F. Dau, O. Polit, and M. Touratier, A efficient c 1 finite element with continuity requirements for multilayered/sandwich shell structures, Comput Struct, vol.82, pp.1889-99, 2004.

O. C. Zienkiewicz and R. L. Taylor, The finite element method, vol.2, 2000.

P. Vidal, D. 'ottavio, M. Thaier, M. B. Polit, and O. , An efficient finite shell element for the static response of piezoelectric laminates, J Intell Mater Syst Struct, vol.22, issue.7, pp.671-90, 2011.
URL : https://hal.archives-ouvertes.fr/hal-01366926

J. N. Reddy, Mechanics of laminated composite plates and shells-theory and analysis, 2004.

A. W. Leissa, Vibration of shells. NASA SP-288, 1973.

K. P. Rao, A rectangular laminated anisotropic shallow thin shell finite element, Comput Methods Appl Mech Eng, vol.15, pp.13-33, 1978.

C. Jeyachandrabose and J. Kirkhope, Explicit formulation of two anisotropic, triangular, thin, shallow shell elements, Comput Struct, vol.25, pp.415-451, 1987.

M. S. Qatu and A. W. Leissa, Bending analysis of laminated plates and shells by different methods, Comput Struct, vol.52, pp.529-568, 1994.

J. N. Reddy, Bending of laminated anisotropic shells by a shear deformable finite element, Fibre Sci Technol, vol.17, pp.9-24, 1982.

D. Chakravorty, J. N. Bandyopadhyay, and P. K. Sinha, Finite element free vibration analysis of doubly curved laminated composite shells, J Sound Vibr, vol.191, pp.491-504, 1996.

S. J. Hossain, P. K. Sinha, and A. H. Sheikh, A finite element formulation for the analysis of laminated composite shells, Comput Struct, vol.82, pp.1623-1661, 2004.

E. Asadi, W. Wang, and M. S. Qatu, Static and vibration analyses of thick deep laminated cylindrical shells using 3d and various shear deformation theories, Compos Struct, vol.94, issue.2, pp.494-500, 2012.

M. Balah and A. Hn, Finite element formulation of a third order laminated finite rotation shell element, Comput Struct, vol.80, pp.1975-90, 2002.

G. Sgambitterra, A. Adumitroaie, E. J. Barbero, and A. Tessler, A robust three-node shell element for laminated composites with matrix damage, Compos B, vol.42, pp.41-50, 2011.

T. Kant and M. P. Menon, Estimation of interlaminar stresses in fibre reinforced composite cylindrical shells, Comput Struct, vol.38, pp.131-178, 1991.

J. N. Reddy and C. F. Liu, A higher-order shear deformation theory of laminated elastic shells, Int J Eng Sci, vol.23, pp.319-349, 1985.

E. Carrera, Theories and finite elements for multilayered, anisotropic, composite plates and shells, Arch Comput Meth Eng, vol.9, pp.87-140, 2002.

M. Cinefra and E. Carrera, Shell finite elements with different through-the-thickness kinematics for the linear analysis of cylindrical multilayered structures, Int J Non-Newt Fluid Mech, vol.93, issue.2, pp.160-82, 2013.

B. Brank, On composite shell models with a piecewise linear warping function, Compos Struct, vol.59, pp.163-71, 2003.

H. S. Jing and K. G. Tzeng, Refined shear deformation theory of laminated shells, AIAA J, vol.31, issue.4, pp.765-73, 1993.

K. Bhaskar and T. K. Varadan, A higher-order theory for bending analysis of laminated shells of revolution, Comput Struct, vol.40, issue.4, pp.815-824, 1991.

M. Ganapathi, B. P. Patel, H. G. Patel, and D. S. Pawargi, Vibration analysis of laminated cross-ply oval cylindrical shells, J Sound Vibr, vol.262, pp.65-86, 2003.

