Mathematics – Numerical Analysis
Scientific paper
2010-08-12
Mathematics
Numerical Analysis
26 pages, 1 figure
Scientific paper
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 $k$ 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 $\ell^2$ strain error for the non-blended QCE energy (QCE), which has low order $\text{O}(\epsilon^{1/2})$ where $\epsilon$ is the atomistic length scale, can be reduced by a factor of $k^{3/2}$ for an optimized blending function where $k$ is the number of atoms in the blending region. The QCE energy has been further shown to suffer from a O$(1)$ 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 $k^2$ 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.
Koten Brian Van
Luskin Mitchell
No associations
LandOfFree
Analysis of Energy-Based Blended Quasicontinuum Approximations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Analysis of Energy-Based Blended Quasicontinuum Approximations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Analysis of Energy-Based Blended Quasicontinuum Approximations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-587531