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