S. Botello, E. Onate, and J. M. Canet, A layer-wise triangle for analysis of laminated composite plates and shells, Comput Struct, vol.70, pp.635-681, 1999.

R. Zinno and E. J. Barbero, A three-dimensional layer-wise constant shear element for general anisotropic shell-type structures, Int J Num Method Eng, vol.37, pp.2445-70, 1994.

P. Seide and R. A. Chaudhuri, Triangular finite element for analysis of thick laminated shells, Int J Num Method Eng, vol.24, issue.8, pp.1563-79, 1987.

Y. Basar and Y. Ding, Interlaminar stress analysis of composites: layer-wise shell finite elements including transverse strains, Comp Eng, vol.5, issue.5, pp.485-99, 1995.

E. I. Grigolyuk and G. M. Kulikov, General direction of development of the theory of multilayered shells, Mech Compos Mater, vol.24, pp.231-272, 1988.

G. M. Kulikov and S. V. Plotnikova, Advanced formulation for laminated composite shells: 3d stress analysis and rigid-body motions, Compos Struct, vol.95, pp.236-282, 2013.

K. P. Soldatos and T. Timarci, A unified formulation of laminated composite, shear deformable, five-degrees-of-freedom cylindrical shell theories, Compos Struct, vol.25, issue.1, pp.165-71, 1993.

M. Cho, K. O. Kim, and M. H. Kim, Efficient higher-order shell theory for laminated composites, Compos Struct, vol.34, issue.2, pp.197-212, 1996.

M. Shariyat, Non-linear dynamic thermo-mechanical buckling analysis of the imperfect laminated and sandwich cylindrical shells based on a global-local theory inherently suitable for non-linear analyses, Int J Non-Linear Mech, vol.46, pp.253-71, 2011.

M. Y. Yasin and S. Kapuria, An efficient layerwise finite element for shallow composite and sandwich shells, Compos Struct, vol.98, pp.202-216, 2013.

X. Shu, A refined theory of laminated shells with higher order transverse shear deformation, Int J Solids Struct, vol.34, issue.6, pp.673-83, 1997.

D. Versino, M. Gherlone, D. Sciuva, and M. , four node shell element for doubly curved multilayered composites based on the refined zigzag theory, Compos Struct, vol.118, pp.392-402, 2014.

W. Zhen and C. Wanji, A global-local higher order theory for multilayered shells and the analysis of laminated cylindrical shell panels, Compos Struct, vol.84, issue.4, pp.350-61, 2008.

U. Icardi and L. Ferrero, Multilayered shell model with variable representation of displacements across the thickness, Compos B, vol.42, pp.18-26, 2011.

E. Carrera, Historical review of zig-zag theories for multilayered plates and shells, Appl Mech Rev, vol.56, issue.3, pp.287-308, 2003.

R. Kapania, A review on the analysis of laminated shells, J Pres Ves Technol, vol.111, pp.88-96, 1989.

A. K. Noor and W. S. Burton, Assessment of computational models for multilayered composite shells, Appl Mech Rev, vol.43, issue.4, pp.67-97, 1990.

W. Gilewski and M. Radwanska, A survey of finite element models for the analysis of moderately thick shells, Finite Elem Anal Des, vol.9, pp.1-21, 1991.

H. Yang, S. Saigal, A. Masud, and R. K. Kapania, A survey of recent shell finite element, Int J Num Method Eng, vol.47, pp.101-128, 2000.

E. Carrera, Theories and finite elements for multilayered, anisotropic, composite plates and shells, Arch Comput Method Eng, vol.9, issue.2, pp.87-140, 2002.

J. N. Reddy and R. A. Arciniega, Shear deformation plate and shell theories: from stavsky to present, Mech Adv Mater Struct, vol.11, pp.535-82, 2004.

J. Hohe and L. Librescu, Advances in the structural modeling of elastic sandwich panels, Mech Adv Mater Struct, vol.11, issue.4-5, pp.395-424, 2004.

