A Regularized Galerkin Boundary Element Method for 3D Potential Flow
The Lagrangian VortexBoundary Element simulation of flow about zero thickness membranes involves the prediction of a corresponding potential flow (via the derivative boundary integral equation for the Neumann problem describing the potential jump distribution on the surface). This could be achieved by the commonly used Panel Method (PM), which assumes piecewise constant variation of the potential jump across the boundary elements/panels. Unfortunately, the velocity evaluations by the Panel Method are severely spiked and of O(1) accuracy for points that are proximal to the membrane surface  thus, adversely affecting the stability and the overall accuracy of the VortexBoundary Element simulations.
To this end, we have formulated a fully Regularized Galerkin Boundary Element Method (RGBEM) for high order prediction of 3D potential flow about zero thickness membranes [16], [17]. Two significant attributes of RGBEM are:
For this project, we have implemented our RGBEM for linear variation of the potential jump across plane triangular elements.
The robustness of RGBEM relative to PM was tested using the example of potential flow normal to a flat square plate (of unit sides). The following view of the velocity vector field at the plane of symmetry depicts the orientation of the plate and the direction of the flow. The plate is on the xy plane and is centered at (x=y=0.5,z=0). The flow is in the positive z direction.
Note that the velocity profiles obtained by PM are severely spiked at points that correspond to the boundary element edges. This is because the potential jump is piecewise constant across the panels and its derivatives at the panel edges would necessarily yield spikes! In contrast, the linear RGBEM leads to piecewise constant tangential velocities; i.e., no spikes as verified in the following plot. However, a log singularity exists in the normal component of the velocity; nevertheless, it is a significant improvement compared to the spikes due to the PM prediction. In the following we note that the RGBEM profiles using only 200 triangular elements are better converged and smoother than the PM profiles using 3042 elements. Indeed, PM profiles using 12482 elements  a 62fold increase in the number of elements  still display significant spikes!
