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Mathematical Problems in Engineering
Volume 2015, Article ID 260641, 8 pages
Research Article

High-Order Spectral Finite Elements in Analysis of Collinear Wave Mixing

1School of Civil Engineering, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
2Key Laboratory of Highway Engineering of Sichuan Province, Southwest Jiaotong University, Chengdu, Sichuan 610031, China
3College of Aerospace Engineering, Chongqing University, Chongqing 400044, China
4Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

Received 8 October 2015; Revised 25 November 2015; Accepted 26 November 2015

Academic Editor: Roman Lewandowski

Copyright © 2015 Changfa Ai et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Implementing collinear wave mixing techniques with numerical methods to detect acoustic nonlinearity due to damage and defects is of vital importance in nondestructive examination engineering. However, numerical simulations in existing literatures are often limited due to the compromise between computational efficiency and accuracy. In order to balance the contradiction, spectral finite element (abbreviated as SFE) with 3 × 3 and 8 × 6 nodes is developed to simulate collinear wave mixing for 1D and 2D cases in this study. The comparisons among analytical solutions, experiments, finite element method (FEM), and spectral finite element method are presented to validate the feasibility, efficiency, and accuracy of the proposed SFEs. The results demonstrate that the proposed SFEs are capable of increasing computational efficiency by as much as 14 times while maintaining the same accuracy in comparison with FEM. In addition, five 3 × 3 nodes’ SFEs or one 8 × 6 nodes’ SFE per the shortest wavelength is sufficient to capture mixing waves. Finally, the proposed 8 × 6 nodes’ SFE is recommended for collinear wave mixing to detect damage, which can offer more accuracy with similar efficiency compared to 3 × 3 nodes’ SFE.