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Abstract Inspired by significant local distortions found near vacancies in a neutron pair distribution function analysis study (G. King et al.,Inorg. Chem.2012, 51, 13060) of Sr2MSbO5.5(M = Ca and Sr), this computational study finds minimum‐energy structures with these and related distortions using density functional theory (DFT) with the Perdew‐Burke‐Ernzerhof (PBE) functional as implemented in the Vienna Ab Initio Simulations Package (VASP) (G. Kresse and J. Furthmüller,Phys. Rev. B, 1996, 54, 11169; G. Kresse and J. Hafner,Phys. Rev. B, 1993, 47, 558; G. Kresse and J. Furthmüller,Comput. Mater. Sci., 1996, 6, 15). All structures were optimized using the conjugate gradient method. The global minima found for both systems featured trigonal bipyramid SbO5structures and edge sharing with M‐centered polyhedra. However, while calcium ions occupied full and partial octahedra, the larger strontium ions were more commonly found in full and partial pentagonal bipyramids. Molecular dynamics with velocity rescaling at1200K revealed movements of the oxygen vacancy via polyhedral rotations. This work highlights the need to consider both square pyramid to trigonal bipyramid rearrangements around small ions and rotational polyhedral movements in simulating oxygen vacancy conduction in oxygen‐deficient double perovskites.
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On the shift‐invert Lanczos method for the buckling eigenvalue problem
Abstract We consider the problem of extracting a few desired eigenpairs of the buckling eigenvalue problem , whereKis symmetric positive semi‐definite,KGis symmetric indefinite, and the pencil is singular, namely,KandKGshare a nontrivial common nullspace. Moreover, in practical buckling analysis of structures, bases for the nullspace ofKand the common nullspace ofKandKGare available. There are two open issues for developing an industrial strength shift‐invert Lanczos method: (1) the shift‐invert operator does not exist or is extremely ill‐conditioned, and (2) the use of the semi‐inner product induced byKdrives the Lanczos vectors rapidly toward the nullspace ofK, which leads to a rapid growth of the Lanczos vectors in norms and causes permanent loss of information and the failure of the method. In this paper, we address these two issues by proposing a generalized buckling spectral transformation of the singular pencil and a regularization of the inner product via a low‐rank updating of the semi‐positive definiteness ofK. The efficacy of our approach is demonstrated by numerical examples, including one from industrial buckling analysis.
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- Award ID(s):
- 1913364
- PAR ID:
- 10452804
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- International Journal for Numerical Methods in Engineering
- Volume:
- 122
- Issue:
- 11
- ISSN:
- 0029-5981
- Page Range / eLocation ID:
- p. 2751-2769
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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