## 5 Conclusions and Future Directions  
In modeling, finding the most-convenient representation of the equations of motion of the studied physical system is an important task, which may have an impact on the resolution methods and the control algorithms. The identification of the Hamiltonian structure of the dynamical model is important, as it yields information about conserved quantities (such as energy function, momentum, etc.) in the system that, from a fundamental point of view, should preferably be respected in coupling simulations. Our goal was to contribute to such a quest for improved analysis procedures. **Main objective. ** More precisely, the intent of this work was to present the basic PHS formulation of [PERSON]'s moderately thick 2D shell model. By their nature, 2D shell equations are approximations to the 3D ones, which result from making specific assumptions determining the range of their applicability. In order to lay down thusly the appropriate PHS expressions for a future numerical work on shell-fluid coupling, we followed a standard variational approach to degenerate the 3D shell continuum description to 2D shell ones exhibiting their effective behaviors. The equations of motion were obtained according to the principle of virtual powers, and the possible intrinsic (but well-known) coupling of kinematic modes due to the often-neglected shell geometric considerations was emphasized. These are relevant for modeling doubly curved shells for example. **Main findings of our investigation. ** The originality of our contribution was to establish then a formal link between an existing mixed (i.e., multi-primary field) variational principle (a complete Hamilton-Hellinger-Prange-Reissner (**HHPR**) one, with a generalized internal force impulse field) and the PHS formulation. This interpretation may seem \"new\" to some research communities less familiar with (computational) continuum mechanics and interested in multiphysics coupling. Although the variational model is known in some publications, for completeness, we included a brief discussion of the numerical integration procedures related to the HHPR principle and its canonical PHS equations. This discussion also briefly sketched out a possible extension to a Hamilton-Pontryagin-Hu-Washizu type of variational principle for the PHS formulation. We believe our contribution allows a better understanding of this PHS formalism from the perspective of continuum mechanics and its computational analysis. It also sheds new light on the limit cases developed recently in [14; 15; 24; 130] for the thin or thick plate theories. The formal variational link, the short discussion on variational integrations, and our PHS formulation of the NR-shell model will be useful for our extensions to other theoretical and numerical applications. **Future extensions of our PHS formulation. ** Below, we collect some extensions that we intend to develop in the near future:  
* _PHS formulations completed for more complex constitutive material laws and shell geometry._ The full formulation of the PHS of the NR shell is nevertheless far from being exposed here. For simplicity, our mathematical treatment of this continuous shell medium was intentionally limited to transversely homogeneous and tangentially isotropic linearly elastic behavior. The model was expressed in the _principal orthogonal curvilinear coordinate system_, which would apply as well to many other shell mechanic models with both constant and variable thicknesses. Moreover, our derivation did not consider as well the singularities and/or discontinuities of any kind (whether fixed or mobile), which can occur, for example, for the cases of shell junctions or impacts. The required regularity can--and must--be relaxed in the variational formalisms so that they include jump conditions for the cases of shell junctions or impacts for instances.