Abstract : Background
We propose an a posteriori estimator of the error of hyper-reduced predictions for elastoviscoplastic problems. For a given fixed mesh, this error estimator aims to forecast the validity domain in the parameter space, of hyper-reduction approximations. This error estimator evaluates if the simulation outputs generated by the hyper-reduced model represent a convenient approximation of the outputs that the finite element simulation would have predicted. We do not account for the approximation error related to the finite element approximation upon which the hyper-reduction approximation is introduced.
Methods
We restrict our attention to generalized standard materials. Upon use of incremental variational principles, we propose an error in constitutive relation. This error is split into three terms including a tailored norm of the hyper-reduction approximation error. This error norm is defined by using the convexity of an incremental potential introduced to state the constitutive equations. The second term of the a posteriori error is related to the stress recovery technique that generates stresses fulfilling the finite element equilibrium equations. The last term is a coupling term between the hyper-reduction approximation error at each time step and the errors committed before this time step. Unfortunately, this last term prevents error certification. In this paper, we restrict our attention to outputs extracted by a Lipschitz function of the displacements.
Results
In the proposed numerical examples, we show very good preliminary results in predicting the validity domain of hyper-reduction approximations. The average computational time of the predictions obtained by hyper reduction, is accelerated by a factor of 6 compared to that of finite element simulations. This speed-up incorporates the computational time devoted to the error estimation.
Conclusions
The numerical implementation of the proposed error estimator is straightforward. It does not require the computation of the incremental potential. In the numerical results, the estimated validity domain of hyper-reduced approximations is inside the reference validity domain. This paper is a first attempt for a posteriori error estimation of hyper-reduction approximations.
