Variational formulation
==========================




Virtual work
---------------


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**Figure 3.1-a: Volumic shell.Subsequent pressure on the initial mean surface and its transforme**

The virtual work of a **follower** :math:`p` pressure (i.e. acting on the transformed mean surface and moving with it) can be expressed in the form:

:math:`\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{{\omega }^{\varphi }}\delta {u}_{p}\text{.}pd{\omega }^{\varphi }`

If we use the isoparametric surface element corresponding to our solid shell modeling, the area :math:`d{\omega }^{\varphi }` is expressed directly in terms of the isoparametric coordinates :math:`d{\xi }_{1}d{\xi }_{2}` and we get the following simple form of the equation above:

:math:`\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\delta {u}_{p}\text{.}p({\xi }_{1},{\xi }_{2}){a}_{1}^{\varphi }({\xi }_{1},{\xi }_{2})\times {a}_{2}^{\varphi }({\xi }_{1},{\xi }_{2})d{\xi }_{1}d{\xi }_{2}`




Tangent operator
-----------------

As the virtual work of the follower pressure depends on the current configuration, its linear variation :math:`\Delta` is not zero and must be taken into account. The tangent operator associated with this virtual work is written in iteration :math:`(i+1)` in the form:

:math:`L\left[\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i+1)}\right]=\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i)}+\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i)}`

where :math:`\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{(i)}` is the increment between two iterations of the virtual work of the subsequent pressure. If the pressure is given in the form:

:math:`p=\lambda {p}_{0}`

:math:`\lambda` being the load level that is fixed during the iterations (load control :math:`\Delta \lambda =0`), we can write:

:math:`\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\delta {u}_{p}\text{.}p({a}_{1}^{\varphi }\times \Delta {a}_{2}^{\varphi }-{a}_{2}^{\varphi }\times \Delta {a}_{1}^{\varphi })d{\xi }_{1}d{\xi }_{2}`

The incremental variations of the vectors of the local base tangent to the transform of the mean surface are given by:

:math:`\begin{array}{}{\Delta a}_{1}^{\varphi }=\frac{\partial }{\partial {\xi }_{1}}{\Delta u}_{P}\\ {\Delta a}_{2}^{\varphi }=\frac{\partial }{\partial {\xi }_{2}}{\Delta u}_{P}\end{array}`

since the initial mean surface doesn't "move" during iterations which results in :math:`\Delta {x}_{P}=0`.

These calculations finally make it possible to establish the expression of the increment of the virtual work of next pressure in the form:

:math:`\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\delta {u}_{P}\cdot P(\left[{a}_{1}^{\varphi }\times \right]\frac{\partial }{\partial {\xi }_{2}}\Delta {u}_{P}-\left[{a}_{2}^{\varphi }\times \right]\frac{\partial }{\partial {\xi }_{2}}\Delta {u}_{P})d{\xi }_{1}d{\xi }_{2}`

where :math:`\left[{a}_{1}^{\varphi }\times \right]\mathrm{et}\left[{a}_{2}^{\varphi }\times \right]` are respectively the antisymmetric matrices of the tangent vectors :math:`{a}_{1}^{\varphi }\mathrm{et}{a}_{2}^{\varphi }` respectively.

**Note:**

In the reference [:ref:`bib2 <bib2>`], one integration per part is undertaken on the expression above. It is shown that the tangent matrix can be decomposed into a symmetric part resulting from integration on the domain and an anti-symmetric part resulting from integration on the contour. It is also shown that the assembly of the antisymmetric parts of the elementary tangent matrices leads to a zero matrix when the pressure is continuous from one finite element to another, due to the existence of a potential associated with the work of the pressure in this case.