دانلود رایگان مقاله جفت کننده ذره گرایانه به زنجیره مبتنی بر انرژی بدون نیروهای سر خوردن هموار

1. Introduction

Atomistic-to-continuum coupling methods (a/c methods) are a class of computational multiscale schemes that combine the accuracy of atomistic models of defects with the computational efficiency of continuum models of elastic far-fields [1–5]. In the present article, we present the first successful implementation of a practical patch test consistent energy based a/c coupling scheme. Previously such schemes were only available for 2-body interactions [6,7]. In recent years a numerical analysis theory of a/c methods has emerged; we refer to [8] for a review. This theory has identified three prototypical classes of a/c schemes: patch test consistent energy-based coupling, force-based coupling (including force-based blending), and energy-based blending. The classical numerical analysis concepts of consistency and stability are applied to precisely quantify the errors committed in these schemes, and clear guidelines are established for their practical implementation including optimisation of approximation parameters. The results in [9,10,8,11,12] indicate that patch test consistent a/c couplings observe (quasi-)optimal error estimates in the energynorm. However, to this date, no general construction and implementation of such schemes has been presented. Instead, one normally compromises by either turning to patch test consistent force-based schemes [13–15,1] or to blending schemes [3,16] which have some control over the consistency error. Quasi-optimal implementations of such schemes are described in [16,15].

4. Conclusion

We have succeeded in presenting the first patch test consistent energy-based atomistic-to-continuum coupling formulation, GRAC, which is applicable to general a/c interface geometries and general (short-ranged) many-body interactions, and demonstrated its potential in a 2D implementation. We have discussed the critical issues of ℓ 1 -minimisation and of stabilisation, and have demonstrated that our final formulations yield an energy-based a/c coupling that is optimal among the energy-based methods we tested, which represent a fairly generic sample, and are even competitive compared against the quasi-optimal force-based coupling schemes. While the construction of the GRAC scheme is involved, it has the advantage that no additional approximation parameters (e.g., the blending function β in the B-QCE and B-QCF schemes [16,15]) must be adapted to the problem at hand. The main challenge that requires additional work is the complexity of the precomputation of the reconstruction parameters, which may become prohibitive for wider interaction stencils, in particular in 3D. It may then become necessary to make further simplifications such as the ones we made in METHOD 2, in order to substantially reduce the computational cost and storage to compute these parameters.