ICAMS / Interdisciplinary Centre for Advanced Materials Simulation


A variational formulation for the passage from atomistic to continuum resolution based on energy-sampling in clusters

Date: 25.01.2008
Place: 7th GAMM-Seminar on Microstructures, Ruhr-Universität Bochum

This contribution presents a novel quasicontinuum (QC) approach aiming at a seamless passage from the atomistic to the continuum description of crystalline solids at zero temperature. The QC method and variants belong to the family of concurrent multiscale methods which perform an upscaling via coarse-graining with adaptive resolution. The present approach heavily draws on the seminal work of Knap and Ortiz [1]. Opposed to Knap and Ortiz, the energy instead of forces is subject to a cluster-based sampling scheme. We show that only the present ansatz endows the QC theory with a variational structure leading to conservative forces and symmetric sti®nesses. Equally, we show that energy-sampling strictly preserves the symmetry in atomic in- teractions, whereas force-sampling does not. This approach allows for the direct application of standard minimization methods and guarantees the ex- istence of an equilibrium state provided that the total potential exhibits a minimum. We choose nanoindentation as a paradigmatic problem for mul- tiscale methods to showcase the performance of the novel QC methodology and to assess it in comparison with lattice statics.We show that the novel QC approach can accurately capture signi¯cant details of the materials response (force-depth curve, load level and locus of dislocation nucleation) but for a small fraction of the fully atomistic variant. Perhaps the most interesting and exciting aspect is a comparison of dislocation microstructures between lattice statics and QC simulations. In view of the nonconvexity of the energy rendering solutions highly nonunique, a good agreement cannot be expected in general, cf. the results in [1]. References 1 Knap, J. and Ortiz, M. (2001): Analysis of the quasicontinuum method. J. Mech. Phys. Solids 49, 1899{1923. 2 Eidel, B. and Stukowski, A. (2007): A variational formulation of the qua- sicontinuum method based on energy sampling in clusters. submitted for publication.

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