A low-order nonconforming method for linear elasticity on general meshes

Abstract

In this work we construct a low-order nonconforming approximation method for linear elasticity problems supporting general meshes and valid in two and three space dimensions. The method is obtained by hacking the Hybrid High-Order method of [18], that requires the use of polynomials of degree k ≥ 1 for stability. Specifically, we show that coercivity can be recovered for k = 0 by introducing a novel term that penalises the jumps of the displacement reconstruction across mesh faces. This term plays a key role in the fulfillment of a discrete Korn inequality on broken polynomial spaces, for which a novel proof valid for general polyhedral meshes is provided. Locking-free error estimates are derived for both the energy-and the L 2 -norms of the error, that are shown to convergence, for smooth solutions, as h and h 2 , respectively (here, h denotes the meshsize). A thorough numerical validation on a complete panel of two-and three-dimensional test cases is provided.

HDGlab: An Open-Source Implementation of the Hybridisable Discontinuous Galerkin Method in MATLAB

“…Variants of this approach achieving optimal secondorder convergence of the primal variable are discussed in [150,262]. Stemming from HHO, lowest-order nonconforming discretisations are proposed in [32] for linear elasticity and in [72] for elliptic obstacle problems. As their highorder counterparts, the above mentioned methodologies allow the use of generic polygonal and polyhedral elements and provide a workaround to the sensitivity issues of FV methods to mesh distortion and stretching [118,119].…”

An unfitted hybrid high-order method for the Stokes interface problem

“…These numerical observations still require a theoretical justification. Notice that a provably well-posed scheme for k = 0 is devised in [7] by penalizing the jumps of the cell unknowns thereby leading to a global coupling of these unknowns as well. Table 2: Test case with pressure jump: pressure errors for various meshes (parameter h), polynomial degrees (parameter k), and geometric resolutions of the interface (parameter n int ).…”

A locking‐free shifted inverse iteration based on multigrid discretization for the elastic eigenvalue problem

“…The linear elasticity discusses how solid objects deform and become internally stressed under prescribed loading conditions and is widely used in structural analysis and engineering design. Because of the wide application background of the elasticity problems, various numerical methods have been developed (see previous studies 1–32,33,34 ). For instance, Bui et al 1,2 studied a meshfree method for the elastodynamic problems.…”

“…Brenner et al 6,7 applied nonconforming Crouzeix–Raviart (CR) finite element to solve the pure displacement and pure traction boundary value problems in two‐dimensional linear elasticity. Botti et al 10 constructed a low‐order nonconforming approximation method for linear elasticity on general meshes. Rui and Sun 13 constructed a finite difference scheme on staggered grids for the linear elasticity problem on nonuniform rectangular grids.…”

“…However, if the displacement fields in nearly incompressible materials are approximated by using standard conforming finite elements, the approximate numerical solutions deteriorate as λ → ∞ (or Poisson ratio ), the so‐called locking phenomenon (see Babuška 35,36 ) will occur. In order to deal with the phenomenon of locking, various numerical methods for the linear elasticity equations/eigenvalue problems have been proposed (e.g., see previous studies 3,5–35 ).…”