Crashworthiness optimization of an automotive front bumper

In automotive industry, structural optimization for crashworthiness criteria is of special importance in the early design stage. Crashing performance of structures under dynamic impact can be investigated using finite element codes. By coupling FE simulation tools with nonlinear mathematical programming procedure and statistical techniques, it is possible to optimize the design with reduced number of analytical evaluations. Optimization methods using statistical techniques are widely used in engineering applications to utilize estimated models which are often referred to metamodels. Meta-modeling optimization is performed through construction of objective functions, design of experiment (DOE) and modeling. Various types of meta-modeling techniques were used for crashworthiness optimization. In this paper the comparative study of Kriging and Radial Basis Function Network (RBFN) was performed in order to improve the crashworthiness effects of a front bumper subsystem subjected to impact. The objective function is the maximization of the specif energy absorption (SEA) and the design variables are geometrical parameters subjected to some design constraints. The optimized solution was achieved interfacing LS-DYNA codes with LS-OPT and using a domain reduction strategy

Mathematical and numerical approach for a crashworthy problem

Vehicle crashworthiness has been improving in recent years with attention mainly directed towards reducing the impact of the crash on the passengers. An optimal way to achieve this target is by exclusive use of specific impact attenuators, such as strategically placed tubular elements. Many of the mechanical devices are designed to absorb impact energy under axial crushing, bending and/or combined loading. An important requirement is that these structural members must be able to dissipate large amount of energy by controlled collapse in the event of a collision. Generally, the total energy dissipated depends on the governing deformation phenomena of all or part of the structural components of simple geometry, such as thin-walled tubes, cones, frames and sections. The energy absorbing capacity differs from one component to the next in a manner which depends on the mode of deformation involved and the material used. During the last decades the attention given to crash energy management has been centred on composite structures. The use of fibre-reinforced plastic composite materials in automotive structures may result in many potential economic and functional benefits due to their improved properties respect to metal ones, ranging from weight reduction to increased strength and durability features. Although significant experimental work on the collapse of fibre-reinforced composite shells has been carried out, studies on the theoretical modelling of the crushing process are quite limited since the complex and brittle fracture mechanisms of composite materials. Most of the studies have been directed towards the axial crush analysis, because it represents more or less the most efficient design. In the present paper, a mathematical approach on the failure mechanisms, pertaining to the stable mode of collapse (Mode I) of thin-walled composite circular tubes subjected to axial loading, was investigated. The analysis was conducted from an energetic point of view; it is therefore necessary to identify the main energy contributions and then equate the total internal energy to the work done by the external load. The average crush load can be obtained minimizing the force contribution, function of several variables, on a domain using a numerical approach. Comparison between theory and experiments concerning crushing loads and total displacements was analysed, showing how the proposed analytical model is efficient for predicting the energy absorption capability of axially collapsing composite shells

Impact behavior of a fully thermoplastic composite

Composites are materials of choice for lightweight structures due to their excellent strength/weight/and stiffness/weight/properties. For several years, the application of composite materials with continuous fiber was limited to those with thermosetting matrix. Recently, interest in composites with thermoplastic matrix is growing thanks to their considerable advantages also in terms of recyclability. The thermoplastic composites appear to be the right alternative to the materials currently used, replacing not only the non-structural parts, but also the structural components located in areas potentially subject to impacts. This paper presents the results of an experimental campaign made on a fully thermoplastic composite, where both the reinforcement and the matrix are made in polypropylene. The target is to analyze its behavior under different impact loading conditions using a drop weight testing machine. The influence of the impact mass and of the velocity on the energy absorption capability of the material have been analyzed and discussed. During the tests, the material showed a ductile behavior and developed extended plasticity without a crack tip. The main observed damage mechanisms were the yarn sliding

Thermosetting and thermoplastic impact attenuator under axial loading

High-performance composites are generally fabricated with continuous fibre and fabric reinforcements embedded in a thermosetting resin. Using thermoplastic matrices, there are substantial reductions in forming time and labour. More recently, the availability of all-polypropylene composites, achieved using the same thermoplastic polymer for both the fibre and the matrix phase, is also increasing because of their recyclability. In this perspective, the work aims to study the mechanical behaviour of a new fully thermoplastic composite, first showing the results of an experimental campaign for the mechanical characterization of the material properties, then examining the behaviour of structures made of such material under axial loading to evaluate their energy absorption capability. The second part of this work is divided into two steps. In the first step, crush tests on simple tubes were performed. In the second step, the behaviour of a specific impact attenuator for a Formula SAE racing car was analysed. Using the same geometry, different material solutions were tested. Beside traditional thermosetting composite structure, a new fully thermoplastic composite and a hybrid solution were used taking into account various feasibility problems in the manufacturing phases. Even if the thermoplastic attenuator does not exhibit the same absorption capability of the thermosetting solutions, an interesting crushing mechanism was noticed: no more brittle failure with formation of debris, but a ductile progression with a load distribution very close to an ideal absorber

