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An Element Decomposition Method for the Helmholtz Equation

Part of: Partial differential equations, boundary value problems Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  02 November 2016

Gang Wang ,
Xiangyang Cui  and
Guangyao Li
Gang Wang*
Affiliation:
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China Collaborative Innovation Center of Intelligent New Energy Vehicle, Shanghai, 200092, P.R. China
Xiangyang Cui*
Affiliation:
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China Collaborative Innovation Center of Intelligent New Energy Vehicle, Shanghai, 200092, P.R. China
Guangyao Li*
Affiliation:
State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, P.R. China Collaborative Innovation Center of Intelligent New Energy Vehicle, Shanghai, 200092, P.R. China
*
*Corresponding author. Email addresses:wanggangny@163.com (G. Wang), cuixy@hnu.edu.cn (X. Y. Cui), gyli@hnu.edu.cn (G. Y. Li)
*Corresponding author. Email addresses:wanggangny@163.com (G. Wang), cuixy@hnu.edu.cn (X. Y. Cui), gyli@hnu.edu.cn (G. Y. Li)
*Corresponding author. Email addresses:wanggangny@163.com (G. Wang), cuixy@hnu.edu.cn (X. Y. Cui), gyli@hnu.edu.cn (G. Y. Li)
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Abstract

It is well-known that the traditional full integral quadrilateral element fails to provide accurate results to the Helmholtz equation with large wave numbers due to the “pollution error” caused by the numerical dispersion. To overcome this deficiency, this paper proposed an element decomposition method (EDM) for analyzing 2D acoustic problems by using quadrilateral element. In the present EDM, the quadrilateral element is first subdivided into four sub-triangles, and the local acoustic gradient in each sub-triangle is obtained using linear interpolation function. The acoustic gradient field of the whole quadrilateral is then formulated through a weighted averaging operation, which means only one integration point is adopted to construct the system matrix. To cure the numerical instability of one-point integration, a variation gradient item is complemented by variance of the local gradients. The discretized system equations are derived using the generalized Galerkin weakform. Numerical examples demonstrate that the EDM can achieves better accuracy and higher computational efficiency. Besides, as no mapping or coordinate transformation is involved, restrictions on the shape elements can be easily removed, which makes the EDM works well even for severely distorted meshes.

Keywords

Helmholtz equationnumerical methodselement decomposition methodone-point integrationquadrilateral elementdispersion error

MSC classification

Secondary: 65C20: Models, numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N22: Solution of discretized equations 68U20: Simulation
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 5 , November 2016 , pp. 1258 - 1282
Copyright
Copyright © Global-Science Press 2016 

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