We consider the Hyper-reduction technique [1], in the framework of parametric structural dynamic problems. Efficient parametric models in hyper-reduction of structural design allow to act towards real time computations. The reduced coordinates are solutions of equations restricted to a subdomain, named the reduced integration domain (RID). Then one significant question is: Can we control the accuracy of the reduced-basis and the RID so they can be used online to reproduce dynamic solutions, which are different than the ones computed to generate the reduced-basis? Concerning the mathematical formulation of the Hyper-reduction, we propose a time-continuous Petrov-Galerkin one. This formulation is more general than the Galerkin projection one [2], since we are able to write our hyper-reduced order model through a reduced basis which is not necessarily orthogonal. We attest the following assumption on the Hyper-reduction of a structural dynamic problem, in this case: While dynamic solutions are regular via parametric evolution and no bifurcation occurs, then the parametric online error relative to the Hyper-reduction by reference reduced elements is also regular with respect to parametric evolution. In particular, a reference hyper-reduction is justified when variations occur within a viscoelastic dynamic problem, for the viscosity parameter. So to answer the preceding question, we show a mathematical and sharp a priori upper bound of the parametric online error relative to a reference hyper-reduction. We show mathematically that the parametric accuracy of the Hyper-reduction is improved when considering enriched reduced elements in association with the reference snapshots expanded by the ones of the parametric derivative of the viscoelastic dynamic solution at the same reference viscosity value [3, 4]. All these findings are generalisations to hyperbolic equations of the ones developed in [4] for parabolic equations in the context of fluid mechanics. Numerical validations of these theoretical results will be shown during the conference for academic applications in structural design
