On pressure robustness and independent determination of displacement and pressure in incompressible linear elasticity

TL;DR

This paper explores how to determine displacement independently or robustly from pressure in incompressible linear elasticity, proposing a Helmholtz decomposition-based method with finite element analysis and validating it through theory and numerics.

Contribution

It introduces a pressure robust method for displacement determination using Helmholtz decomposition and finite element techniques, applicable to convex domains with specific boundary conditions.

Findings

Independent displacement determination is possible only in specific boundary conditions.

Pressure robust computation of displacement is feasible with Helmholtz decomposition.

The proposed three-step method effectively separates force components and solves elasticity problems.

Abstract

We investigate the possibility to determine the divergence-free displacement $\mathbf{u}$ \emph{independently} from the pressure reaction $p$ for a class of boundary value problems in incompressible linear elasticity. If not possible, we investigate if it is possible to determine it \emph{pressure robustly}, i.e. pollution free from the pressure reaction. For convex domains there is but one variational boundary value problem among the investigated that allows the independent determination. It is the one with essential no-penetration conditions combined with homogeneous tangential traction conditions. Further, in most but not all investigated cases, the weakly divergence-free displacement can be computed pressure robustly provided the total body force is decomposed into its direct sum of divergence- and rotation-free components using a Helmholtz decomposition. The elasticity problem is solved using these components as separate right-hand sides. The total solution is obtained using the superposition principle. We employ a $(\mathbf{u},p)$ higher-order finite element formulation with discontinuous pressure elements. It is \emph{inf-sup} stable for polynomial degree $p\ge 2$ but not pressure robust by itself. We propose a three step procedure to solve the elasticity problem preceded by the Helmholtz decomposition of the total body force. The extra cost for the three-step procedure is essentially the cost for the Helmholtz decomposition of the assembled total body force, and the small cost of solving the elasticity problem with one extra right-hand side. The results are corroborated by theoretical derivations as well as numerical results.