Mixed finite element methods for linear elasticity with weakly imposed symmetry

By Douglas N. Arnold, Richard S. Falk, Ragnar Winther. Mathematics of Computation, 2007

323 citations recorded by the source corpus.

In this paper, we construct new finite element methods for the approximation of the equations of linear elasticity in three space dimensions that produce direct approxima- tions to both stresses and displacements. The methods are based on a modified form of the Hellinger-Reissner variational principle that only weakly imposes the symmetry condition on the stresses. Although this approach has been previously used by a number of authors, a key new ingredient here is a constructive derivation of the elasticity complex starting from the de Rham complex. By mimicking this construction in the discrete case, we derive new mixed finite elements for elasticity in a systematic manner from known discretizations of the de Rham complex. These elements appear to be simpler than the ones previously derived. For example, we construct stable discretizations which use only piecewise linear elements to approximate the stress field and piecewise constant functions to approximate the displacement field.

Subfields: Civil and Structural Engineering, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/47c9efa6e22c9105d1187c6d0e24e552c26d701a

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In Computational Mechanics

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Citations in the corpus

Reduced symmetry elements in linear elasticity (2008, methodology): “Our first example (and main example of this sub section) will be the Arnold-Falk-Winther element [4].”
Finite Element Methods for Linear Elasticity (2008, methodology): “An analogous argument works for general r when n =2 , and the basic outline of the proof is the same when n =3 , although in this case the operator S1 is more complicated. The details can be found in [ 6 ].”
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, 28, 35, 36], and [8, 12, 18, 20, 21].”
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 Second Elasticity Element Using the Matrix Bubble (2012, background): “Method V (K) W (K) A(K) Order Stenberg [24] P(K) + curl (P(K)λ0λ1λ2λ3) P (K) A(K) k ≥ 1 peers [4, 12] RT (K) + curl (P(K)λ0λ1λ2λ3) P (K) CA(1)(K) k = 0 afw [5] P(K) P(K) A(K) k ≥ 0 Our 1st element [15] RT (K) + curl ((curlÃ(K))bK) P (K) A(K) k ≥ 1 Section 2 (this paper) P(K) + curl ((curlÃ(K))bK) P ”
Finite element exterior calculus, homological techniques, and applications (2006, methodology): “In Arnold et al. (2006c) (two dimensions) and Arnold et al. (2005) (three dimensions), we developed a discrete version of the BGG construction, which allowed us to derive stable mixed finite elements for elasticity in a systematic manner based on the finite element subcomplexes of the de Rham…”
A Mimetic discretization method for linear elasticity (2010, methodology): “In this paper we extend the ideas of [4,5,15] in order to construct an MFD method that is uniformly stable and convergent.”