Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

Abstract

SUMMARYThis paper presents a bubble‐enhanced smoothed finite element formulation for the analysis of volume‐constrained problems in two‐dimensional linear elasticity. The new formulation is derived based on the variational multi‐scale approach in which unequal order displacement‐pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine‐scale displacement approximation. An area‐weighted averaging scheme is employed for the two‐scale smoothed strain calculation under the framework of edge‐based smoothed FEM. The smoothed fine‐scale solution is shown to naturally contain the stress field jump of the smoothed coarse‐scale solution across the boundary of edge‐based smoothing domain and thus provides the possibility to stabilize the global solution for volume‐constrained problems. A global monolithic solution strategy is employed, and the fine‐scale solution is solved without the consideration of approximating the strong form of the fine‐scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd.

An adaptive selective ES-FEM for plastic collapse analysis

“…Also, an alternative approach based on the usual piecewise linear displacements enriched with a cubic bubble function and an edge-based strain smoothing was investigated for voumetric locking (Nguyen-Xuan and Liu, 2013;Ong et al, 2015). A variational multi-scale analysis (Wu et al, 2014) was presented to help elucidate the origins of two-level mesh repartitioning scheme for the volumetric locking-free analysis of nearincompressible problems as well. In the following, we generalize the two-level mesh repartitioning scheme in the selective ES-FEM for correctly computing the plastic limit loads of cohesive and cohesive-frictional material problems.…”

An edge-based/node-based selective smoothed finite element method using tetrahedrons for cardiovascular tissues

“…What is worth to be mentioned is that selective FS/NS-FEM-T4 performs well only for nearly incompressible material and shows slight oscillation of hydrostatic pressure field just like the selective integration technique in FEM. To reduce the hydrostatic pressure oscillation, readers can seek to the help of bubble function implemented in S-FEM which has been detail illustrated in reference [26][27][28][29].…”

Modification of the quadratic 10‐node tetrahedron for thin structures and stiff materials under large‐strain hyperelastic deformation

“…This approach covers introducing (i) nodal quantities, as, for instance, in the averaged nodal pressure approach [3][4][5] or in the uniform-strain setting, [6][7][8][9][10][11] and (ii) face-based or edge-based smoothed strains. 12,13 These methods can be seen as assumed-deformation-gradient methods derivable from weighted-residual equations. 8,10 To address just the response under conditions of near-incompressibility of isotropic materials, one can pass from the irreducible displacement-based methods to mixed methods with the mechanical pressure as one of the independent variables.…”