Approximating Weighted Duo-Preservation in Comparative Genomics

Motivated by comparative genomics, Chen et al. [9] introduced the Maximum Duo-preservation String Mapping (MDSM) problem in which we are given two strings $s_1$ and $s_2$ from the same alphabet and the goal is to find a mapping $\pi$ between them so as to maximize the number of duos preserved. A duo is any two consecutive characters in a string and it is preserved in the mapping if its two consecutive characters in $s_1$ are mapped to same two consecutive characters in $s_2$. The MDSM problem is known to be NP-hard and there are approximation algorithms for this problem [3, 5, 13], but all of them consider only the "unweighted" version of the problem in the sense that a duo from $s_1$ is preserved by mapping to any same duo in $s_2$ regardless of their positions in the respective strings. However, it is well-desired in comparative genomics to find mappings that consider preserving duos that are "closer" to each other under some distance measure [19]. In this paper, we introduce a generalized version of the problem, called the Maximum-Weight Duo-preservation String Mapping (MWDSM) problem that captures both duos-preservation and duos-distance measures in the sense that mapping a duo from $s_1$ to each preserved duo in $s_2$ has a weight, indicating the "closeness" of the two duos. The objective of the MWDSM problem is to find a mapping so as to maximize the total weight of preserved duos. In this paper, we give a polynomial-time 6-approximation algorithm for this problem.Comment: Appeared in proceedings of the 23rd International Computing and Combinatorics Conference (COCOON 2017

