Dispersion Error Analysis of Stable Node-Based Finite Element Method for the Helmholtz Equation (Communications in Computational Physics, 2018)
Numerical dispersion error is inevitable when the finite element method isemployed to simulate acoustic problems. Studies have shown that the dispersion erroris essentially rooted at the "overly-stiff" property of the standard FEM model. Toreduce the dispersion error effectively, a discrete model that provides a proper softeningeffects is needed. Thus, the stable node-based smoothed finite element method(SNS-FEM) which contains a stable item is presented. In this paper, the SNS-FEM isinvestigated in details with respect to the pollution effect. Different kinds of meshesare employed to analyze the relationship between the dispersion error and the parameterinvolved in the stable item. To ensure the SNS-FEM can be applied to the practicalengineering problems effectively, the relationship is finally constructed based on thehexagonal patch for the commonly used unstructured mesh is very similar to it. Byminimizing the discretization error, an optimal parameter equipped in this novel SNSFEMis formulated. Both theoretical analysis and numerical examples demonstratethat the SNS-FEM with the optimal parameter reduces the dispersion error significantly compared with the FEM and the well performed GLS, especially for mid- andhigh-wave number problems.
Keywords: Acoustic; numerical method; the stable node-based smoothed finite element method (SNS-FEM); dispersion error; hexagonal patches.