Title: A Regularized Galerkin Boundary Element Method (RGBEM) for Simulating Potential Flow About Zero Thickness Bodies

The prediction of potential flow about zero thickness membranes by the boundary element method constitutes an integral component of the Lagrangian vortex-boundary element simulation of flow about parachutes. To this end, the vortex loop (or the panel) method has been used, for some time now, in the aerospace industry with relative success [1, 2]. Vortex loops (with constant circulation) are equivalent to boundary elements with piecewise constant variation of the potential jump. In this case, extending the analysis in [3], the near field potential velocity evaluations can be shown to be {Omicron}(1). The accurate evaluation of the potential velocity field very near the parachute surface is particularly critical to the overall accuracy and stability of the vortex-boundary element simulations. As we will demonstrate in Section 3, the boundary integral singularities, which arise due to the application of low order boundary elements, may lead to severely spiked potential velocities at vortex element centers that are near the boundary. The spikes in turn cause the erratic motion of the vortex elements, and the eventual loss of smoothness of the vorticity field and possible numerical blow up. In light of the arguments above, the application of boundary elements with (at least) a linear variation of the potential jump--or, equivalently, piecewise constant vortex sheets--would appear to be more appropriate for vortex-boundary element simulations. For this case, two strategies are possible for obtaining the potential flow field. The first option is to solve the integral equations for the (unknown) strengths of the surface vortex sheets. As we will discuss in Section 2.1, the challenge in this case is to devise a consistent system of equations that imposes the solenoidality of the locally 2-D vortex sheets. The second approach is to solve for the unknown potential jump distribution. In this case, for commonly used C{sup o} shape functions, the boundary integral is singular at the collocation points. Unfortunately, the development of elements with C{sup 1} continuity for the potential jumps is quite complicated in 3-D. To this end, the application of Galerkin ''smoothing'' to the boundary integral equations removes the singularity at the collocation points; thus allowing the use of C{sup o} elements and potential jump distributions [4]. Successful implementations of the Galerkin Boundary Element Method to 2-D conduction [4] and elastostatic [5] problems have been reported in the literature. Thus far, the singularity removal algorithms have been based on a posterior and mathematically complex reasoning, which have required Taylor series expansion and limit processes. The application of these strategies to 3-D is expected to be significantly more complicated. In this report, we develop the formulation for a ''Regularized'' Galerkin Boundary Element Method (RGBEM). The regularization procedure involves simple manipulations using vector calculus to reduce the singularity of the hypersingular boundary integral equation by two orders for C{sup o} elements. For the case of linear potential jump distributions over plane triangles the regularized integral is simplified considerably to a double surface integral of the Green function. This is the case implemented and tested in this report. Using the example problem of flow normal to a square flat plate, the linear RGBEM predictions are demonstrated here to be more accurate, to converge faster, and to be significantly less spiked than the solutions obtained by the vortex loop method.