Continuum finite-element region

For hybrid methodologies dealing with dynamical processes, the need to connect different computational models creates a second, very significant, problem Wave reflections may occur at the artificial boundary. Such a reflection arises because of the mismatch in wave spectra between the regions a classical atomistic region, for instance, emits waves that are significantly shorter than those that can be captured by a continuum finite-element (FE) region. 

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 Schematic representation of the coupling between the atomistic and continuum domains. The finite-element method (FEM) is used in the continuum region. The arrows represent the static load on the system. The shaded area indicates the interface domain.

Figure 5 Schematic representation of the continuum/atomistic coupling in the CADD method. In (a) the system is portrayed The atomic region Q,a and the continuum one fie are joined at an interface 00/ defined by a line of atoms (not necessarily straight) and are subject to a prescribed traction, To, and initial displacements, uq. In (b) the continuum/ atomistic interface is displayed in detail Solid circles represent real atoms in the bulk, gray circles represent real atoms on the interface, and unfilled circles represent pad atoms. Pad atoms are situated in the continuum region. The finite-element mapping is shown in the continuum region as well.

In this section, we discuss hybrid methodologies that explore the dynamical evolution of systems composed of a continuum region (usually described using finite-element methods) coupled to a discrete one [modeled using molecular dynamics (MD) algorithms and semiempirical classical potentials]. 

Due to the sheer number, it is not possible to include in this review all of the proposed hybrid methodologies that deal with the coupling of continuum and atomistic regions in dynamical simulations. However, before leaving this section, we want to mention a few more types of methodologies. First, the multigrid methods,of which the recent work of Waisman and Fish is a good example. Also, the MPM/MD method, where the material point method (MPM) is used instead of the finite-element method (FEM) to couple continuum mechanics to conventional molecular dynamics. 

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