M. S. Qatu, E. Asadi, and W. Wang, Review of recent literature on static analyses of composite shells: 2000-2010, Open J Compos Mater, vol.2, pp.61-86, 2012.

M. F. Caliri, A. Ferreira, and V. Tita, A review on plate and shell theories for laminated and sandwich structures highlighting the finite element method, Compos Struct, vol.156, pp.63-77, 2016.

M. Cho and J. S. Kim, A postprocess method for laminated shells with a doubly curved nine-noded finite element, Compos B, vol.31, issue.1, pp.65-74, 2000.

R. Tanov and A. Tabiei, Adding transverse normal stresses to layered shell finite elements for the analysis of composite structures, Compos Struct, vol.76, issue.4, pp.338-382, 2006.

E. Viola, F. Tornabene, and N. Fantuzzi, Static analysis of completely doubly-curved laminated shells and panels using general higher-order shear deformation theories, Compos Struct, vol.101, pp.59-93, 2013.

T. Pian and K. Sumihara, State-of-the-art development of hybrid/mixed finite element method, Finite Elem Anal Des, vol.21, pp.5-20, 1995.

D. J. Haas and S. W. Lee, A nine-node assumed-strain finite element for composite plates and shells, Comput Struct, vol.26, issue.3, pp.445-52, 1987.

M. L. Liu and C. To, Free vibration analysis of laminated composite shell structures using hybrid strain based layerwise finite elements, Finite Elem Anal Des, vol.40, pp.83-120, 2003.

Y. Yong and Y. Cho, Higher-order, partial hybrid stress, finite element formulation for laminated plate and shell analyses, Comput Struct, vol.57, pp.817-844, 1995.

S. Di and E. Ramm, Hybrid stress formulation for higher-order theory of laminated shell analysis, Comput Methods Appl Mech Eng, vol.109, pp.359-76, 1993.

L. Vu-quoc and X. G. Tan, Efficient hybrid-eas solid element for accurate stress prediction in thick laminated beams, plates, and shells, Comput Methods Appl Mech Eng, vol.253, pp.337-55, 2013.

K. Rah, W. V. Paepegem, and J. Degrieck, An optimal versatile partial hybrid stress solid-shell element for the analysis of multilayer composites, Int J Num Method Eng, vol.93, issue.2, pp.201-224, 2013.

W. Feng and S. V. Hoa, A partial hybrid degenerated plate/shell element for the analysis of laminated composites, Int J Num Method Eng, vol.39, pp.3625-3664, 1996.

A. Noor and C. M. Andersen, Mixed isoparametric finite element models of laminated composite shells, Comput Methods Appl Mech Eng, vol.11, issue.3, pp.255-80, 1977.

H. Jing and M. L. Liao, Partial hybrid stress element for the analysis of thick laminated composite plates, Int J Num Method Eng, vol.28, issue.12, pp.2813-2840, 1989.

C. Wu and C. Liu, Mixed finite-element analysis of thick doubly curved laminated shells, J Aerosp Eng, vol.8, issue.1, pp.43-53, 1995.

E. Reissner, On a certain mixed variational theorem and a proposed application, Int J Num Method Eng, vol.20, pp.1366-74, 1984.

E. Reissner, On a mixed variational theorem and on a shear deformable plate theory, Int J Num Method Eng, vol.23, pp.193-201, 1986.

H. Murakami, Laminated composite plate theory with improved in-plane responses, J Appl Mech ASME, vol.53, pp.661-667, 1986.

A. Toledano and H. Murakami, A high-order laminated plate theory with improved in-plane responses, Int J Solids Struct, vol.23, pp.111-142, 1987.

A. Toledano and H. Murakami, A composite plate theory for arbitrary laminate configurations, J Appl Mech ASME, vol.24, pp.181-190, 1987.

E. Carrera, Evaluation of layerwise mixed theories for laminated plates analysis, AIAA J, vol.36, issue.5, pp.830-839, 1998.

