On stability, convergence and accuracy of bES-FEM and bFS-FEM for nearly incompressible elasticity

Abstract

We present in this paper a rigorous theoretical framework to show stability, convergence and accuracy of improved edge-based and face-based smoothed finite element methods (bES-FEM and bFS-FEM) for nearly-incompressible elasticity problems. The crucial idea is that the space of piecewise linear polynomials used for the displacements is enriched with bubble functions on each element, while the pressure is a piecewise constant function. The meshes of triangular or tetrahedral elements required by these methods can be generated automatically. The enrichment induces a softening in the bilinear form allowing the weakened weak (W 2 ) procedure to produce a high-quality solution, free from locking and that does not oscillate. We prove theoretically that both methods confirm the uniform inf-sup and convergence conditions. Four numerical examples are given to validate the reliability of the bES-FEM and bFS-FEM.

Postbuckling of carbon nanotube reinforced functionally graded plates with edges elastically restrained against translation and rotation under axial compression

“…It is apparent that, apart from those analytical methods which are limited to only certain problems, numerical methods provide the best solution to the above problem; it can be approximated by using the traditional discretization techniques, for example the finite element method [34][35][36] and the finite difference method [37]. Meanwhile, the more recently proposed approximation technique, the element-free or mesh-free method, has become a powerful numerical tool for approximating solutions for many boundary value problems.…”

An adaptive selective ES-FEM for plastic collapse analysis

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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].…”