Axisymmetric polyharmonic spline approximation in the dual reciprocity method

Abstract

AbstractThe paper is devoted to developing the dual reciprocity boundary element method for axisymmetric problems based on usage of axisymmetric polyharmonic splines. The axisymmetric polyharmonic spline is introduced as a linear combination of polyharmonic radial basic functions and polynomial terms. The analytical expressions for proposed axisymmetric polyharmonic radial basic functions are obtained for splines with arbitrary degrees. These expressions include special elliptic integrals that are analyzed and calculated for the first time. The relationships between radial basic functions with positive and negative degree numbers are obtained that allows us to receive the recurrence formulae for specific elliptic integrals with nearest indexes. It reveals the possibility of calculating radial basic functions with arbitrary orders by using the combination of only the first two members in recursive sequence. Implementation of the Gauss well‐known arithmetic‐geometric mean technique provides calculation of the specific elliptic integrals and axisymmetric polyharmonic splines with any given accuracy. Numerical examples for solving two axisymmetric problems in potential theory using the proposed dual reciprocity boundary element method demonstrate high calculation accuracy with low computational costs.