A. G. Michell and &. Lviii, The limits of economy of material in frame-structures, Philos. Mag. Ser, vol.6, issue.47, pp.589-597, 1904.

G. I. Rozvany, Grillages of maximum strength and maximum stiffness, Int. J. Mech. Sci, vol.14, issue.10, pp.651-666, 1972.

G. I. Rozvany and W. Prager, Optimal design of partially discretized grillages, J. Mech. Phys. Solids, vol.24, issue.2-3, pp.125-136, 1976.

M. P. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng, vol.71, issue.2, pp.197-224, 1988.

M. P. Bendsøe, Optimal shape design as a material distribution problem, Struct. Optim, vol.1, issue.4, pp.193-202, 1989.

G. I. Rozvany and M. Zhou, The COC algorithm, part I: Cross-section optimization or sizing, Comput. Methods Appl. Mech. Eng, vol.89, issue.1-3, pp.281-308, 1991.

D. N. Dyck and D. A. Lowther, Automated design of magnetic devices by optimizing material distribution, IEEE Trans. Magn, vol.32, issue.3, pp.1188-1193, 1996.

, Scopus -Document search, p.12, 2014.

M. P. Bendsøe and O. Sigmund, Topology Optimization -Theory, Methods and Applications, 2004.

Y. Wang, A Study on Microstructures of Homogenization for Topology Optimization, 2003.

F. Belblidia and S. Bulman, A hybrid topology optimization algorithm for static and vibrating shell structures, Int. J. Numer. Methods Eng, vol.54, issue.6, pp.835-852, 2002.

B. Hassani and E. Hinton, Homogenization and Structural Topology Optimization, 1999.

J. Folgado, H. Rodrigues, and J. M. Guedes, Layout Design of Plate Reinforcement with a Buckling Load Criterion, 1995.

F. Campelo, J. A. Ram´?rez, and H. Igarashi, A survey of topology optimization in electromagnetics: considerations and current trends, p.46, 2010.

J. Yoo and N. Kikuchi, Topology optimization in magnetic fields using the homogenization design method, Int. J. Numer. Methods Eng, vol.48, issue.10, pp.1463-1479, 2000.

G. I. Rozvany, M. Zhou, and T. Birker, Generalized shape optimization without homogenization, Struct. Optim, vol.4, issue.3-4, pp.250-252, 1992.

K. Song-yop-hahn, H. Choi, I. Lee, J. Park, J. Lee et al., Inverse problem application of topology optimization method with mutual energy concept and design sensitivity, IEEE Trans. Magn, vol.36, issue.4, pp.1144-1147, 2000.

S. Wang and J. Kang, Topology optimization of nonlinear magnetostatics, IEEE Trans. Magn, vol.38, issue.2, pp.1029-1032, 2002.

M. Stolpe and K. Svanberg, On the trajectories of penalization methods for topology optimization, Struct. Multidiscip. Optim, vol.21, issue.2, pp.128-139, 2001.

A. Rietz, Sufficiency of a finite exponent in SIMP (power law) methods, Struct. Multidiscip. Optim, vol.21, issue.2, pp.159-163, 2001.

J. S. Choi and J. Yoo, Structural optimization of ferromagnetic materials based on the magnetic reluctivity for magnetic field problems, Comput. Methods Appl. Mech. Eng, vol.197, pp.4193-4206, 2008.

S. Sanogo, Conception Optimale de Circuits Magnétiques dédiés à la Propulsion Spatiale Electrique par des Méthodes d'Optimisation Topologique, 2016.

O. Sigmund, Design of multiphysics actuators using topology optimization -Part II: Two-material structures, Comput. Methods Appl. Mech. Eng, vol.190, pp.6605-6627, 2001.

T. Labbe and B. Dehez, Convexity-Oriented Mapping Method for the Topology Optimization of Electromagnetic Devices Composed of Iron and Coils, IEEE Trans. Magn, vol.46, issue.5, pp.1177-1185, 2010.

