Abstract
In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. The edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; then, the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced. To effectively solve the transient wave propagation problems, the Bathe time integration scheme is employed to perform the involved time integration. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; then, the temporal numerical dispersion error in the calculated solutions can also be notably controlled. A number of supporting numerical examples are considered to examine the capabilities of the present approach. The numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM.
Highlights
In this work, the edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. e edge-based gradient smoothing technique (GST) can properly soften the “overly soft” system matrices from the standard finite element approach; the spatial numerical dispersion error of the calculated solutions for wave problems can be significantly reduced
A number of supporting numerical examples are considered to examine the capabilities of the present approach. e numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM
Much research effort has been made and a variety of advanced or modified finite element schemes have been proposed, including the smoothed finite element method (S-FEM). e S-FEM is developed by combining the classical finite element concepts and the generalized gradient smoothing technique (GGST) which is frequently employed in the meshless methods
Summary
Received 1 May 2020; Revised 3 July 2020; Accepted 8 July 2020; Published 31 July 2020. The edge-based smoothed finite element method (ES-FEM) is incorporated with the Bathe time integration scheme to solve the transient wave propagation problems. Due to the appropriate “numerical dissipation effects” from the Bathe time integration method, the spurious oscillations in the relatively large wave numbers (high frequencies) can be effectively suppressed; the temporal numerical dispersion error in the calculated solutions can be notably controlled. E numerical results show that ES-FEM works very well with the Bathe time integration technique, and much more numerical solutions can be reached for solving transient wave propagation problems compared to the standard FEM. The finite element solutions for wave problems usually suffer from the numerical dispersion error issue for large wave numbers (namely, high frequencies) [23,24,25,26,27]. Quite importantly, when the low-order linear elements (such as triangular or quadrilateral element) are used, this troublesome dispersion issue will be even more severe
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