Mapping of C* Elements in Finite Element Method using Transformation Matrix

Abstract

Mapping between local and global coordinates is an important issue in finite element method, as all calculations are performed in local coordinates. The concern arises when subparametric are used, in which the shape functions of the field variable and the geometry of the element are not the same. This is particularly the case for * C elements in which the extra degrees of freedoms added to the nodes make the elements sub-parametric. In the present work, transformation matrix for 1* C (an 8-noded hexahedron element with 12 degrees of freedom at each node) is obtained using equivalent 0 C elements (with the same number of degrees of freedom). The convergence rate of 8-noded 1* C element is nearly equal to its equivalent 0 C element, while it consumes less CPU time with respect to the 0 C element. The existence of derivative degrees of freedom at the nodes of 1* C element along with excellent convergence makes it superior compared with it equivalent 0 C element. Keywords—Mapping, Finite element method, * C elements, Convergence, 0 C elements. I. MAPPING CONCEPT LEMENTS are divided into 3 categories in finite element method. These are: iso-parametric, sub-parametric and super-parametric elements. Iso-parametric elements are those in which the shape functions for both the field variable and the geometry of the element are the same. For two dimensional iso-parametric elements we have: 1 1 1 , , = n n n i i i i i i i i i x N x y N y N φ φ = = = = = ∑ ∑ ∑ (1) All calculations in finite element methods are usually carried out in local coordinates, ξ and η . The transformation between derivatives of shape functions in global and local coordinates for two dimensional iso-parametric elements is performed using Jacobian matrix which is defined as follow: