Analysis of Energy-Based Blended Quasi-Continuum Approximations

The development of patch test consistent quasi-continuum energies for multidimensional crystalline solids modeled by many-body potentials remains a challenge. The original quasi-continuum energy (QCE) [R. Miller and E. Tadmor, Model. Simul. Mater. Sci. Eng., 17 (2009), 053001] 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 blended quasi-continuum 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 quasi-continuum 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 nonblended QCE energy, which has low order $O(\varepsilon^{1/2})$, where $\varepsilon$ is the atomistic length scale [M. Dobson and M. Luskin, SIAM J. Numer. Anal., 47 (2009), pp. 2455-2475, P. Ming and J. Z. Yang, Multiscale Model. Simul., 7 (2009), pp. 1838-1875], 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 [M. Dobson, M. Luskin, and C. Ortner, J. Mech. Phys. Solids, 58 (2010), pp. 1741-1757]. 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.

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