Stabilized finite elements for time‐harmonic waves in incompressible and nearly incompressible elastic solids

Abstract

Summary The propagation of waves in elastic solids at or near the incompressible limit is of interest in many current and emerging applications. Standard low‐order Galerkin finite element discretization struggles with both incompressibility and wave dispersion. Galerkin least squares stabilization is known to improve computational performance of each of these ingredients separately. A novel approach of combined pressure‐curl stabilization is presented, facilitating the use of continuous, equal‐order interpolation of displacements and pressure. The pressure stabilization parameter is determined by stability considerations, while the curl stabilization parameter is determined by dispersion considerations. The proposed pressure‐curl–stabilized scheme provides stable and accurate results on a variety of numerical tests for incompressible and nearly incompressible elastic waves computed with linear elements.

Modal analysis of elastic vibrations of incompressible materials using a pressure-stabilized finite element method

“…Note that we have applied P ⊥ h to ∇q h to highlight the symmetry of the formulation, even if it has no effect for quasi-uniform meshes. Finally, instead of Problem (27) we consider: find…”

“…As opposed to the application in various structural models involving compressible materials, the incompressible media necessitate the incorporation of the pressure, or mean stress, into the model. In the standard Galerkin formulations, the displacement and pressure interpolations are required to satisfy the classical Babuška-Brezzi inf-sup condition [14,22,[27][28][29]. These considerations apply to both the classical boundary value problem defining steady state incompressible elasticity (equivalently incompressible fluid flows) and the eigenproblem to be described in the sequel.…”

“…Stabilization strategies based on mesh-free polynomial projection methods are considered in [32,33]. In [27], a pressure-curl stabilization approach is proposed in which the determination of the pressure stabilization parameter is based on stability concerns, and the curl stabilization parameter is determined on the account of dispersion.…”