Nonstationary statistical theory for multipactor

[EN] This work presents a new and general approach to the real dynamics of the multipactor process: the nonstationary statistical multipactor theory. The nonstationary theory removes the stationarity assumption of the classical theory and, as a consequence, it is able to adequately model electron exponential growth as well as absorption processes, above and below the multipactor breakdown level. In addition, it considers both double-surface and single-surface interactions constituting a full framework for nonresonant polyphase multipactor analysis. This work formulates the new theory and validates it with numerical and experimental results with excellent agreement. (C) 2010 American Institute of Physics.The authors would like to thank ESA/ESTEC for having funded this research activity through the Contract "Study of High Order Modes and Fringing Fields in Multipactor Effect" (Contract No. 1-5918/08/NL/GLC) and to the Ministerio de Ciencia e Innovacion (Spain) for the support through the "Programa Torres Quevedo" Contract No. PTQ05-02-02759.Anza, S.; Vicente, C.; Gil, J.; Boria Esbert, VE.; Gimeno, B.; Raboso, D. (2010). Nonstationary statistical theory for multipactor. Physics of Plasmas. 17(6):1-11. https://doi.org/10.1063/1.3443128S111176Farnsworth, P. T. (1934). Television by electron image scanning. Journal of the Franklin Institute, 218(4), 411-444. doi:10.1016/s0016-0032(34)90415-4Starting potentials of high-frequency gas discharges at low pressure. (1948). Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 192(1030), 446-463. doi:10.1098/rspa.1948.0018Vaughan, J. R. M. (1988). Multipactor. IEEE Transactions on Electron Devices, 35(7), 1172-1180. doi:10.1109/16.3387Hatch, A. J., & Williams, H. B. (1954). The Secondary Electron Resonance Mechanism of Low‐Pressure High‐Frequency Gas Breakdown. Journal of Applied Physics, 25(4), 417-423. doi:10.1063/1.1721656Hatch, A. J., & Williams, H. B. (1958). Multipacting Modes of High-Frequency Gaseous Breakdown. Physical Review, 112(3), 681-685. doi:10.1103/physrev.112.681H. M. Wachowski, El Segundo Technical Operations Aerospace Corporation, Technical Report No. TDR-269(9990)-5, El Segundo, California, 1964.Furman, M., & Pivi, M. (2002). Probabilistic model for the simulation of secondary electron emission. Physical Review Special Topics - Accelerators and Beams, 5(12). doi:10.1103/physrevstab.5.124404A. Woode and J. Petit, ESTEC Technical Report No. 1556, Noordwijk, 1989.Riyopoulos, S., Chernin, D., & Dialetis, D. (1995). Theory of electron multipactor in crossed fields. Physics of Plasmas, 2(8), 3194-3213. doi:10.1063/1.871151Kishek, R. A., Lau, Y. Y., Ang, L. K., Valfells, A., & Gilgenbach, R. M. (1998). Multipactor discharge on metals and dielectrics: Historical review and recent theories. Physics of Plasmas, 5(5), 2120-2126. doi:10.1063/1.872883Gilardini, A. L. (1992). New breakdown modes of the multipacting discharge. Journal of Applied Physics, 71(9), 4629-4631. doi:10.1063/1.350767Kryazhev, A., Buyanova, M., Semenov, V., Anderson, D., Lisak, M., Puech, J., … Sombrin, J. (2002). Hybrid resonant modes of two-sided multipactor and transition to the polyphase regime. Physics of Plasmas, 9(11), 4736-4743. doi:10.1063/1.1514969Semenov, V. E., Rakova, E., Udiljak, R., Anderson, D., Lisak, M., & Puech, J. (2008). Conformal mapping analysis of multipactor breakdown in waveguide irises. Physics of Plasmas, 15(3), 033501. doi:10.1063/1.2884712Semenov, V. E., Rakova, E. I., Anderson, D., Lisak, M., & Puech, J. (2007). Multipactor in rectangular waveguides. Physics of Plasmas, 14(3), 033501. doi:10.1063/1.2480678Woo, R. (1968). Multipacting Discharges between Coaxial Electrodes. Journal of Applied Physics, 39(3), 1528-1533. doi:10.1063/1.1656390Vdovicheva, N. K., Sazontov, A. G., & Semenov, V. E. (2004). Statistical Theory of Two-Sided Multipactor. Radiophysics and Quantum Electronics, 47(8), 580-596. doi:10.1023/b:raqe.0000049556.18329.e9Vdovicheva, N. K., Sazontov, A. G., Sazontov, V. A., & Semenov, V. E. (2006). Influence of the angular anisotropy of secondary emission on the characteristics of a two-sided multipactor. Radiophysics and Quantum Electronics, 49(5), 368-376. doi:10.1007/s11141-006-0069-2Sazontov, A. G., Sazontov, V. A., & Vdovicheva, N. K. (2008). Multipactor Breakdown Prediction in a Rectangular Waveguide: Statistical Theory and Simulation Results. Contributions to Plasma Physics, 48(4), 331-346. doi:10.1002/ctpp.200810057Sazontov, A., Buyanova, M., Semenov, V., Rakova, E., Vdovicheva, N., Anderson, D., … Lapierre, L. (2005). Effect of emission velocity spread of secondary electrons in two-sided multipactor. Physics of Plasmas, 12(5), 053102. doi:10.1063/1.1881532Kossyi, I. A., Lukyanchikov, G. S., Semenov, V. E., Rakova, E. I., Anderson, D., Lisak, M., & Puech, J. (2008). Polyphase (non-resonant) multipactor in rectangular waveguides. Journal of Physics D: Applied Physics, 41(6), 065203. doi:10.1088/0022-3727/41/6/065203Vaughan, J. R. M. (1989). A new formula for secondary emission yield. IEEE Transactions on Electron Devices, 36(9), 1963-1967. doi:10.1109/16.34278C. Vicente, M. Mattes, D. Wolk, H. L. Hartnagel, J. R. Mosig, and D. Raboso, Proceedings of the 5th International Workshop on Multipactor, RF and DC Corona and Passive Intermodulation in Space RF Hardware (ESTEC, Noordwijk, The Netherlands, 2005), pp. 11–17.Gilardini, A. L. (1995). Multipacting discharges: Constant‐ktheory and simulation results. Journal of Applied Physics, 78(2), 783-795. doi:10.1063/1.360336A. Kryazhev, M.S. thesis, Chalmers University of Technology, Göteborg, Sweden, 2002.Anza, S., Vicente, C., Gimeno, B., Boria, V. E., & Armendáriz, J. (2007). Long-term multipactor discharge in multicarrier systems. Physics of Plasmas, 14(8), 082112. doi:10.1063/1.2768019P. Zuccarello, A. González, G. Piñero, and M. de Diego, Proceedings of the 4th International Workshop on Multipactor, RF and DC Corona and Passive Intermodulation in Space RF Hardware (ESTEC, Noordwijk, The Netherlands, 2003), pp. 469–473.Polyanin, A. (1998). Handbook of Integral Equations. doi:10.1201/9781420050066S. Anza, C. Vicente, D. Raboso, J. Gil, B. Gimeno, and V. E. Boria, IEEE International Microwave Symposium (IEEE, Atlanta, 2008), pp. 1095–1098.C. Vicente, M. Mattes, D. Wolk, H. L. Hartnagel, J. R. Mosig, and D. Raboso, Microwave Symposium Digest, IEEE MTT-S International (IEEE, Long Beach, California, 2005), Vol. 2, pp. 1055–1058

