A variational formulation of dissipative quasicontinuum methods
O. Rokos, L. A. A. Beex, J. Zeman and R. H. J. Peerlings
International Journal of Solids and Structures, 102, 214-229 (2016).

ABSTRACT
Lattice systems and discrete networks with dissipative interactions are
successfully employed as meso-scale models of heterogeneous solids. As the
application scale generally is much larger than that of the discrete links,
physically relevant simulations are computationally expensive. The
QuasiContinuum (QC) method is a multiscale approach that reduces the
computational cost of direct numerical simulations by fully resolving complex
phenomena only in regions of interest while coarsening elsewhere. In previous
work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally
conservative QC methodology was generalized to a virtual-power-based QC approach
that includes local dissipative mechanisms. In this contribution, the
virtual-power-based QC method is reformulated from a variational point of view,
by employing the energy-based variational framework for rate-independent
processes (Mielke and Roubicek, Rate-Independent Systems: Theory and
Application, Springer-Verlag, 2015). By construction it is shown that the QC
method with dissipative interactions can be expressed as a minimization problem
of a properly built energy potential, providing solutions equivalent to those of
the virtual-power-based QC formulation. The theoretical considerations are
demonstrated on three simple examples. For them we verify energy consistency,
quantify relative errors in energies, and discuss errors in internal variables
obtained for different meshes and two summation rules.