Local coordinate system and transition matrix ================================== It is necessary to build a coordinate system local to the element to define the displacement jump :math:`\delta` (input data for the laws of behavior: see [:external:ref:`R7.02.11 `] and [:external:ref:`R7.01.25 `]). Moreover, the transition matrix :math:`R` is defined from the global coordinate system to the local coordinate system. This part is valid for joint models in pure mechanics and for coupled hydromechanical models. 2D case ------ Let :math:`(X,Y)` be the global benchmark. The direction given by the long sides [:ref:`12 <12>`] and [:ref:`34 <34>`] of the 2D joint element makes it possible to define a local coordinate system :math:`(n,t)` to the joint element (see figure 1): :math:`t=\frac{\overrightarrow{12}}{\parallel \overrightarrow{12}\parallel }`, :math:`n=t\wedge (X\wedge Y)` The transition matrix from the global coordinate system to the local coordinate system is expressed: :math:`R=\left[\begin{array}{cc}{n}_{x}& {n}_{y}\\ {t}_{x}& {t}_{y}\end{array}\right]` 3D case ------ The global benchmark is :math:`(X,Y,Z)`. To build the coordinate system local to the joint element, the covariant base of the corresponding surface element is used. If we note :math:`s({\xi }^{\mathrm{1,}}{\xi }^{2})` the parameterized position of a point on the surface element: :math:`s({\xi }^{\mathrm{1,}}{\xi }^{2})=\sum _{n=1}^{\mathrm{Nb}}{N}_{n}({\xi }^{1},{\xi }^{2}){s}^{n}` where :math:`{N}_{n}` and :math:`{s}^{n}` respectively designate the shape function and the geometric position of the node :math:`n`, and :math:`\mathrm{Nb}` the number of nodes of the surface element. The local covariant base :math:`({a}_{\mathrm{1,}}{a}_{2})` is defined as follows: :math:`{a}_{1}=\frac{\partial s}{\partial {\xi }_{1}}=\sum _{n=1}^{\mathrm{Nb}}\frac{\partial {N}_{n}}{\partial {\xi }^{1}}{s}^{n}` :math:`{a}_{2}=\frac{\partial s}{\partial {\xi }_{2}}=\sum _{n=1}^{\mathrm{Nb}}\frac{\partial {N}_{n}}{\partial {\xi }^{2}}{s}^{n}` These two vectors are in fact vectors tangent to the element at a given point. The local direct orthonormal base :math:`(n,t,\tau )` is then constructed in the following manner: :math:`t=\frac{{a}_{1}}{\parallel {a}_{1}\parallel }` :math:`n=\frac{t\wedge {a}_{2}}{\parallel {a}_{2}\parallel }` :math:`\tau =n\wedge t` The transition matrix from the global coordinate system to the local coordinate system is given by: :math:`R=\left[\begin{array}{ccc}{n}_{x}& {n}_{y}& {n}_{z}\\ {t}_{x}& {t}_{y}& {t}_{z}\\ {\tau }_{x}& {\tau }_{y}& {\tau }_{z}\end{array}\right]` .. _RefHeading__34511398: