4. Discretization

At points \(P\) of the mean surface, the interpolation of the virtual displacement is written as:

\(\delta u({\xi }_{\mathrm{1,}}{\xi }_{2})=\sum _{I=1}^{\text{NB}1}{N}_{I}^{(1)}({\xi }_{\mathrm{1,}}{\xi }_{2}){(\begin{array}{}\delta u\\ \delta v\\ \delta w\end{array})}_{I}\)

and the interpolation of the incremental displacement between two iterations is written as:

\(\delta u({\xi }_{\mathrm{1,}}{\xi }_{2})=\sum _{I=1}^{\text{NB}1}{N}_{I}^{(1)}({\xi }_{\mathrm{1,}}{\xi }_{2}){(\begin{array}{}\Delta u\\ \Delta v\\ \Delta w\end{array})}_{I}\)

We rewrite the previous two equations in matrix form:

\(\begin{array}{}\delta u({\xi }_{1},{\xi }_{2})=\left[N\right]{\left\{\delta u\right\}}^{e}\\ \Delta u({\xi }_{1},{\xi }_{2})=\left[N\right]{\left\{\Delta u\right\}}^{e}\end{array}\)

where \(\left[N\right]\) is the matrix of translational form functions at the mean surface, whose expression is:

\(\left[N\right]=\left[\text{...}{\left[{N}_{I}^{(1)}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\right]}_{I=\mathrm{1,}\text{NB}1}\text{...}{\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}_{\text{NB}2}\right]\)

The form functions \({N}_{I}^{(1)}\text{et}{N}_{I}^{(2)}\) (used later) are given in the appendix to [R3.07.04]. Nodes \(I=\mathrm{1,}\text{NB}1\) are the vertex nodes and the midpoints of the sides (for the quadrangle and the triangle). Node \(\text{NB}2\) is at the barycenter of the element.

Vector \({\left\{\delta u\right\}}^{e}\) is the nodal vector for virtual movements given by:

\({\left\{\delta u\right\}}^{e}=(\begin{array}{}\mathrm{.}\\ \mathrm{.}\\ \mathrm{.}\\ {(\begin{array}{}\delta w\\ \delta {\theta }_{x}\\ \delta {\theta }_{y}\\ \delta {\theta }_{z}\end{array})}_{I}\\ \mathrm{.}\\ \mathrm{.}\\ \mathrm{.}\\ I=\mathrm{1,}\mathrm{NB1}\\ \\ {(\begin{array}{}\delta {\theta }_{x}\\ \delta {\theta }_{x}\\ \delta {\theta }_{x}\end{array})}_{\mathrm{NB2}}\end{array})\)

Vector \({\left\{\Delta u\right\}}^{e}\) is the nodal vector for incremental movements between two iterations.

\({\left\{\Delta u\right\}}^{e}=(\begin{array}{}\mathrm{.}\\ \mathrm{.}\\ \mathrm{.}\\ {(\begin{array}{}\Delta u\\ \Delta v\\ \Delta w\\ \Delta {\theta }_{x}\\ \Delta {\theta }_{y}\\ \Delta {\theta }_{z}\end{array})}_{I}\\ \mathrm{.}\\ \mathrm{.}\\ \mathrm{.}\\ I=\mathrm{1,}\mathrm{NB1}\\ \\ {(\begin{array}{}\Delta {\theta }_{x}\\ \Delta {\theta }_{x}\\ \Delta {\theta }_{x}\end{array})}_{\mathrm{NB2}}\end{array})\)

This discretization also allows us to establish the expression of the derivatives of the incremental displacement of the mean surface with respect to the isoparametric surface coordinates in the form:

\(\begin{array}{}\frac{\partial }{\partial {\xi }_{1}}\Delta u({\xi }_{\mathrm{1,}}{\xi }_{2})=\left[\frac{\partial }{\partial {\xi }_{2}}N\right]{\left\{\Delta u\right\}}^{e}\\ \frac{\partial }{\partial {\xi }_{2}}\Delta u({\xi }_{\mathrm{1,}}{\xi }_{2})=\left[\frac{\partial }{\partial {\xi }_{2}}N\right]{\left\{\Delta u\right\}}^{e}\end{array}\)

where \(\left[\frac{\partial }{\partial {\xi }_{1}}N\right]\text{et}\left[\frac{\partial }{\partial {\xi }_{2}}N\right]\) are the matrices derived from the functions of forms of translation at the mean surface, whose expressions are:

\(\begin{array}{}\left[\frac{\partial }{\partial {\xi }_{1}}N\right]=\left[\mathrm{...}{\left[\frac{\partial {N}_{I}^{(1)}}{\partial {\xi }_{1}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\right]}_{I=\mathrm{1,}\text{NB}1}\mathrm{...}{\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}_{\text{NB}2}\right]\\ \left[\frac{\partial }{\partial {\xi }_{2}}N\right]=\left[\mathrm{...}{\left[\frac{\partial {N}_{I}^{(1)}}{\partial {\xi }_{2}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]\right]}_{I=\mathrm{1,}\text{NB}1}\mathrm{...}{\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}_{\text{NB}2}\right]\end{array}\)

Thus, the virtual work of the following pressure can be expressed in the following matrix form:

\(\delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=\left\{\delta {u}^{e}\right\}\mathrm{.}\left\{{f}_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{e}\right\}\)

with \(\left\{{f}_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{e}\right\}\) the nodal vector of external forces which can be expressed in the following way:

\(\left\{{f}_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}^{e}\right\}={\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}{\left[N\right]}^{T}({a}_{1}^{\varphi }\times {a}_{1}^{\varphi })d{\xi }_{1}d{\xi }_{2}\)

It is important to note that with our parametrization of the mean surface transform, the Jacobian \(\text{det}(\left[J({\xi }_{3}=0)\right])\) of this surface is not involved in the calculation of finite element objects.

It should also be noted that the pressure is discretized with an isoparametric interpolation of the values at the NB2 nodes:

\(p({\xi }_{1},{\xi }_{2})=\sum _{I=1}^{\text{NB}2}{N}_{I}^{(2)}({\xi }_{1},{\xi }_{2}){p}_{I}\)

We can also express the increment between two iterations of the virtual work of the following pressure in matrix form:

\(\Delta \delta {\pi }_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}=-\{\delta {u}^{e}\}\{{K}_{T}^{e}{}_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}\}\{\Delta {u}^{e}\}\)

where \(\{{K}_{T}^{e}{}_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}\}\) is the contribution in the tangent stiffness matrix of external forces which can be expressed as:

\(\{{K}_{T}^{e}{}_{\begin{array}{}\mathrm{pression}\\ \mathrm{suiveuse}\end{array}}\}={\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\left[N\right]p\left[{a}_{1}^{\varphi }\times \right]\left[\frac{\partial }{\partial {\xi }_{2}}N\right]d{\xi }_{1}d{\xi }_{2}-{\int }_{\left[-\mathrm{1,}+1\right]\times \left[-\mathrm{1,}+1\right]}\left[N\right]p\left[{a}_{2}^{\varphi }\times \right]\left[\frac{\partial }{\partial {\xi }_{1}}N\right]d{\xi }_{1}d{\xi }_{2}\)

Note:

We note that the finite element formulations resulting from this approach do not involve the degrees of freedom of rotation. The treatment is therefore also valid for the finite element facets of three-dimensional elasticity.