4. Resolution algorithm#
4.1. Nonlinear algorithm for solving balance equations#
In the general case of modeling (variable coefficients, desaturation, convection) the variational problem presented above is non-linear with respect to the displacement fields, pressure and temperature. After discretization by finite elements, a nonlinear matrix system is obtained. The resolution matrix also contains a non-symmetric term and is treated as such (no symmetrization of this matrix to use minimum methods). In all modeling cases, the non-linear solver of Code_Aster STAT_NON_LINE based on a Newton-Raphson method, described in [R5.03.01], is used. Its principle is as follows (the equations corresponding to the treatment by dualization of boundary conditions are not explicitly indicated here).
The thermo-poro-mechanical equilibrium equation at the moment \({t}^{\text{+}}\), knowing \(({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})\) at the previous moment, as well as the possible internal variables, is written:
\({F}_{i}({\text{u}}_{\text{+}},{P}_{\text{+}},{T}_{\text{+}})={L}_{e}({t}^{\text{+}})-G({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})\),
To find the solution to this nonlinear equation, we construct a sequence:
initialized by a prediction that gives \(({\text{u}}_{\text{0}},{P}_{\text{0}},{T}_{\text{0}})=({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})+(\Delta {\text{u}}_{\text{0}},\Delta {P}_{\text{0}},\Delta {T}_{\text{0}})\):
\({\mathit{DF}}_{i({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})}\circ (\Delta {\text{u}}_{\text{0}},\Delta {P}_{\text{0}},\Delta {T}_{\text{0}})={L}_{e}({t}^{\text{+}})-{L}_{e}({t}^{\text{-}})\)
corrected by recurrence giving:
\(({\text{u}}_{n+1},{P}_{n+1},{T}_{n+1})=({\text{u}}_{n},{P}_{n},{T}_{n})+(\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\)
\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{F}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})+{L}_{e}({t}^{\text{+}})-G({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})\)
The following notations have been adopted:
\({F}_{i}(\text{u},P,T)\) contains the deformation work, the contributions to the current moment of the hydraulic and thermal dissipation terms expressed within the \(\theta\) ‑method, and the variations in fluid mass supply and entropy;
\({\mathit{DF}}_{i}\) refers to the tangent operator, which may not be updated at each iteration in \(({\text{u}}_{n},{P}_{n},{T}_{n})\), according to a compromise between calculation cost and convergence speed; the convergence is verified by a test on the relative norm of the difference in successive iterations (via the keyword INCO_GLOB_RELA);
\(G({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})\) contains the contributions to the previous moment of the terms of hydraulic and thermal dissipation expressed within the \(\theta\) - method, and of the variations in fluid mass supply and entropy;
\({L}_{e}(t)\) refers to the virtual work of external « dead » forces and external hydraulic and heat inputs expressed by the \(\theta\) -method.
At convergence at iteration \(n+1\), the fields are updated. \(({\text{u}}_{\text{+}},{P}_{\text{+}},{T}_{\text{+}})=({\text{u}}_{n+1},{P}_{n+1},{T}_{n+1})\)
In the current version of algorithm THM, we decided to group all the terms including those due to the following forces and those of the time minus:
By posing:
\(-{R}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})=-{F}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})-G({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})\)
,
So \({\mathit{DF}}_{i}={\mathit{DR}}_{i}\)
we finally have:
