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Due to the strong reliance on element quality, there exist some inherent shortcomings of the traditional finite element method (FEM). The model of FEM behaves overly stiff, and the solutions of automated generated linear elements are generally of poor accuracy about especially gradient results. The proposed cell-based smoothed point interpolation method (CS-PIM) aims to improve the results accuracy of the thermoelastic problems via properly softening the overly-stiff stiffness.

This novel approach is based on the newly developed G space and weakened weak (w²) formulation, and of which shape functions are created using the point interpolation method and the cell-based gradient smoothing operation is conducted based on the linear triangular background cells.

Owing to the property of softened stiffness, the present method can generally achieve better accuracy and higher convergence results (especially for the temperature gradient and thermal stress solutions) than the FEM does by using the simplest linear triangular background cells, which has been examined by extensive numerical studies.

The CS-PIM is capable of producing more accurate results of temperature gradients as well as thermal stresses with the automated generated and unstructured background cells, which make it a better candidate for solving practical thermoelastic problems.

It is the first time that the novel CS-PIM was further developed for solving thermoelastic problems, which shows its tremendous potential for practical implications.

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.