Reformulating the Finite Element Method Based on Complex Fourier Elements in increasing the solution accuracy of the Navier-Stokes and Laplace Equations

Document Type : Original Research

Authors
S. Farmani
M. Ghaeini-Hessaroeyeh
S. Hamzehei-Javaran
Shahid Bahonar University of Kerman
Abstract
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.

Keywords

Subjects

[1] Reddy, J.N. 2004 An introduction to nonlinear finite element analysis, Oxford University Press Inc, New York.
[2] Zienkiewicz, O.C. 1971 The Finite Element Method in Engineering Science. McGraw-Hill, New York.
[3] Hood, P. 1970 A finite element solution of the Navier-Stokes equations for incompressible contained flow, M. SC. Thesis, University of Wales, Swansea.
[4] Oden, J.T. 1971 The finite element method in fluid mechanics. Lecture for NATO Advanced Study Institute on Finite Element Methods in Continuum Mechanics, Lisbon.
[5] Taylor, C. and Hood, P. 1973 A numerical solution of the Navier-Stokes equations using the finite element technique. Computers & Fluids, 1, 73-100.
[6] Reddy, J.N. 2010 The finite element method in heat transfer and fluid dynamics. third edition, CRC Press.
[7] Nardini D, Brebbia CA. 1983 A new approach to free vibration analysis using boundary elements. Applied Mathematical Modelling. 7(3), 157–62.
[8] Golberg MA, Chen CS. 1994 The theory of radial basis functions applied to the BEM for inhomogeneous partial differential equations”. Boundary Elements Communications. 5(2), 57–61.
[9] Chen CS. 1995 The method of fundamental solution for non-linear thermal explosions. Communications in Numerical Methods in Engineering. 11, 675–81.
[10] Karur SR, Ramachandran PA. 1995 Augmented thin plate spline approximation in DRM. Boundary Elements Communications. 6(2), 55–58.
[11] Agnantiaris JP, Polyzos D, Beskos DE. 1996 Some studies on dual reciprocity BEM for elastodynamic analysis. Computational Mechanics. 17, 270–277.
[12] Rashed, YF. 2008 Free Vibration of Structures with Trigonometric SIN(R) Function in the Dual Reciprocity Boundary Element Analysis. Advances in Structural Engineering. 11(4), 397–409.
[13] Samaan MF, Rashed YF. 2009 Free vibration multiquadric boundary elements applied to plane elasticity. Applied Mathematical Modelling. 33, 2421–32.
[14] Hamzehei Javaran S, Khaji, N. 2012 Inverse multiquadric (IMQ) function as radial basis function for plane dynamic analysis using dual reciprocity boundary element method. 15th World Conference on Earthquake Engineering, Lisboa, Portugal.
[15] Hamzeh Javaran S, Khaji N, Moharrami H. 2011A dual reciprocity BEM approach using new Fourier radial basis functions applied to 2D elastodynamic transient analysis. Engineering Analysis with Boundary Elements. 35,85–95.
[16] Hamzeh Javaran S, Khaji N, Noorzad A. 2011 First kind Bessel function (J-Bessel) as radial basis function for plane dynamic analysis using dual reciprocity boundary element method. Acta Mech. 218, 247–58.
[17] Hamzehei Javaran, S., Shojaee, S. 2017. The solution of elasto static and dynamic problems using the boundary element method based on spherical Hankel element framework. International Journal of Numerical Methods in Engineering, 112(13), 2067-2086.
[18] Wendland H. 1995 Piecewise polynomial, positive definite and compactly supported radial basis functions of minimal degree. Advances in Computational Mathematics. 4, 389–96.
[19] Reddy, J.N. 1986 Applied Functional Analysis and Variational Methods in Engineering, McGraw Hill, New York.
[20] Oden, J. T. and Carey, G. F. 1983 Finite Elements, Mathematical Aspects, Vol. IV,Prentice.Hall, Englewood Cliffs, NJ.
[21] Kreyszig E. 2006 Advanced engineering mathematics. JohnWiley & Sons, Inc.
[22] Wang, J.G and Liu, G.R. 2002 On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Computer Methods in Applied Mechanics and Engineering.191, 2611-2630.
[23] Khaji, N, Hamzehei Javaran S. 2013 New complex Fourier shape functions for the analysis of two-dimensional potential problems using boundary element method. Engineering Analysis with Boundary Elements. 37, 260-272.
[24] Reddy, J.N. 2013 An introduction to continuum mechanics, Cambridge University Press.
Modares Civil Engineering journal
Volume 18, Issue 6
November and December 2018
Pages 181-192