\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{R}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})+{L}_{e}({t}^{\text{+}})\)
The general equilibrium algorithm will then be written, for a step of time:
Initializations:
Calculating \({L}_{e}({t}^{\text{+}})\) (option CHAR_MECA)
Calculating \({\mathit{DF}}_{i({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})}\) (option RIGI_MECA - TANG)
Calculation of \((\Delta {\text{u}}_{\text{0}},\Delta {P}_{\text{0}},\Delta {T}_{\text{0}})\) by: \({\mathit{DF}}_{i({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})}\circ (\Delta {\text{u}}_{\text{0}},\Delta {P}_{\text{0}},\Delta {T}_{\text{0}})={L}_{e}({t}^{\text{+}})-{L}_{e}({t}^{\text{-}})\)
Newton balance iterations n
If option FULL_MECA:
Calculating \({\mathit{DF}}_{i({\text{u}}^{\text{+}},{P}^{\text{+}},{T}^{\text{+}})}\) and \(-{R}_{i}({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})\):
Tangent Matrix Update: \({\mathit{DF}}_{i}={\mathit{DF}}_{i({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})}\)
If option RAPH_MECA
Calculation of
Calculation of \((\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\) by:
\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{R}_{i}({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})+{L}_{e}({t}^{\text{+}})\)
Updated:
\(({\text{u}}_{n+1}^{\text{+}},{P}_{n+1}^{\text{+}},{T}_{n+1}^{\text{+}})=({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})+(\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\)
IF convergence test*OK
End Newton: no next step
Otherly
n = n+1
4.2. Loop on the elements, the Gauss points#
As in all finite element codes, the terms are calculated by looping over the elements and looping over the Gauss points:
\(\begin{array}{c}{R}_{i}({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})=\sum _{\mathit{el}}\sum _{g}{w}_{g}^{\mathit{el}}{R}_{gi}^{\mathit{el}}({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})\\ {\mathit{DF}}_{i({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})}=\sum _{\mathit{el}}\sum _{g}{w}_{g}^{\mathit{el}}{\mathit{DF}}_{gi({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})}^{\mathit{el}}\end{array}\)
Note: \(\{{X}^{\mathit{el}}\}\) the vector of nodal unknowns, on a finite element el
for example \(\{{X}^{\mathit{el}}\}=\begin{array}{c}u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\\ u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\\ u\\ v\\ w\\ {p}_{1}\\ {p}_{2}\\ T\end{array}\begin{array}{c}\begin{array}{c}\\ \\ \\ \\ \\ \end{array}\}\text{noeud 1}\\ \begin{array}{c}\\ \\ \\ \\ \\ \end{array}\}\text{noeud 2}\\ \begin{array}{c}\\ \\ \\ \\ \\ \end{array}\}\text{noeud 3}\end{array}\)
In this paragraph, to simplify the presentation, we assume that we are dealing with a finite element supporting displacement ddls, two pressure ddls, and one temperature ddl.
Note \(\{{\text{E}}_{g}^{\mathit{el}}\}\) the vector of generalized deformations at the Gauss point g of the element*el*
For example:
\(\{{\text{E}}_{g}^{\mathit{el}}\}=\left\{\begin{array}{c}\text{u}\\ \epsilon (\text{u})\\ {p}_{1}\\ \nabla {p}_{1}\\ {p}_{2}\\ \nabla {p}_{2}\\ T\\ \nabla T\end{array}\right\}\)
We note \(\{{\Sigma }_{g}^{\mathit{el}}\}\) the generalized stress vector for the Gauss point g of the element*el*
For example, and always in the most complete case:
\(\{{\Sigma }_{g}^{\mathit{el}}\}=\left\{\begin{array}{c}\underline{\underline{\sigma \text{'}}}\\ {\sigma }_{p}\\ {m}_{1}^{1}\\ {\text{M}}_{1}^{1}\\ {h}_{\mathrm{1m}}^{1}\\ {m}_{1}^{2}\\ {\text{M}}_{1}^{2}\\ {h}_{\mathrm{1m}}^{2}\\ {m}_{2}^{1}\\ {\text{M}}_{2}^{1}\\ {h}_{\mathrm{2m}}^{1}\\ {m}_{2}^{2}\\ {\text{M}}_{2}^{2}\\ {h}_{\mathrm{2m}}^{2}\\ Q\text{'}\\ \text{q}\end{array}\right\}\)
