The solution of elastostatic and dynamic problems using the boundary element method based on spherical Hankel element framework

Abstract

In this paper, new spherical Hankel shape functions are used to reformulate boundary element method for 2-dimensional elastostatic and elastodynamic problems. To this end, the dual reciprocity boundary element method is reconsidered by using new spherical Hankel shape functions to approximate the state variables (displacements and tractions) of Navier's differential equation. Using enrichment of a class of radial basis functions (RBFs), called spherical Hankel RBFs hereafter, the interpolation functions of a Hankel boundary element framework has been derived. For this purpose, polynomial terms are added to the functional expansion that only uses spherical Hankel RBF in the approximation. In addition to polynomial function fields, the participation of spherical Bessel function fields has also increased robustness and efficiency in the interpolation. It is very interesting that there is no Runge phenomenon in equispaced Hankel macroelements, unlike equispaced classic Lagrange ones. Several numerical examples are provided to demonstrate the effectiveness, robustness and accuracy of the proposed Hankel shape functions and in comparison with the classic Lagrange ones, they show much more accurate and stable results. KEYWORDS 2D elastostatic and elastodynamic problems, boundary element method, dual reciprocity method, equispaced macroelements, Runge phenomenon, spherical Hankel shape functions | INTRODUCTIONElastostatics and elastodynamics illustrate a wide range of phenomena in engineering and physical problems such as force equilibrium of special structures and analysis of structures under vibratory motor, earthquake, explosion, and collision loads, where wave propagation is expressed by a governing linear partial differential equation, associated with suitable boundary and initial conditions. Generally, finding the solution of elastostatic and elastodynamic problems for analysis and design purposes through analytical approaches can be hard and time-consuming. Also, with a little complexity in boundary conditions, analytical solution may even become impossible. Therefore, in most practical engineering cases, solving them numerically seems to be inevitable. One of the numerical methods that have been considered significantly by researchers is boundary element method (BEM). 1,2 As it is clear from its name, boundary element only needs the boundary to be discretized, and it is not necessary to discretize the domain in this method. This matter leads to the reduction of unknowns to be stored. Therefore, less computational cost and storage space will be spent. For problems such as stress concentration or infinite domains, BEM can

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

“…By substituting a and b from relation (13) into Equation (9), it can result to the following relation (for more details, see the works of Hamzehei Javaran and Shojaee 30 and Wang and Liu 34 ):…”

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.…”