Higher Order Global Differentiability Local Approximations for 2-D Distorted Triangular Elements

Abstract

The paper presents development of higher order global differentiability local approximations for two dimensional distorted triangular elements. We consider seven node (3 vertex nodes, 3 mid-side nodes and a center node) p-version C 00 hierarchical triangular element. The distorted element geometry in x y space is mapped into ξ η space in a two unit equilateral triangle with seven nodes (master element). For the master triangular element, 2-D C 00 p-version hierarchical local approximation functions are considered in ξ η space. The degrees of freedom and the approximation functions from the mid-side nodes and/or center node are borrowed to derive the derivative degrees of freedom at the corner nodes in ξ η space for various higher order global differentiability approximations in ξ η space. The derivative degrees of freedom at the corner nodes in ξ η space are transformed from the natural coordinate space to the physical coordinate space (x, y) using Jacobians of transformations to obtain the desired higher order global differentiability local approximations in the (x y) coordinate space. A pascal triangle is used to establish a systematic procedure for the selection of degrees of freedom and the corresponding approximation functions from C 00 p-version hierarchical element for global differentiability of any desired order. A quadrature procedure for integrating over triangular domain is also presented. The procedure integrates algebraic polynomials over triangular domains in ξ η space exactly. Numerical studies will be presented in a subsequent companion paper.