The finite element routines calculate the matrix: \({[B]}_{g}^{\mathit{el}}\) defined by:
\(\{{E}_{g}^{\mathit{el}}\}={[B]}_{g}^{\mathit{el}}\{X\}\)
The algorithm will then become:
Initializations:
Calculating \({L}_{e}({t}^{\text{+}})\) (option CHAR_MECA)
Calculating \({\mathit{DF}}_{i({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})}\) (option RIGI_MECA - TANG)
Calculation of \((\Delta {\text{u}}_{\text{0}},\Delta {P}_{\text{0}},\Delta {T}_{\text{0}})\) by: \({\mathit{DF}}_{i({\text{u}}_{\text{-}},{P}_{\text{-}},{T}_{\text{-}})}\circ (\Delta {\text{u}}_{\text{0}},\Delta {P}_{\text{0}},\Delta {T}_{\text{0}})={L}_{e}({t}^{\text{+}})-{L}_{e}({t}^{\text{-}})\)
Newton balance iterations n
Loop elements el
Gauss point loop*g
Calculation \({[B]}_{g}^{\mathit{el}}\)
Calculus \(\left\{{E}_{g}^{\mathit{el}\text{-}}\right\}={\left[B\right]}_{g}^{\mathit{el}}\left\{{X}^{\text{-}}\right\}\) and \(\left\{{E}_{\mathit{gn}}^{\mathit{el}\text{+}}\right\}={\left[B\right]}_{g}^{\mathit{el}}\left\{{X}_{n}^{\text{+}}\right\}\)
Calculation \(\left\{{\Sigma }_{\mathit{gn}}^{\mathit{el}\text{+}}\right\}\), \(-{R}_{\mathit{ig}}^{\mathit{el}}({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})\) and \({\mathit{DF}}_{gi({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})}^{\mathit{el}}\) (depending on options) from:
\(\left\{{E}_{g}^{\mathit{el}\text{-}}\right\},\left\{{E}_{g}^{\mathit{el}\text{+}}\right\},\left\{{\Sigma }_{g}^{\mathit{el}\text{-}}\right\},\left\{{E}_{g}^{\mathit{el}\text{+}}\right\},{\left[B\right]}_{g}^{\mathit{el}}\)
Calculation of \((\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\) by:
\({\mathit{DF}}_{i}\circ (\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})=-{R}_{i}({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})+{L}_{e}({t}^{\text{+}})\)
Updated:
\(({\text{u}}_{n+1}^{\text{+}},{P}_{n+1}^{\text{+}},{T}_{n+1}^{\text{+}})=({\text{u}}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})+(\delta {\text{u}}_{n+1},\delta {P}_{n+1},\delta {T}_{n+1})\)
IF convergence test*OK
End Newton: no next step
Otherly
n = n+1
4.3. Vectors and matrices depending on the options: routine EQUTHM#
The central framed part of the algorithm presented above is carried out by a generic routine EQUTHM. A graphical representation of the call of this routine is given in the appendix.
This routine is configured according to the equations present (mechanical, hydraulic with 1 or 2 pressures, thermal). The work done by this routine is set by the option.
The term \(-{R}_{i}({\text{u}}_{n},{P}_{n},{T}_{n})\) will be calculated by options RAPH_MECA and FULL_MECA. This term includes the following volume forces: it will be considered that the following forces will be considered to be integrated into options RAPH_MECA, FULL_MECA, and RIGI_MECA_TANG. In the case where the user data does not have volume forces, the vector \({\text{F}}^{{m}^{\text{+}}}\) will simply be zero.
The presentations made in the following two paragraphs are made in the most general case where we have a mechanical equation, two hydraulic equations and a thermal equation. Routine EQUTHM will calculate or not the various terms depending on how one describes the equations present.
The indices g and el are now omitted, but it is clear that what is described applies to each Gauss point in each element.
Note:
As part of saturated permanent HM modeling, a routine similar to routine EQUTHM has been implemented (routine EQUTHP), which takes into account the specificities of the permanent modeling equations (no mass input) .