S. Sanogo, F. Messine, C. Henaux, and R. Vilamot, Topology Optimization for Magnetic Circuits Dedicated to Electric Propulsion, IEEE Trans. Magn, vol.50, issue.12, pp.1-13, 2014.

H. Igarashi and K. Watanabe, Complex Adjoint Variable Method for Finite-Element Analysis of Eddy Current Problems, IEEE Trans. Magn, vol.46, issue.8, pp.2739-2742, 2010.

G. Allaire, A review of adjoint methods for sensitivity analysis, uncertainty quantification and optimization in numerical codes, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01242950

S. Jameson and A. Jameson, Adjoint Formulations for Topology, Shape and Discrete Optimization, 2007.

S. Wang, S. Park, and J. Kang, Multi-domain topology optimization of electromagnetic systems, COMPEL -Int. J. Comput. Math. Electr. Electron. Eng, vol.23, issue.4, pp.1036-1044, 2004.

J. S. Choi and J. Yoo, Simultaneous structural topology optimization of electromagnetic sources and ferromagnetic materials, Comput. Methods Appl. Mech. Eng, vol.198, pp.2111-2121, 2009.

Y. Okamoto, Y. Matsubayashi, S. Wakao, and S. Sato, Material-Density-Based Topology Optimization With Magnetic Nonlinearity by Means of Stabilized Sequential Linear Programming: SLPSTAB', IEEE Trans. Magn, vol.51, issue.3, pp.1-4, 2015.

J. Byun, I. Park, and S. Hahn, Topology optimization of electrostatic actuator using design sensitivity, IEEE Trans. Magn, vol.38, issue.2, pp.1053-1056, 2002.

S. Wang, J. Kang, and J. Noh, Topology Optimization of a Single-Phase Induction Motor for Rotary Compressor, IEEE Trans. Magn, vol.40, issue.3, pp.1591-1596, 2004.

D. Kim, J. K. Sykulski, and D. A. Lowther, The Implications of the Use of Composite Materials in Electromagnetic Device Topology and Shape Optimization, IEEE Trans. Magn, vol.45, issue.3, pp.1154-1157, 2009.

H. Shokrollahi and K. Janghorban, Soft magnetic composite materials (SMCs), J. Mater. Process. Technol, vol.189, issue.1-3, pp.1-12, 2007.

T. Labbe, B. Dehez, M. Markovic, and Y. Perriard, Torque-to-weight ratio maximization in PMSM using topology optimization, pp.1-5, 2010.

J. S. Choi, K. Izui, S. Nishiwaki, A. Kawamoto, and T. Nomura, Topology Optimization of the Stator for Minimizing Cogging Torque of IPM Motors, IEEE Trans. Magn, vol.47, issue.10, pp.3024-3027, 2011.

G. Kiziltas, Automated Topology Design for Electromagnetic Devices, vol.712, pp.2155-2160, 2004.

R. C. Carbonari, E. C. Silva, and S. Nishiwaki, Design of piezoelectric multiactuated microtools using topology optimization, Smart Mater. Struct, vol.14, issue.6, pp.1431-1447, 2005.

G. Anagnostou, E. M. Rønquist, and A. T. Patera, A computational procedure for part design, Comput. Methods Appl. Mech. Eng, vol.97, issue.1, pp.33-48, 1992.

Y. Okamoto, K. Akiyama, and N. Takahashi, 3-D topology optimization of singlepole-type head by using design sensitivity analysis', IEEE Trans. Magn, vol.42, issue.4, pp.1087-1090, 2006.

J. Seok-choi and J. Yoo, Structural Topology Optimization of Magnetic Actuators Using Genetic Algorithms and ON/OFF Sensitivity, IEEE Trans. Magn, vol.45, issue.5, pp.2276-2279, 2009.

C. Im, H. Jung, and Y. Kim, Hybrid genetic algorithm for electromagnetic topology optimization, IEEE Trans. Magn, vol.39, issue.5, pp.2163-2169, 2003.

