An implicit gradient model by a reproducing kernel strain regularization in strain localization problems

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An accelerated, convergent, and stable nodal integration in Galerkin meshfree methods for linear and nonlinear mechanics (2016, methodology): “x/oj ̨j6n are determined by enforcing (25), consequently, the functions in (26) are constructed as [38]:”
Nonlocal Operators with Applications to Image Processing (2008, methodology): “…can be generalized to any order, e.g. second order derivatives can be estimated by We construct w such that the unit vectors are orthonormal: that is: A similar framework for approximating partial derivatives for the purpose of strain localization was found by Chen, Zhang and Belytschko in [18].”
Meshfree Methods: Progress Made after 20 Years (2017, unknown): “Implicit gradients have been employed for regularization in strain localization problems (Chen et al. 2004) to avoid the need of ambiguous boundary conditions associated with the standard gradient-type regularization methods, and easing the computational cost of meshfree collocation methods (Chi et…”