Distortion Measures and Inverse Mapping for Isoparametric 8-Node Plane Finite Elements with Curved Boundaries

Communications in Numerical Methods in Engineering, Wiley, Vol 14, pp.87-101, 1998

MESHING
RESEARCH
CORNER

Department of Structural Mechanics, Chalmers University of Technology. S-412 96 Goteborg, Sweden

Abstract
Utilizing systematically differential geometry the paper describes a method which substantially improves results obtained by Yuan et al. (1994), though the same technique is used in both articles. An 8-node isoparametric element wath curved boundaries is analysed as an object of differential geometry. Inverse transformations between normal (geodesic) co-ordinates and natural (isoparametric) co-ordinates are derived in terms of a Taylor series which is convergent and does not need many terms to give an excellent approximation of the element shape with four curved sides. The concept of local normal co- ordinates results in the definition of distortion measures of a plane element. It is shown, by exploring the theory of geodesic curves, that the distortion parameters of a chord quadrilateral, spanned on the corner nodes of the 8-node element with curved boundaries, are the basic distortion measures for this 8- node element. Thus, significant reduction of the number of these parameters, from 12 to 4, from previous works is obtained. For the purpose of the finite element method, which is very sensitive to a shape of quadrilateral elements, only basic deviation measures from a regular form of a plane element are of interest. The distortion measures due to curvatures of sides seem to be of secondary significance in the analysis if straight sides of the chord quadrilateral and curved boundaries are isomorphic. The mathematical analysis used is quite general and relies strongly on differential geometry. The results are independent of co-ordinate systems. The meaning of element distortion measures is suggested. This analysis can be extended to curved surfaces in R3.