A multiscale quasicontinuum method for dissipative lattice models and discrete networks

Abstract

Lattice models and discrete networks naturally describe mechanical phenomena at the mesoscale of fibrous materials. A disadvantage of lattice models is their computational cost. The quasicontinuum (QC) method is a suitable multiscale approach that reduces the computational cost of lattice models and allows the incorporation of local lattice defects in large-scale problems. So far, all QC methods are formulated for conservative (mostly atomistic) lattice models. Lattice models of fibrous materials however, often require non-conservative interactions. In this article, a QC formulation is derived based on the virtual-power of a non-conservative lattice model. By using the virtual-power statement instead of force-equilibrium, errors in the governing equations of the force-based QC formulations are avoided. Nevertheless, the non-conservative interaction forces can still be directly inserted in the virtualpower QC framework. The summation rules for energy-based QC methods can still be used in the proposed framework as shown by two multiscale examples.

Higher‐order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending

“…Note also that if p and q run over all components of u in a QC method, this results in an energetically-consistent method (as was made clear in [19,31,32]), in contrast to a force-based QC method such as the one of [33]. The fact that all b lattice interactions must be visited to construct the governing equations (i.e.…”

“…Furthermore, conforming triangulations are generally used to achieve a smooth transition from fully resolved domains to coarse-grained domains (see e.g. [17,[19][20][21][22][23][24][25][26][31][32][33]). …”

“…Note that not all QC approaches are equation-free; see e.g. [19][20][21] for equation-free methods and e.g. [17,22] for non-equation-free formulations.…”

“…The recent work of [27] on a (conservative) lattice for human erythrocyte membranes forms one of the few exceptions. Recently however, a virtual-power-based QC method [19] was constructed that allows it to be used for non-conservative lattice models (i.e. including dissipation).…”

“…As dissipation is included, a standard QC method cannot be used and the virtual-power-based QC variant [19] is adopted. The two multiscale examples that are used to evaluate the different higher-order QC methods focus on uniaxial deformation and pure bending.…”