### Introduce A New Multiscale Method: Bridging Scale Method

By Albert C. To

In the last decade, great effort has been put forth to develop simulation methods which can resolve material behavior across the multiple length scales present in many physical systems (Liu et al., 2004). The bulk of this work has been motivated by new design paradigms that require an understanding of the phenomena at the continuum level brought on by the underlying micro- or nanoscale physics, for applications such as MEMS/NEMS devices or novel alloy design. Recently, attention has also been directed toward biological systems, many of which exhibit a strongly coupled hierarchical nature. In systems design, the traditional engineering simulation approach has made extensive use of continuum level modeling via empirical constitutive relations and numerical methods such as the Finite Element Method (FEM) while atomistic level methods such as molecular dynamics (MD) have been used to study detailed phenomena such as dislocation nucleation and propagation or the details of failure mechanisms.

Figure 1. Schematic of domain type considered in bridging scale method

Each of these approaches has limitations: in the case of FEM, the resolution is limited to the size of the continuum element for which the constitutive relation employed remains valid. For MD, the enormous number of degrees of freedom required for a continuum level simulation makes its use intractable for system sizes greater than hundred of nanometers. Thus, it is desirable to seek a method which can be used over large length scales, but maintain atomistic or near-atomistic resolution in regions of interest.

Figure 2. Bridging scale simulation of a Mode I crack propagation. (Left panel: bridging scale method; right panel: full molecular dynamics simulation) .

Figure 3. Bridging scale simulation of Mode II crack propagation. Center region with high resolution contains MD, lower resolution region denotes FEM only region. Note that the waves propagate naturally within the overlap region, however at the THK interface, the fine scale part is dissipated and the coarse scale propagates.

Bridging scale method is a dynamic multiscale method that couples atomic modeling with continuum modeling without introducing spurious reflections at the interface (Wagner and Liu 2003, Park et al 2005a, Park et al 2005b). The method assumes molecular dynamics and finite elements (FE) solutions exist in the whole domain, but MD is performed only in a small localized domain as shown in Fig. 1. By minimizing the Lagrangians of the MD and FE, a linear projector is derived to decompose the solutions into fine and coarse scale solutions. At each time step, the projector projects the fine scale part of the MD solutions to the FE region and is added to the FE solutions to obtain the total solutions. At the interface, a time history kernel is derived to filter out the high frequency content of outgoing phonons such that spurious reflections will be minimized. This approach has a distinct advantage that the continuum domain do not need to be meshed down to the atomic resolution near the atomistic/continuum interface, saving the need for special mesh design. The formulation also allows the finite element solutions to be run at much larger time step than at in MD, thus making the method computationally efficient. Bridging scale method has been used to simulate elastic wave propagation and dynamic brittle fractures and has shown very good performances compared to full molecular dynamics simulations (Wagner and Liu 2003, Park et al 2005a, Park et al 2005b).

Figure 4. Bridging scale simulation of a Mode II crack propagation. Left panel shows the final mesh distortion after the solid has completely fractured. Right panel shows the complete fracture.

Figure 2 shows a dynamic crack propagation in a brittle solid, and the results for bridging scale method and molecular dynamics simulation are almost identical. Figure 3 demonstrates the wave propagation/dissipation abilities of the THK interface. Notice that the coarse scale (long wavelength) waves propagate out of the high resolution region, while the fine scale (short wavelength) waves are dissipated at the interface such that there are no spurious reflections. Figure 4 demonstrates the ability for BSM to capture fracture without resorting to remeshing or special continuum fracture elements. This aspect is a unique strength of BSM. It is due to the use of a projection of interatomic forces in the FEM+MD overlap region to obtain the FEM internal forces. This projection avoids the computation of the deformation gradient or Jacobian for the finite elements in the overlap region, thus allowing mesh distortions that would not be possible in a standard FEM simulation.

The cited publications can be found in this webpage:

http://www.tam.northwestern.edu/wkl/_wkl/htm/publications.htm

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