A Variational Formulation of Dissipative Quasicontinuum Methods

TL;DR

This paper introduces a variational formulation of dissipative quasicontinuum methods, enabling energy-based analysis and improved understanding of multiscale simulations involving dissipative interactions in lattice systems.

Contribution

It reformulates the virtual-power-based QC method from a variational perspective, allowing energy minimization principles to be applied to dissipative multiscale modeling.

Findings

Energy consistency verified in examples

Quantified errors in energies and internal variables

Demonstrated equivalence of formulations

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 Roub\'i\v{c}ek, 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.