Error Control for Molecular Statics Problems

Abstract

In this paper, we present an extension of goal-oriented error estimation and adaptation to the simulation of multi-scale problems of molecular statics. Computable error estimates for the quasicontinuum method are developed with respect to specific quantities of interest and an adaptive strategy based upon these estimates is proposed for error control. The theoretical results are illustrated on a nanoindentation problem in which the quantity of interest is the force acting on the indenter. The promising capability of such error estimates and adaptive procedure for the solution of multi-scale problems is demonstrated on numerical examples.

An analysis of the effect of ghost force oscillation on quasicontinuum error

“…In such simulations, a few localized regions require the accuracy and high computational expense of atomistic scale resolution, but the rest of the material has a slowly varying deformation gradient which can be more efficiently computed using the continuum approximation without loss of the desired accuracy. Adaptive algorithms have been developed for QC to determine which regions require the accuracy of atomistic modeling and how to coarsen the finite element mesh in the continuum region [1][2][3][16][17][18]20]. The atoms retained in the atomistic region and the atoms at nodes of the piecewise linear finite element mesh in the continuum region are collectively denoted as representative atoms.…”

A new adaptive modeling strategy based on optimal control for atomic-to-continuum coupling simulations

“…Numerous works on modeling error estimation and modeling adaptation have been performed in recent years (see e.g. [19][20][21]4,23]). We shall recall here only the main aspects of the theory.…”

“…We can, for instance, consider a larger particle domain X d than that used to compute (u, w, k) and solve for the Arlequin adjoint solution ðp u ;p w ;p k Þ in this new configuration. The latter solution could then be employed to derive an approximate adjoint solutionp 0 of p 0 using (19) (full details of the approach can be found in [21,4]). We now present an adaptive algorithm that aims at controlling the modeling error E within some preset tolerance c tol .…”

On the application of the Arlequin method to the coupling of particle and continuum models

“…One major difference between the two papers is that we are interested here in coupling highly heterogeneous particle models with homogeneous continuum models. Our ultimate objective in the investigation of such coupling algorithms is to extend ideas of goal-oriented error estimation and adaptation [15,16] to control the size and position of the overlapping region so as to deliver highly accurate simulations.…”