The quasicontinuum (QC) method was developed with the goal of greatly reducing the cost of molecular dynamics (MD) simulations. In the QC method, a subset of particles is selected to be representative particles to which the remaining particles are slaved via piecewise linear interpolation. In regions where the deformation is smooth, relatively few representative particles are use whereas, in regions where defects occur, one may want most or all the particles to be representative particles.
Although it is true that the QC approximation drastically reduces the number of equations and degrees of freedom, both proportional to the number of representative particles, the work needed to assemble the equations still depends on the total number of particles.
Several studies have been devoted to further reducing the complexity of the QC method. The simplest such approximation is a node-based summation rule; unfortunately, in some cases, this approach results in a rank-deficiency problem. To overcome this problem, a cluster summation method (QC-CS) was developed that resulted in a complexity, in the case of short-range interactions, depending only on the number of representative particles. Unfortunately, for the long-range interaction case, the
complexity, still depended on the total number of particles.
We have developed a new approximation of the QC method that replaces the sums appearing in that method by shorter sums using a "quadrature"-rule. We have shown analytically and computationally that, for one-dimensional implementations, the new method requires less work than the QC-CS method for the short-range interaction case and, unlike the QC-CS method, the new method has complexity depending only on the number of representative particles for the long-range interaction case.
Obviously, for QC-type methods, the higher the density of representative particles, the closer one is to a full atomistic model; this allows for the resolution of defects at the atomistic level simply by increasing that density where needed. On the other hand, away from defects, one can use relative few representative atoms which is how QC-type methods effect their efficiency gains over, e.g., MD. Thus QC and its variants, including the new QC-QR method, are in of themselves multiscale methods, with the local scale determined by the local spacing between representative particles.
Current work on the QC-QR method focuses on extending the analyses and implementation of the method to 2D and 3D problems and also on the use of the QC-QR method as a replacement for MD in atmomistic to continuum coupled materials model, with the peridynamics model replacing classical elasticity (CE). In the latter effort, 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.
Comparison the cluster summation and "quadrature"-rule approximations to the QC method for short-range interactions
Comparison QC method with the "quadrature"-rule approximation to the QC method for long-range interactions
Y. Zhang and M. Gunzburger; Quadrature-rule type approximations to the quasicontinuum method for short and long-range interatomic interactions, to appear in Comp. Meth. Appl. Mech. Engrg.
Last updated: 12/29/09 by Max Gunzburger