期刊
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
卷 342, 期 -, 页码 224-239出版社
ELSEVIER SCIENCE SA
DOI: 10.1016/j.cma.2018.07.042
关键词
Quantum mechanics; Partition of unity method; Bloch boundary conditions; Variational mass lumping; Enrichment functions; Stability
资金
- U.S. Department of Energy [DE-AC52-07NA27344]
Quantum mechanical calculations require the repeated solution of a Schrodinger equation for the wavefunctions of the system, from which materials properties follow. Recent work has shown the effectiveness of enriched finite element type Galerkin methods at significantly reducing the degrees of freedom required to obtain accurate solutions. However, time to solution has been adversely affected by the need to solve a generalized rather than standard eigenvalue problem and the ill-conditioning of associated system matrices. In this work, we address both issues by proposing a stable and efficient orbital-enriched partition of unity method to solve the Schrodinger boundary-value problem in a parallelepiped unit cell subject to Bloch-periodic boundary conditions. In the proposed partition of unity method, the three-dimensional domain is covered by overlapping patches, with a compactly-supported weight function associated with each patch. A key ingredient in our approach is the use of non-negative weight functions that possess the flat-top property, i.e., each weight function is identically equal to unity over some finite subset of its support. This flat-top property provides a pathway to devise a stable approximation over the whole domain. On each patch, we use pth degree orthogonal (Legendre) polynomials that ensure pth order completeness, and in addition include eigenfunctions of the radial Schrodinger equation. Furthermore, we adopt a variational lumping approach to construct a (block-)diagonal overlap matrix that yields a standard eigenvalue problem for which there exist efficient eigensolvers. The accuracy, stability, and efficiency of the proposed method is demonstrated for the Schrodinger equation with a harmonic potential as well as a localized Gaussian potential. We show that the proposed approach delivers optimal rates of convergence in the energy, and the use of orbital enrichment significantly reduces the number of degrees of freedom for a given desired accuracy in the energy eigenvalues while the stability of the enriched approach is fully maintained. (C) 2018 Elsevier B.V. All rights reserved.
作者
我是这篇论文的作者
点击您的名字以认领此论文并将其添加到您的个人资料中。
推荐
暂无数据