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インパクトファクター: 1.016 5年インパクトファクター: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN 印刷: 1543-1649
ISSN オンライン: 1940-4352

# International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v4.i1.20
pages 3-18

## Variational Principle and Mechanical Computation for Energy Bands of Periodic Materials

Hongwu Zhang
Department of Engineering Mechanics, Faculty of Vehicle Engineering and Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, P. R. China
Q. Gao
Department of Engineering Mechanics, State Key Laboratory of Structural Analysis of Industrial Equipment, Dalian University of Technology, Dalian 116024, P. R. China
Wanxie Zhong
Dalian University of Technology

### 要約

Based on the Bloch theorem and tight-binding theory, a variational principle is applied to analyze the energy bands of crystals. The stiffness matrix used in the finite element method (FEM) is introduced for the expression of the energy of the unit cell of the crystal, and thus the coordinate transformation technique in FEM is applied in the assembly of the total energy and the stiffness matrix of the crystal. The periodical boundary conditions are given, and the energy bands of the three-dimensional crystal are computed. Using the dynamic substructure model and introducing the dual variables, the energy band analysis of the free vibration of the atomic chain is transformed into symplectic eigenvalue problems. The potential energy and mixed energy are computed by combining segments recursively until the shortest periodical length of the chain is assembled. Finally, the pass-band eigenvalues of the energy bands are calculated using the Wittrick-Williams algorithm. The numerical results are given to illustrate the potential of the theory and algorithm developed.

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