Consistent Diffuse Derivative Approximation in Particle Methods for Weak and Strong Formulations (1): Mathematical Foundations and Discretizations

Consistent diffuse derivative approximation (CDDA) which can be implemented in weak and strong form meshfree methods are presented. CDDA is equivalently derived from the diffuse derivative of moving least squares (MLS) approximation, the SPH approximation with derivative correction and the Taylor polynomial based on the MLS method. Various features of CDDA such as consistency and various mathematical properties are explored. In the weak formulation, the method suffers from passing the patch test due to the lack of accuracy coming from the discrepancy between the exact derivative and diffuse derivative of test function. However, employing non-differentiable weight functions with sharp peak dramatically improves the accuracy. Specially, the strong formulation based on the collocation scheme takes advantage of the inherited merits of meshfree method and shows the best efficiency with sufficient accuracy in discretizing partial differential equations.