Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES‐FEM)

The paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge-based smoothed finite element method (ES-FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the 'overly stiff' feature of the discrete model. In such an 'overly stiff' FEM model, the wave propagates with an artificially higher 'numerical' speed, and hence the numerical wave-number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge-based gradient smoothing operations, the ES-FEM model, however, behaves much softer than the standard FEM model, leading to the so-called very 'close-to-exact' stiffness. Therefore the ES-FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES-FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model.

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