An Unfitted Hybrid High-Order Method for the Elastodynamics Problem with Imperfect Interface

Abstract

We design and analyse an unfitted hybrid high-order (HHO) method for the elastic wave equation in a medium made of two components separated by an imperfect interface of linear slip type, across which the traction is continuous and the displacement jump is proportional to the traction through a compliancy tensor $\bK=\alpha\bI+(\beta-\alpha)\bn\otimes\bn$. The mesh is not fitted to the interface: the discrete unknowns are doubled in the cut cells, the small cuts are cured by a cell agglomeration procedure, and no unknown is attached to the interface. The two specific ingredients of the method are a local symmetric strain reconstruction in each cut subcell, which incorporates the interface condition through the regularised interface stiffness $\bS_h=(h_T\delta^{-1}\bI+\bK)^{-1}$ in the spirit of Hansbo and Hansbo {\em{A finite element method for the simulation of strong and weak discontinuities in solid mechanics.}} {Comput. Methods Appl. Mech. Engrg.}, 193, 2004, and an interface stabilisation built from the same matrix. For the space semi-discrete problem we prove that the discrete bilinear form is coercive and continuous, and we derive an energy-error estimate of order $h^{k+1}$ and an $L^2$-error estimate of order $h^{k+2}$, with constants independent of the compliancy parameters and of how the interface cuts the mesh. The scheme is combined either with the Newmark scheme, which conserves a discrete energy exactly, or with singly diagonally implicit Runge--Kutta schemes of order up to four. Numerical experiments in two dimensions confirm the predicted convergence rates for $k\in\{1,2,3\}$, the robustness with respect to the compliancy over sixteen orders of magnitude, and illustrate the propagation of elastic waves across an unresolved slipping interface.