TY  - EJOU
AU  - Devaraj, G  
AU  - Narayan, Shashi  
AU  - Roy, Debasish  

TI  - A Smooth Discretization Bridging Finite Element and Mesh-free Methods Using Polynomial Reproducing Simplex Splines
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2014
VL  - 102
IS  - 1
SN  - 1526-1506

AB  - This work sets forth a 'hybrid' discretization scheme utilizing bivariate simplex splines as kernels in a polynomial reproducing scheme constructed over a conventional Finite Element Method (FEM)-like domain discretization based on Delaunay triangulation. Careful construction of the simplex spline knotset ensures the success of the polynomial reproduction procedure at all points in the domain of interest, a significant advancement over its precursor, the DMS-FEM. The shape functions in the proposed method inherit the global continuity (<i>C<sup>p<sup>-1</sup></sup></i>) and local supports of the simplex splines of degree <i>p</i>. In the proposed scheme, the triangles comprising the domain discretization also serve as background cells for numerical integration which here are near-aligned to the supports of the shape functions (and their intersections), thus considerably ameliorating an oft-cited source of inaccuracy in the numerical integration of mesh-free (MF) schemes. Numerical experiments show the proposed method requires lower order quadrature rules for accurate evaluation of integrals in the Galerkin weak form. Numerical demonstrations of optimal convergence rates for a few test cases are given and the method is also implemented to compute crack-tip fields in a gradient-enhanced elasticity model.
KW  - globally smooth shape function
KW  -  hybrid method
KW  -  polynomial reproduction
KW  -  bivariate simplex splines
KW  -  knot construction
KW  -  moment matrix invertibility

DO  - 10.3970/cmes.2014.102.001