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

Abstract

Summary In this paper, finite element method is reformulated using new shape functions to approximate the state variables (ie, displacement field and its derivatives) and inhomogeneous term (ie, inertia term) of Navier's differential equation. These shape functions and corresponding elements are called spherical Hankel hereafter. It is possible for these elements to satisfy the polynomial and the first and second kind of Bessel function fields simultaneously, while the classic Lagrange elements can only satisfy polynomial ones. These shape functions are so robust that with least degrees of freedom, much better results can be achieved in comparison with classic Lagrange ones. It is interesting that no Runge phenomenon exists in the interpolation of proposed shape functions when going to higher degrees of freedom, while it may occur in classic Lagrange ones. Moreover, the spherical Hankel shape functions have a significant robustness in the approximation of folded surfaces. Five numerical examples related to the usage of suggested shape functions in finite element method in solving problems are studied, and their results are compared with those obtained from classic Lagrange shape functions and analytical solutions (if available) to show the efficiency and accuracy of the present 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.…”

Complex Fourier‐based stress intensity factor analysis of plane elasticity using boundary element theories

“…For example, the conical functions, 35–37 the thin‐plate splines, 38–40 the Gaussian functions, 41 multiquadrics, 42,43 and compact supported functions 44–47 are in the nonoscillatory class while the real and complex Fourier RBF 48–53 and J‐Bessel RBF 54,55 belong to the oscillatory class. The combination of the first and second kind of Bessel function in complex space is utilized in Hankel function and its application can be seen in various problems 56–59 …”

“…The combination of the first and second kind of Bessel function in complex space is utilized in Hankel function and its application can be seen in various problems. [56][57][58][59] In this article, to demonstrate the accuracy and efficiency of the proposed shape functions, six numerical examples are solved and the results are compared with the analytical results as well as those obtained by the classic Lagrange shape functions. The numerical results show that the proposed Fourier shape functions represent more accurate solutions in comparison with the classic Lagrange shape functions.…”