4.3.1. Residue or nodal force: options RAPH_MECA and FULL_MECA#
The terms of the variational formulation will be distributed according to the following principle:
If \({\text{E}}_{g}^{\text{*}\mathit{el}}\) designates a virtual deformation field, \({\text{E}}_{g}^{\text{*}\mathit{elT}}=(\text{v},\epsilon (\text{v}),{\pi }_{\mathrm{1,}}\nabla {\pi }_{\mathrm{1,}}{\pi }_{\mathrm{2,}}\nabla {\pi }_{\mathrm{2,}}\tau ,\nabla \tau )\) calculated from a virtual nodal displacement vector: \(\left\{{X}^{\text{*}\mathit{el}}\right\}\)
\({\text{E}}_{g}^{\text{*}\mathit{elT}}\cdot {R}_{\mathit{ig}}^{\mathit{el}}({\text{u}}_{\text{+}},{P}_{\text{+}},{T}_{\text{+}})={R}_{1}\text{v}+{R}_{2}\epsilon (\text{v})+{R}_{3}{\pi }_{1}+{R}_{4}\nabla {\pi }_{1}+{R}_{5}{\pi }_{2}+{R}_{6}\nabla {\pi }_{2}+{R}_{7}\tau +{R}_{8}\nabla \tau\)
We then have:
Index |
R |
associated with |
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From there we will define the nodal residue vector \(\left\{{V}_{g}^{\mathit{el}}\right\}\) such that:
\({\left\{{X}^{\text{*}\mathit{el}}\right\}}^{T}\cdot \left\{{V}_{g}^{\mathit{el}}\right\}={E}_{g}^{\text{*}{\mathit{el}}^{T}}\cdot {R}_{\mathit{ig}}^{\mathit{el}}({\text{u}}_{\text{+}},{P}_{\text{+}},{T}_{\text{+}})\)
\(\left\{{V}_{g}^{\mathit{el}}\right\}\) will be calculated by:
\(\left\{{V}_{g}^{\mathit{el}}\right\}={\left[{B}_{g}^{\mathit{el}}\right]}^{T}\cdot \{R\}\)
Note:
In the context of saturated permanent HM modeling, routine EQUTHP never combines the R3 and R5 terms.
4.3.2. Loading: options CHAR_MECA#
This chapter is only here for the record because routine EQUTHM won’t take care of these terms.
The terms of the variational formulation will be distributed according to the following principle:
\({\text{E}}_{g}^{\text{*}{\mathit{el}}^{T}}\cdot {L}_{\mathit{eg}}^{\mathit{el}}(t\text{+})={L}_{1}\text{v}+{L}_{2}\epsilon (\text{v})+{L}_{3}{\pi }_{1}+{L}_{4}\nabla {\pi }_{1}+{L}_{5}{\pi }_{2}+{L}_{6}\nabla {\pi }_{2}+{L}_{7}\tau +{L}_{8}\nabla \tau\)
Index |
L |
element type |
associated with |
1 |
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edge |
\(\text{v}\) |
3 |
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edge |
\({\pi }_{1}\) |
5 |
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edge |
\({\pi }_{2}\) |
7 |
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volume edge |
\(\tau\) |
4.3.3. Tangent operator: options FULL_MECA, RIGI_MECA_TANG#
Note on matrix notations:
In the following, if \(X\) designates a component vector \({X}^{i}\) and \(Y\) a component vector \({Y}^{i}\) , :math:`left[frac{partial X}{partial Y}right]`*will refer to a matrix whose element\((\mathit{ligne}\mathrm{:}i,\mathit{colonne}\mathrm{:}j)`*is*is* :math:\)frac{partial {X}^{i}}{partial {Y}^{j}}` .
