A quasi-implicit characteristic-based penalty finite-element method for incompressible laminar viscous flows

In this paper, a novel characteristic-based penalty (CBP) scheme for the finite-element method (FEM) is proposed to solve 2-dimensional incompressible laminar flow. This new CBP scheme employs the characteristic-Galerkin method to stabilize the convective oscillation. To mitigate the incompressible constraint, the selective reduced integration (SRI) and the recently proposed selective node-based smoothed FEM (SNS-FEM) are used for the 4-node quadrilateral element (CBP-Q4SRI) and the 3-node triangular element (CBP-T3SNS), respectively. Meanwhile, the reduced integration (RI) for Q4 element (CBP-Q4RI) and NS-FEM for T3 element (CBP-T3NS) with CBP scheme are also investigated. The quasi-implicit CBP scheme is applied to allow a large time step for sufficient large penalty parameters. Due to the absences of pressure degree of freedoms, the quasi-implicit CBP-FEM has higher efficiency than quasi-implicit CBS-FEM. In this paper, the CBP-Q4SRI has been verified and validated with high accuracy, stability, and fast convergence. Unexpectedly, CBP-Q4RI is of no instability, high accuracy, and even slightly faster convergence than CBP-Q4SRI. For unstructured T3 elements, CBP-T3SNS also shows high accuracy and good convergence but with pressure oscillation using a large penalty parameter; CBP-T3NS produces oscillated wrong velocity and pressure results. In addition, the applicable ranges of penalty parameter for different proposed methods have been investigated.