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 hyperreduction 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.

International audienceThe following study is motivated by the need to capture the elasto-viscoplastic behavior of a “real” industrial power module lead-free solder joint. In this work, we carried out a numerical design of experiments in order to forecast the ability of an experimental bending system to identify the specimen material properties. As a proof of principle, the micro-mechanical elastic behavior of power module copper substrates was then characterized thanks to the development of an innovative in-situ micro-mechanical bending test under an optical profilometer. An inverse Finite-Element Method has been applied in order to identify the material properties of test specimens designed directly out of industrial assemblies and not from bulk solder for good representativity. The results show that identified copper Young's modulus values are lower than that of a bulk material. It will be defined as such in the next identificatio n step targeting the solder joint