On singularities in the solution of three‐dimensional Stokes flow and incompressible elasticity problems with corners

Abstract

SUMMARYIn this paper, a numerical procedure is presented for the computation of corner singularities in the solution of three-dimensional Stokes flow and incompressible elasticity problems near corners of various shape. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of this problem is approximated using a mixed u, p Galerkin-Petrov finite element method. Additionally, a separation of variables is used to reduce the dimension of the original problem. As a result, the quadratic eigenvalue problem (P + Q + 2 R)d = 0 is obtained, where the saddle-point-type matrices P, Q, R are defined explicitly. For a numerical solution of the algebraic eigenvalue problem an iterative technique based on the Arnoldi method in combination with an Uzawa-like scheme is used. This technique needs only one direct matrix factorization as well as few matrix-vector products for finding all eigenvalues in the interval R( ) ∈ (−0.5, 1.0), as well as the corresponding eigenvectors. Some benchmark tests show that this technique is robust and very accurate. Problems from practical importance are also analysed, for instance the surface-breaking crack in an incompressible elastic material and the three-dimensional viscous flow of a Newtonian fluid past a trihedral corner.

Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures

“…The extension of Theorem 2.1 is more delicate due to the lack of regularity in the vicinity of the interface tip (see, e.g. [26]).…”

“…, w h Σ (27) for all w h ∈ W h and n > r. Owing to (26) and (27), the semi-implicit scheme (24)-(25) can be reformulated as shown in Algorithm 3.…”

The Stationary Navier-Stokes System with No-Slip Boundary Condition on Polygons: Corner Singularity and Regularity

“…In [14,15] polyhedral domains are investigated on the weighted Sobolev spaces where different boundary conditions are arbitrarily combined, and extended to the Navier-Stokes system. Regarding the numerical applications for the corner singularities, one may refer to [2,8] for the Stokes or Navier-Stokes flows in the plane polygons and [4] for the computation of corner singularities in the solution of three-dimensional Stokes flow near corners of various shape.…”