A Hybrid High-Order Method for Nonlinear Elasticity

Abstract

In this work we propose and analyze a novel Hybrid High-Order discretization of a class of (linear and) nonlinear elasticity models in the small deformation regime which are of common use in solid mechanics. The proposed method is valid in two and three space dimensions, it supports general meshes including polyhedral elements and nonmatching interfaces, enables arbitrary approximation order, and the resolution cost can be reduced by statically condensing a large subset of the unknowns for linearized versions of the problem. Additionally, the method satisfies a local principle of virtual work inside each mesh element, with interface tractions that obey the law of action and reaction. A complete analysis covering very general stress-strain laws is carried out, and optimal error estimates are proved. Extensive numerical validation on model test problems is also provided on two types of nonlinear models.

A Hybrid High-Order Discretization Method for Nonlinear Poroelasticity

“…According to [9,Theorem 7], the nonlinear elasticity problem (33) admits a solution. Once the initial displacement u 0 h is computed, we set p 0 h such that, for all T ∈ T h , (p 0…”

“…Before proving the convergence of the scheme, we recall two preliminary approximation results for the projector I k h and the projection p n h that have been proved in [9,Theorem 16] and [6,Lemma 11], respectively. There is a strictly positive constant C pj depending only on Ω, k, and the mesh regularity parameter, such that,…”

A Hybrid High‐Order method for finite elastoplastic deformations within a logarithmic strain framework

“…In this case, we slightly abuse the notation by denoting again by (p, q) L 2 Q (T) , the quantity equal to the right-hand side of (33). The discrete energy functional  n h ∶ U k,l,n h,D ×…”

“…The discrete problem (35) expresses the principle of virtual work at the global level, and adapting the ideas introduced in the work of Cockburn et al 39 (see also the works of Abbas et al 1 and Botti et al 33 ), it is possible to infer a local principle of virtual work in terms of face-based discrete tractions that comply with the law of action and reaction.…”

“…The devising of HHO methods hinges on two key ideas: (i) a local reconstruction operator acting on the face and cell unknowns that builds a tensor‐valued polynomial representing the displacement gradient in the polynomial space , where T is a generic mesh cell and d is the space dimension; and (ii) a local stabilization operator that weakly enforces on each mesh face the consistency between the local face unknowns and the trace of the cell unknowns . Furthermore, HHO methods offer several advantages: (i) general meshes (including fairly general polyhedral mesh cells and nonmatching interfaces) are supported; (ii) a local formulation using equilibrated fluxes is available; (iii) computational benefits owing to the static condensation of the cell unknowns and the higher‐order convergence rates; and (iv) the construction is dimension‐independent.…”