Free and Forced Vibration Analysis of Two-Dimensional Linear Elastic Solids Using the Finite Element Methods Enriched by Interpolation Cover Functions

Abstract

In this paper, a novel enriched three-node triangular element with the augmented interpolation cover functions is proposed based on the original linear triangular element for two-dimensional solids. In this enriched triangular element, the augmented interpolation cover functions are employed to enrich the original standard linear shape functions over element patches. As a result, the original linear approximation space can be effectively enriched without adding extra nodes. To eliminate the linear dependence issue of the present method, an effective scheme is used to make the system matrices of the numerical model completely positive-definite. Through several typical numerical examples, the abilities of the present enriched three node triangular element in forced and free vibration analysis of two-dimensional solids are studied. The results show that, compared with the original linear triangular element, the present element can not only provide more accurate numerical results, but also have higher computational efficiency and convergence rate.

Enriched Finite Element Method Based on Interpolation Covers for Structural Dynamics Analysis

“…If the damping effects (C = 0) are ignored, it is feasible to rewrite Equation (25) for free vibration analysis by…”

“…The second-order time-dependent dynamic issues, regulated by the matrix equation indicated in Equation (25), should be solved to perform forced vibration analysis [30,31]. Numerous other direct time integration strategies have been established in practice to solve structural dynamic issues.…”

“…Furthermore, a variety of typical MEE-based specific examples have been used to reveal how efficiently the current E-FEM handles multi-physical coupling issues, unlike the conventional FEM. Li [25] and colleagues employed the supplementary interpolation cover functions, which are constructed using appropriate polynomial bases, to improve the performances of the traditional FEM at the same time handling 2D dynamic scenarios. As a result, the gradient field of the issue under examination can be more precisely described, and the initial linear approximation space of the conventional FEM can be considerably enriched [18][19][20][21][22][23][24][25][26][27].…”

“…Li [25] and colleagues employed the supplementary interpolation cover functions, which are constructed using appropriate polynomial bases, to improve the performances of the traditional FEM at the same time handling 2D dynamic scenarios. As a result, the gradient field of the issue under examination can be more precisely described, and the initial linear approximation space of the conventional FEM can be considerably enriched [18][19][20][21][22][23][24][25][26][27]. In summary, E-FEM has extensive application prospects.…”

epSFEM: A Julia-Based Software Package of Parallel Incremental Smoothed Finite Element Method (S-FEM) for Elastic-Plastic Problems

“…Currently, numerical methods are the most important tools for solving various scientific and engineering problems [1]. For example, the Finite Element Method (FEM), one of the most successful numerical methods, has been widely employed in different scientific and engineering fields because of its mathematically rigorous proof and satisfactory efficiency [2][3][4]. However, the shortcomings and deficiencies of FEM are becoming increasingly significant [2,[5][6][7][8].…”