To calculate the tangent operator \({\mathit{DF}}_{i}\), we will calculate the following quantities:
\(\left[\text{DRDE}\right]\) =
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DR7U |
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Where we noted:
\(\begin{array}{c}\mathit{DRiU}=\underline{\frac{\partial {F}_{i}}{\partial u}}\\ \mathit{DRiE}=\underline{\underline{\frac{\partial {F}_{i}}{\partial \epsilon }}}\\ \mathit{DRiP1}=\frac{\partial {F}_{i}}{\partial {p}_{1}}\\ \mathit{DRiP2}=\frac{\partial {F}_{i}}{\partial {p}_{2}}\\ \mathit{DRiGP1}=\underline{\frac{\partial {F}_{i}}{\partial \nabla {p}_{1}}}\\ \mathit{DRiGP2}=\underline{\frac{\partial {F}_{i}}{\partial \nabla {p}_{2}}}\\ \mathit{DRiT}=\frac{\partial {F}_{i}}{\partial T}\\ \mathit{DRiGT}=\underline{\frac{\partial {F}_{i}}{\partial \nabla T}}\end{array}\)
To do these calculations it is considered that the laws of behavior will provide, for the corresponding options, all of the following derivatives:
\(\left[\text{DSDE}\right]=\left[\begin{array}{cccccccc}\frac{\partial \sigma \text{'}}{\partial \text{u}}& \frac{\partial \sigma \text{'}}{\partial \epsilon }& \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 \text{u}}& \frac{\partial {\sigma }_{p}}{\partial \epsilon }& \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 \text{u}}& \frac{\partial {m}_{1}^{1}}{\partial \epsilon }& \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 {\text{M}}_{1}^{1}}{\partial \text{u}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \epsilon }& \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial T}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla T}\\ \frac{\partial {m}_{1}^{2}}{\partial \text{u}}& \frac{\partial {m}_{1}^{2}}{\partial \epsilon }& \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 {\text{M}}_{1}^{2}}{\partial \text{u}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \epsilon }& \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial T}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla T}\\ \frac{\partial {m}_{2}^{1}}{\partial \text{u}}& \frac{\partial {m}_{2}^{1}}{\partial \epsilon }& \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 {\text{M}}_{2}^{1}}{\partial \text{u}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \epsilon }& \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial T}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla T}\\ \frac{\partial {m}_{2}^{2}}{\partial \text{u}}& \frac{\partial {m}_{2}^{2}}{\partial \epsilon }& \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 {\text{M}}_{2}^{2}}{\partial \text{u}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \epsilon }& \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial T}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \text{u}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \epsilon }& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla T}\\ \frac{\partial Q\text{'}}{\partial \text{u}}& \frac{\partial Q\text{'}}{\partial \epsilon }& \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 \text{q}}{\partial \text{u}}& \frac{\partial \text{q}}{\partial \epsilon }& \frac{\partial \text{q}}{\partial {p}_{1}}& \frac{\partial \text{q}}{\partial \nabla {p}_{1}}& \frac{\partial \text{q}}{\partial {p}_{2}}& \frac{\partial \text{q}}{\partial \nabla {p}_{2}}& \frac{\partial \text{q}}{\partial T}& \frac{\partial \text{q}}{\partial \nabla T}\end{array}\right]\)
In fact, in these expressions, the derivatives with respect to u are all zero, but we keep the writing given the definition of matrices \({[B]}_{g}^{\mathit{el}}\) that we adopted.
The call to the laws of behavior will provide the pieces of the matrix \(\left[\text{DSDE}\right]\) according to the equations present:
\(\left[\text{DMECDE}\right]=\left[\begin{array}{c}\frac{\partial \sigma \text{'}}{\partial \epsilon }\\ \frac{\partial {\sigma }_{p}}{\partial \epsilon }\end{array}\right];\left[\text{DMECP1}\right]=\left[\begin{array}{cc}\frac{\partial \sigma \text{'}}{\partial {p}_{1}}& \frac{\partial \sigma \text{'}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\sigma }_{p}}{\partial {p}_{1}}& \frac{\partial {\sigma }_{p}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DMECP2}\right]=\left[\begin{array}{cc}\frac{\partial \sigma \text{'}}{\partial {p}_{2}}& \frac{\partial \sigma \text{'}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\sigma }_{p}}{\partial {p}_{2}}& \frac{\partial {\sigma }_{p}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DMECDT}\right]=\left[\begin{array}{cc}\frac{\partial \sigma \text{'}}{\partial T}& \frac{\partial \sigma \text{'}}{\partial \nabla T}\\ \frac{\partial {\sigma }_{p}}{\partial T}& \frac{\partial {\sigma }_{p}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP11DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \epsilon }\end{array}\right];\left[\text{DP11P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP11P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP11DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{1}}{\partial T}& \frac{\partial {m}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{1}}{\partial T}& \frac{\partial {\text{M}}_{1}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{1}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP12DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \epsilon }\end{array}\right];\left[\text{DP12P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP12P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP12DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{1}^{2}}{\partial T}& \frac{\partial {m}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{1}^{2}}{\partial T}& \frac{\partial {\text{M}}_{1}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{1m}}^{2}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP21DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \epsilon }\end{array}\right];\left[\text{DP21P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP21P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP21DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{1}}{\partial T}& \frac{\partial {m}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{1}}{\partial T}& \frac{\partial {\text{M}}_{2}^{1}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{1}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DP22DE}\right]=\left[\begin{array}{c}\frac{\partial {m}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial \epsilon }\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \epsilon }\end{array}\right];\left[\text{DP22P1}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{1}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{1}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{1}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DP22P2}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {m}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial {p}_{2}}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla {p}_{2}}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial {p}_{2}}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DP22DT}\right]=\left[\begin{array}{cc}\frac{\partial {m}_{2}^{2}}{\partial T}& \frac{\partial {m}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {\text{M}}_{2}^{2}}{\partial T}& \frac{\partial {\text{M}}_{2}^{2}}{\partial \nabla T}\\ \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial T}& \frac{\partial {h}_{\mathrm{2m}}^{2}}{\partial \nabla T}\end{array}\right]\)
\(\left[\text{DTDE}\right]=\left[\begin{array}{c}\frac{\partial Q\text{'}}{\partial \epsilon }\\ \frac{\partial \text{q}}{\partial \epsilon }\end{array}\right];\left[\text{DTDP1}\right]=\left[\begin{array}{cc}\frac{\partial Q\text{'}}{\partial {p}_{1}}& \frac{\partial Q\text{'}}{\partial \nabla {p}_{1}}\\ \frac{\partial \text{q}}{\partial {p}_{1}}& \frac{\partial \text{q}}{\partial \nabla {p}_{1}}\end{array}\right];\left[\text{DTDP2}\right]=\left[\begin{array}{cc}\frac{\partial Q\text{'}}{\partial {p}_{2}}& \frac{\partial Q\text{'}}{\partial \nabla {p}_{2}}\\ \frac{\partial \text{q}}{\partial {p}_{2}}& \frac{\partial \text{q}}{\partial \nabla {p}_{2}}\end{array}\right];\left[\text{DTDT}\right]=\left[\begin{array}{cc}\frac{\partial Q\text{'}}{\partial T}& \frac{\partial Q\text{'}}{\partial \nabla T}\\ \frac{\partial \text{q}}{\partial T}& \frac{\partial \text{q}}{\partial \nabla T}\end{array}\right]\)
Moreover, by deriving the expression of the residue in relation to the constraints, we define:
\(\left[\text{DRDS}\right]=\left[\begin{array}{cccccccccccccccc}\frac{\partial {R}_{1}}{\partial \sigma \text{'}}& \frac{\partial {R}_{1}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{1}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{1}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{1}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{1}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{1}}{\partial Q\text{'}}& \frac{\partial {R}_{1}}{\partial \text{q}}\\ \frac{\partial {R}_{2}}{\partial \sigma \text{'}}& \frac{\partial {R}_{2}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{2}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{2}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{2}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{2}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{2}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{2}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{2}}{\partial Q\text{'}}& \frac{\partial {R}_{2}}{\partial \text{q}}\\ \frac{\partial {R}_{3}}{\partial \sigma \text{'}}& \frac{\partial {R}_{3}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{3}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{3}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{3}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{3}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{3}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{3}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{3}}{\partial Q\text{'}}& \frac{\partial {R}_{3}}{\partial \text{q}}\\ \frac{\partial {R}_{4}}{\partial \sigma \text{'}}& \frac{\partial {R}_{4}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{4}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{4}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{4}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{4}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{4}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{4}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{4}}{\partial