Dispersion Error Analysis of Stable Node-Based Finite Element Method for the Helmholtz Equation

Abstract

Numerical dispersion error is inevitable when the finite element method is employed to simulate acoustic problems. Studies have shown that the dispersion error is essentially rooted at the "overly-stiff" property of the standard FEM model. To reduce the dispersion error effectively, a discrete model that provides a proper softening effects 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 is investigated in details with respect to the pollution effect. Different kinds of meshes are employed to analyze the relationship between the dispersion error and the parameter involved in the stable item. To ensure the SNS-FEM can be applied in the practical engineering problems effectively, the relationship is finally constructed based on the hexagonal patch for the commonly used unstructured mesh is very similar to it. By minimizing the discretization error, an optimal parameter equipped in this novel SNS-FEM is formulated. Both theoretical analysis and numerical examples demonstrate that the SNS-FEM with the optimal parameter reduces the dispersion error significantly compared with the FEM and the well performed GLS, especially for mid-and high-wave number problems. AMS subject classifications: 35J05, 65C20, 65N30, 65N22, 68U20, 81U30 Key words: Acoustic, numerical method, the stable node-based smoothed finite element method (SNS-FEM), dispersion error, hexagonal patches.

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