Stabilized conforming nodal integration: exactness and variational justification

Abstract

In most Galerkin mesh-free methods, background integration cells partitioning the problem domain are required to evaluate the weak form. It is therefore worthwhile to consider these methods using the notions of domain decomposition with the integration cells being the subdomains. Presuming that the analytical solution is admissible in the trial solution, domain and boundary integration exactness, which depend on the orders of the employed trial solution and the required solution exactness, are identified for the strict satisfaction of traction reciprocity and natural boundary condition in the weak form. Unfortunately, trial solutions constructed by many mesh-free approximants contain non-polynomial terms which cannot be exactly integrated by Gaussian quadratures. Recently, stabilized conforming (SC) nodal integration for Galerkin mesh-free methods was proposed and illustrated to be linearly exact. This paper will discuss how linear exactness is ensured and how spurious oscillation encountered by direct nodal integration is suppressed in SC nodal integration from a domain decomposition point of view. Moreover, it will be shown that SC nodal integration can be formulated by the Hellinger–Reissner Principle and thus justified in the classical variational sense. Applications of the method to straight beam, plate and curved beam problems are presented.