4. Resolution strategy#

4.1. Newton-Raphson method#

The resolution strategy is nothing more than a simple Newton method. Unlike the case of touch-friction [R5.03.52] or [R5.03.54], there are no fixed point loops or sign fields. The only operation to be performed in addition to Newton’s iterations is therefore to update the internal variable.

For a step of time:
- Newton iterations Calculation of the tangent matrix and the second limb
- End of Newton iterations

Updating the internal variable \(\alpha\)

One could legitimately ask why the internal variable is not updated during Newton’s iterations. In fact, as it is a parameter measuring irreversibility, and determined by a maximum over time, it should only be updated at each converged time step. In fact, in the opposite case, if this parameter exceeds its equilibrium value during a Newton’s iteration, Newton’s algorithm will then be unable to reduce it to find the equilibrium value.

In the one-dimensional case, Newton’s method is an iterative process that makes it possible to approach the zeros of a continuous and differentiable function. We’re getting back to the \(F(x)=0\) resolution. We build a sequence of points \({x}^{k}\) by doing a Taylor expansion of \(F\) in the vicinity of \({x}^{k}\), which gives the first order:

\(F({x}^{k+1})\approx F({x}^{k})+{F}^{\text{'}}({x}^{k})({x}^{k+1}-{x}^{k})\)

By noting \({\mathrm{\delta x}}^{k}={x}^{k+1}-{x}^{k}\) the increment between two successive iterations, the equation linearized at iteration \(k+1\) is then as follows:

\({F}^{\text{'}}({x}^{k}){\mathrm{\delta x}}^{k}=-F({x}^{k})\)

In the case of the finite element method, \({F}^{\text{'}}({x}^{k})\) is similar to the tangent matrix, which can be calculated at each iteration if necessary, \({\mathrm{\delta x}}^{k}\) is the vector of the increments of the nodal unknowns, and \(F({x}^{k})\) is the second member. Note that \({F}^{\text{'}}({x}^{k})\) and \(F({x}^{k})\) only involve quantities from iteration \(k\), which are therefore known quantities.

4.2. Differentiation of cohesive law#

To give an example of the differentiation of the cohesive law coupling the modes, consider the regularized cohesive law CZM_EXP_REG. The differential of \({t}_{c}(\mathrm{〚}u\mathrm{〛})\) will be \(\frac{\mathrm{\partial }{t}_{c}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}\mathrm{\cdot }\mathrm{〚}\delta u\mathrm{〛}\) with:

Gold \(\frac{\mathrm{\partial }{t}_{c}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}\mathrm{=}H({\mathrm{〚}u\mathrm{〛}}_{\text{èq}}\mathrm{-}\alpha )\frac{\mathrm{\partial }{\sigma }_{\mathit{lin}}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}+(1\mathrm{-}H({\mathrm{〚}u\mathrm{〛}}_{\text{èq}}\mathrm{-}\alpha ))\frac{\mathrm{\partial }{\sigma }_{\mathit{dis}}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}+\frac{\mathrm{\partial }{\sigma }_{\mathit{pen}}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}\)

We reuse the expressions of these three partial derivatives that are given in [R7.02.11], with \(\delta =⟦u⟧\). In the expression for \({\sigma }_{\mathit{dis}}\), you must write \(\alpha \mathrm{=}{\mathrm{〚}u\mathrm{〛}}_{\mathit{èq}}\), which thus becomes a variable to be taken into account in the derivation. In practice, in the code, four cases are distinguished for clarity of reading:

