Discretized writing ==================== Ratings --------- Remember that elements CABLE_GAINE have 3 knots and that only the end nodes have Lagrange degrees of freedom (cf.). * Displacement shape functions are noted :math:`{n}_{m}` with :math:`m\mathrm{\in }\mathrm{\{}\mathrm{1,}\mathrm{2,}3\mathrm{\}}`. * The shape functions of the Lagrange multipliers are noted :math:`{l}_{p}` with :math:`p\mathrm{\in }\mathrm{\{}\mathrm{1,}2\mathrm{\}}`. We note (the quantities are evaluated at each Gauss point): * :math:`{s}_{g}` the curvilinear abscissa * :math:`{\omega }_{g}` the weight of Gauss points * :math:`{\sigma }_{g}` the stress tensor * :math:`{N}_{g}^{3}` matrix of the values of the shape functions at the Gauss point :math:`g` discretizing the movements of the sheath (3 components) :math:`{N}_{g}^{3}\mathrm{=}(\begin{array}{ccccccccc}{n}_{1}({s}_{g})& 0& 0& {n}_{2}({s}_{g})& 0& 0& {n}_{3}({s}_{g})& 0& 0\\ 0& {n}_{1}({s}_{g})& 0& 0& {n}_{2}({s}_{g})& 0& 0& {n}_{3}({s}_{g})& 0\\ 0& 0& {n}_{1}({s}_{g})& 0& 0& {n}_{2}({s}_{g})& 0& 0& {n}_{3}({s}_{g})\end{array})` * :math:`{N}_{g}^{1}` matrix of the values of the shape functions at the Gauss point :math:`g` discretizing the relative displacements of the cable (1 component): :math:`{N}_{g}^{1}\mathrm{=}(\begin{array}{ccc}{n}_{1}({s}_{g})& {n}_{2}({s}_{g})& {n}_{3}({s}_{g})\end{array})` * :math:`{T}_{g}` the tangent vector to the Gauss point :math:`g` * :math:`\mathrm{\nabla }{N}_{g}^{1}` the derivatives of form functions discretizing the relative movements of the cable * :math:`\mathrm{\nabla }{N}_{g}^{3}` the derivatives of the shape functions discretizing the movements of the sheath * We also ask to simplify the notations: :math:`{B}_{g}\mathrm{=}{T}_{g}^{T}\mathrm{\nabla }{N}_{g}^{3}` * :math:`{L}_{g}` matrix of values of form functions discretizing the Lagrange multiplier at the Gauss point :math:`{L}_{g}\mathrm{=}(\begin{array}{cc}{l}_{1}({s}_{g})& {l}_{2}({s}_{g})\end{array})` * :math:`\mathrm{\{}U\mathrm{\}}` nodal movements (duct + cable) * :math:`\mathrm{\{}{U}_{\mathit{ga}}\mathrm{\}}` the nodal movements of the sheath * :math:`\{{G}_{c}\}` the relative nodal displacement of the cable * :math:`\{\Lambda \}` the nodal Lagrange multiplier * :math:`\mathrm{\{}{F}_{\mathit{ext}}^{\mathit{ga}}\mathrm{\}}` the nodal external force vector, dual of sheath movements * :math:`\{{F}_{\mathit{ext}}^{c}\}` the nodal external force vector, dual of the nodal relative cable displacement * :math:`A` the cable section Discretization of optimality conditions ------------------------------------------ The equations 1.1, 1.2 and 1.3 (Eq. 1.4 being treated at the level of the law of behavior) that characterize the saddle point are discretized in the form: :math:`\mathrm{\sum }{\omega }_{g}({B}_{g}^{T}\mathrm{:}{\sigma }_{g})\mathrm{=}\mathrm{\{}{F}_{\mathit{ext}}^{\mathit{ga}}\mathrm{\}}` :math:`\mathrm{\sum }{\omega }_{g}\mathrm{[}{\mathrm{[}\mathrm{\nabla }{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{:}{\sigma }_{g}+{\mathrm{[}{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{L}_{g}\mathrm{]}\mathrm{\{}\Lambda \mathrm{\}}+r\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}r\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}\mathrm{]}\mathrm{=}\mathrm{\{}{F}_{\mathit{ext}}^{c}\mathrm{\}}` :math:`\mathrm{\sum }{\omega }_{g}\mathrm{[}{\mathrm{[}{L}_{g}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}\mathrm{]}\mathrm{=}\mathrm{\{}0\mathrm{\}}` Internal forces --------------- The notations are introduced: :math:`({f}_{u}^{g})\mathrm{=}({B}_{g}^{T}\mathrm{:}{\sigma }_{g})` :math:`({f}_{{g}_{c}}^{g})\mathrm{=}{\mathrm{[}\mathrm{\nabla }{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{:}{\sigma }_{g}+{\mathrm{[}{N}_{g}^{1}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{L}_{g}\mathrm{]}\mathrm{\{}\Lambda \mathrm{\}}+r\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}r\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}` :math:`({f}_{\lambda }^{g})\mathrm{=}{\mathrm{[}{L}_{g}\mathrm{]}}^{T}\mathrm{\cdot }\mathrm{[}\mathrm{[}{N}_{g}^{1}\mathrm{]}\mathrm{\{}{G}_{c}\mathrm{\}}\mathrm{-}\delta (\mathrm{\{}{G}_{c}\mathrm{\}},\mathrm{\{}\Lambda \mathrm{\}})\mathrm{]}` The contributions to the forces, for :math:`i\mathrm{\in }\mathrm{\{}\mathrm{1,}\mathrm{2,}3\mathrm{\}}`, :math:`m\mathrm{\in }\mathrm{\{}\mathrm{1,}\mathrm{2,}3\mathrm{\}}`, and :math:`p\mathrm{\in }\mathrm{\{}\mathrm{1,}2\mathrm{\}}`, are then written: :math:`{({f}_{u})}_{i,m}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{({f}_{{u}_{\mathit{gaine}}}^{g})}_{i,m}` :math:`{({f}_{{g}_{c}})}_{m}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{({f}_{{u}_{\mathit{cable}}}^{g})}_{m}` :math:`{({f}_{\lambda })}_{p}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{({f}_{\lambda }^{g})}_{p}` where :math:`{({f}_{u}^{g})}_{i,m}` is the :math:`(3(m\mathrm{-}1)+i)` component of :math:`({f}_{u}^{g})`, :math:`{({f}_{{g}_{c}}^{g})}_{m}` is the :math:`m` component of :math:`({f}_{{g}_{c}}^{g})` and :math:`{({f}_{\lambda }^{g})}_{p}` is the :math:`p` component of :math:`({f}_{\lambda }^{g})` Stiffness matrix ------------------- Recall the Gauss one-point notations: * the relative displacement of the cable: :math:`{g}_{g}\mathrm{=}{N}_{g}^{1}\mathrm{\{}{G}_{c}\mathrm{\}}` * cable deformation: :math:`{\varepsilon }_{g}\mathrm{=}{B}_{g}\mathrm{\{}{U}_{\mathit{ga}}\mathrm{\}}+(\mathrm{\nabla }{N}_{g}^{1})\mathrm{\{}{G}_{c}\mathrm{\}}` * the Lagrange multiplier: :math:`{\lambda }_{g}\mathrm{=}({L}_{g})\mathrm{\{}\Lambda \mathrm{\}}` Assuming :math:`{\tau }_{g}\mathrm{=}{\lambda }_{g}+{\mathit{rg}}_{g}`, the following two paragraphs give the contributions to the tangent matrix for the adherent and slippery cases and for the rubbing case. .. _RefNumPara__1499_1597736730: Adherent and slippery case ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In adherent and slippery cases, the tangent matrix is symmetric (minimization of a saddle point): :math:`{K}_{uu}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{B}_{g}^{T}\frac{d\sigma }{d\varepsilon }{B}_{g}` :math:`{K}_{u{g}_{c}}\mathrm{=}{({K}_{{g}_{c}u})}^{T}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}{B}_{g}^{T}\frac{d\sigma }{d\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})` :math:`{K}_{u\lambda }\mathrm{=}{({K}_{\lambda u})}^{T}\mathrm{=}0` :math:`{K}_{{g}_{c}{g}_{c}}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{(\mathrm{\nabla }{N}_{g}^{1})}^{T}\frac{d\sigma }{d\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})+{({N}_{g}^{1})}^{T}r({N}_{g}^{1})\mathrm{-}{({N}_{g}^{1})}^{T}\mathit{r²}\frac{d{\delta }_{g}}{d{\tau }_{g}}({N}_{g}^{1})\mathrm{]}` :math:`{K}_{{g}_{c}\lambda }\mathrm{=}{({K}_{\lambda {g}_{c}})}^{T}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{({N}_{g}^{1})}^{T}({L}_{g})+{({N}_{g}^{1})}^{T}r\frac{d{\delta }_{g}}{d{\tau }_{g}}({L}_{g})\mathrm{]}` :math:`{K}_{\lambda \lambda }\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}\mathrm{-}{({L}_{g})}^{T}\frac{d{\delta }_{g}}{d{\tau }_{g}}({L}_{g})\mathrm{]}` It is specified that :math:`\frac{d{\delta }_{g}}{d{\tau }_{g}}` is obtained from the law of friction behavior. Rubbing case ~~~~~~~~~~~~ In the case of friction, there is a dependence of the slip on the tension in the cable. The following expressions are changed as a result: :math:`{K}_{{g}_{c}u}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{(\mathrm{\nabla }{N}_{g}^{1})}^{T}\frac{d\sigma }{d\varepsilon }{B}_{g}\mathrm{-}r{({N}_{g}^{1})}^{T}\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }{B}_{g}\mathrm{]}` :math:`{K}_{\lambda u}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}\mathrm{-}{({L}_{g})}^{T}\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }{B}_{g}\mathrm{]}` :math:`{K}_{{g}_{c}{g}_{c}}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{(\mathrm{\nabla }{N}_{g}^{1})}^{T}\frac{d\sigma }{d\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})+{({N}_{g}^{1})}^{T}r({N}_{g}^{1})\mathrm{-}{({N}_{g}^{1})}^{T}\mathit{r²}\frac{d{\delta }_{g}}{d{\tau }_{g}}({N}_{g}^{1})\mathrm{-}{({N}_{g}^{1})}^{T}r\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})\mathrm{]}` :math:`{K}_{\lambda {g}_{c}}\mathrm{=}{\mathrm{\sum }}_{g}{\omega }_{g}\mathrm{[}{({L}_{g})}^{T}({N}_{g}^{1})+{({L}_{g})}^{T}r\frac{d{\delta }_{g}}{d{\tau }_{g}}({N}_{g}^{1})\mathrm{-}{({L}_{g})}^{T}\frac{\mathrm{\partial }{\delta }_{g}}{\mathrm{\partial }N}A\frac{\mathrm{\partial }\sigma }{\mathrm{\partial }\varepsilon }(\mathrm{\nabla }{N}_{g}^{1})\mathrm{]}` (The other expressions are unchanged from :ref:`5.4.1 `) It can be seen that the matrix is no longer symmetric.