269 research outputs found
A Γ-Convergence Analysis of the Quasicontinuum Method
We present a Γ-convergence analysis of the quasicontinuum method focused on the behavior of the approximate energy functionals in the continuum limit of a harmonic and defect-free crystal. The analysis shows that, under general conditions of stability and boundedness of the energy, the continuum limit is attained provided that the continuum---e.g., finite-element---approximation spaces are strongly dense in an appropriate topology and provided that the lattice size converges to zero more rapidly than the mesh size. The equicoercivity of the quasicontinuum energy functionals is likewise established with broad generality, which, in conjunction with Γ-convergence, ensures the convergence of the minimizers. We also show under rather general conditions that, for interatomic energies having a clusterwise additive structure, summation or quadrature rules that suitably approximate the local element energies do not affect the continuum limit. Finally, we propose a discrete patch test that provides a practical means of assessing the convergence of quasicontinuum approximations. We demonstrate the utility of the discrete patch test by means of selected examples of application
Formulation and optimization of the energy-based blended quasicontinuum method
We formulate an energy-based atomistic-to-continuum coupling method based on blending the quasicontinuum method for the simulation of crystal defects. We utilize theoretical results from Van Koten and Luskin [32] and Ortner and Van Koten [24] to derive optimal choices of approximation parameters (blending function and finite element grid) for microcrack and di-vacancy test problems and confirm our analytical predictions in numerical tests
Formulation and optimization of the energy-based blended quasicontinuum method
We formulate an energy-based atomistic-to-continuum coupling method based on
blending the quasicontinuum method for the simulation of crystal defects. We
utilize theoretical results from Ortner and Van Koten (manuscript) to derive
optimal choices of approximation parameters (blending function and finite
element grid) for microcrack and di-vacancy test problems and confirm our
analytical predictions in numerical tests
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
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