Evaluation of gradients on the boundary using fully regularized hypersingular boundary integral equations

The evaluation of the gradient of the primary variable on the boundary for the Laplace problem, and the stress for the elasticity problem, involves hypersingular boundary integrals (HBIEs). To obtain any meaningful results from these integrals, an appropriate regularization scheme needs to be developed. We present an elegant way of calculation of gradients on the boundary of a body, starting from HBIEs regularized by using "simple solutions" or "modes". Our method is currently limited to the calculation of gradients at regular points on the boundary at which the gradients of the primary variables are continuous. The iterative scheme developed in this paper is shown to be extremely robust for the calculations of the gradients. The method is tested on two Laplace problems and two problems in linear elasticity. This method does not involve any limiting process and can be easily extended to 3-dimensions. The approach developed in this paper can also be extended to other problems like acoustics and elastodynamics.