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This paper presents the higher order extension of Particle Discretization Scheme (PDS) and its implementation in the FEM framework (PDS-FEM). The main advantage of PDS-FEM[1][2] is that it provides simple and numerically efficient treatment for modelling propagating cracks.  PDS uses conjugate tessellations to approximate functions and their derivatives, and in this paper we use the tessellation pair Voronoi  and Delaunay. In the original formulation[1][2], only the characteristic functions of Voronoi and Delaunay tessellations were used for approximating functions and derivatives, respectively.

In the higher order extension of PDS, we approximates functions and their derivatives as the union of local polynomial expansions within the support of each tessellation element.  Note that the use of characteristic function with local supports makes the base functions ’s  to be discontinuous along the boundary of tessellation element, hence the approximated function  has numerous discontinuities along the boundaries of tessellation elements. It is to define a bounded derivative approximation, PDS uses conjugate tessellation in approximating the derivative.

The proposed higher order extension of PDS is implemented in FEM framework for simulating the deformation of linear elastic bodies and propagating cracks. The numerous discontinuities present in the function approximation are exploited to model brittle cracks numerically efficiently. Several benchmark problems are presented to demonstrate the improvement in accuracy[3]. Further, J-integral about a mode-I crack tip field is estimated to demonstrate the improvement in accuracy of crack tip stress fields. It is shown that the singular crack tip stress field also has higher order accuracy and convergence rates, in addition to improved crack surface traction [4].