## 5 Conclusions and Future Directions
For more general engineering applications coupled with various physical modeling, our work can therefore be modified for a transversely isotropic material [77] and in non-principal curvilinear coordinate forms by formulating it in invariant tensor notation. This may seem very complex at first sight, but is very efficient for a large variety of more complex (composite) shell structure shapes appearing in many industries, including the automotive, aircraft, and civil engineering industries. Giving the formulations of some of these extensions will be one of our future objectives. * _Multiphysics coupling._ The use of the PHS formalism in multiphysics coupling analyses constitutes for us another main interest. In particular, the generic variational framework used to derive the PHS formulation for the NR shell model is broadly applicable as well to more complex physical models. By unifying the expressions of the coupled systems of conservationlaws, the so-derived multiphysics PHS models can further be used for controller design in a wide variety of applications such as inflatable space structures, launcher tank vibration damping, payload vibration protection with smart materials, and many other related applications. Interestingly, the PHS formalism can be used quite readily and directly in certain vibro-acoustical or thermo-mechanical applications, based, for instance, on the thermo-elasticity theory in [110; 112; 113; 114; 139]. With the proper definition of the operators, which we shall also present elsewhere, the PHS formulation can cover these coupling cases with the infinitesimal deformation cases of linear elastic, moderately thick shells and linear elastic thin shells (with or without rotatory inertia), as well as either deep or shallow shells (including the degenerate case of plates) [70; 71]. An interesting coupling can also be with some enhanced shell models, such as those in [54; 59; 98]. These models include more complex features than the NR shell model, such as Cosserat's additional microrotations in [54; 140] and the micromorphic-like stress field featuresacross the shell thickness in [59; 98]. Their micromorphic shell models were obtained variationally, by applying the Hellinger-Prange-Reissner functional directly to the 3D elastic shell structure with the polynomial approximations of the stress field. As a result, the shell model [98] proved to be very accurate in the evaluation of stresses, since 3D equilibrium equations and boundary conditions at the faces of the shell are (or can be approximatively) satisfied. As regards the fluid-shell structure interaction problems, the application of our variational description of the PHS equations turns out to be also very close to the one in [115]. Nevertheless, in that work, as in others, some features required for the PHS formulation of the shell were missing. * _Numerical simulations._ Once the PHS formulation has been obtained at the continuous level, a very interesting topic is the structure-preserving discretization of the system. Indeed, obtaining a finite-dimensional PHS from of an infinite-dimensional one in a systematic way is very useful for simulation and control purposes. We discussed in Section 4.2 the possible numerical solution strategies (and mentioned some of their illustrations) that can be applied to our mixed variational formulation of NR shell elastodynamics by the (weak or strong) PHS formalisms ([91; 92]). Investigations of the weak time PHS formalism have been applied quite recently in [30; 41; 42], but have not yet been applied to shells. Therefore, regarding the benefit of the current symplectic computational developments, benchmarking comparisons with the other GNI strategies should be also performed and will be presented for beams, plates, and shells. Simulations including the multiphysics enrichments mentioned above will also be carried out. Conceptualization, M.C., I.F.N. and D.M. ; Methodology, M.C. and Y.G. ; Validation, M.C., I.F.N. and Y.G. ; Formal analysis, M.C. and I.F.N. ; Investigation, M.C., I.F.N. and D.M.