Locking-Free and Gradient-Robust \({\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})\)-Conforming HDG Methods for Linear Elasticity

Abstract

Robust discretization methods for (nearly-incompressible) linear elasticity are free of volume-locking and gradient-robust. While volume-locking is a well-known problem that can be dealt with in many different discretization approaches, the concept of gradient-robustness for linear elasticity is new: it assures that dominant gradient fields in the momentum balance do not lead to spurious displacements. We discuss both aspects and propose novel Hybrid Discontinuous Galerkin (HDG) methods for linear elasticity. The starting point for these methods is a divergence-conforming discretization. As a consequence of its well-behaved Stokes limit the method is gradient-robust and free of volume-locking. To improve computational efficiency, we additionally consider discretizations with relaxed divergence-conformity and a modification which re-enables gradient-robustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

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G. Fu gratefully acknowledge the partial support of this work from U.S. National Science Foundation through Grant DMS-2012031.

Appendix. The BDM Interpolator for Discontinuous Functions

Appendix. The BDM Interpolator for Discontinuous Functions

The BDM interpolator for discontinuous functions is defined element-by-element for \(\varvec{v}_T \in {\varvec{H}}^1(T)\) through

$$\begin{aligned} (\Pi _V \varvec{v}_T \! \cdot \! \varvec{n}, \varphi )_{F}&= ( \{\!\!\{ \varvec{v}_T \! \cdot \! \varvec{n} \}\!\!\}_*, \varphi )_{F}&\forall ~ \varphi \!\in \! {\mathbb {P}}^{k}\!(F), F \!\in \! \partial T, \end{aligned}$$
(41a)
$$\begin{aligned} (\Pi _V \varvec{v}_T, \varphi )_{T}&= (\varvec{v}_T, \varphi )_{T}&\forall ~ \varphi \in \varvec{{\mathcal {N}}}^{k-2}(T), \end{aligned}$$
(41b)

with \(\varvec{{\mathcal {N}}}^{k-2}:=[{\mathbb {P}}^{k-2}(T)]^d + [{\mathbb {P}}^{k-2}(T)]^d \times x\) and \(\{\!\!\{ \cdot \}\!\!\}_*\) the usual DG average operator, cf. [21, 39].

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Fu, G., Lehrenfeld, C., Linke, A. et al. Locking-Free and Gradient-Robust \({\varvec{H}}({{\,\mathrm{{\text {div}}}\,}})\)-Conforming HDG Methods for Linear Elasticity. J Sci Comput 86, 39 (2021). https://doi.org/10.1007/s10915-020-01396-6

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Keywords

  • Linear elasticity
  • Nearly incompressible
  • Locking phenomenon
  • Volume-locking
  • Gradient-robustness
  • Discontinuous Galerkin
  • \({\varvec{H}}\) (div)-conforming HDG

Mathematics Subject Classification

  • 65N30
  • 65N12
  • 76S05
  • 76D07