Computers, Materials & Continua DOI:10.32604/cmc.2022.020889 

Article 
Polygonal Finite Element for TwoDimensional LidDriven Cavity Flow
1Faculty of Civil Engineering, Vietnam Maritime University, 484 Lach Tray Str., Hai Phong, Vietnam
2Faculty of Civil Engineering and Electricity, Ho Chi Minh City Open University, Vietnam
3CIRTech Institute, Ho Chi Minh City University of Technology (HUTECH), Ho Chi Minh City, Vietnam
4Department of Architectural Engineering, Sejong University, 209 Neungdongro, Gwangjingu, Seoul, 05006, Republic of Korea
5Soete Laboratory, Faculty of Engineering and Architecture, Ghent University, Technologiepark Zwijnaarde, 903, B9052, Zwijnaarde, Belgium
*Corresponding Author: M. AbdelWahab. Emails: magd.a.w@hutech.edu.vn; magd.abdelwahab@UGent.be
Received: 12 June 2021; Accepted: 14 July 2021
Abstract: This paper investigates a polygonal finite element (PFE) to solve a twodimensional (2D) incompressible steady fluid problem in a cavity square. It is a wellknown standard benchmark (i.e., liddriven cavity flow)to evaluate the numerical methods in solving fluid problems controlled by the Navier–Stokes (N–S) equation system. The approximation solutions provided in this research are based on our developed equalorder mixed PFE, called Pe1Pe1. It is an exciting development based on constructing the mixed scheme method of two equalorder discretisation spaces for both fluid pressure and velocity fields of flows and our proposed stabilisation technique. In this research, to handle the nonlinear problem of NS, the Picard iteration scheme is applied. Our proposed method’s performance and convergence are validated by several simulations coded by commercial software, i.e., MATLAB. For this research, the benchmark is executed with various Reynolds numbers up to the maximum
Keywords: Liddriven cavity; incompressible; steady; Navier–Stokes equations; polygonal finite element method
This research, instead of using widely used numerical approaches such as the finite difference method (FDMfinite volume method (FVM)), finite element method (FEM), e.g., [1–3], etc., proposes an advanced PFE method (PFEM) to solve the 2D liddriven cavity problem controlled by the incompressible steady NS equations. As known, PFEM is a robust numerical method offering a wide range of distinguished advantages, especially flexibility and the benefit of Voronoi algorithms to generate meshes with arbitrary element shapes [4,5]. Furthermore, PFEM offers better accuracy without the need for a sizeable overall mesh scale compared to its triangular and quadrilateral counterparts [6,7]. It means that PFEs can provide better solutions than triangular and quadrilateral elements [7,8]. However, the fact is that developments of PFEM in the fluid field is still too modest compared to the enormous potential of the method. The most current research of PFEM for fluid analysis, hitherto, was only given by Talischi et al. [6] in 2014, see and then is our research in 2019, see [9–12]. However, such research only focuses on the performance, stability and convergence of the PFEs for incompressible steady Stokes problems. Therefore, this research aims to use our recently developed PFE to solve 2D incompressible steady NS cavity problems.
As that goal, this study adopts the equalorder mixed PFE, named Pe1Pe1. The primary advantage of our developed element is the computational ability for fluid flows on all kinds of mesh families, e.g., triangular, quadrilateral, hexagonal, random and centroidal Voronoi meshes. It is constructed by the mixed scheme between two equalorder discretisation spaces for both fluid pressure and velocity fields of flows. In this research, Wachspress basis shape functions are utilised to represent the approximation spaces of velocity and pressure fields. Furthermore, this research executes our novel stabilisation technique to eliminate the instability of the equalorder mixed scheme [9–12]. It is an innovation of the local polynomial pressure projection method introduced by Bochev et al. [13] in 2004. It automatically adapts the local stabilisation term for each arbitrary shape of element on a polygonal mesh. Our advanced stabilisation method adds a term to penalise pressure deviations from the ‘consistent’ polynomial order. And it helps to avoid the residual terms of the penalty method to maintain the symmetry of the numerical system.
