7. Resolution algorithm#
7.1. Nonlinear algorithm for solving balance equations#
In the general case of modeling (variable coefficients, desaturation, convection) the variational problem presented above [éq 4.1-1] to [éq 4.3-1] is non-linear with respect to the displacement fields, pressure and temperature. After discretization by finite elements, a nonlinear matrix system is obtained. The tangent operator matrix also contains a non-symmetric term treated as such. In all modeling cases, the nonlinear solver STAT_NON_LINE based on a Newton-Raphson method, described in [bib5], is used. We introduce the vector functional:
\(F(U)=R(U)-{L}^{\mathrm{meca}}\) eq 7.1-1
The associated tangent operator is noted: \(DF=\frac{\partial F}{\partial U}\).
For modules THM, which are the subject of this note, the \({L}^{\text{meca}}\) operator does not depend on generalized movements. All the terms that depend on generalized movements have been introduced in \(R\), and that is precisely why displacements are found in generalized deformations. In this regard, note the very particular treatment of the term \({\int }_{\Omega }r{F}^{m}\text{.}v\) in equation [éq 4.1-1].
Based on [éq 6.3.4-1], \({\int }_{\Omega }r{F}^{m}\text{.}vd\Omega ={\int }_{\Omega }({r}_{0}+{m}_{1}^{1+}+{m}_{1}^{2+}+{m}_{2}^{1+}+{m}_{2}^{2+}){F}^{m}\text{.}vd\Omega\).
We have chosen to split this term in two:
The term \({\int }_{\Omega }{r}_{0}{F}^{m}\text{.}vd\Omega\) is a contribution to \({L}^{\mathrm{meca}}\) if the user has entered the operand PESANTEUR of the load used (defined by the command AFFE_CHAR_MECA), while the term \({\int }_{\Omega }({m}_{1}^{1+}+{m}_{1}^{2+}+{m}_{2}^{1+}+{m}_{2}^{2+}){F}^{m}\text{.}vd\Omega\), which depends on generalized constraints, is a contribution to \(\mathrm{R}\).
7.2. Transition from nodal values to values at Gauss points#
As in all finite element codes, the terms are calculated by looping over the elements and looping over the gauss points. By noting \({R}_{g}^{\mathrm{el}}\) and \(D{F}_{g}^{\mathrm{el}}\) the values at the Gauss point \(g\) of the element \(\mathrm{el}\) of the nodal forces and the tangent operator, and \({w}_{g}^{\mathrm{el}}\) the integration weight related to this Gauss point, we have:
\(R(U)=\sum _{\mathrm{el}}\sum _{g}{w}_{g}^{\mathrm{el}}{R}_{g}^{\mathrm{el}}(U)\) eq 7.2-1
\(DF(U)=\sum _{\mathrm{el}}\sum _{g}{w}_{g}^{\mathrm{el}}D{F}_{g}^{\mathrm{el}}(U)\) eq 7.2-2
Let us then denote \({U}^{\mathrm{el}}\) the vector of nodal unknowns on a finite element \(\mathrm{el}\). We can thus have:
\(\text{par exemple}{U}^{\mathrm{el}}=\begin{array}{c}\begin{array}{c}u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\end{array}\}\text{noeud 1}\\ \begin{array}{c}u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\end{array}\}\text{noeud 2}\\ \begin{array}{c}u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\end{array}\}\text{noeud 3}\end{array}\).
