A family of symmetric mixed finite elements for linear elasticity on tetrahedral grids

Abstract

Abstract. A family of stable mixed finite elements for the linear elasticity on tetrahedral grids are constructed, where the stress is approximated by symmetric H(div)-P k polynomial tensors and the displacement is approximated by C −1 -P k−1 polynomial vectors, for all k ≥ 4. Numerical tests are provided.

Topology optimization with mixed finite elements on regular grids

“…The relevant degrees of freedom for the tangential component are the shear stresses evaluated at the vertices of the rectangular element K. Further details on shape functions and degrees of freedom for the considered family of mixed finite elements can be found in [25,26,27].…”

Nonconforming mixed finite elements for linear elasticity on simplicial grids

“…Analogous results on tetrahedral grids can be found in [9,10], and on rectangular and cuboid grids in [11][12][13]. Recently, Hu and Zhang [14,15] and Hu [16] proposed a new family of conforming mixed elements on simplicial grids and the latter is for any dimension. This new class of elements has fewer degrees of freedom than those in the earlier literature.…”

“…The lower order conforming stress spaces for 1 ≤ k ≤ n − 1 in [8,10,17] include some proper high order piecewise polynomials and they add to the difficulty of implementation. On the other hand, as mentioned in [23], since the conforming stress spaces in [8][9][10] involve vertex degrees of freedom, techniques like hybridization usually available for mixed methods are not available for these methods, as well as for those elements in [14,15,17]. Apart from the weakly symmetric mixed finite element methods [24][25][26][27], strongly symmetric nonconforming stress approximations are used to overcome these drawbacks, see [23,[28][29][30][31][32] for some nonconforming mixed elements on simplicial grids, and [33][34][35][36][37] on rectangular and cuboid grids.…”

A least squares method for linear elasticity using a patch reconstructed space

“…We solve the linear elasticity problem on the unit cube Ω = [0, 1] 3 and we set Γ N is the boundary with y = 1 and Γ D = ∂\Γ N . mesh level 1/10 1/20 1/40 1/80 1/160 Let the exact displacement u(see [17]) be…”

“…Therefore, there are many efforts devoted to the mixed finite element method based on the weak form of the stress-displacement formulation. We refer to [1,5,3,15,17] for conforming mixed elements and [16,4,24] for non-conforming elements and the references therein. The main challenge of the mixed finite element method is the construction of proper finite element spaces since the requirement of the stable combination of approximation spaces and the symmetric constraint of the stress tensor.…”