Apollonius unilateral transducer constant power gain circles on 3D Smith charts

[EN] Unilateral transducer constant power gain circles play an essential role in the design of radio-frequency amplifiers and active modulators, as they help to determine optimal impedance matching conditions to meet gain and stability specifications. It is shown that these gain circles are a subfamily of Apollonius circles. For better visualisation, unilateral transducer constant gain power circles have been plotted for the first time on the three-dimensional (3D) Smith chart. To this end, a natural relationship from an inversive geometry was required, in order to relate the gain circles with cutting planes for the 3D Smith chart.This work has been partly funded by the FP7 PCIG11-2012-322162 Marie Curie CIG, POSDRU/159/1.5/S/134398 and by DGCYT grant MTM2012-33073.Muller, AA.; Sanabria Codesal, E.; Moldoveanu, A.; Asavei, V.; Soto Pacheco, P.; Boria Esbert, VE.; Lucyszyn, S. (2014). Apollonius unilateral transducer constant power gain circles on 3D Smith charts. Electronics Letters. 50(21):1531-1533. https://doi.org/10.1049/el.2014.2695S153115335021Ciccognani, W., Longhi, P. E., Colangeli, S., & Limiti, E. (2013). Constant Mismatch Circles and Application to Low-Noise Microwave Amplifier Design. IEEE Transactions on Microwave Theory and Techniques, 61(12), 4154-4167. doi:10.1109/tmtt.2013.2288696Mukherjee, B., Patel, P., & Mukherjee, J. (2014). Hemispherical Dielectric Resonator Antenna Based on Apollonian Gasket of Circles—A Fractal Approach. IEEE Transactions on Antennas and Propagation, 62(1), 40-47. doi:10.1109/tap.2013.2287011Muller, A. A., Soto, P., Dascalu, D., Neculoiu, D., & Boria, V. E. (2011). A 3-D Smith Chart Based on the Riemann Sphere for Active and Passive Microwave Circuits. IEEE Microwave and Wireless Components Letters, 21(6), 286-288. doi:10.1109/lmwc.2011.2132697Lucyszyn, S., & Robertson, I. D. (1994). Monolithic narrow-band filter using ultrahigh-Q tunable active inductors. IEEE Transactions on Microwave Theory and Techniques, 42(12), 2617-2622. doi:10.1109/22.339805Caporal Del Barrio, S., Pedersen, G. F., Bahramzy, P., Jagielski, O., & Svendsen, S. (2014). Thermal loss in high-Q antennas. Electronics Letters, 50(13), 917-919. doi:10.1049/el.2014.122

Multipactor radiation analysis within a waveguide region based on a frequency-domain representation of the dynamics of charged particles

