In this work, an SFEM is proposed for solving acoustic problems by redistributing the entries in the mass matrix to “tune” the balance between “stiffness” and “mass” of discrete equation systems, aiming to minimize the dispersion error. The paper aims to discuss this issue.
This is done by simply shifting the four integration points’ locations when computing the entries of the mass matrix in the scheme of SFEM, while ensuring the mass conservation. The proposed method is devised for bilinear quadratic elements.
The balance between “stiffness” and “mass” of discrete equation systems is critically important in simulating wave propagation problems such as acoustics. A formula is also derived for possibly the best mass redistribution in terms of minimizing dispersion error reduction. Both theoretical and numerical examples demonstrate that the present method possesses distinct advantages compared with the conventional SFEM using the same quadrilateral mesh.
After introducing the mass-redistribution technique, the magnitude of the leading relative dispersion error (the quadratic term) of MR-SFEM is bounded by (5/8), which is much smaller than that of original SFEM models with traditional mass matrix (13/4) and consistence mass matrix (2). Owing to properly turning the balancing between stiffness and mass, the MR-SFEM achieves higher accuracy and much better natural eigenfrequencies prediction than the original SFEM does.
The authors wish to thank the support of National Natural Science Foundation of China (Grant No. 11202074, Grant No. 61232014), and Research Project of State Key Laboratory of Structural Analysis for Industrial Equipment (Grant No. GZ1403). It is also supported by the Recruitment Program of Global Young Experts (No. D1007001), the Fundamental Research Funds for the Central Universities (Grant No. DUT14RC(3)002) and 2014 annual open project of the State Key Laboratory of Ocean Engineering (Grant No. 1401). The work by the fourth author is supported by the United States NSF Grant under the award No. 1214188.
He, Z., Li, G., Zhang, G., Liu, G.-R., Gu, Y. and Li, E. (2015), "Acoustic analysis using a mass-redistributed smoothed finite element method with quadrilateral mesh", Engineering Computations, Vol. 32 No. 8, pp. 2292-2317. https://doi.org/10.1108/EC-10-2014-0219
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