.. _linear_elasticity-elastic_shifted_periodic: linear_elasticity/elastic_shifted_periodic.py ============================================= **Description** Linear elasticity with linear combination constraints and periodic boundary conditions. The linear combination constraints are used to apply periodic boundary conditions with a shift in the second axis direction. Find :math:\ul{u} such that: .. math:: \int_{\Omega} D_{ijkl}\ e_{ij}(\ul{v}) e_{kl}(\ul{u}) = - \int_{\Gamma_{bottom}} \ul{v} \cdot \ull{\sigma} \cdot \ul{n} \;, \quad \forall \ul{v} \;, \ul{u} = 0 \mbox{ on } \Gamma_{left} \;, u_1 = u_2 = 0 \mbox{ on } \Gamma_{right} \;, \ul{u}(\ul{x}) = \ul{u}(\ul{y}) \mbox{ for } \ul{x} \in \Gamma_{bottom}, \ul{y} \in \Gamma_{top}, \ul{y} = P_1(\ul{x}) \;, \ul{u}(\ul{x}) = \ul{u}(\ul{y}) + a(\ul{y}) \mbox{ for } \ul{x} \in \Gamma_{near}, \ul{y} \in \Gamma_{far}, \ul{y} = P_2(\ul{x}) \;, where .. math:: D_{ijkl} = \mu (\delta_{ik} \delta_{jl}+\delta_{il} \delta_{jk}) + \lambda \ \delta_{ij} \delta_{kl} \;, and the traction :math:\ull{\sigma} \cdot \ul{n} = \bar{p} \ull{I} \cdot \ul{n} is given in terms of traction pressure :math:\bar{p}. The function :math:a(\ul{y}) is given (the shift), :math:P_1 and :math:P_2 are the periodic coordinate mappings. View the results using:: \$ ./postproc.py block.vtk --wireframe -b --only-names=u -d'u,plot_displacements,rel_scaling=1,color_kind="scalars",color_name="von_mises_stress"' .. image:: /../doc/images/gallery/linear_elasticity-elastic_shifted_periodic.png :download:source code  .. literalinclude:: /../examples/linear_elasticity/elastic_shifted_periodic.py