Let’s also note \({E}_{g}^{\mathrm{el}}\) the vector of generalized deformations at the Gauss point \(g\) of the element \(\mathrm{el}\) and \({\Sigma }_{g}^{\mathrm{el}}\) the generalized stress vector for the Gauss point \(g\) of the element \(\mathrm{el}\). In the most complete case we thus have:
\({E}_{g}^{\mathrm{el}}={\left\{\begin{array}{c}u\\ \varepsilon (u)\\ {p}_{1}\\ \nabla {p}_{1}\\ {p}_{2}\\ \nabla {p}_{2}\\ T\\ \nabla T\end{array}\right\}}_{g}^{\mathrm{el}};{\Sigma }_{g}^{\mathrm{el}}={\left\{\begin{array}{c}\sigma \text{'}\\ {\sigma }_{p}\\ {m}_{1}^{1}\\ {M}_{1}^{1}\\ {h}^{{m}_{1}^{1}}\\ {m}_{1}^{2}\\ {M}_{1}^{2}\\ {h}^{{m}_{1}^{2}}\\ {m}_{2}^{1}\\ {M}_{2}^{1}\\ {h}^{{m}_{2}^{1}}\\ {m}_{2}^{2}\\ {M}_{2}^{2}\\ {h}^{{m}_{2}^{2}}\\ Q\text{'}\\ q\end{array}\right\}}_{g}^{\mathrm{el}}\)
The finite element shape functions then make it possible to calculate the matrix \({Q}_{g}^{\mathrm{el}}\) for the transition from nodal unknowns to generalized deformations at Gauss points defined by:
\({E}_{g}^{\mathrm{el}}={Q}_{g}^{\mathrm{el}}\text{.}{U}_{g}^{\mathrm{el}}\) eq 7.2-3
7.3. Vectors and matrices according to the options#
The presentations of the following two paragraphs are made in the most general case where we have a mechanical equation, two hydraulic equations and a thermal equation. Clues \(g\) and \(\mathrm{el}\) are now omitted, but it’s clear that what’s being described applies to every Gauss point in each element.
7.3.1. Residue or nodal force: options RAPH_MECA and FULL_MECA#
The terms of the variational formulation are divided according to the following principle:
If \({E}_{g}^{\text{*}\mathrm{el}}\) designates a virtual deformation field, \({E}_{g}^{\text{*}\mathrm{el}}=(v,\varepsilon (v),{\pi }_{1},\nabla {\pi }_{1},{\pi }_{2},\nabla {\pi }_{2},\tau ,\nabla \tau )\) calculated from a virtual nodal displacement vector \({U}^{\text{*}\mathrm{el}}\), we can define: \({E}_{g}^{\text{*}\mathrm{el}T}\text{.}{\overline{\Sigma }}_{g}^{\mathrm{el}}(U)={\overline{\Sigma }}_{1}v+{\overline{\Sigma }}_{2}(v)+{\overline{\Sigma }}_{3}{\pi }_{1}+{\overline{\Sigma }}_{4}\nabla {\pi }_{1}+{\overline{\Sigma }}_{5}{\pi }_{2}+{\overline{\Sigma }}_{6}\nabla {\pi }_{2}+{\overline{\Sigma }}_{7}\tau +{\overline{\Sigma }}_{8}\nabla \tau\). We then use the discrete variational formulations [éq 5.1-1], [éq 5.2-1], [éq 5.3-1], and replace the integrals \({\int }_{\Omega }fd\Omega\) by \(\sum _{\mathrm{el}}\sum _{g}{w}_{g}^{\mathrm{el}}{f}_{g}^{\mathrm{el}}\) for all integrants \(f\). We distinguish the terms multiplying \(v\), \(\varepsilon (v)\), respectively, \({\pi }_{1}\),,, \(\nabla {\pi }_{1}\),, \({\pi }_{2}\), \(\nabla {\pi }_{2}\), \(\tau\) and \(\nabla \tau\), and we find:
Index |
\(\overline{\Sigma }\) |
associated with |
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Note:
The first term \({\stackrel{ˉ}{\Sigma }}_{1}\) does not include the term \(-{r}_{0}{F}^{m}\) because it is put into the external load \({L}^{\mathrm{meca}}\) and calculated by the external gravity load calculation option.
Using the [éq 7.2-1] definition of \({R}_{g}^{\mathrm{el}}\), we have:
\({U}^{\text{*}{\mathrm{el}}^{T}}\text{.}{R}_{g}^{\mathrm{el}}={E}_{g}^{\text{*}{\mathrm{el}}^{T}}\text{.}{\overline{\Sigma }}_{g}^{\mathrm{el}}\), which still gives us:
\({R}_{g}^{\mathrm{el}}={Q}_{g}^{{\mathrm{el}}^{T}}\text{.}{\overline{\Sigma }}_{g}^{\mathrm{el}}\)
This last equality is only the local form at a Gauss point of [éq 6.2-2].