[EN] A technique for the accurate computation of the electromagnetic fields radiated by a charged particle moving within a parallel-plate waveguide is presented. Based on a transformation of the time-varying current density of the particle into a time-harmonic current density, this technique allows the evaluation of the radiated electromagnetic fields both in the frequency and time domains, as well as in the near- and far-field regions. For this purpose, several accelerated versions of the parallel-plate Green's function in the frequency domain have been considered. The theory has been successfully applied to the multipactor discharge occurring within a two metal-plates region. The proposed formulation has been tested with a particle-in-cell code based on the finite-difference time-domain method, obtaining good agreement.The authors would like to thank ESA/ESTEC for having funded this research activity through the Contract "RF Breakdown in Multicarrier Systems" (Contract No. 19918/06/NL/GLC).Gimeno, B.; Sorolla, E.; Anza, S.; Vicente, C.; Gil, J.; Pérez, AM.; Boria Esbert, VE.... (2009). Multipactor radiation analysis within a waveguide region based on a frequency-domain representation of the dynamics of charged particles. Physical review. E, Statistical, nonlinear, and soft matter physics. 79(4):1-9. https://doi.org/10.1103/PhysRevE.79.046604S19794Figueroa, H., Gai, W., Konecny, R., Norem, J., Ruggiero, A., Schoessow, P., & Simpson, J. (1988). Direct Measurement of Beam-Induced Fields in Accelerating Structures. Physical Review Letters, 60(21), 2144-2147. doi:10.1103/physrevlett.60.2144Ng, K.-Y. (1990). Wake fields in a dielectric-lined waveguide. Physical Review D, 42(5), 1819-1828. doi:10.1103/physrevd.42.1819Rosing, M., & Gai, W. (1990). Longitudinal- and transverse-wake-field effects in dielectric structures. Physical Review D, 42(5), 1829-1834. doi:10.1103/physrevd.42.1829Gai, W., Kanareykin, A. D., Kustov, A. L., & Simpson, J. (1997). Numerical simulations of intense charged-particle beam propagation in a dielectric wake-field accelerator. Physical Review E, 55(3), 3481-3488. doi:10.1103/physreve.55.3481Burov, A., & Danilov, V. (1999). Suppression of Transverse Bunch Instabilities by Asymmetries in the Chamber Geometry. Physical Review Letters, 82(11), 2286-2289. doi:10.1103/physrevlett.82.2286Xiao, L., Gai, W., & Sun, X. (2001). Field analysis of a dielectric-loaded rectangular waveguide accelerating structure. Physical Review E, 65(1). doi:10.1103/physreve.65.016505Jing, C., Liu, W., Xiao, L., Gai, W., Schoessow, P., & Wong, T. (2003). Dipole-mode wakefields in dielectric-loaded rectangular waveguide accelerating structures. Physical Review E, 68(1). doi:10.1103/physreve.68.016502Stupakov, G., Bane, K. L. F., & Zagorodnov, I. (2007). Optical approximation in the theory of geometric impedance. Physical Review Special Topics - Accelerators and Beams, 10(5). doi:10.1103/physrevstab.10.054401Hatch, A. J., & Williams, H. B. (1954). The Secondary Electron Resonance Mechanism of Low‐Pressure High‐Frequency Gas Breakdown. Journal of Applied Physics, 25(4), 417-423. doi:10.1063/1.1721656Hatch, A. J., & Williams, H. B. (1958). Multipacting Modes of High-Frequency Gaseous Breakdown. Physical Review, 112(3), 681-685. doi:10.1103/physrev.112.681Vaughan, J. R. M. (1988). Multipactor. IEEE Transactions on Electron Devices, 35(7), 1172-1180. doi:10.1109/16.3387Gilardini, A. L. (1995). Multipacting discharges: Constant‐ktheory and simulation results. Journal of Applied Physics, 78(2), 783-795. doi:10.1063/1.360336Riyopoulos, S. (1997). Multipactor saturation due to space-charge-induced debunching. Physics of Plasmas, 4(5), 1448-1462. doi:10.1063/1.872319Kryazhev, A., Buyanova, M., Semenov, V., Anderson, D., Lisak, M., Puech, J., … Sombrin, J. (2002). Hybrid resonant modes of two-sided multipactor and transition to the polyphase regime. Physics of Plasmas, 9(11), 4736-4743. doi:10.1063/1.1514969Udiljak, R., Anderson, D., Ingvarson, P., Jordan, U., Jostell, U., Lapierre, L., … Sombrin, J. (2003). New method for detection of multipaction. IEEE Transactions on Plasma Science, 31(3), 396-404. doi:10.1109/tps.2003.811646De Lara, J., Perez, F., Alfonseca, M., Galan, L., Montero, I., Roman, E., & Garcia-Baquero, D. R. (2006). Multipactor prediction for on-board spacecraft RF equipment