Advances in multiple scale kernel particle methods

A novel approach to multiresolution analysis based on reproducing kernel particle methods (RKPM) and wavelets is presented. The concepts of reproducing conditions, discrete convolutions, and multiple scale analysis are described. By means of a newly proposed semidiscrete Fourier analysis, RKPM is further elaborated in the frequency domain, and the interpolation estimate and the convergence of Galerkin solutions are given. The elimination of a mesh, combined with the properties of the dilation and translation of a window function, multiresolution analysis, wavelet-based error estimators, and edge detection brings about a new generation of hp adaptive methods. In addition, this class of multiple scale reproducing kernel particle methods is particularly suitable for problems with large deformations, high gradients, and high modal density. The current application areas of RKPM include structural acoustics, structural dynamics, elastic-plastic deformation, computational fluid dynamics and hyperelasticity.