Locking-free hybrid high-order method for linear elasticity

TL;DR

This work develops a locking-free hybrid-high-order method for linear elasticity that uses a single reconstruction operator for the linear Green strain, avoiding the classical deviatoric/spherical split. It provides a λ-robust a priori analysis yielding quasi-best approximation with constants independent of the Lamé parameter $\lambda$ and stabilization-free, data-oscillation-tolerant a posteriori error estimators, verified on simplicial meshes. A key contribution is the local equivalence of stabilizations, enabling stabilization-free a posteriori control and extending the framework to incompressible limits, i.e., Stokes-like behavior. The methodology is supported by numerical benchmarks demonstrating optimal adaptive convergence rates and robustness in near-incompressible regimes, underscoring its practical impact for reliable, high-order solid-mechanics simulations.

Abstract

The hybrid-high order (HHO) scheme has many successful applications including linear elasticity as the first step towards computational solid mechanics. The striking advantage is the simplicity among other higher-order nonconforming schemes and its geometric flexibility as a polytopal method on the expanse of a parameter-free refined stabilization. This paper utilizes just one reconstruction operator for the linear Green strain and therefore does not rely on a split in deviatoric and spherical behaviour as in the classical HHO discretization. The a priori error analysis provides quasi-best approximation with $λ$-independent equivalence constants. The reliable and (up to data oscillations) efficient a posteriori error estimates are stabilization-free and $λ$-robust. The error analysis is carried out on simplicial meshes to allow conforming piecewise polynomials finite elements in the kernel of the stabilization terms. Numerical benchmarks provide empirical evidence for optimal convergence rates of the a posteriori error estimator in some associated adaptive mesh-refining algorithm also in the incompressible limit, where this paper provides corresponding assertions for the Stokes problem.

Figures (5)

• Figure 1: (a) Cook's membrane and (b) convergence history plot of $\eta$ for $k = 1, \dots, 5$ in \ref{['sec:Cooks']}
• Figure 2: (a) Adaptive triangulation into 592 triangles generated with $k = 2$ and (b) comparison between different $\nu$ with $k = 2$ in \ref{['sec:Cooks']}
• Figure 3: (a) Rotated L-shaped domain and (b) convergence history plot of $\widetilde{\eta}$ for $k = 1,\dots,5$ in \ref{['sec:Lshape']}
• Figure 4: (a) Convergence history plot of $\|\sigma - \sigma_h\|$ for $k = 1, \dots, 5$ and (b) adaptive triangulation into 545 triangles generated with $k = 2$ in \ref{['sec:Lshape']}
• Figure 5: (a) Efficiency index $\widetilde{\eta}/\|\sigma - \sigma_h\|$ for $k = 1,\dots,5$ and (b) comparison between different $\nu$ with $k = 2$ in \ref{['sec:Lshape']}

Theorems & Definitions (40)

• Theorem 2.1: equivalence of stabilizations
• proof
• Corollary 2.2: kernel of $|\bullet|_\mathrm{s}$
• proof
• Lemma 2.3: commuting diagram
• proof
• Lemma 2.4: best approximation of $\mathcal{R} \circ \mathrm{I}$
• proof
• Remark 2.5: $\mathcal{\varepsilon}_h \circ \mathrm{I} = \varepsilon$ in $S^{k+1}_\mathrm{D}(\mathcal{T})$
• Lemma 2.6: $\varepsilon_\mathrm{pw} \circ \mathcal{R}$ vs $\varepsilon_h$