Evalauation of shape functions for nodes on the boundary in weak form

I am doing my Masters Thesis on Application of Mesh free method on Laser cutting.

I am using Radial point interpolation method for evaluation of the shape functions.I have successfully applied it to the interior nodes and I am using weak form for the nodes on the natural boundaries.

I am facing difficulty in application of numerical integration for the functions in the weak form .I am using circular subdomain and applying Gauss quadrature formula for the numerical integration.

I would be glad if I can get some help in this regard.

Thank you

Regards

Sharmistha Sarangi