Composite multiscale models (consisting of two or more individual models and a coupling strategy that connects them) have been proposed to handle a wide range of scales. If we define ``a wide range of scales'' by comparing to the range of scales over which molecular dynamics (MD) is tractable and classical elasticiy (CE) is valid, it is natural to seek composite or multiscale atomistic-to-continuum (AtC) models that involve MD and CE. Direct coupling of MD and CE has been studied extensively in molecular statics and quasi-static settings.
Many different approaches to AtC coupling have been proposed. Broadly speaking, most AtC coupling methods can be divided into two categories: use domain decomposition for uncoupling the component models or blend the two models within an overlap region. For example, alternating Schwarz methods have been applied to the AtC setting and error and convergence analyses were obtained. Blending methods are multi-model multiscale methods in which two models are coupled by blending the energies or forces in an overlap region. Of course, blending methods may also be viewed as a class of overlapping domain decomposition methods. For example, a partition-of-unity function defined over the overlap region can be used to define a global energy that is the weighted sum of the energies of the two models, thus avoiding over counting the energy in the overlap region.
In our work on AtC coupling methods, we focused on constructing a blended MD-CE model explicitly designed to satisfy Newton's third law. This is a desirable property of a blended model, given that the underlying atomistic and continuum models themselves obey Newton's third law. Fully three-dimensional numerical experiments were conducted coupling an atomistic EAM aluminum model with a finite element model of the same material. Although the numerical results obtained using the model appear reasonable, the error in patch test results were several orders of magnitude larger than can be explained by roundoff errors. We were able to prove that this method is in fact not consistent: it fails a simple linear patch test.
Ideally, one would like a model that is both consistent and satisfies Newton's third law. In our subsequent work, we sought such a model. We first extended our previous results by developing a canonical blended AtC model for MD-CE coupling. This framework led us to realize there are four classes of blending models. We can choose to blend the atomistic and continuum material models directly or define the blended material models by changing their internal definitions. This leads to four possible combinations, including our previous method and other previously defined ``bridging domain'' methods as well as some new methods. The results of our analyses are striking: no blending method simultaneously satisfies Newton's third law and consistency. Thus, although some of the models we developed and studied yielded reasonable results and are potentially useful in some practical nanoscale settings, question remain regarding their stability and robustness.
We also explored several issues (e.g., application of constraints, choice of blending function, and grid resolution effects) related to the implementation of AtC blending methods.
We also investigated the application of optimization-based domain decomposition methods to AtC coupling problems that allows for a black-box implementation using existing codes for each of the component models. In this approach, an optimization problem is solved in which the discrepancy between the solutions of the two models in an overlap region or along an interface boundary is minimized.
Current work on the AtC coupling methods using the quadrature rule-quasicontinuum method as a replacement for MD method as a replacement for MD and the peridynamics model as a replacement for CE in AtC models. Our goal is to help eliminate the gap that seems to exist between the validity of CE and the tractability of MD in AtC coupled models based on those two constituents.
J. Fish, M. Nuggehally, M. Shephard, C. Picu, S. Badia, M. Parks, and M. Gunzburger ; Concurrent AtC coupling based on a blend of the continuum stress and the atomistic force, Comp. Meth. Appl. Mech. Engrg. 196 2007, 4548-4560.
S. Badia, P. Bochev, J. Fish, M. Gunzburger, R. Lehoucq, M. Nuggehally, and M. Parks; A force-based blending model for atomistic-to-continuum coupling, Inter. J. Multiscale Comp. Engrg. 5 2007, 387-406.
S. Badia, P. Bochev, R. Lehoucq, and M. Parks; Bridging methods for coupling atomistic and continuum models, Large Scale Scientific Computing: 6th International Conference, LSSC 2007, Sozopol, Bulgaria, June 5-9, 2007, Springer, Berlin, 2008, 16-27.
S. Badia, M. Parks, P. Bochev, M. Gunzburger, and R. Lehoucq; On atomistic-to-continuum coupling by blending, Multiscale Model. Simul. 7 2008, 381-406.
J. Reese and M. Gunzburger; 1D atomistic-to-continuum coupling via optimization, Proc. 4th Inter. Conf. Multiscale Materials Modeling, 53-56.
P. Bochev, R. Lehoucq, M. Parks, S. Badia, and M. Gunzburger; Blending methods for coupling atomistic and continuum models, Multiscale Methods: Bridging the Scales in Science and Engineering, Oxford, Oxford, 165-189, 2010.
P. Seleson and M. Gunzburger; Bridging methods for atomistic-to-continuum coupling and their implementation, Comm. Comp. Phys. 7 2010, 831-876.
Last updated: 12/29/09 by Max Gunzburger