A hybrid high-order locking-free method for linear elasticity on general meshes

Abstract

We devise an arbitrary-order locking-free method for linear elasticity. The method relies on a pure-displacement (primal) formulation and leads to a symmetric, positive definite system matrix with compact stencil. The degrees of freedom are vector-valued polynomials of arbitrary order k≥1 on the mesh faces, so that in three space dimensions, the lowest-order scheme only requires 9 degrees of freedom per mesh face. The method can be deployed on general polyhedral meshes. The key idea is to reconstruct the symmetric gradient and divergence inside each mesh cell in terms of the degrees of freedom by solving inexpensive local problems. The discrete problem is assembled cell-wise using these operators and a high-order stabilization bilinear form. Locking-free error estimates are derived for the energy norm and for the L2-norm of the displacement, with optimal convergence rates of order (k+1) and (k+2), respectively, for smooth solutions on general meshes. The theoretical results are confirmed numerically, and the CPU cost is evaluated on both standard and polygonal meshes.