Resolution strategy ======================= Newton-Raphson method ------------------------- The resolution strategy is nothing more than a simple Newton method. Unlike the case of touch-friction [:ref:`R5.03.52 `] or [:ref:`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 :math:`\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 :math:`F(x)=0` resolution. We build a sequence of points :math:`{x}^{k}` by doing a Taylor expansion of :math:`F` in the vicinity of :math:`{x}^{k}`, which gives the first order: :math:`F({x}^{k+1})\approx F({x}^{k})+{F}^{\text{'}}({x}^{k})({x}^{k+1}-{x}^{k})` By noting :math:`{\mathrm{\delta x}}^{k}={x}^{k+1}-{x}^{k}` the increment between two successive iterations, the equation linearized at iteration :math:`k+1` is then as follows: :math:`{F}^{\text{'}}({x}^{k}){\mathrm{\delta x}}^{k}=-F({x}^{k})` In the case of the finite element method, :math:`{F}^{\text{'}}({x}^{k})` is similar to the tangent matrix, which can be calculated at each iteration if necessary, :math:`{\mathrm{\delta x}}^{k}` is the vector of the increments of the nodal unknowns, and :math:`F({x}^{k})` is the second member. Note that :math:`{F}^{\text{'}}({x}^{k})` and :math:`F({x}^{k})` only involve quantities from iteration :math:`k`, which are therefore known quantities. .. _RefNumPara__63020_859574599: 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 :math:`{t}_{c}(\mathrm{〚}u\mathrm{〛})` will be :math:`\frac{\mathrm{\partial }{t}_{c}}{\mathrm{\partial }\mathrm{〚}u\mathrm{〛}}\mathrm{\cdot }\mathrm{〚}\delta u\mathrm{〛}` with: Gold :math:`\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 [:ref:`R7.02.11 `], with :math:`\delta =⟦u⟧`. In the expression for :math:`{\sigma }_{\mathit{dis}}`, you must write :math:`\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: * * :math:`{⟦u⟧}_{\mathit{èq}}\ge \alpha \mathrm{et}⟦{u}_{n}⟧\ge 0` (non-contacting dissipative). So we have: :math:`\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)` * :math:`{⟦u⟧}_{\mathit{èq}}<\alpha \mathrm{et}⟦{u}_{n}⟧<0` (rubber band contacting). With :math:`({\tau }_{1},{\tau }_{2})` a tangential plane base, we have: :math:`\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)` * :math:`{⟦u⟧}_{\mathit{èq}}\ge \alpha \mathrm{et}⟦{u}_{n}⟧<0` (dissipative contacting). By similar reasoning by replacing :math:`{\sigma }_{\mathit{lin}}` with :math:`{\sigma }_{\mathit{dis}}`, we get: :math:`\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]` * :math:`{⟦u⟧}_{\mathit{èq}}<\alpha \mathrm{et}⟦{u}_{n}⟧\ge 0` (non-contacting elastic). :math:`\frac{\partial {t}_{c}}{\partial ⟦u⟧}=\frac{{\sigma }_{c}}{\alpha }\mathrm{exp}(\frac{-{\sigma }_{c}}{{G}_{c}}\alpha )\text{Id}` Linearization of the problem ------------------------- Integral writing for a formulation with regularized cohesive law ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The linear system of the three equations at Newton's iteration :math:`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 :math:`\delta` in front, and the test fields are now noted with a star): Find :math:`\left(\mathit{\delta u},{\mathit{\delta \lambda }}_{n},{\mathit{\delta \lambda }}_{\tau }\right)\in {V}_{0}\times H\times H` such as: :math:`\forall \left({u}^{\text{*}},{\lambda }_{n}^{\text{*}},{\lambda }_{\tau }^{\text{*}}\right)\in {V}_{0}\times H\times H` .. csv-table:: "Equation of balance", ":math:`\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", ":math:`{\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", ":math:`{\int }_{\mathit{\Gamma c}}{\lambda }_{\tau }^{\text{*}}\left({\lambda }_{\tau }+\delta {\lambda }_{\tau }-{t}_{c,\tau }\right){\mathit{d\Gamma }}_{c}=0`" Integral writing for a formulation with mixed cohesive law for quadratic elements ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The linear system of the three equations at Newton's iteration :math:`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 :math:`\delta` in front, and the test fields are now noted with a star): Find :math:`(\mathit{\delta u},\mathit{\delta \lambda })\mathrm{\in }{V}_{0}\mathrm{\times }H` such as: :math:`\mathrm{\forall }({u}^{\text{*}},{\lambda }^{\text{*}})\mathrm{\in }{V}_{0}\mathrm{\times }H` .. csv-table:: "Equation of balance", ":math:`\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", ":math:`\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}`" 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 :math:`\left(\mathit{\delta u},\delta \mu ,\mathit{\delta w},\mathit{\delta \lambda }\right)\in {V}_{0}\times H\times H\times H` such as: :math:`\forall \left({u}^{\text{*}},{\mu }^{\text{*}},{w}^{\text{*}},{\lambda }^{\text{*}}\right)\in {V}_{0}\times H\times H\times H` .. csv-table:: "Equation of balance", ":math:`\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", ":math:`{\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", ":math:`\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", ":math:`\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}`" .. _Ref1192115711: