The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

Abstract

Summary In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.

Utilizing new spherical Hankel shape functions to reformulate the deflection, free vibration, and buckling analysis of Mindlin plates based on finite element method

“…According to the literature, shape parameters are constants used in RBFs to increase the accuracy [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. It can be said that any specific problem requires the most suitable shape parameter by its own nature provided that high accuracy is demanded.…”

“…By way of illustration, conical, multiquadric, inverse multiquadric, Gaussian, and J-Bessel [12] has just one shape parameter, whilst complex Fourier [11] and Hankel RBFs [21][22][23] have two of them. For the Hankel shape functions, n and e are the shape parameters that belong to the set of whole numbers and positive real numbers, respectively.…”

“…Accordingly, an oscillatory RBF based on spherical Hankel functions is applied to derive new spherical Hankel shape functions. The Hankel function is created by combining the first and second kind of Bessel functions in complex space [21][22][23]. Using spherical Hankel shape functions results in profiting from the advantages of complex number space in functional space, which leads to a reduction in both algebraic manipulations and formulations.…”

Improvement of numerical modeling in the solution of static and transient dynamic problems using finite element method based on spherical Hankel shape functions

“…The spherical Hankel radial basis function (RBF) with all of its advantageous features plays the main role here. Recently, the implementations of these functions in solving elastostatic and elastodynamic problems using boundary element method and incompressible viscous flow problems using FEM were presented by Hamzehei Javaran et al There are two kinds of RBFs reported in the literature, including oscillatory and nonoscillatory ones. The conical functions, thin plate splines, Gaussian functions, multiquadric, inverse multiquadric, and compact supported functions can be mentioned as nonoscillatory RBF, while real and complex Fourier RBF and J‐Bessel RBF belong to the oscillatory class.…”