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

摘要

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{sup}(00) hierarchical triangular element. The distorted element geometry in xy space is mapped into ξη space in a two unit equilateral triangle with seven nodes (master element). For the master triangular element, 2-D C{sup}(00) p-version hierarchical local approximation functions are considered in ξη 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 ξη space for various higher order global differentiability approximations in ξη space. The derivative degrees of freedom at the corner nodes in ξη 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 (xy) 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{sup}(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 ξη space exactly. Numerical studies will be presented in a subsequent companion paper.