with the MEST software tool. IEEE Transactions on Plasma Science, 34(2), 476-484. doi:10.1109/tps.2006.872450Torregrosa, G., Coves, A., Vicente, C. P., Perez, A. M., Gimeno, B., & Boria, V. E. (2006). Time evolution of an electron discharge in a parallel-plate dielectric-loaded waveguide. IEEE Electron Device Letters, 27(7), 619-621. doi:10.1109/led.2006.877284Udiljak, R., Anderson, D., Lisak, M., Semenov, V. E., & Puech, J. (2007). Multipactor in a coaxial transmission line. I. Analytical study. Physics of Plasmas, 14(3), 033508. doi:10.1063/1.2710464Semenov, V. E., Zharova, N., Udiljak, R., Anderson, D., Lisak, M., & Puech, J. (2007). Multipactor in a coaxial transmission line. II. Particle-in-cell simulations. Physics of Plasmas, 14(3), 033509. doi:10.1063/1.2710466Anza, S., Vicente, C., Gimeno, B., Boria, V. E., & Armendáriz, J. (2007). Long-term multipactor discharge in multicarrier systems. Physics of Plasmas, 14(8), 082112. doi:10.1063/1.2768019Udiljak, R., Anderson, D., Lisak, M., Puech, J., & Semenov, V. E. (2007). Multipactor in a Waveguide Iris. IEEE Transactions on Plasma Science, 35(2), 388-395. doi:10.1109/tps.2007.892737Burton, R. J., de Jong, M. S., & Funk, L. W. (1991). Vacuum and multipactor performance of the hadron electron ring accelerator 52 MHz cavities. Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films, 9(3), 2081-2084. doi:10.1116/1.577417Yamaguchi, S., Saito, Y., Anami, S., & Michizono, S. (1992). Trajectory simulation of multipactoring electrons in an S-band pillbox RF window. IEEE Transactions on Nuclear Science, 39(2), 278-282. doi:10.1109/23.277497Kishek, R., & Lau, Y. Y. (1995). Interaction of Multipactor Discharge and rf Circuit. Physical Review Letters, 75(6), 1218-1221. doi:10.1103/physrevlett.75.1218Lay-Kee Ang, Lau, Y. Y., Kishek, R. A., & Gilgenbach, R. M. (1998). Power deposited on a dielectric by multipactor. IEEE Transactions on Plasma Science, 26(3), 290-295. doi:10.1109/27.700756Kishek, R. A., Lau, Y. Y., Ang, L. K., Valfells, A., & Gilgenbach, R. M. (1998). Multipactor discharge on metals and dielectrics: Historical review and recent theories. Physics of Plasmas, 5(5), 2120-2126. doi:10.1063/1.872883Neuber, A., Hemmert, D., Krompholz, H., Hatfield, L., & Kristiansen, M. (1999). Initiation of high power microwave dielectric interface breakdown. Journal of Applied Physics, 86(3), 1724-1728. doi:10.1063/1.370953Chojnacki, E. (2000). Simulations of a multipactor-inhibited waveguide geometry. Physical Review Special Topics - Accelerators and Beams, 3(3). doi:10.1103/physrevstab.3.032001Cimino, R., Collins, I. R., Furman, M. A., Pivi, M., Ruggiero, F., Rumolo, G., & Zimmermann, F. (2004). Can Low-Energy Electrons Affect High-Energy Physics Accelerators? Physical Review Letters, 93(1). doi:10.1103/physrevlett.93.014801Abe, T., Kageyama, T., Akai, K., Ebihara, K., Sakai, H., & Takeuchi, Y. (2006). Multipactoring zone map of an rf input coupler and its application to high beam current storage rings. Physical Review Special Topics - Accelerators and Beams, 9(6). doi:10.1103/physrevstab.9.062002Sorolla, E., Anza, S., Gimeno, B., Perez, A. M. P., Vicente, C., Gil, J., … Boria, V. E. (2008). An Analytical Model to Evaluate the Radiated Power Spectrum of a Multipactor Discharge in a Parallel-Plate Region. IEEE Transactions on Electron Devices, 55(8), 2252-2258. doi:10.1109/ted.2008.926271Harrington, R. F. (2001). Time-Harmonic Electromagnetic Fields. doi:10.1109/9780470546710Hanson, G. W., & Yakovlev, A. B. (2002). Operator Theory for Electromagnetics. doi:10.1007/978-1-4757-3679-3Ewald, P. P. (1921). Die Berechnung optischer und elektrostatischer Gitterpotentiale. Annalen der Physik, 369(3), 253-287. doi:10.1002/andp.19213690304Myun-Joo Park, & Sangwook Nam. (1998). Rapid summation of the Green’s function for the rectangular waveguide. IEEE Transactions on Microwave Theory and Techniques, 46(12), 2164-2166. doi:10.1109/22.739301Capolino, F., Wilton, D. R., & Johnson, W. A. (2005). Efficient computation of the 2-D Green’s function for 1-D periodic structures using the Ewald method. IEEE Transactions on Antennas and Propagation, 53(9), 2977-2984. doi:10.1109/tap.2005.854556Kustepeli, A., & Martin, A. Q. (2000). On the splitting parameter in the Ewald method. IEEE Microwave and Guided Wave Letters, 10(5), 168-170. doi:10.1109/75.85036