## 4 Hamiltonian Variational Formalisms versus Canonical Ones
### Some Remarks on These Variational Principles and Their Numerical Treatments
#### 4.2.2 Time-Differential versus Time-Variational GNI Treatments  
**Strong time integration. ** In many numerical strategies of resolution involving space-time variables, a spatial discretization (e.g., finite-element method) is customarily performed first to obtain a system of ordinary differential equations in the time variable. The classical Hamiltonian canonical equations in Equation (76) and the PHS canonical ones in Equation (87) are often solved following this time-differential numerical methodology. This strategy does not require the use of the impulse \(\overline{\mathbf{F}}^{\rm int}\) of the generalized internal forces \(\overline{\mathbf{F}}^{\rm int}\equiv\frac{\partial\ \overline{\mathbf{F}}^{\rm int}}{ \partial t}\), but the latter itself. This classical partially variational strategy of the PHS formalism can be considered as a strong time-integration approach [91]. After discretization in space, the symplectic integration schemes (which are designed for numerical solutions of canonical equations) can be used. Nevertheless, in order to benefit from some advantages of the mixed variational principles, the resolution must be naturally and systematically based on mixed finite elements for the discretization on the space domain at least, but not on the time domain. This computational strategy was applied in [30] called the _partitioned finite-element method_ (PFEM) in [129]. It was also applied to linear Reissner-Mindlin and Kirchhoff-Love plates in [14; 15; 130] (which can be considered as different natural limit cases of the NR shell model, the Kirchhoff-Love plate limit being numerically troublesome due to the occurrence of the _shear-locking phenomenon_). It has also been applied to the nonlinear shallow water equation in 2D [13] and extended to dissipative systems in [131]; the implementation details can be found in [132], and the numerical analysis of the convergence of the PFEM was presented in [133]. **Partial- versus full-variational, weak time integrations. ** Besides, discrete-time versions of the foregoing Hamiltonian-like variational principles, i.e., which approximate the integrals of action in Equations (74), (77) and (85), are also of great interest. In general, these weak discrete approaches follow consistently the chosen variational principle. Thus, the works in [46; 47; 48; 134] (see also [135]) developed integrators that are based on a discretization of the Lagrange-d'Alembert principle (with action integrals like \(\overline{\mathcal{A}}_{\mathrm{L}}[\mathfrak{q},t_{f}]\) in Equation (74)), as well as on a variational formulation of dissipation. They mentioned satisfactory results. As illustrated by the numerical experiments in [39; 46; 136], by adopting a space-time view of variational integrators of the integrals of action, one can have integrators that preserve the energy, momentum, and symplectic structure. Illustrating the problems of linear elastodynamics that are solved with such a time-variational strategy, multi-field variational principles in which displacement, velocity, and momentum fields are taken to be independent (with action integrals like \(\overline{\mathcal{A}}_{\mathrm{H}}[\mathfrak{q},\boldsymbol{\pi},t_{f}]\) in Equation (77)) are, for instance, in [91; 92; 119] and treated with space-time finite elements in [137]. An analogous discrete methodology was also developed for the other variational formulations with similar success, while using their primary fields (including the time variable). For instance, the work on linear shell elastodynamics in [91] (see also [92] for beam elastodynamics and [138] for (2D) plane elastodynamics/elasticity) provided such a weak time-variational integration strategy both for the extended Hamilton-(Hellinger-Prange-Reissner and Pontryagin-Hu-Washizu) variational formulations.