A posteriori constitutive relation error estimators for nonlinear finite element analysis and adaptive control
Résumé
Today quite a lot of a posteriori error estimators are available to control finite element calculations. We describe here error estimators based on constitutive relation residuals which have been developed for 20 years, in particular at Cachan. This approach has a strong physical meaning and is quite general. Different errors on constitutive relation can be easily introduced to measure the quality of finite element computations of plastic and viscoplastic structures whose behavior is described by internal variables. These measures take into account, over the studied time interval, all the classical sources of error involved in the computation: the space discretization (the mesh), the time discretization and the iterative technique used to solve the nonlinear discrete problem. To quantify more specifically each source of error, we introduce quantities called indicators. Numerical experiments show that the indicators are linked to the error in a limit sense. The errors and the indicators may then used to simultaneously adapt the mesh, the time discretization and the stopping criteria to meet a prescribed accuracy. The adaptive control is illustrated on two plane stress problems using the Prandtl-Reuss plastic model and its viscoplastic version.