Smooth Conformal α-NEM for Gradient Elasticity

Amirtham Rajagopal, M. Scherer, Paul Steinmann, N. Sukumar

Abstract


Strain gradient theory for continuum analysis has kinematic relations, which include terms from second gradients of the deformation map. This results in balance equations that have fourth order spatial derivatives, supplemented with higher order boundary conditions. The weak formulation of fourth order operators stipulate that the basis functions must be globally

 

 

C1 continuous; very few finite elements in two dimensions meet this requirement. In this paper, we propose a meshfree methodology for the analysis of gradient continua using a conformal α-shape based natural element method (NEM). The conformal α-NEM allows the construction of models entirely in terms of nodes, and ensures quadratic precision of the interpolant over convex and non-convex boundaries. Smooth natural neighbor interpolants are achieved by a transformation of Farin’s C1 interpolant, which are obtained by embedding Sibson’s natural neighbor coordinates in the Bernstein-B´ezier surface representation of a cubic simplex. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.


Full Text: PDF