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

Abstract

This paper presents development of higher order global differentiability local approximations for two dimensional quadrilateral elements of distorted geometries. The distorted quadrilateral elements in physical coordinate space x y are mapped into a master element in ξ η natural coordinate space in a two unit square with the origin at the center of the element. For the master element, 2-D C 00 p-version hierarchical local approximations are considered. The degrees of freedom and the approximation functions from the mid-side nodes and/or center node are borrowed to derive desired derivative degrees of freedom at the corner nodes in the ξ η space for various higher order global differentiability approximations in the ξ η space. These derivative degrees of freedom at the corner nodes in ξ η space are then transformed from the natural coordinate space (ξ, η) 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. Pascal rectangle 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 the global differentiability of any desired order in x y space. Numerical studies have been conducted to demonstrate the performance of the elements in terms of accuracy and convergence rates for the boundary value problems described by self-adjoint, non-self-adjoint and non-linear differential operators using undistorted as well as distorted discretizations (1). The numerical studies in this paper are only presented for self-adjoint operator