- \({⟦u⟧}_{\mathit{èq}}\ge \alpha \mathrm{et}⟦{u}_{n}⟧\ge 0\) (non-contacting dissipative). So we have: \(\frac{\partial {t}_{c}}{\partial ⟦u⟧}={\sigma }_{c}\mathrm{exp}\left(-\frac{{\sigma }_{c}}{{G}_{c}}{⟦u⟧}_{\mathit{èq}}\right)\left(\frac{\text{Id}}{{⟦u⟧}_{\mathit{èq}}}-\frac{⟦u⟧}{{⟦u⟧}_{\mathit{èq}}}\otimes \frac{⟦u⟧}{{⟦u⟧}_{\mathit{èq}}}\left(\frac{{\sigma }_{c}}{{G}_{c}}+\frac{1}{{⟦u⟧}_{\mathit{èq}}}\right)\right)\)
\({⟦u⟧}_{\mathit{èq}}<\alpha \mathrm{et}⟦{u}_{n}⟧<0\) (rubber band contacting). With \(({\tau }_{1},{\tau }_{2})\) a tangential plane base, we have: \(\frac{\partial {t}_{c}}{\partial ⟦u⟧}=\frac{\partial {\sigma }_{\mathit{lin}}(⟦{u}_{\tau }⟧)}{\partial ⟦u⟧}+\frac{\partial {\sigma }_{\mathit{pen}}(⟦{u}_{n}⟧)}{\partial ⟦u⟧}=Cn\otimes n+\frac{{\sigma }_{c}}{\alpha }\mathrm{exp}(\frac{-{\sigma }_{c}}{{G}_{c}}\alpha )\left({\tau }_{1}\otimes {\tau }_{1}+{\tau }_{2}\otimes {\tau }_{2}\right)\)
\({⟦u⟧}_{\mathit{èq}}\ge \alpha \mathrm{et}⟦{u}_{n}⟧<0\) (dissipative contacting). By similar reasoning by replacing \({\sigma }_{\mathit{lin}}\) with \({\sigma }_{\mathit{dis}}\), we get: \(\frac{\partial {t}_{c}}{\partial ⟦u⟧}=Cn\otimes n+\mathrm{exp}\left(\frac{-{\sigma }_{c}}{{G}_{c}}{⟦u⟧}_{\mathit{èq}}\right)\left[\frac{{\sigma }_{c}}{{⟦u⟧}_{\mathit{èq}}}\left({\tau }_{1}\otimes {\tau }_{1}+{\tau }_{2}\otimes {\tau }_{2}\right)-\left(\frac{{\sigma }_{c}^{2}}{{G}_{c}}+\frac{{\sigma }_{c}}{{⟦u⟧}_{\mathit{èq}}}\right)\frac{{⟦u⟧}_{\tau }\otimes {⟦u⟧}_{\tau }}{{{⟦u⟧}_{\mathit{èq}}}^{2}}\right]\)
\({⟦u⟧}_{\mathit{èq}}<\alpha \mathrm{et}⟦{u}_{n}⟧\ge 0\) (non-contacting elastic). \(\frac{\partial {t}_{c}}{\partial ⟦u⟧}=\frac{{\sigma }_{c}}{\alpha }\mathrm{exp}(\frac{-{\sigma }_{c}}{{G}_{c}}\alpha )\text{Id}\)

4.3. Linearization of the problem#

4.3.1. Integral writing for a formulation with regularized cohesive law#

The linear system of the three equations at Newton’s iteration \(k+1\) is written as follows (to avoid making it heavier, the references to Newton’s iteration are omitted, because obviously, the unknowns are noted with a \(\delta\) in front, and the test fields are now noted with a star):

Find \(\left(\mathit{\delta u},{\mathit{\delta \lambda }}_{n},{\mathit{\delta \lambda }}_{\tau }\right)\in {V}_{0}\times H\times H\) such as:

\(\forall \left({u}^{\text{*}},{\lambda }_{n}^{\text{*}},{\lambda }_{\tau }^{\text{*}}\right)\in {V}_{0}\times H\times H\)

Equation of balance	\(\begin{array}{c}{\int }_{\Omega }\sigma (\mathit{\delta u})\mathrm{:}\epsilon ({u}^{\text{}})d\Omega \\ +{\int }_{{\Gamma }_{c}}\left[\frac{\partial {t}_{c,n}}{\partial {⟦u⟧}_{n}}{⟦\delta u⟧}_{n}+\frac{\partial {t}_{c,n}}{\partial {⟦u⟧}_{\tau }}{⟦\delta u⟧}_{\tau }\right]{⟦{u}^{\text{}}⟧}_{n}{\mathit{d\Gamma }}_{c}\\ +{\int }_{{\Gamma }_{c}}\left[\frac{\partial {t}_{c,\tau }}{\partial {⟦u⟧}_{n}}{⟦\delta u⟧}_{n}+\frac{\partial {t}_{c,\tau }}{\partial {⟦u⟧}_{\tau }}{⟦\delta u⟧}_{\tau }\right]{⟦{u}^{\text{}}⟧}_{\tau }{\mathit{d\Gamma }}_{c}\\ =-{\int }_{\Omega }\sigma (u)\mathrm{:}\epsilon ({u}^{\text{}})d\Omega +{\int }_{\Omega }f\cdot {u}^{\text{}}d\Omega +{\int }_{{\Gamma }_{t}}t\cdot {u}^{\text{}}{\mathit{d\Gamma }}_{t}\\ -{\int }_{{\Gamma }_{c}}\left({t}_{c,n}{⟦{u}^{\text{}}⟧}_{n}+{t}_{c,\tau }{⟦{u}^{\text{}}⟧}_{\tau }\right){\mathit{d\Gamma }}_{c}\end{array}\)
Interface: normal part	\({\int }_{\mathit{\Gamma c}}{\lambda }_{n}^{\text{*}}\left({\lambda }_{n}+{\text{δλ}}_{n}-{t}_{c}\cdot n\right){\mathit{d\Gamma }}_{c}=0\)
Interface: tangential part	\({\int }_{\mathit{\Gamma c}}{\lambda }_{\tau }^{\text{*}}\left({\lambda }_{\tau }+\delta {\lambda }_{\tau }-{t}_{c,\tau }\right){\mathit{d\Gamma }}_{c}=0\)

