Evaluation of the integral terms in reproducing kernel methods

Reproducing kernel method (RKM) has its origins in wavelets and it is based on convolution theory. Being their continuous version, RKM is often referred as the general framework for meshless methods. In fact, since in real computation discretization is inevitable, these integrals need to be evaluated numerically, leading to the creation of reproducing kernel particle method RKPM and moving least squares MLS approximation. Nevertheless, in this paper the integrals in RKM are explicitly evaluated for polynomials basis function and simple geometries in one, two and three dimensions even with conforming holes. Moreover, a general formula is provided for complicated shapes also for multiple connected domains. This is possible through a boundary formulation where domain integrals involved in RKM are transformed by Gauss theorem in circular or flux integrals. Parallelization is readily enabled since no preliminary arrangements of nodes is needed for the moments matrix. Furthermore, using symbolic inversion, computation of shape functions in RKM is considerably speeded up.