7.3.2. Tangent operator: options FULL_MECA, RIGI_MECA_TANG#
In the following, if \(X\) designates a vector of components \({X}^{i}\) and \(Y\) a vector of components \({Y}^{j}\), \(\left[\frac{\partial X}{\partial Y}\right]\) will designate a matrix whose element occupying row \(i\) and column \(j\) is \(\frac{\partial {X}^{i}}{\partial {Y}^{j}}\).
To calculate the tangent operator, we will calculate the following quantities:
\(\left[\text{DRDE}\right]=\)
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Where we noted:
\(\begin{array}{cc}\text{DRiU}=\frac{\partial {F}_{i}}{\partial u}& \text{DRiGP}1=\frac{\partial {F}_{i}}{\partial \nabla {p}_{1}}\\ \text{DRiE}=\frac{\partial {F}_{i}}{\partial \varepsilon }& \text{DRiGP}2=\frac{\partial {F}_{i}}{\partial \nabla {p}_{2}}\\ \text{DRiP}1=\frac{\partial {F}_{i}}{\partial {p}_{1}}& \text{DRiT}=\frac{\partial {F}_{i}}{\partial T}\\ \text{DRiP}2=\frac{\partial {F}_{i}}{\partial {p}_{2}}& \text{DRiDT}=\frac{\partial {F}_{i}}{\partial \nabla T}\end{array}\)
To do these calculations, it is considered that the laws of behavior provide, for the corresponding options, all of the following derivatives:
\(D\Sigma \mathrm{DE}=\left[\begin{array}{cccccccc}\frac{\partial \sigma \text{'}}{u}& \frac{\partial \sigma \text{'}}{\partial \varepsilon }& \frac{\partial \sigma \text{'}}{\partial {p}_{1}}& \frac{\partial \sigma \text{'}}{\partial \nabla {p}_{1}}& \frac{\partial \sigma \text{'}}{\partial {p}_{2}}& \frac{\partial \sigma \text{'}}{\partial \nabla {p}_{2}}& \frac{\partial \sigma \text{'}}{\partial T}& \frac{\partial \sigma \text{'}}{\partial \nabla T}\\ \frac{\partial {\sigma }_{p}}{\partial u}& \frac{\partial {\sigma }_{p}}{\partial \varepsilon }& \frac{\partial {\sigma }_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma }_{p}}{\partial \nabla {p}_{1}}& \frac{\partial {\sigma }_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma }_{p}}{\partial \nabla {p}_{2}}& \frac{\partial {\sigma }_{p}}{\partial T}& \frac{\partial {\sigma }_{p}}{\partial \nabla T}\\ \frac{\partial {m}_{1}^{1}}{\partial u}& \frac{\partial {m}_{1}^{1}}{\partial \varepsilon }& \frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {M}_{1}^{1}}{\partial u}& \frac{\partial {M}_{1}^{1}}{\partial \varepsilon }& \frac{\partial {M}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {M}_{1}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {M}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {M}_{1}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {M}_{1}^{1}}{\partial T}& \frac{\partial {M}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{1}^{1}}}{\partial u}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \varepsilon }& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \nabla {p}_{1}}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \nabla {p}_{2}}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial T}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \nabla T}\\ \frac{\partial {m}_{1}^{2}}{\partial u}& \frac{\partial {m}_{1}^{2}}{\partial \varepsilon }& \frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {M}_{1}^{2}}{\partial u}& \frac{\partial {M}_{1}^{2}}{\partial \varepsilon }& \frac{\partial {M}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {M}_{1}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {M}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {M}_{1}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {M}_{1}^{2}}{\partial T}& \frac{\partial {M}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{1}^{2}}}{\partial u}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \varepsilon }& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \nabla {p}_{1}}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \nabla {p}_{2}}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial T}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \nabla T}\\ \frac{\partial {m}_{2}^{1}}{\partial u}& \frac{\partial {m}_{2}^{1}}{\partial \varepsilon }& \frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {M}_{2}^{1}}{\partial u}& \frac{\partial {M}_{2}^{1}}{\partial \varepsilon }& \frac{\partial {M}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {M}_{2}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {M}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {M}_{2}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {M}_{2}^{1}}{\partial T}& \frac{\partial {M}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{2}^{1}}}{\partial u}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \varepsilon }& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \nabla {p}_{1}}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \nabla {p}_{2}}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial T}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \nabla T}\\ \frac{\partial {m}_{2}^{2}}{\partial u}& \frac{\partial {m}_{2}^{2}}{\partial \varepsilon }& \frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {M}_{2}^{2}}{\partial u}& \frac{\partial {M}_{2}^{2}}{\partial \varepsilon }& \frac{\partial {M}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {M}_{2}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {M}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {M}_{2}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {M}_{2}^{2}}{\partial T}& \frac{\partial {M}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{2}^{2}}}{\partial u}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \varepsilon }& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \nabla {p}_{1}}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \nabla {p}_{2}}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial T}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \nabla T}\\ \frac{\partial Q\text{'}}{\partial u}& \frac{\partial Q\text{'}}{\partial \varepsilon }& \frac{\partial Q\text{'}}{\partial {p}_{1}}& \frac{\partial Q\text{'}}{\partial \nabla {p}_{1}}& \frac{\partial Q\text{'}}{\partial {p}_{2}}& \frac{\partial Q\text{'}}{\partial \nabla {p}_{2}}& \frac{\partial Q\text{'}}{\partial T}& \frac{\partial Q\text{'}}{\partial \nabla T}\\ \frac{\partial q}{\partial u}& \frac{\partial q}{\partial \varepsilon }& \frac{\partial q}{\partial {p}_{1}}& \frac{\partial q}{\partial \nabla {p}_{1}}& \frac{\partial q}{\partial {p}_{2}}& \frac{\partial q}{\partial \nabla {p}_{2}}& \frac{\partial q}{\partial T}& \frac{\partial q}{\partial \nabla T}\end{array}\right]\)
Note:
In these expressions, the derivatives with respect to \(u\) are all zero, but we keep the writing given the definition of the matries \({Q}_{g}^{\text{el}}\) that we adopted.
The call to the laws of behavior will provide the pieces of the matrix \(D\Sigma \text{DE}\) according to the equations present:
\(\left[\text{DMECDE}\right]=\left[\begin{array}{c}\frac{\partial \sigma \text{'}}{\partial \varepsilon }\\ \frac{\partial {\sigma }_{p}}{\partial \varepsilon }\end{array}\right];\left[\text{DMECP1}\right]=\left[\begin{array}{cc}\frac{\partial \sigma \text{'}}{\partial {p}_{1}}& \frac{\partial \sigma \text{'}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\sigma }_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma }_{p}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DMECP}2\right]=\left[\begin{array}{cc}\frac{\partial \sigma \text{'}}{\partial {p}_{2}}& \frac{\partial \sigma \text{'}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\sigma }_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma }_{p}}{\partial \nabla {p}_{2}}\end{array}\right]\left[\text{DMECDT}\right]=\left[\begin{array}{cc}\frac{\partial \sigma }{\partial T}& \frac{\partial \sigma }{\partial \nabla T}\\ \frac{\partial {\sigma }_{p}}{\partial T}& \frac{\partial {\sigma }_{p}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP11DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{1}^{1}}{\partial \varepsilon }\\ \frac{\partial {M}_{1}^{1}}{\partial \varepsilon }\\ \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \varepsilon }\end{array}\right];\left[\text{DP11P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {M}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {M}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}^{{m}_{1}^{1}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP11P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {M}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {M}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}^{{m}_{1}^{1}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \nabla {p}_{2}}\end{array}\right]\left[\text{DP11DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {M}_{1}^{1}}{\partial T}& \frac{\partial {M}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{1}^{1}}}{\partial T}& \frac{\partial {h}^{{m}_{1}^{1}}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP12DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{1}^{2}}{\partial \varepsilon }\\ \frac{\partial {M}_{1}^{2}}{\partial \varepsilon }\\ \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \varepsilon }\end{array}\right];\left[\text{DP12P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {M}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {M}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}^{{m}_{1}^{2}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP12P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {M}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {M}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}^{{m}_{1}^{2}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \nabla {p}_{2}}\end{array}\right]\left[\text{DP12DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {M}_{1}^{2}}{\partial T}& \frac{\partial {M}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{1}^{2}}}{\partial T}& \frac{\partial {h}^{{m}_{1}^{2}}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP21DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{2}^{1}}{\partial \varepsilon }\\ \frac{\partial {M}_{2}^{1}}{\partial \varepsilon }\\ \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \varepsilon }\end{array}\right];\left[\text{DP21P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {M}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {M}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}^{{m}_{2}^{1}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP21P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {M}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {M}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}^{{m}_{2}^{1}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \nabla {p}_{2}}\end{array}\right]\left[\text{DP21DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {M}_{2}^{1}}{\partial T}& \frac{\partial {M}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{2}^{1}}}{\partial T}& \frac{\partial {h}^{{m}_{2}^{1}}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP22DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{2}^{2}}{\partial \varepsilon }\\ \frac{\partial {M}_{2}^{2}}{\partial \varepsilon }\\ \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \varepsilon }\end{array}\right];\left[\text{DP22P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {M}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {M}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}^{{m}_{2}^{2}}}{\partial {p}_{1}}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP22P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {M}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {M}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}^{{m}_{2}^{2}}}{\partial {p}_{2}}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \nabla {p}_{2}}\end{array}\right]\left[\text{DP22DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {M}_{2}^{2}}{\partial T}& \frac{\partial {M}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}^{{m}_{2}^{2}}}{\partial T}& \frac{\partial {h}^{{m}_{2}^{2}}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DTDE}\right]=\left[\begin{array}{c}\frac{\partial Q\text{'}}{\partial \varepsilon }\\ \frac{\partial q}{\partial \varepsilon }\end{array}\right];\left[\text{DTDP1}\right]=\left[\begin{array}{cc}\frac{\partial Q\text{'}}{\partial {p}_{1}}& \frac{\partial Q\text{'}}{\partial \nabla {p}_{1}}\\ \frac{\partial q}{\partial {p}_{1}}& \frac{\partial q}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DTDP2}\right]=\left[\begin{array}{cc}\frac{\partial Q\text{'}}{\partial {p}_{2}}& \frac{\partial Q\text{'}}{\partial \nabla {p}_{2}}\\ \frac{\partial q}{\partial {p}_{2}}& \frac{\partial q}{\partial \nabla {p}_{2}}\end{array}\right]\left[\text{DTDT}\right]=\left[\begin{array}{cc}\frac{\partial Q\text{'}}{\partial T}& \frac{\partial Q\text{'}}{\partial T}\\ \frac{\partial q}{\partial T}& \frac{\partial q}{\partial T}\end{array}\right]\)
Moreover, by deriving the expression of the residue in relation to the constraints, we define:
\(D\overline{\Sigma }D\Sigma =\left[\begin{array}{cccccccccccccccc}\frac{\partial \overline{{\Sigma }_{1}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{1}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{2}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{2}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{3}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{3}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{4}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{4}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{5}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{5}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{6}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{6}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{7}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{7}}}{\partial q}\\ \frac{\partial \overline{{\Sigma }_{8}}}{\partial \sigma \text{'}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {\sigma }_{p}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {m}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {M}_{1}^{1}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {h}^{{m}_{1}^{1}}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {m}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {M}_{1}^{2}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {h}^{{m}_{1}^{2}}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {m}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {M}_{2}^{1}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {h}^{{m}_{2}^{1}}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {m}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {M}_{2}^{2}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial {h}^{{m}_{2}^{2}}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial Q\text{'}}& \frac{\partial \overline{{\Sigma }_{8}}}{\partial q}\end{array}\right]\)
Since all these quantities are not necessarily calculated, we will note, for \(i\) from 1 to 8:
\(\begin{array}{cc}\left[D\overline{\Sigma }iD\sigma \right]=\left[\frac{\partial {\overline{\Sigma }}_{i}}{\partial \sigma \text{'}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {\sigma }_{p}}\right]& \left[D\overline{\Sigma }i\text{DP}\text{21}\right]=\left[\frac{\partial {\overline{\Sigma }}_{i}}{\partial {m}_{2}^{1}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {M}_{2}^{1}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {h}^{{m}_{2}^{1}}}\right]\\ \left[D\overline{\Sigma }i\text{DP}\text{11}\right]=\left[\frac{\partial {\overline{\Sigma }}_{i}}{\partial {m}_{1}^{2}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {M}_{1}^{1}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {h}^{{m}_{1}^{2}}}\right]& \left[D\overline{\Sigma }i\text{DP}\text{22}\right]=\left[\frac{\partial {\overline{\Sigma }}_{i}}{\partial {m}_{2}^{2}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {M}_{2}^{2}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {h}^{{m}_{2}^{2}}}\right]\\ \left[D\overline{\Sigma }i\text{DP}\text{12}\right]=\left[\frac{\partial {\overline{\Sigma }}_{i}}{\partial {m}_{1}^{2}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {M}_{1}^{1}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial {h}^{{m}_{1}^{2}}}\right]& \left[D\overline{\Sigma }i\text{DT}\right]=\left[\frac{\partial {\overline{\Sigma }}_{i}}{\partial Q\text{'}},\frac{\partial {\overline{\Sigma }}_{i}}{\partial q}\right]\end{array}\)
It is then clear that:
\(D\overline{\Sigma }\mathrm{DE}=D\overline{\Sigma }D\Sigma \text{.}D\Sigma \mathrm{DE}\)
And the contribution of the Gauss point to the tangent matrix \(D{F}_{g}^{\text{el}}\) is obtained by:
\(D{F}_{g}^{\text{el}}={Q}_{g}^{{\text{el}}^{T}}\text{.}D\overline{\Sigma }\text{DE}\text{.}{Q}_{g}^{\text{el}}\)
7.4. Global algorithm#
The algorithm then becomes:
Initializations:
Calculating \({L}^{{\text{meca}}^{+}}\) (option CHAR_MECA)
Calculating \(D{F}^{-}\) (option RIGI_MECA_TANG)
Calculation of \(\Delta {U}_{0}\) by: \(D{F}^{-}\text{.}\Delta {U}_{0}={L}^{{\text{meca}}^{+}}-{L}^{{\text{meca}}^{-}}\)
Newton balance iterations
El element loop
G Gauss point buckle
Calculation \({Q}_{g}^{\text{el}}\)
Calculation \({E}_{g}^{{\text{el}}^{-}}={Q}_{g}^{\text{el}}\text{.}{U}^{{\text{el}}^{-}}\) and \({E}_{g}^{{\text{el}}^{+}}={Q}_{g}^{\text{el}}\text{.}{U}^{{\text{el}}^{+}}\)
calculation of: \({\Sigma }_{gn}^{{\text{el}}^{+}},{\alpha }_{g}^{{\text{el}}^{+}},\frac{\partial {\Sigma }_{gn}^{{\text{el}}^{+}}}{\partial {E}_{gn}^{{\text{el}}^{+}}}\) (depending on option) from \({E}_{g}^{{\text{el}}^{-}},{\Sigma }_{g}^{{\text{el}}^{-}},{\alpha }_{g}^{{\text{el}}^{-}},{E}_{gn}^{{\text{el}}^{+}}\)
calculating \({\overline{\Sigma }}_{gn}^{{\text{el}}^{+}}\) from \({\Sigma }_{gn}^{{\text{el}}^{+}}\); \({R}_{gn}^{{\text{el}}^{+}}={Q}_{g}^{{\text{el}}^{T}}\text{.}{\overline{\Sigma }}_{gn}^{{\text{el}}^{+}}\)
calculation of \(\frac{\partial {\overline{\Sigma }}_{gn}^{{\text{el}}^{+}}}{\partial {\Sigma }_{gn}^{{\text{el}}^{+}}}\) from \({\Sigma }_{gn}^{{\text{el}}^{+}}\); \(D{F}_{gn}^{{\text{el}}^{+}}={Q}_{g}^{{\text{el}}^{T}}\text{.}\frac{\partial {\overline{\Sigma }}_{gn}^{{\text{el}}^{+}}}{\partial {\Sigma }_{gn}^{{\text{el}}^{+}}}\text{.}\frac{\partial {\Sigma }_{gn}^{{\text{el}}^{+}}}{\partial {E}_{gn}^{{\text{el}}^{+}}}\text{.}{Q}_{g}^{\text{el}}\) (depending on option)
Calculation of \(\delta {U}_{n+1}\) by:
\(D{F}_{n}^{+}\text{.}\delta {U}_{n+1}=-{R}_{n}^{+}+{L}^{{\text{meca}}^{+}}\)
Updated:
\(\Delta {U}_{n+1}=\Delta {U}_{n}+\rho \delta {U}_{n+1}\)
If convergence test OK
End Newton: no next step
Sinon
\(n=n+1\)