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  4. Novel Quasicontinuum Theory and Computationalethod for Coupling Atoms and Finite Elements
 
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Novel Quasicontinuum Theory and Computationalethod for Coupling Atoms and Finite Elements

Date Issued
2008
Date
2008
Author(s)
Chang, Shu-Wei
URI
http://ntur.lib.ntu.edu.tw//handle/246246/187588
Abstract
The theoretical and computational limitations of continuous description and atomistic modeling on capturing phenomena at the micro- or nano-scale have called for the need to fuse atoms with finite elements. Atomistic modeling has been used to address a wide variety of deformation processes in solids in the past decades. Nevertheless, the length scale restriction of atomistic modeling has long been a substantial obstacle in making useful prediction. In the past few years, a very promising method called the quasicontinuum (QC) method has been developed to circumvent this length scale problem. y using kinematic constraints through finite element interpolation, the QC method with kinematic constraints (QCKC) allows for developing fully atomistic scale resolution near defects while exploiting coarser description further away to reduce redundant degrees of freedom. Furthermore, utilization of kinematic constraints makes it possible to develop approximated energy and force formulations in an efficient way. Two approximated methods have been developed. One is based on energy approximation (QCE) and the other on force approximation (QCF). The QCE was developed by Tadmor et al. (1996) and Shenoy et al. (1999a) by using the Cauchy-Born rule to approximate the energy functional. The QCF was developed by Knap and Ortiz (2001) by introducing the cluster summation rule to approximate force calculations. The objective of this thesis is to revisit current practice in QC, analyze variants of QC to reproduce QCKC and derive novel formulations for improvement. or the interface of atomistic region and finite elements, improperly applying Cauchy-Born rule and cluster summation rule at the interface have been identified and the deficiency is removed by introducing transition entities to the QC method. New description of the force formulation that allows transition elements to correctly transmit inhomogeneity from the atomistic to the continuum regions is proposed. The formulation is verified through reproducing the QCKC.he errors of force formulation for finite element region of both the QCE and QCF methods are analyzed. For the QCE method, errors are mainly governed by the size of elements but not by the variation of deformation gradients. The error mainly results from the inherent “local” assumption of the Cauchy-Born rule and thus it is unavoidable. For the QCF method, errors are mainly affected by the ratio of the number of sampling atoms to the number of total slaved atoms. To control the error within 10% the sampling ratio should be greater than 90%, an indication of a very inefficient scheme. Generally speaking, the QCE method is only suitable for large elements while the QCF method is only suitable for small elements. y considering the property of central symmetry potential and by analyzing the distribution of atomic forces of represented atoms within an element, a novel summation rule based on decomposition of force contributions from slaved atoms in exterior cluster is developed. The new summation rule approximates the forces for finite element region accurately and efficiently. It is shown that for a constant sampling cutoff (20 angstroms), the error of computing force is controlled under 10% for all sizes of elements. Therefore, the force formulation using the summation rule proposed herein reproduces the QCKC method efficiently.inally, QC with accurate and efficient formulations to reproduce the QCKC method at the atomistic, transition and finite element regions is compiled together and used to study indentation size effect of Au thin film with spherical indenters. The results show that for a spherical indenter, the hardness is not affected by indentation depth but by the sphere radius. The hardness decreases with the increase of the sphere radius.
Subjects
quasicontinuum
transition
molecular dynamics
finite elements
ghost force
Type
thesis
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