Q\text{'}}& \frac{\partial {R}_{4}}{\partial \text{q}}\\ \frac{\partial {R}_{5}}{\partial \sigma \text{'}}& \frac{\partial {R}_{5}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{5}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{5}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{5}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{5}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{5}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{5}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{5}}{\partial Q\text{'}}& \frac{\partial {R}_{5}}{\partial \text{q}}\\ \frac{\partial {R}_{6}}{\partial \sigma \text{'}}& \frac{\partial {R}_{6}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{6}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{6}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{6}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{6}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{6}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{6}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{6}}{\partial Q\text{'}}& \frac{\partial {R}_{6}}{\partial \text{q}}\\ \frac{\partial {R}_{7}}{\partial \sigma \text{'}}& \frac{\partial {R}_{7}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{7}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{7}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{7}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{7}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{7}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{7}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{7}}{\partial Q\text{'}}& \frac{\partial {R}_{7}}{\partial \text{q}}\\ \frac{\partial {R}_{8}}{\partial \sigma \text{'}}& \frac{\partial {R}_{8}}{\partial {\sigma }_{p}}& \frac{\partial {R}_{8}}{\partial {m}_{1}^{1}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{1}^{1}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{1m}}^{1}}& \frac{\partial {R}_{8}}{\partial {m}_{1}^{2}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{1}^{2}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{1m}}^{2}}& \frac{\partial {R}_{8}}{\partial {m}_{2}^{1}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{2}^{1}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{2m}}^{1}}& \frac{\partial {R}_{8}}{\partial {m}_{2}^{2}}& \frac{\partial {R}_{8}}{\partial {\text{M}}_{2}^{2}}& \frac{\partial {R}_{8}}{\partial {h}_{\mathrm{2m}}^{2}}& \frac{\partial {R}_{8}}{\partial Q\text{'}}& \frac{\partial {R}_{8}}{\partial \text{q}}\end{array}\right]\)
Since all these quantities are not necessarily calculated, we will note:
\(\left[\text{DR1DS}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {\sigma }^{\text{'}\text{+}}}& \frac{\partial {R}_{1}}{\partial {\sigma }_{p}^{\text{+}}}\end{array}\right];\left[\text{DR1P11}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{1\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1\text{+}}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{1\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{1\text{+}}}& \frac{\partial {R}_{1}}{\partial {\sigma }_{\mathrm{1m}}^{1\text{+}}}\end{array}\right]\)
\(\left[\text{DR1P12}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{2\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2\text{+}}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{1}^{2\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{1}^{2\text{+}}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{1m}}^{2\text{+}}}\end{array}\right]\)
\(\left[\text{DR1P21}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{1\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1\text{+}}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{1\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{1\text{+}}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{1\text{+}}}\end{array}\right]\)
\(\left[\text{DR1P22}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{2\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2\text{+}}}\end{array}\right]\mathit{ou}\left[\begin{array}{ccc}\frac{\partial {R}_{1}}{\partial {m}_{2}^{2\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{M}}_{2}^{2\text{+}}}& \frac{\partial {R}_{1}}{\partial {h}_{\mathrm{2m}}^{2\text{+}}}\end{array}\right]\)
\(\left[\text{DR1DT}\right]=\left[\begin{array}{cc}\frac{\partial {R}_{1}}{\partial {Q}^{\text{'}\text{+}}}& \frac{\partial {R}_{1}}{\partial {\text{q}}^{\text{+}}}\end{array}\right]\)
Likewise:
\(\left[\text{DR8DS}\right],\left[\text{DR8P11}\right],\left[\text{DR8P12}\right],\left[\text{DR8P21}\right],\left[\text{DR8P22}\right],\left[\text{DR8DT}\right]\)
It is then clear that:
\(\left[\text{DRDE}\right]=\left[\text{DRDS}\right]\cdot \left[\text{DSDE}\right]\)
And the contribution of the Gauss point to the tangent matrix \({{\text{DF}}_{g}^{\mathit{el}}}_{i({u}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})}\) is obtained by:
\(\left[{{\text{DF}}_{g}^{\mathit{el}}}_{i({u}_{n}^{\text{+}},{P}_{n}^{\text{+}},{T}_{n}^{\text{+}})}\right]={\left[{\text{B}}_{g}^{\mathit{el}}\right]}^{T}\cdot \left[\text{DRDE}\right]\cdot \left[{\text{B}}_{g}^{\mathit{el}}\right]\)