Quasi-continuum method

The QC method which presents a relationship between the deformations of a continuum with that of its crystal lattice uses the classical Cau-chy-Bom rule and representative atoms. The quasi-continuum method mixes atomistic-continuum formulation and is based on a finite element discretization of a continuum mechanics variation principle. 

Extensions of the QC Method Because of its versatility, the QC method has been widely applied and, naturally, extended as well. While its original formulation was for zero-temperature static problems only, several groups have modified it to allow for finite-temperature investigations of equilibrium properties as well. A detailed discussion of some of these methodologies is presented in the discussion of finite-temperature methods below. Also, Dupoy et al. have extended it to include a finite-temperature alternative to molecular dynamics (see below). Lastly, the quasi-continuum method has also been coupled to a DFT description of the system in the OFDFT-QC (orbital-free DFT-QC) methodology discussed below. 

Shenoy et al. in 1999 proposed the quasi-continuum Monte Carlo (QCMC) method as a way to extend the quasi-continuum method to the study of equilibrium properties of materials at finite temperature. The objective of this treatment is to construct a computationally manageable expression for a temperature-dependent effective energy for a system maintained at fixed temperature. Such an energy would then be used instead of the zero-temperature effective energy (e.g., Eq. [10]) in a Monte Carlo formulation of the QC method. 

Prediction of Dislocation Nucleation During Nanoindentation by the Orbital-Free Density Functional Theory Local Quasi-Continuum Method. 

Time-stepper-based methods are, in effect, alternative ensembles for performing microscopic (molecular dynamics, kMC, Brownian dynamics) simulations. limovative multiscale/multilevel techniques proposed over the last decade that can be integrated in an equation-free, time-stepper-based framework include the quasi-continuum methods of Phillips and coworkers (Phillips, 2001 Ortiz and Philhps, 1999) and the optimal prediction methods of Chorin and cowoikers (Chorin et al., 1998, 2000) (see the discussion in Kevrekidis et al., 2003). 

Many multiscale methods have been developed across different disciplines. Consequently, much needs to be done in the fundamental theory of multiscale numerical methods that applies across these disciplines. One method is famous in structural materials problems the quasi-continuum method of Tadmor, Ortiz, and Philips. It links the atomistic and continuum models through the finite element method by doing a separate atomist structural relaxation calculation on each cell of the finite element method mesh, rather than using empirical constitutive information. Thus, it directly and dynamically incorporates atomistic-scale information into the deterministic scale finite element method. It has been nsed mainly to predict observed mechanical properties of materials on the basis of their constituent defects. 

In this method, the respective positions of about 100 ions (inner zone) are calculated under various assumptions for the inter-ionic potentials, assuming the rest of the lattice as a quasi-continuum. A polarization per unit cell is attributed from measured values of the static dielectric constant. The configuration within the iimer zone (which may generate a cluster ) is varied until the total energy of the lattice is minimized ... 

Others then went on to study various aspects of quasi-continuum concurrent multiscale methods. Lidirokas et al. [57] studied local stress states around Si nanopixels using this method. Bazant [58] argued that these atomistic-finite element multiscale methods cannot really capture inelastic behavior like fracture because the softening effect requires a regularization of the local region that is not resolved. 

Figure 1.2 illustrates the resonance ) decaying into a quasi-continuum (left side) and into a true continuum (right side). The black rectangle is the useful part of the continuum implied in the dynamics. It corresponds to the wavefunction H (p) (a doorway state in spectroscopy), which is the second term in the method of moments (see Eq. (17)). The physical results are obtained at the limit 5 -> 0 while tF/5 remains constant. The transition rate r to the continuum is equal to... 

Fig. 11. The Mott non-metal to metal transition as a function of the average sei>aration between the atoms of a cluster of 91 atoms. Shown are the computed energies in units of t (logarithmic scale) of the excited electronic states relative to the ground state. Metallic behavior requires that there is a quasi-continuum of states. In the lowest approximation, shown as dashed lines, states where an electron has moved have excess energies of U above the covalent states, t measures the strength of the exchange coupling between adjacent atoms and hence decreases exponentially with the spacing between atoms. When tsiU, the exchange coupling can overcome the Coulomb repulsion. See Refs. 229 and 334 for details of the computational method.

There are numerous coupled atomistic-continuum modeling strategies, for example, peridynamics, a family of quasi-continuum (QC) methods, finite element adaptive techniques (FEAt), coupling-of-length scales (CLS) method, and coupled atomistic-and-discrete-dislocalion (CADD) method. The grand challenge of multiscale modeling is to connect atomic-scale processes to mesoscale and bulk continuum models. 

The quasi-continuum (QC) method was first introduced in 1996 by Tadmor et al. for the investigation of deformation in solids. Ever since, this method has been one of the most powerful and widely applied hybrid methodologies. Its primary applications include the study of dislocation nucleation, cracks, interfaces, grain boundary structure and deformation, nanoindentation phenomena, and so on. Various applications are discussed in more detail below. Since its appearance, the model has been improved and expanded, " and these more complete versions are briefly presented here. If additional details are needed, several specialized reviews are available. 

An alternative approach to the calculation of the Helmholtz free energy is the k-space quasi-harmonic model (QC-QHMK) ° " introduced in 2001 by Aluru et al. This method, still a generalization of the quasi-continuum... 

To bypass the limitations of the Cauchy-Born rule, in 2006, Lu et al. proposed a more involved scheme to couple standard DFT to quasi-continuum calculations. In their method, the part of the system far away from the zone of interest is described using a classical (nonquantum) quasi-continumn approach (see discussion above on QC for details), i.e., considering both local (continuum) and nonlocal (atomistic) terms. Classical potentials (EAM in the applications presented) are used to evaluate the energy within the QC calculations. A third region is considered as well, covering the part of the system that needs a more detailed description. It is in this region that density functional theory is used. 

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