Volume 18, Issue 6 (2018)
MCEJ 2018, 18(6): 181-192 |
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Farmani S, Ghaeini-Hessaroeyeh M, Hamzehei-Javaran S. Reformulating the Finite Element Method Based on Complex Fourier Elements in increasing the solution accuracy of the Navier-Stokes and Laplace Equations. MCEJ 2018; 18 (6) :181-192
URL: http://mcej.modares.ac.ir/article-16-20993-en.html
URL: http://mcej.modares.ac.ir/article-16-20993-en.html
1- Shahid Bahonar University of Kerman
2- Shahid Bahonar University of Kerman ,mghaeini@uk.ac.ir
2- Shahid Bahonar University of Kerman ,
Abstract: (5427 Views)
In this paper, the Navier-Stokes and Laplace equations are solved using the Finite Element Method (FEM) based on complex Fourier elements. The FEM is considered by two types of shape functions: Lagrange shape functions and new complex Fourier functions. The proposed interpolation functions are derived using enrichment of complex Fourier radial basis functions in the form of 
. The present functions have properties of Gaussian and real Fourier radial basis functions. These useful properties have provided the robustness of the proposed method. Also, these functions have the simultaneously functions field such as trigonometric, exponential, and polynomial; while the classic Lagrange functions satisfied only polynomial functions field. In other words, these features provide an improvement in the solution accuracy with number of elements which are equal or lower than the ones used by the classic finite element method.
Solving the Navier-Stokes and Laplace equations is the important challenge in the fluids mechanics problems. The most problems cannot be solved by the analytical methods. For this reason, the numerical methods are developed. Generally, the numerical methods are divided to two classes: the methods based on the mesh and meshless methods. In the first class, the computational domain are meshed and the governing equations are solved based them the finite element method, Finite Difference Method (FDM) and finite volume method (FVM) are placed in this category. While, in the second category methods, the computational domain is divided to moving particles. In these methods, there is no needed to any grid and the equations are solved on the particles. The smoothed particles hydrodynamic (SPH) method, Moving Particles Semi-implicit method and Discrete Least Squares Meshless method are in this class. The FEM is capable to solving the problems with complicated geometry. Also, the Neumann boundary conditions are applied properly.
Generally, the numerical methods such as finite element and finite difference methods are based on the mesh for solving the equations. For obtaining the results with high accuracy, it is needed to have enough elements. On the other side, when the number of elements (or number of degrees of freedom) is enhanced, the CPU time and storage space are also increased. For this reason, in this paper, the complex Fourier shape functions have been developed, which using them, both the number of elements can be reduced and also the suitable results can be obtained.
In the present paper, at first, the governing equations and boundary conditions are expressed. Then, the FEM formulation and solution procedure are stated. Next, the complex Fourier shape functions and their enrichment process are described. Finally, three benchmark numerical examples are used in solving the Navier-Stokes and Laplace equations for the application of the proposed functions in the finite element method. These tests include Couette flow, flow of a viscous lubricant in a slider bearing and steady state heat transfer in rectangular region. In order to show the efficiency and accuracy of the present method, the results of the proposed method are compared with the classic functions and also the analytical solutions. The results of this comparison indicate the high accuracy of the proposed method.
. The present functions have properties of Gaussian and real Fourier radial basis functions. These useful properties have provided the robustness of the proposed method. Also, these functions have the simultaneously functions field such as trigonometric, exponential, and polynomial; while the classic Lagrange functions satisfied only polynomial functions field. In other words, these features provide an improvement in the solution accuracy with number of elements which are equal or lower than the ones used by the classic finite element method.Solving the Navier-Stokes and Laplace equations is the important challenge in the fluids mechanics problems. The most problems cannot be solved by the analytical methods. For this reason, the numerical methods are developed. Generally, the numerical methods are divided to two classes: the methods based on the mesh and meshless methods. In the first class, the computational domain are meshed and the governing equations are solved based them the finite element method, Finite Difference Method (FDM) and finite volume method (FVM) are placed in this category. While, in the second category methods, the computational domain is divided to moving particles. In these methods, there is no needed to any grid and the equations are solved on the particles. The smoothed particles hydrodynamic (SPH) method, Moving Particles Semi-implicit method and Discrete Least Squares Meshless method are in this class. The FEM is capable to solving the problems with complicated geometry. Also, the Neumann boundary conditions are applied properly.
Generally, the numerical methods such as finite element and finite difference methods are based on the mesh for solving the equations. For obtaining the results with high accuracy, it is needed to have enough elements. On the other side, when the number of elements (or number of degrees of freedom) is enhanced, the CPU time and storage space are also increased. For this reason, in this paper, the complex Fourier shape functions have been developed, which using them, both the number of elements can be reduced and also the suitable results can be obtained.
In the present paper, at first, the governing equations and boundary conditions are expressed. Then, the FEM formulation and solution procedure are stated. Next, the complex Fourier shape functions and their enrichment process are described. Finally, three benchmark numerical examples are used in solving the Navier-Stokes and Laplace equations for the application of the proposed functions in the finite element method. These tests include Couette flow, flow of a viscous lubricant in a slider bearing and steady state heat transfer in rectangular region. In order to show the efficiency and accuracy of the present method, the results of the proposed method are compared with the classic functions and also the analytical solutions. The results of this comparison indicate the high accuracy of the proposed method.
Article Type: Original Research |
Subject:
Hydraulical Structures
Received: 2018/05/18 | Accepted: 2018/12/4 | Published: 2019/03/15
Received: 2018/05/18 | Accepted: 2018/12/4 | Published: 2019/03/15
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