Imposition of boundary conditions for elliptic equations in the context of non boundary fitted meshless methods

This paper describes a novel method to treat Neumann and Dirichlet boundary conditions in meshless discretizations of elliptic equations using nodal integration. The usual meshless framework for boundary fitted discretizations is first presented. Then, the possibility of dealing with non boundary fitted clouds of points is integrated into this framework. With this new method, the discretization of the boundary can automatically be generated from a covering point cloud in such a fashion that the degrees of freedom of the final discrete problem are associated with interior nodes only. This process minimally depends on the description of the simulation domain as it only needs to test whether a point is inside or outside of the domain. The emphasis is strongly set on building an adequate discrete structure, which allows a convenient interpretation of necessary conditions to pass the linear patch test.