K. Watanabe, F. Campelo, and H. Igarashi, Topology Optimization Based on Immune Algorithm and Multigrid Method, IEEE Trans. Magn, vol.43, issue.4, pp.1637-1640, 2007.

F. Campelo, K. Watanabe, and H. Igarashi, 3D topology optimization using an immune algorithm', COMPEL -Int, J. Comput. Math. Electr. Electron. Eng, vol.26, issue.3, pp.677-688, 2007.

F. Campelo, S. Ota, K. Watanabe, and H. Igarashi, Generating Parametric Design Models Using Information From Topology Optimization, IEEE Trans. Magn, vol.44, issue.6, pp.986-989, 2008.

W. Wang, Y. Lu, J. S. Fu, and Y. Xiong, Particle swarm optimization and finite-element based approach for microwave filter design, IEEE Trans. Magn, vol.41, issue.5, pp.1800-1803, 2005.

S. Sanogo and F. Messine, Design of space thrusters: a topology optimization problem solved via a Branch and Bound method, J. Glob. Optim, vol.64, issue.2, pp.273-288, 2016.

K. Akiyama, D. Miyagi, and N. Takahashi, Design of CF-SPT Head Having Large Recording Field and Small Stray Field Using 3-D ON/OFF Method, IEEE Trans. Magn, vol.42, issue.10, pp.2431-2433, 2006.

N. Takahashi, T. Yamada, and D. Miyagi, Optimization of rotor of actual IPM motor using ON/OFF method, pp.1-1, 2010.

M. Alexe and A. Sandu, Forward and adjoint sensitivity analysis with continuous explicit Runge-Kutta schemes, Appl. Math. Comput, vol.208, issue.2, pp.328-346, 2009.

Y. Okamoto, Y. Tominaga, and S. Sato, Topological Design for 3-D Optimization Using the Combination of Multistep Genetic Algorithm with Design Space Reduction and Nonconforming Mesh Connection, IEEE Trans. Magn, vol.48, issue.2, pp.515-518, 2012.

Y. Okamoto, Y. Tominaga, S. Wakao, and S. Sato, Topology Optimization of Rotor Core Combined With Identification of Current Phase Angle in IPM Motor Using Multistep Genetic Algorithm, IEEE Trans. Magn, vol.50, issue.2, pp.725-728, 2014.

S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys, vol.79, issue.1, pp.12-49, 1988.

G. Allaire, F. Jouve, and A. Toader, A level-set method for shape optimization, Comptes Rendus Math, vol.334, issue.12, pp.1125-1130, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01336301

S. Zhou, W. Li, and Q. Li, Level-set based topology optimization for electromagnetic dipole antenna design, J. Comput. Phys, vol.229, issue.19, pp.6915-6930, 2010.

Y. Okamoto, S. Wakao, and S. Sato, Topology Optimization Based on Regularized Level-Set Function for Solving 3-D Nonlinear Magnetic Field System With Spatial Symmetric Condition, IEEE Trans. Magn, vol.52, issue.3, pp.1-4, 2016.

Y. Hidaka, T. Sato, K. Watanabe, and H. Igarashi, Stochastic topology optimization based on level-set method', COMPEL -Int, J. Comput. Math. Electr. Electron. Eng, vol.33, issue.6, pp.1904-1919, 2014.

H. Lombaert, Level set method: Explanation, 2006.

S. Osher and R. P. Fedkiw, Level set methods and dynamic implicit surfaces, 2003.

J. A. Sethian and J. A. Sethian, Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, 1999.

S. Park and S. Min, Magnetic Actuator Design for Maximizing Force Using Level Set Based Topology Optimization, IEEE Trans. Magn, vol.45, issue.5, pp.2336-2339, 2009.

H. Shim, V. Ho, S. Wang, and D. A. Tortorelli, Level Set-Based Topology Optimization for Electromagnetic Systems, IEEE Trans. Magn, vol.45, issue.3, pp.1582-1585, 2009.

