Formulation and Application of Spherical Hankel Elements in Numerical Modelling of Potential Problems using Boundary Element Method

In this paper, a new boundary element analysis for the modeling of two-dimensional potential problems is proposed. The boundary element method is reformulated here based on spherical Hankel elements for the purpose of approximation of the state variables of the Poisson and Laplace differential equations (potentials and fluxes). Spherical Hankel function is obtained by combing Bessel function of the first (similar to J-Bessel ones) and second (also called Neumann functions) kind so that the properties of both mentioned functions will be combined and result in a robust interpolation tool. The interpolation functions of the boundary element method are obtained using the enrichment of the spherical Hankel radial basis functions. To this end, the expansion of a function in which only the spherical Hankel radial basis functions approximations are used have been given polynomial terms. Generally, radial basis function (RBF) is an efficient tool in finding the solution of non-homogeneous partial differential equations. Its main idea is the expansion of non-homogeneous term by its values in interpolation nodes, based on Euclidean norm that leads to obtaining a particular solution. Although the J-Bessel RBF contains the features of the first kind of Bessel function, it usually cannot represent the full properties of a physical phenomenon. Therefore, using the combination of the first and second kind of Bessel function in complex space (Hankel function) may lead to more accurate and robust results. In other words, the solution of Bessel equation can be referred as a prominent usage of both first and second kind of Bessel, which shows that using them together may result in more accuracy and robustness. The aforementioned discussion brings this matter to mind whether it is possible to present RBFs that benefit from both Bessel functions of the first and second kind. Therefore, by the idea of combining spherical Hankel in imaginary space, enrichment of them for a three-node element in the natural coordinate system is explained in this paper. Moreover, the algebraic manipulations and formulations are reduced because of profiting from the advantages of complex number space in functional space. It is also possible for the proposed shape function to satisfy both Bessel function fields and polynomial functions, unlike classic Lagrange shape functions that only satisfy the polynomial function fields. Moreover, the proposed shape functions benefit from the infinite piecewise continuous property, which does not exist in the classic Lagrange shape functions with limited continuity. The spherical Hankel function of the first kind has a strong singularity in its imaginary part, the spherical Neumann function. This issue results in the fact that when the Euclidean norm tends to zero, the limit does not exist. In the following, an extra term with power  is applied to remove this singularity. After the elimination of the singularity, the limit state of coinciding source point and field point is calculated. In the end, to demonstrate the accuracy and efficiency of the proposed shape functions, several numerical examples are solved and compared with the analytical results as well as those obtained by classic Lagrange shape functions. The numerical results show that the proposed Hankel shape functions represent more accurate solutions, using fewer degrees of freedom, in comparison with classic Lagrange shape functions.