Reproducing Kernel Partition of Unity: from Continuum to Quantum

Through three classes of problems at continuum macro-scale, meso-scale, and quantum-scale, we demonstrate how the reproducing kernel (RK) approximation can be constructed to alleviate the numerical difficulties associated with the standard finite element methods (FEM) in solving the partial differential equations of the three demonstration problems. A typical problem at macro-scale that cannot be effectively analyzed by FEM is the problem involving large material distortion and damage. Employment of the mesh based methods often encounters mesh entanglement type ill-conditioning or solution divergence in large deformation and fragment-impact problems. With the proper selection of continuity and locality in the RK approximation, this class of problems can be modeled with desired regularity. At meso-scale, surface energy of materials becomes critical, and the numerical method that can accurately approximate moving discontinuities on the evolving material interfaces is essential. For example, modeling of microstructure evolution and topological changes requires a continuous remeshing using FEM. This difficulty is resolved by the RK approximation by introducing an interface enrichment function to adequately capture meso-scale moving material interfaces. In quantum mechanics, Schrödinger equation exhibits a highly nonlinear behavior near nuclei. We demonstrate how p-refinement and enrichment of orbital functions can be formulated under the RK approximation framework for effective numerical solution of Schrödinger equation.