As mentioned, this research employs a wellknown benchmark (i.e., liddriven cavity flow)a classical standard to evaluate the numerical methods in solving the NS equations. This benchmark’s main advantage is the simplicity of the geometry, leading to applying numerical methods on this flow in terms of coding is relatively easy and straightforward. Despite its simple geometry, the driven cavity flow occupies a rich fluid physics flow [14,15]. The cavity problem is early and widely utilised by many researchers, e.g., Ghia et al. [16]; Botella et al. [14]; Bruneau et al. [17], etc. This study, hence, applies the 2D liddriven cavity benchmark with various Reynolds numbers (i.e.,
N–S equations, as known, are a system of the nonlinear term of convection problems [18]. Nonlinear equations cannot, in general, be solved analytically [19,20]. Thus, in this case, nonlinear equations must be handled by iteration progress. The concept behind such iteration techniques starts at an arbitrary point–the closest possible point to the solution sought–and gradually reach the solution through a series of sequent tests [19,21,22]. Because of the advantages of the Picard iteration method (i.e., a huge ball of convergence [23–25], efficiency and simplicity [26], etc.), Picard is chosen to handle the nonlinear term in NS equations in this study. In addition, to take advantage of the PFEM, the mesh generation algorithms and Voronoi diagrams' properties [4–6] are utilised. Besides, the Wachspress coordinates [27,28] are also adopted. In addition, the advanced techniques developed in [5,7,29], which handle the bottleneck of generating quality polygonal meshes, are applied.
The paper is organised as follows: Section 2 presents the incompressible steady NS flow problems. In Section 3, we illustrate the iteration progress to solve the NS equation system’s nonlinear problem. Then, Section 4 presents the mixed discretisation scheme. Section 5 shows the numerical tests’ results. Finally, the conclusions and future works are given in Section 6.
2 Incompressible Navier–Stokes Equations
As known, the steadystate NS equation system for incompressible fluid flow is generally written as following [25,30,31]:
where
where
where
where
where
Then, determine the bilinear form
The fundamental feature is the skewsymmetric of convection term:
So, the continuity becomes:
The experience shows that the wellposedness of the weak formulations of N–S systems is a complicated problem because of the nonlinearity. To explain it, we define
It means that it is impossible to apply the homogeneous problem to establish uniqueness for Eqs. (7) and (8). To ensure the uniqueness of the weak solution, it can base on wellposedness conditions between the forcing function
Alternatively, a wellknown condition for uniqueness is [33]:
where
To solve N–S equations, we need a linearised process to deal with the nonlinear problem at every computation step. So, we first need an “initial guess” that commonly is the solution of the corresponding Stokes problem
The solution of Eqs. (7) and (8) at the
where
The solution of Eqs. (21) and (22) is the socalled Newton correction. The previous iterate is updated by
The second approach of linearisation is the Picard method. It bases on the dropping of the quadratic term
4 Polygonal Mixed Finite Discretisation
A discretisation based on two finite spaces
Based on the Newton iteration, find the correction terms
Here,
To state the approximated solution, we introduce a set of vectorvalued basis functions
In which, as mentioned, the polygonal basis shape functions for both velocity and pressure field in this research are constructed by Wachspress coordinates as:
and their gradients are:
where,
in which
Substituting Eqs. (29) and (30) into Eqs. (27) and (28) gives a system of linear equations as:
Here,
The righthand side vectors in Eq. (34) are the nonlinear residuals associated with the current discrete solution estimates
For Picard iteration, by omitting the Newton derivative matrix, the mixed finite discretisation system of Eqs. (27) and (28) becomes:
Because of the efficiency and simplicity, the Picard iterative scheme is applied in this research to handle the nonlinear term in NS equations. In case of unstable problems, we need to eliminate the zero block in the system Eq. (34) as well as the system Eq. (41) by a stabilisation matrix to get stable results. Therefore, the stabilised analogue of the system Eq. (41) is [11,25]:
Here,
for all
where
