A Hybrid High-Order Method for the Elasticity Problem with Linear Slip Interface

Abstract

We design and analyse a hybrid high-order (HHO) method for the linear elasticity problem with a linear slip interface, i.e.\ with interface conditions of spring type, in which the jump of the displacement across the interface is proportional to the traction through a compliancy tensor $\bK=α\bI+(β-α)\udl{n}\otimes\udl{n}$. The mesh is fitted to the interface, the discrete unknowns are polynomials of degree $k\ge 1$ on the mesh faces and of degree $k+1$ in the mesh cells, and general polytopal cells are allowed. The two novelties of the method lie in the local symmetric strain reconstruction, which incorporates the interface condition, and in the interface stabilisation, which is built from a regularised interface stiffness $\bS_h$ in the spirit of Hansbo and Hansbo \cite{HH04}. As a consequence, a single formulation covers the whole range of compliancies, from the perfectly bonded interface $α=β=0$, where the interface stabilisation acts as a Nitsche-type penalty, to the traction-free interface obtained as $α,β\to+\infty$, and no unknown is attached to the jump. We prove that the discrete bilinear form is coercive and that the errors converge as $h^{k+1}$ in the energy norm and as $h^{k+2}$ in the $L^2$ norm, with constants that are independent of the compliancy parameters and of the Lamé coefficient $λ$, so that the method is also locking free. Various numerical examples and comparisons are provided to confirm the theoretical results.