Acoustic analysis using a mass-redistributed smoothed finite element method with quadrilateral mesh

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.