Multi-scale methods

In this paper four multiple scale methods are proposed. The meshless hierarchical partition of unity is used as a multiple scale basis. The multiple scale analysis with the introduction of a dilation parameter to perform multiresolution analysis is discussed. The multiple eld based on a 1-D gradient plasticity theory with material length scale is also proposed to remove the mesh dependency di culty in softening=localization problems. A non-local (smoothing) particle integration procedure with its multiple scale analysis are then developed. These techniques are described in the context of the reproducing kernel particle method. Results are presented for elastic-plastic one-dimensional problems and 2-D large deformation strain localization problems to illustrate the e ectiveness of these methods