazar-Palma are with the Depart- mento de Teoria de la Senal y Communicaciones, Universidad Carlos III de Madrid, 28911 Madrid, Spain (e-mail: alamparez@tsc.uc3m.es; m.salazar-palma@ieee.org). 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.2011.2165482 thogonal functions defined in the interval [13], [14]. By choosing the associated Laguerre functions to represent the transient part of the response, the time variable can be ana- lytically eliminated from the final computational equations in a Galerkin formulation. Hence this methodology does not re- quire a Courant-Friedrich-Levy sampling criteria relating the time samples to the spatial samples. As the time variable can be eliminated analytically from the final computational equations, stability of the computational system is no longer governed by the nature of the spatial sampling. In the previous work, the associated Laguerre functions are chosen as the temporal basis functions. But it has a drawback, namely, the derivative of the associated Laguerre function forms a summation over its lower degrees. In this way, the final equa- tions that are conventionally used will contain a large number of summations, making the computational procedure inefficient. Therefore, it is proposed to use a new basis function set, which is a combination of associated Laguerre functions. This new basis function set will retain all the advantage of the associated La- guerre functions while its derivative will now be another com- bination of polynomials instead of a summation, which will re- duce the computation time by a factor of about 10 to 20. Also in the previous publications of the MOD method, the lower de- gree temporal basis functions with different retarded time are no longer orthogonal to each other. When calculating the coeffi- cients for each degree, the integration over the previous polyno- mial orders is needed to eliminate the components of the lower degree basis functions [15], [16]. This takes long CPU time and results in a computationally inefficient procedure. Therefore, a new computational form of the Green’s function is used to treat the retarded time component related to the basis functions of a lower degree to reduce the operation count, resulting in a fur- ther increase in the computational efficiency over the original formulation. In this paper, first, the new transient basis function is de- scribed and a modified computational form of the Green’s func- tion in the MOD method is introduced. Numerical results are then presented to show the efficiency and accuracy of the im- proved formulas. With the improvement in efficiency, ordinary PCs are able to calculate transient responses from full-sized air- crafts and their results are shown in this paper. II. TIME-DOMAIN ELECTRIC FIELD INTEGRAL EQUATIONS By enforcing, the total tangential electric field to be zero on the surface for all time, the TD-EFIE is obtained as (1) 0018-926X/$26.00 © 2011 IEEE