E. Carrera and L. Demasi, Classical and advanced multilayered plate elements based upon PVD and RMVT. part 1: derivation of finite element matrices, Int J Num Method Eng, vol.55, pp.191-231, 2002.

E. Carrera and L. Demasi, Classical and advanced multilayered plate elements based upon PVD and RMVT. part 2: numerical implementations, Int J Num Method Eng, vol.55, pp.253-91, 2002.

B. Brank and E. Carrera, Multilayered shell finite element with interlaminar continuous shear stresses: a refinement of the reissner-mindlin formulation, Int J Num Method Eng, vol.48, issue.6, pp.843-74, 2000.

B. Brank and E. Carrera, A family of shear-deformable shell finite elements for composite structures, Comput Struct, vol.76, issue.1, pp.287-97, 2000.

M. Cinefra, C. Chinosi, L. D. Croce, and E. Carrera, Refined shell finite elements based on rmvt and mitc for the analysis of laminated structures, Compos Struct, vol.113, pp.492-499, 2014.

E. Carrera, Developments, ideas and evaluations based upon the reissner's mixed theorem in the modeling of multilayered plates and shells, Appl Mech Rev, vol.54, pp.301-330, 2001.

O. Allix and P. Vidal, A new multi-solution approach suitable for structural identification problems, Comput Methods Appl Mech Eng, vol.191, pp.2727-58, 2002.
URL : https://hal.archives-ouvertes.fr/hal-01366984

A. Ammar, B. Mokdada, F. Chinesta, and R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids, J Non-Newton Fluid Mech, vol.139, pp.153-76, 2006.
URL : https://hal.archives-ouvertes.fr/hal-01004909

F. Chinesta, A. Ammar, and E. Cueto, Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models, Arch Comput Methods Eng, vol.17, issue.4, pp.327-50, 2010.
URL : https://hal.archives-ouvertes.fr/hal-01007235

M. Savoia and J. N. Reddy, A variational approach to three-dimensional elasticity solutions of laminated composite plates, J Appl Mech ASME, vol.59, pp.166-75, 1992.

B. Bognet, A. Leygue, and F. Chinesta, Separated representations of 3D elastic solutions in shell geometries, Adv Model Simul Eng Sci, vol.1, p.4, 2014.

P. Vidal, L. Gallimard, and O. Polit, Proper generalized decomposition and layer-wise approach for the modeling of composite plate structures, Int J Solids Struct, vol.50, pp.2239-50, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01366917

P. Vidal, L. Gallimard, and O. Polit, Shell finite element based on the proper generalized decomposition for the modeling of cylindrical composite structures, Comput Struct, vol.132, pp.1-11, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01366963

P. Vidal, L. Gallimard, and O. Polit, Modeling of composite plates based on Reissners Mixed Variational Theorem with variables separation, Compos B, vol.86, pp.229-271, 2016.

J. G. Ren, Exact solutions for laminated cylindrical shells in cylindrical bending, Comp Sci Technol, vol.29, pp.169-87, 1987.

K. Bhaskar and T. K. Varadan, Exact elasticity solution for laminated anisotropic cylindrical shells, J Appl Mech ASME, vol.60, pp.41-48, 1993.

E. Carrera and S. Brischetto, A comparison of various kinematic models for sandwich shell panels with soft core, J Comp Mater, vol.43, issue.20, pp.2201-2222, 2009.

M. Bernadou, Finite element methods for thin shell problems, 1996.

E. Carrera, The effects of shear deformation and curvature on buckling and vibrations of cross-ply laminated composite shells, J Sound Vibr, vol.151, pp.405-438, 1991.

M. D'ottavio, D. Ballhause, T. Wallmersperger, and B. Kröplin, Considerations on higher-order finite elements for multilayered plates based on a unified formulation, Comput Struct, vol.84, pp.1222-1257, 2006.

, Publisher's Note

, Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations