A variational formulation of dissipative quasicontinuum methods. (15th December 2016)

• Record Type: Journal Article
• Authors: Rokoš, O.; Beex, L.A.A.; Zeman, J.; Peerlings, R.H.J.
• Is Part Of: International journal of solids and structures. Volume 102/103(2016)
• Page Start: 214
• Page End: 229
• Publication Date: 2016-12-15
• Subjects: Lattice model -- Quasicontinuum method -- Variational formulation -- Plasticity -- Multiscale modelling; Mechanics, Applied -- Periodicals; Structural analysis (Engineering) -- Periodicals; Elastic solids -- Periodicals; Mécanique appliquée -- Périodiques; Constructions, Théorie des -- Périodiques; Solides élastiques -- Périodiques; Elastic solids; Mechanics, Applied; Structural analysis (Engineering); Periodicals; 624.18
• DOI: 10.1016/j.ijsolstr.2016.10.003
• Languages: English
• ISSNs: 0020-7683

Abstract

Highlights: The virtual-power-based QC for regular lattice systems is reformulated from a variational point of view. This framework consistently encompasses the entire class of rate-independent interactions. Relative errors in energies are low (4% error vs. 50-fold reduction in the number of dofs). 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 Roubíček, 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.