As mentioned, this research applies the equalorder PFE, e.g., Pe1Pe1, to analyse the extensive wellknown liddriven cavity flow (see Fig. 1) of incompressible steady NS flows. The detail of this benchmark is that its domain is a unit square
The initial results of this paper are presented in Fig. 2, indicating the velocity contours at the different
For more profound validation, the velocities components (i.e.,
Tab. 2 contains more information about the apparent proof for the convergence and precision of the new approach in this study. Moreover, our models computed on increasingly finer meshes steadily exceed the reference and convergence solution, as seen in Fig. 4 and Tab. 2. Furthermore, readers will see that our new results are greater than Ghia et al. [16] and Vanka [36], i.e., the results of
The subsequent validation provided in this research is the results of fluid pressure. These results are firstly presented in Figs. 5 and 6 In which, Fig. 5 depicts the fluid pressures on the entire cavity using contour plots. These results indicate that as
The indications of pressure on the cavity centrelines are then seen in Fig. 6 to allow a detailed comparison with the literature. As the figure, it is evident that the current pressure results, i.e., the pressure at
All the above results already exhibited a perfect convergence as well as the precision of velocity and pressure. Thus, the remaining part is to illustrate another result of streamlines. The first streamlines regarding the liddriven cavity flow at
Besides, the streamlined results of the liddriven cavity simulation at
In conclusion, all the above results reveal that the present polygonal method is entirely good enough to deal with the 2D incompressible steady flows of liddriven cavity benchmark. It is reinforced by numerous extensive comparisons to previously published research that used various highly precise techniques. For example, Botella et al. [14] in 1998 used the highly accurate Chebyshev collocation method associated with the subtraction method of the leading terms of the asymptotic expansion to obtain a highly accurate spectral solution with a maximum of grid mesh of
To summarise, this research’s commitment to designing new PFEM to solve 2D incompressible steady fluid flows controlled by NS equations is practical. This contribution is focused on the development of a mixedorder equalorder system. This idea then applies Wachspress basis shape functions associated with our advanced stabilisation technique to generate the equalorder mixed PFE, called Pe1Pe1 [10]. Furthermore, in this article, we successfully address the Picard iteration technique for our proposed PFE to deal with the nonlinear convection term of NS equations. This paper incorporates a widely mathematical benchmark to evaluate the accuracy and performance of our developed PFE. It is the liddriven cavity benchmark executed to measure the efficiency of numerical methods in solving NS problems. Our established method shows an excellent agreement with the highly accurate solutions found in the literature regarding current research solutions. It means that the efficacy of our proposed technique for solving incompressible steady NS fluid flows has been proven without any reasonable doubt.
Furthermore, the production of higherorder PFEs is a highpotential direction for future research in this field. For example, Rand et al. [43] suggest a novel quadratic serendipity shape function in 2014, a solid foundation for this direction. Alternatively, the Taylor–Hood elements, as described in [44], could be a good start. Additionally, extending our proposed PFEs to computations for transient fluid flow problems is a significant strategic work of this study. The proposed method's application to 3D problems is, therefore, a priority task. Other promising ideas for future research are free surface and fluidstructure interaction problems. In addition, in recent years, Deep Neural Networks (DNNs) developments are becoming a mathematical option to solve the partial differential equations (PDEs) of different phenomena in science and engineering [45,46]. Thus, the efficiency of DNNs is an exciting direction for this research.
Funding Statement: This work was supported by the VLIRUOS TEAM Project, VN2017TEA454A 103, ‘An innovative solution to protect Vietnamese coastal riverbanks from floods and erosion’ funded by the Flemish Government.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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