Computational Modeling of Discrete Mechanical Systems and Complex Networks: Where We are and Where We are Going

WHERE WE ARE IN MODELING MECHANICAL, TRAFFIC, AND SOCIO-ECONOMICAL DISCRETE SYSTEMS Discrete systems have been firstly introduced in physics (Fisher and Wiodm, 1969; Noor and Nemeth, 1980; Adhikari et al., 1996; Kornyak, 2009) to simulate materials at the microand nano-scales where a continuum description of matter breaks down. The constituents can be atoms or molecules and their interactions are usually modeled by force fields resulting from chemical potentials or weak van der Waals interactions, depending on the type of bonding. These models have been further exploited in mechanics with the aim of predicting macroscopic properties such as strength and toughness from the non-linear interactions taking place between the constituents at the different scales. Pioneering attempts in mechanics to model discrete mechanical systems are those using lattice models (van Mier et al., 1995; Schlangen and Garboczi, 1997) characterized by a network of nodes connected by links modeled by beams. Although proven to suffer from meshdependency, they have been broadly used for studying the non-linear fracture behavior of concrete at the meso-scale. Another line of research regards the generalizedBorn approach (Pellegrini and Field, 2002; Marenich et al., 2013), firstly used in chemistry to model a solute represented as a set of three-dimensional spheres into a continuum medium solvent, then applied in molecular mechanics (called MM/GBSA) to investigate contact and fracture of bodies at the micro-scale. The high-computational power and the development of powerful open source software allow nowadays the design of wide discrete scalable models composed of up to millions of particles or molecules whose equations of motions and mutual interactions are described by highly nonlinear interatomic potential laws. This is the field of molecular dynamics (MD), which led to the development of specific explicit time integration schemes (like the velocity-verlet integration scheme) to solve large systems of equations with a reduced computational cost. Car and Parrinello (1985) proposed a minimization of the total energy of the system by applying a dynamical simulated annealing based on MD. MD computations can also be coupled with continuum simulations by multi-scale methods. Among the many strategies available in the literature, (Shenoy et al., 1999; Knap and Ortiz, 2001) developed an approach based on the Tadmor’s quasi-continuum method (Tadmor et al., 1996) operating on a representative atomistic zone with a reduced number of degrees of freedom. Local minima of the whole system potential energy are determined via the total energy from a cluster of atoms, avoiding the complete calculation of the full atomistic force field. The MD enriched continuum method by Belytschko and Xiao (2003) and Xiao and Belytschko (2003) was also another pioneering approach to couple a potential energy Hamiltonian calculation conducted on a fine scale MD domain with a Lagrangian calculation on a coarse scale continuum domain with an overlapped bridging domain among the two representations. Recently, an implementation of interatomic potential laws within a displacement-based finite element (FE) formulation has also been proposed in Nasdala et al. (2010), with a rigorous implicit solution scheme, aiming at generating models where non-linear discrete and continuous systems can be suitably combined. Discrete systems made of nodes and links are also used in other disciplines than mechanics to model transport or socio-economical networks (Caldarelli and Vespignani, 2007; Whrittle, 2012). Based on a continuum version of traffic conservation along a link, Lighthill and Whitham (1955) and Whitham (1974) and independently Richards (1956) proposed the LWR model where the governing equation describing the dependency of the traffic flux on time and on location along a link is a nonlinear hyperbolic partial differential equation (PDE) analogous to that describing the propagation of the front of a wave inside a medium. For a homogenous link where the velocity of traffic is the same at any location and no shocks on the traffic flow are present, the integration of the LWR PDE leads to a non-linear relation between the traffic flow and the density of vehicles, which fully represents the traffic state. Also, in economics, it is of great interest to quantify the effect of breaking a link over the whole network response by simulating the dynamics of flow redistribution. Again a flow model can be used as suggested in Zhou et al. (2010) to decode a huge human crowd without distinction between