NURBS-Divergence-Meshless (NDM) method

In this study, we present a new formulation that applies the divergence theorem within the NURBS parameter space to discretize governing equations using meshless techniques. We refer to this method as the NURBS-Divergence-Meshless (NDM) method. In this method, NURBS not only represents the domain exactly but also simplifies both the expression and the process of line integration, benefiting from the rectangular structure of the parameter space.
The primary advantage of NDM lies in its use of the divergence theorem, which simplifies the discretization of governing equations by ensuring that the line integral yields the required number of equations. It also enables the direct imposition of boundary conditions in interpolation schemes lacking the Kronecker delta property, such as moving least squares (MLS).
To illustrate the foundational concept, the discretization and solution of a simple scalar problem are presented. The equation is discretized using either the MLS or radial basis function (RBF) interpolation, with the divergence theorem providing the resulting algebraic equations. Convergence studies show that the non-overlapping NDM-MLS performs best, exhibiting all the anticipated favorable characteristics.