A Quadratic Basis Function, Quadratic Geometry, High Order Panel Method

Abstract

Most panel method implementations use both low order basis function representations of the solution and flat panel representations of the body surface. Although several implementations of higher order panel methods exist, diculties in robustly computing the self term integrals remain. In this paper, methods for integrating the single and double layer self term integrals are presented. The approaches are conceptually simple and robust. The paper considers quadratic basis functions to represent the solution, while the geometry of the body is approximated using piecewise parametric quadratic patches. Increased convergence rates are demonstrated for cases where the quadratic basis functions on quadratic curved panels are used. The quadratic panel method converges at a rate proportional to the cube of the panel side length (or NP 3 2, where NP is the number of panels).