Locking free and gradient robust H(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. 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 gradientrobustness, yielding a robust and quasi-optimal discretization also in the sense of HDG superconvergence.

A monolithic divergence-conforming HDG scheme for a linear fluid-structure interaction model

“…Similar results are observed for the k = 2 case, which needs roughly about 285 iterations to converge for the compressible case in Table 3 and about 210 iterations for the nearly incompressible case. However, it is also clear that the preconditioner is not robust with respect to polynomial degree k. We finally point out that the k-dependency on the iteration counts is due to the auxiliary space velocity preconditioner (22) since if we replace iA by the exact inverse A −1 , the iteration counts are then observed to be quite insensitive to the polynomial degree: about 30-40 iterations are needed in the compressible cases, and about 20-30 iterations in the nearly incompressible cases for polynomial degree k = 1, 2, 3, 4. This is expected as the polynomial degree in the pressure block is kept to be 0 regardless of the velocity polynomial degree k in the global linear system due to static condensation; see Remark 3.2.…”

“…Semi-discrete divergence-conforming HDG scheme. In this subsection, we present the divergence-conforming HDG spatial discretization [22,30,31] of the linear FSI system (1). We use the globally divergence-conforming finite element space V r h in (2b) to approximate the global velocity…”

“…The additional spring term β s η s in the structure equation does not alter the form of the resulting global linear system. Hence we still apply the preconditioned MinRes solver using the preconditioner (19) with AMG blocks (22) and (20). Due to different boundary conditions, we shall add the boundary contribution…”

“…We stop the MinRes iteration when residual norm is decreased by a factor of 10 −6 . The average number of iterations for convergence for k = 1 with h = 0.05 is 60 and that for k = 2 with h = 0.1 is 52 when the edge-block Gauss-Seidel smoother R e is used in the velocity preconditioner (22). If we instead use the point Gauss-Seidel smoother, the numbers would be 360 for k = 1 and 246 for k = 2.…”

“…In this paper, we present a novel monolithic divergence-conforming HDG scheme for a linear FSI problem with a thick structure. The fluid Stokes problem is discretized using the divergencefree HDG scheme of Lehrenfeld and Schöberl [30,31], and the structure linear elasticity problem is discretized using the divergence-conforming HDG scheme of Fu et al [22]. We approximate the fluid and structure velocities together using a single H(div)-conforming finite element space, and we also introduce a global (hybrid) unknown that approximate the tangential component of the velocities on the mesh skeleton for the purpose of efficient implementation.…”