THE INVERSE MAPPING AND DISTORTION MEASURES FOR 8-NODE HEXAHEDRAL ISOPARAMETRIC ELEMENTS

The inverse relations of the isoparametric mapping for the 8-node hexa hedra are derived by using the theory of geodesics in differential geo metry. Such inverse relations assume the form of infinite power series in the element geodesic coordinates, which are shown to be the skew C artesian coordinates determined by the Jacobian of the mapping evaluat ed at the origin. By expressing the geodesic coordinates in turn in te rms of the isoparametric coordinates, the coefficients in the resulted polynomials are suggested to be the distortion parameters of the elem ent. These distortion parameters can be used to completely describe th e inverse relations and the determinant of the Jacobian of the mapping . The meanings of them can also be explained geometrically and mathema tically. These methods of defining the distortion measures and derivin g the inverse relations of the mapping are completely general, and can be applied to any other two- or three-dimensional isoparametric eleme nts.