  The purpose of this work is to present an improved energy conservation method
for hyperelastodynamic contact problems based on specific normal compliance
conditions. In order to determine this Improved Normal Compliance (INC) law, we
use a Moreau--Yosida $\alpha$-regularization to approximate the unilateral
contact law. Then, based on the work of Hauret--LeTallec
\cite{hauret2006energy}, we propose in the discrete framework a specific
approach allowing to respect the energy conservation of the system in adequacy
with the continuous case. This strategy (INC) is characterized by a conserving
behavior for frictionless impacts and admissible dissipation for friction
phenomena while limiting penetration. Then, we detail the numerical treatment
within the framework of the semi-smooth Newton method and primal-dual active
set strategy for the normal compliance conditions with friction. We finally
provide some numerical experiments to bring into light the energy conservation
and the efficiency of the INC method by comparing with different classical
methods from the literature throught representative contact problems.
