Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method
- PDF / 891,637 Bytes
- 11 Pages / 547.087 x 737.008 pts Page_size
- 16 Downloads / 211 Views
(0123456789().,-volV) (0123456789().,-volV)
ORIGINAL PAPER
Approximation of the Stokes eigenvalue problem on triangular domains using a stabilized finite element method ¨ nder Tu¨rk O
Received: 25 May 2020 / Accepted: 11 September 2020 Ó Springer Nature B.V. 2020
Abstract In this paper, we consider a stabilized finite element method for the approximation of the Stokes eigenvalue problem on triangular domains. The method depends on orthogonal subscales that has proved to be an appropriate means for approximating eigenvalue problems in the framework of residual based approaches. We consider several isosceles triangular domains with various apex angles to investigate the characteristics of the eigensolutions in regards to the variation of the domain properties. This study presents the first finite element approximation to the solutions of the Stokes eigenvalue problem on triangular domains, to the best of our knowledge. We provide plots of several velocity and pressure fields corresponding to the fundamental eigenmodes to analyze the flow characteristics in detail. Furthermore, we consider the problem on triangular domains including a crack, and investigate the influence of the length of the slit on the fundamental mode to some extent. The results reveal the correlation between the domain properties and the eigenpairs, and the fact that there are various critical lengths of the slit where the eigenspace is notably affected. ¨ . Tu¨rk O Department of Mathematics, Gebze Technical University, 41400 Kocaeli, Turkey ¨ . Tu¨rk (&) O Institute of Applied Mathematics, Middle East Technical University, 06800 Ankara, Turkey e-mail: [email protected]
Keywords Stokes eigenvalue problem Stabilized finite elements Triangular domains Domains with a crack
1 Introduction Approximating the Stokes eigenvalue problem (EVP) is a subject of an ongoing research in fluid dynamics for many reasons. The flows concerning rheological models and materials processing are governed by the Stokes equations which are also form-identical to the incompressible elasticity equations. In addition, the Stokes problem arises naturally in the theoretical study of the complicated models describing physical phenomena such as turbulence; its eigenfunctions span the function spaces in which the solutions of the full Navier-Stokes equations are sought [1] (see also [2, 3]). Furthermore, the Stokes EVP is used as a benchmark problem in the analysis of the algorithms designed in computational fluid dynamics [4, 5]. Therefore, solving both the Stokes source and eigenvalue problems has been playing an important role in the analysis of fluid and solid mechanics problems. On the other hand, the eigenpairs of the Stokes operator (requiring vanishing divergence) can be acquired analytically (in view of Fourier theory) only on some restrictive cases where its boundary conditions are periodic in all, or in all but one, spatial directions
123
Meccanica
[1, 6]. Thus, in general one has to rely on numerical approximations to determine the eigenmodes accompa
Data Loading...
