  The introduction of Physics-informed Neural Networks (PINNs) has led to an
increased interest in deep neural networks as universal approximators of PDEs
in the solid mechanics community. Recently, the Deep Energy Method (DEM) has
been proposed. DEM is based on energy minimization principles, contrary to PINN
which is based on the residual of the PDEs. A significant advantage of DEM, is
that it requires the approximation of lower order derivatives compared to
formulations that are based on strong form residuals. However both DEM and
classical PINN formulations struggle to resolve fine features of the stress and
displacement fields, for example concentration features in solid mechanics
applications. We propose an extension to the Deep Energy Method (DEM) to
resolve these features for finite strain hyperelasticity. The developed
framework termed mixed Deep Energy Method (mDEM) introduces stress measures as
an additional output of the NN to the recently introduced pure displacement
formulation. Using this approach, Neumann boundary conditions are approximated
more accurately and the accuracy around spatial features which are typically
responsible for high concentrations is increased. In order to make the proposed
approach more versatile, we introduce a numerical integration scheme based on
Delaunay integration, which enables the mDEM framework to be used for random
training point position sets commonly needed for computational domains with
stress concentrations. We highlight the advantages of the proposed approach
while showing the shortcomings of classical PINN and DEM formulations. The
method is offering comparable results to Finite-Element Method (FEM) on the
forward calculation of challenging computational experiments involving domains
with fine geometric features and concentrated loads.
