Reproducing Kernel Particle Method for Solving Partial Differential Equations

This chapter gives a state of the art review of the weak-form- and strong-form-based reproducing kernel particle method (RKPM). The construction of the continuous and discrete reproducing kernel (RK) approximations and their approximation properties are first presented. Recent developments and their connection to other meshfree methods, as well as an overview of the inherent multiresolution capabilities of the RKPM, are then given. The Galerkin-based method is discussed in detail, along with several recent advances in quadrature and stability. A review of strong form collocation with RK approximation as well as implicit gradient enhancement is presented. Formulations for large deformation and contact problems for which the RKPM is particularly well suited are also discussed.