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

Abstract

Summary 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.

A stabilized finite element method based on characteristic‐based polynomial pressure projection scheme for incompressible flows

“…Selective reduced integration is only applicable for bilinear or higher‐order elements. Our previous study 35,36 has shown that selective gradient smoothing using linear elements still exhibits pressure oscillation for incompressible Stokes flow. The element pair of linear velocity and constant pressure (P1‐P0) is consistent with the methodology of Taylor–Hood element pairs but with pressure instability 37 .…”

A semi-implicit characteristic-based polynomial pressure projection for FEM to solve incompressible flows

“…This stabilization should help FEM to satisfy the Ladyzhenskaya-Babuška-Breezi (LBB) or inf-sup condition. Many techniques have been developed to overcome incompressible conditions, such as selective reduced integration (Malkus and Hughes, 1978), selective gradient smoothing (Jiang et al, 2015(Jiang et al, , 2014Li et al, 2015;Nguyen et al, 2007), average nodal pressure/strain (Bonet et al, 2001;Jiang et al, 2017b), Taylor-Hood element pairs (Brezzi and Fortin, 1991), bubble function enriched element (MINI element) (Arnold et al, 1984;Arnold and Qin, 1992;Franca and Farhat, 1995;Nguyen-Xuan and Liu, 2015), F-bar method (Neto et al, 2005;Onishi et al, 2016), fractional time stepping or split (Han et al, 2014;He, 2015;Jiang et al, 2017d;Nithiarasu et al, 2006), artificial compressibility (Nithiarasu, 2003;Xu et al, 2012), pressure stabilized Petrov-Galerkin (PSPG) (Tezduyar and Osawa, 2000) and so on. However, these techniques have limitations, especially for FEM models that use the simplest linear three-node triangular (T3) and fournode tetrahedral (T4) elements.…”

“…Selective reduced integration is only applicable to bilinear or higher order elements. Selective gradient smoothing (Jiang et al, 2017a(Jiang et al, , 2017c for T3 elements still suffers from the pressure oscillation. The Taylor-Hood element pair of linear velocity and constant pressure (P1-P0) is consistent but still with pressure oscillation (Dohrmann and Bochev, 2004).…”

A cell-based smoothed finite element method for incompressible turbulent flows

“…Thus, the Ladyzhenskaya-Babuška-Breezi or inf-sup condition is required to restrain the instability of pressure. In an attempt to address the instability of pressure, relevant stabilization was proposed and widely used, for example, the artificial compressibility method (He et al, 2002;Madsen and Schaeffer, 2006;Nithiarasu, 2003;Thomas and Nithiarasu, 2005;Xu et al, 2012), the stabilization method of the F-bar (Onishi et al, 2016(Onishi et al, , 2017Onishi and Amaya, 2015), the bubble shape functions (Arnold et al, 1984;Franca and Farhat, 1995;Franca and Oliveira, 2003;Nguyen-Xuan and Liu, 2015;Turner et al, 2010), a fractional-step or time-splitting scheme (Jiang et al, 2018a(Jiang et al, , 2018b(Jiang et al, , 2018cHe, 2015;Nithiarasu et al, 2010), the average nodal pressure (Bonet et al, 2001;Jiang et al, 2018a), the pressure stabilized Petrov-Galerkin formulation (Tezduyar, 1991), stabilized pressure gradient projection (SPGP) method (Codina et al, 2010;Codina and Blasco, 2000) and so on. For synthesizing the advantages of SUPG and SPGP, SUPG and SPGP methods were combined in this work (Liu et al, 2021), which was also coupled with the fractional step method (Kim and Moin, 1985) to deal with spatial oscillations and the instability of pressure.…”

“…As the restriction of isoparametric mapping is relieved in S-FEM, the accuracy loss is effectively reduced when dealing with distorted meshes. In the calculation of solid mechanics, S-FEM has better performances in accuracy, convergence and other indicators than that of standard FEM (Jiang, et al, 2018b;. S-FEM could combine the advantages of FEM and also overcome deficiencies of FEM.…”