4.3.2. Integral writing for a formulation with mixed cohesive law for quadratic elements#

Find \((\mathit{\delta u},\mathit{\delta \lambda })\mathrm{\in }{V}_{0}\mathrm{\times }H\) such as:

\(\mathrm{\forall }({u}^{\text{*}},{\lambda }^{\text{*}})\mathrm{\in }{V}_{0}\mathrm{\times }H\)

Equation of balance

\(\begin{array}{c}{\int }_{\Omega }\sigma \left(\delta u\right)\mathrm{:}ϵ\left({u}^{\text{*}}\right)d\Omega +{\int }_{\Gamma }\left(\mathit{Id}-r\frac{\partial \delta }{\partial p}\right)\cdot \delta \lambda \cdot ⟦{u}^{\text{*}}⟧d\Gamma \\ +{\int }_{\Gamma }r\left(\mathit{Id}-r\frac{\partial \delta }{\partial p}\right)\cdot ⟦\delta u⟧\cdot ⟦{u}^{\text{*}}⟧d\Gamma \\ =-{\int }_{\Omega }\sigma \left(u\right)\mathrm{:}ϵ\left({u}^{\text{*}}\right)d\Omega +{\int }_{\Omega }f\cdot {u}^{\text{*}}d\Omega +{\int }_{{\Gamma }_{t}}t\cdot {u}^{\text{*}}d\Gamma \\ -{\int }_{{\Gamma }_{c}}\left[\lambda +r\left(⟦u⟧-\delta (p)\right)\right]\cdot ⟦{u}^{\text{*}}⟧d\Gamma \end{array}\)

Interface law

\(\begin{array}{c}{\int }_{{\Gamma }_{c}}\left(1-r\frac{\partial \delta }{\partial p}\right)\cdot ⟦u⟧\cdot {\lambda }^{\text{*}}d\Gamma -{\int }_{\Gamma }\frac{\partial \delta }{\partial p}\cdot \delta \lambda \cdot \lambda \text{*}d\Gamma \\ =-{\int }_{\Gamma }\left(⟦u⟧-\delta \right)\cdot {\lambda }^{\text{*}}d\Gamma \end{array}\)

4.3.3. Integral writing for a formulation with mixed cohesive law for linear elements#

The linear system to be solved for a Newtonian iteration is written as:

Find \(\left(\mathit{\delta u},\delta \mu ,\mathit{\delta w},\mathit{\delta \lambda }\right)\in {V}_{0}\times H\times H\times H\) such as:

\(\forall \left({u}^{\text{*}},{\mu }^{\text{*}},{w}^{\text{*}},{\lambda }^{\text{*}}\right)\in {V}_{0}\times H\times H\times H\)

Equation of balance	\(\begin{array}{c}{\int }_{\Omega }\sigma (\delta u)\mathrm{:}ϵ({u}^{\text{}})d\Omega +{\int }_{\Gamma }\delta \mu \cdot ⟦{u}^{\text{}}⟧d\Gamma \\ =-{\int }_{\Omega }\sigma (u)\mathrm{:}ϵ({u}^{\text{}})d\Omega +{\int }_{{\Gamma }_{g}}g\cdot {u}^{\text{}}d{\Gamma }_{g}-{\int }_{\Gamma }\mu \cdot ⟦{u}^{\text{*}}⟧d\Gamma \end{array}\)
Move jump projection	\({\int }_{\Gamma }\left(⟦\delta u⟧-\delta w\right)\cdot {\mu }^{\text{}}d\Gamma =-{\int }_{\Gamma }\left(⟦u⟧-w\right)\cdot {\mu }^{\text{}}d\Gamma\)
Cohesive constraint	\(\begin{array}{c}-{\int }_{\Gamma }\left[\delta \mu -\frac{\partial {t}_{c}}{\partial (\lambda +rw)}\cdot \left(\delta \lambda +r\delta w\right)\right]\cdot {w}^{\text{}}d\Gamma \\ ={\int }_{\Gamma }\left[\mu -{t}_{c}(\lambda +rw)\right]\cdot {w}^{\text{}}d\Gamma \end{array}\)
Interface law	\(\begin{array}{c}-{\int }_{\Gamma }\left[\frac{\delta \lambda }{r}-\frac{\partial {t}_{c}}{\partial (\lambda +rw)}\cdot \left(\frac{\delta \lambda }{r}+\delta w\right)\right]\cdot {\lambda }^{\text{}}d\Gamma \\ ={\int }_{\Gamma }\frac{\left[\lambda -{t}_{c}(\lambda +rw)\right]}{r}\cdot {\lambda }^{\text{}}d\Gamma \end{array}\)