Hybrid Fundamental Solution based Finite Element Method for Axisymmetric Potential Problems

A hybrid fundamental solution based finite element method is proposed for analyzing axisymmetric potential problems with axisymmetric or arbitrary boundary conditions. In the proposed method, the intra-element potential field is established by using a linear combination of fundamental solutions as interior trial functions. The frame potential field is independently defined on the element boundary. To save computational time and memory, the original three-dimensional problem needs to be reduced to a two dimensional one. Both the axisymmetric geometry and the boundary conditions are expressed in the form of the Fourier series for the solution of those problems where arbitrary boundary conditions are concerned. In doing so, the two assumed fields of an element are expanded into a series of symmetric and asymmetric components. Based on the axisymmetric form of Hellinger-Reissner functional, the resultant element stiffness equation for each Fourier term involves integrals along the element boundary only. The principle of superposition is finally exploited for the final solution. Several numerical examples demonstrate high-efficiency and insensitivity to mesh distortion of the proposed method.

Keywords: Axisymmetric potential problem, Arbitrary boundary condition, Hellinger-Reissner functional, Finite element method, Fundamental solution, Fourier series expansion.