Generalized multiple scale reproducing kernel particle methods

An approach to unify reproducing kernel methods under one large umbrella and an extension to include time and spatial shifting are proposed. The study is divided into three major topics. The groundwork is set by revisiting the Fourier analysis of discrete systems. The multiresolution concept and its significance in devising the reproducing kernel methods and its discrete counterpart, reproducing kernel particle methods, are explained. An edge detection technique based on multiresolution analysis is developed. This wavelet approach, together with particle methods, gives rise to a straightforward h-adaptivity algorithm. By using this framework, a Hermite reproducing kernel method is also proposed, and its relation to wavelet methods is presented. It is also shown that the new approach generalizes existing kernel methods, and it can easily be degenerated into other widely used methods such as partition of unity, moving least-square interpolants, smooth particle hydrodynamics, scaling functions and wavelets, and multiple scale analysis. Furthermore, the Hermite reproducing kernel particle method, a particle based discrete version of the Hermite reproducing kernel method is developed. Finally, multiple-scale methods based on frequency and wave number shifting techniques are presented. A stability analysis is also presented for Newmark time-integration schemes for the low frequency equation. Numerical examples are presented throughout the paper to illustrate the flexibility and accuracy of this class of multiple scale methods.