A Second Elasticity Element Using the Matrix Bubble

By Jay Gopalakrishnan, Johnny Guzmán. 2012

107 citations recorded by the source corpus.

We presented a family of finite elements that use a polynomial space augmented by certain matrix bubbles in [Math. Comp., 79 (2010), 1331–1349]. In this sequel, we exhibit a second family of elements that use the same matrix bubble. This second element uses a stress space smaller than the first, while maintaining the same space for rotations (which are the Lagrange multipliers corresponding to a weak symmetry constraint). The space of displacements are of one degree less than the first method. The analysis, while similar to the first, requires a few new adjustments as the new Fortin projector may not preserve weak symmetry, but we are able to prove optimal convergence for all the variables. Finally, we present a sufficient condition wherein a mixed method with weakly imposed stress symmetry in fact yields an exactly symmetric stress tensor approximation.

Subfields: Biomedical Engineering, Computational Mechanics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/2c8acc2c9f500c72fe342ffbfa9e01cfeee2bfab

Citations in the corpus

• Finite element approximations of symmetric tensors on simplicial grids in Rn: the lower order case (2014, unknown): “We refer interested readers to [2, 5, 6, 10, 12, 18, 11, 22, 27, 33, 39, 40, 9, 13, 19, 23, 24, 29, 28], for recent progress on mixed finite elements for linear elasticity.”
• A family of conforming mixed finite elements for linear elasticity on triangular grids (2014, unknown): “From then on, various stable mixed elements have been constructed, see [2, 4, 5, 9, 11, 17], [10, 19, 23, 27, 33, 34], and [8, 12, 18, 20, 21].”