Singularity cancellation in boundary integral equations for axisymmetric Stokes flow

Abstract

The boundary element method (BEM) provides a scheme for solving boundary integral equations numerically by dividing the boundary into elements and approximating the unknown variables over these elements by means of shape functions. In recent years this method has developed into a powerful tool for solving engineering problems. One difficulty that arises in solving problems by the boundary element method is the computation of integrals that exist only in the sense of the Cauchy principal value. Singular kernels appear in integrals that are on or near the diagonal of the matrices that result from the discretization of the boundary, and the solution is very sensitive to the accurate calculation of these integrals. For axisymmetric creeping flow problems (i.e., at zero Reynolds numbers’s2) the singularities are of order l/r and logr. Integration through the logarithmic singularity is easily performed by logarithmic Gauss quadrature. In Cartesian co-ordinates the calculation of the l/r-singular integrals has been overcome by the use of the rigid body motion m e t h ~ d . ~ However, in axisymmetric creeping flow, rigid body motion cannot be imposed in the radial direction. Other modes of motion, analogous to rigid body motion, have been suggested by Sarihan and Mukherjee? Bakr’ and Rizzo and Shippy.6 For certain problems the principal value of singular integral can even be evaluated analytically.’ However, because of the complexity of the kernel functions (combinations of elliptic integrals of the first and second kinds), general analytical integration cannot be performed. Recently, Guiggiani and Cassilini’ reported the direct calculation of Cauchy principal value integrals using standard and logarithmic Gauss quadrature. Telles’ used a mapping technique to eliminate the singularity. We will show here that these singularities cancel out directly in axisymmetric BEM by means of algebraic simplification for arbitrary elements and mappings. It is more practical to cancel out the l/r-singularities and obtain non-singular expressions rather than to calculate them individually using other techniques. We begin by presenting the boundary integral equations that describe creeping flow of an incompressible Newtonian fluid. Then we show how these equations ‘simplify’ for axisymmetric flow. Finally we show how the singularities cancel for any order elements.