S. Park and S. Min, Optimal Topology Design of Magnetic Devices Using Level-Set Method, IEEE Trans. Magn, vol.45, issue.3, pp.1610-1613, 2009.

M. Masmoudi, J. Pommier, and B. Samet, The topological asymptotic expansion for the Maxwell equations and some applications, Inverse Probl, vol.21, issue.2, pp.547-564, 2005.

A. A. Gomes and A. Suleman, Application of spectral level set methodology in topology optimization, Struct. Multidiscip. Optim, vol.31, issue.6, pp.430-443, 2006.

F. Campelo, K. Watanabe, and H. Igarashi, A spectral topology optimization approach to the design of magnetic devices, presented at the 13th IEEE conference on Electromagnetic Field Computation (CEFC), 2008.

E. Arvas and L. Sevgi, A Tutorial on the Method of Moments, IEEE Antennas Propag. Mag, vol.54, issue.3, pp.260-275, 2012.

Y. Hidaka, T. Sato, and H. Igarashi, Topology Optimization Method Based on OnOff Method and Level Set Approach, IEEE Trans. Magn, vol.50, issue.2, pp.617-620, 2014.

T. A. Poulsen, A simple scheme to prevent checkerboard patterns and one-node connected hinges in topology optimization, Struct. Multidiscip. Optim, vol.24, issue.5, pp.396-399, 2002.

, Présentation générale | code_Carmel, p.11, 2017.

, Sophemis by L2EP, p.11, 2017.

R. Bargallo, Finite Elements For Electrical Engineering, 2006.

Y. and L. Menach, Contribution à la modélisation numérique des phénomènes électromagnétiques 3D en basse fréquence, 2012.

E. Delgado-da-carvalheira, Finite Elemnt Method for Maxwell's Equations, 2007.

A. Bacchus, Représentativité de la modélisation aux élémetns finis pour le diagnostic de machines synchrones de grande puissance, 2016.

A. Marrocco, Analyse Numérique des Problèmes d'Électrotechniques', Ann. Sci. Math. Quèbec, vol.1, issue.2, pp.217-296, 1977.

Z. Ren and A. Razek, Local force computation in deformable bodies using edge elements, IEEE Trans. Magn, vol.28, issue.2, pp.1212-1215, 1992.

J. Coulomb, A methodology for the determination of global electromechanical quantities from a finite element analysis and its application to the evaluation of magnetic forces, torques and stiffness, IEEE Trans. Magn, vol.19, issue.6, pp.2514-2519, 1983.

B. Boualem, Contribution à la modélisation des systèmes électrotechniques à l'aide des formulations en potentiels: application à la machine asynchrone, 1997.

M. Cavazzuti, Design of Experiments, Optimization Methods, pp.13-42, 2013.

&. Salome',

O. Sigmund and K. Maute, Topology optimization approaches: A comparative review, Struct. Multidiscip. Optim, 2013.

P. Venkataraman, Applied optimization with MATLAB programming, 2009.

T. Young and M. J. Mohlenkamp, Introduction to Numerical Methods and Matlab Programming for Engineers

T. Labbe, Topology optimization methods for the design of electromagnetic actuators, 2011.

T. Ishikawa, T. Watanabe, S. Mizuno, and N. Kurita, Effect of Cleaning Level on Topology Optimization of Permanent Magnet Synchronous Generator, the 19th International Conference on Electrical Machines and Systems (ICEMS), 2016.

P. Gangl and U. Langer, Topology optimization of electric machines based on topological sensitivity analysis, Comput. Vis. Sci, vol.15, issue.6, pp.345-354, 2012.

Y. Okamoto, M. Ohtake, and N. Takahashi, Magnetic shield design of perpendicular magnetic recording head by using topology optimization technique, IEEE Trans. Magn, vol.41, issue.5, pp.1788-1791, 2005.

A. Ammar, Modélisation et Optimisation d'un Générateur Synchrone à Double Excitation de Forte Puissance, 2013.

M. Ployard, Efficacité énergétique des machines de production d'électricité, 2017.