21 research outputs found
Analysis of Energy-Based Blended Quasicontinuum Approximations
The development of patch test consistent quasicontinuum energies for
multi-dimensional crystalline solids modeled by many-body potentials remains a
challenge. The original quasicontinuum energy (QCE) has been implemented for
many-body potentials in two and three space dimensions, but it is not patch
test consistent. We propose that by blending the atomistic and corresponding
Cauchy-Born continuum models of QCE in an interfacial region with thickness of
a small number of blended atoms, a general quasicontinuum energy (BQCE) can
be developed with the potential to significantly improve the accuracy of QCE
near lattice instabilities such as dislocation formation and motion. In this
paper, we give an error analysis of the blended quasicontinuum energy (BQCE)
for a periodic one-dimensional chain of atoms with next-nearest neighbor
interactions. Our analysis includes the optimization of the blending function
for an improved convergence rate. We show that the strain error for
the non-blended QCE energy (QCE), which has low order
where is the atomistic length scale, can
be reduced by a factor of for an optimized blending function where
is the number of atoms in the blending region. The QCE energy has been
further shown to suffer from a O error in the critical strain at which the
lattice loses stability. We prove that the error in the critical strain of BQCE
can be reduced by a factor of for an optimized blending function, thus
demonstrating that the BQCE energy for an optimized blending function has the
potential to give an accurate approximation of the deformation near lattice
instabilities such as crack growth.Comment: 26 pages, 1 figur
Consistent Energy-Based Atomistic/Continuum Coupling for Two-Body Potentials in One and Two Dimensions
This paper addresses the problem of consistent energy-based coupling of
atomistic and continuum models of materials, limited to zero-temperature
statics of simple crystals. It has been widely recognized that the most
practical coupled methods exhibit large errors on the atomistic/continuum
interface (which are often attributed to spurious forces called "ghost
forces"). There are only few existing works that propose a coupling which is
sufficiently accurate near the interface under certain limitations. In this
paper a novel coupling that is free from "ghost forces" is proposed for a
two-body interaction potential under the assumptions of either (i) one spatial
dimension, or (ii) two spatial dimensions and piecewise affine finite elements
for describing the continuum deformation. The performance of the proposed
coupling is demonstrated with numerical experiments. The coupling strategy is
based on judiciously defining the contributions of the atomistic bonds to the
discrete and the continuum potential energy. The same method in one dimension
has been independently developed and analyzed in Li and Luskin
(arXiv:1007.2336).Comment: 31 page
The role of the patch test in 2D atomistic-to-continuum coupling methods
For a general class of atomistic-to-continuum coupling methods, coupling
multi-body interatomic potentials with a P1-finite element discretisation of
Cauchy--Born nonlinear elasticity, this paper adresses the question whether
patch test consistency (or, absence of ghost forces) implies a first-order
error estimate.
In two dimensions it is shown that this is indeed true under the following
additional technical assumptions: (i) an energy consistency condition, (ii)
locality of the interface correction, (iii) volumetric scaling of the interface
correction, and (iv) connectedness of the atomistic region. The extent to which
these assumptions are necessary is discussed in detail.Comment: Version 2: correction of some minor mistakes, added discussion of
multiple connected atomistic region, minor improvements of styl
Consistent Energy-based Atomistic/Continuum Coupling for Two-body Potentials in Three Dimensions
Very few works exist to date on development of a consistent energy-based
coupling of atomistic and continuum models of materials in more than one
dimension. The difficulty in constructing such a coupling consists in defining
a coupled energy whose minimizers are free from uncontrollable errors on the
atomistic/continuum interface. In this paper a consistent coupling in three
dimensions is proposed. The main achievement of this work is to identify and
efficiently treat a modified Cauchy-Born continuum model which can be coupled
to the exact atomistic model. The convergence and stability of the method is
confirmed with numerical tests.Comment: 29 pages, 1 Matlab code. Typos corrected, exposition improve
Positive definiteness of the blended force-based quasicontinuum method
The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M2AN Math. Model. Numer. Anal., 42 (2008), pp. 113--139] remains an open problem. In one and two dimensions, we show that by blending atomistic and Cauchy--Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width
Positive definiteness of the blended force-based quasicontinuum method
The development of consistent and stable quasicontinuum models for multidimensional crystalline solids remains a challenge. For example, proving the stability of the force-based quasicontinuum (QCF) model [M. Dobson and M. Luskin, M2AN Math. Model. Numer. Anal., 42 (2008), pp. 113--139] remains an open problem. In one and two dimensions, we show that by blending atomistic and Cauchy--Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width