DISTORTION MEASURES AND INVERSE MAPPING FOR ISOPARAMETRIC 8-NODE PLANE FINITE-ELEMENTS WITH CURVED BOUNDARIES

Utilizing systematically differential geometry the paper describes a m ethod which substantially improves results obtained by Yuan et al. (19 94), though the same technique is used in both articles. An 8-node iso parametric element with curved boundaries is analysed as an object of differential geometry. Inverse transformations between normal (geodesi c) co-ordinates and natural (isoparametric) co-ordinates are derived i n terms of a Taylor series which is convergent and does not need many terms to give an excellent approximation of the element shape with fou r curved sides. The concept of local normal co-ordinates results in th e definition of distortion measures of a plane element. It is shown, b y exploring the theory of geodesic curves, that the distortion paramet ers of a chord quadrilateral, spanned on the corner nodes of the 8-nod e element with curved boundaries, are the basic distortion measures fo r this 8-node element. Thus, significant reduction of the number of th ese parameters, from 12 to 4, from previous works is obtained. For the purpose of the finite element method, which is very sensitive to a sh ape of quadrilateral elements, only basic deviation measures from a re gular form of a plane element are of interest. The distortion measures due to curvatures of sides seem to be of secondary significance in th e analysis if straight sides of the chord quadrilateral and curved bou ndaries are isomorphic. The mathematical analysis used is quite genera l and relies strongly on differential geometry. The results are indepe ndent of co-ordinate systems. The meaning of element distortion measur es is suggested. This analysis can be extended to curved surfaces in R -3. (C) 1998 John Wiley & Sons, Ltd.