Abstract
The inverse relations of the isoparametric mapping for the 8-node hexahedra are derived by using the theory of geodesics in differential geometry. Such inverse relations assume the form of infinite power series in the element geodesic coordinates, which are shown to be the skew Cartesian coordinates determined by the Jacobian of the mapping evaluated at the origin. By expressing the geodesic coordinates in turn in terms of the isoparametric coordinates, the coefficients in the resulted polynomials are suggested to be the distortion parameters of the element. These distortion parameters can be used to sompletely describe the inverse relations and the determinant of the Jacobian of the mapping. The meanings of them can also be explained geometrically and mathematically. These methods of defining the distortion measures and deriving the inverse relations of the mapping are completely general, and can be applied to any other two-or three-dimensional isoparametric elements.
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References
Bishop, R. L.; Goldberg, S. I. 1968: Tensor Analysis on Manifolds, Macmillan
Eisenhart, L. P. 1949: Riemannian Geometry, Princeton University Press
Hua, C. 1990: An inverse transformation for quadrilateral isoparametric elements, analysis and application. Finite Elements in Analysis and Design 7: 159–166
Knupp, P. M. 1990: On the invertibility of the isoparametric map. Comput. Meths. Appl. Mech. Eng. 78: 313–329
Murti, V.; Valliappan, S. 1986. Numerical inverse isoparametric mapping in remeshing and nodal quantity contouring. Comput. Struct. 22: 1011–1021
Murti, V.; Wang, Y.; Valliappan, S. 1988: Numerical inverse isoparametric mapping in 3D FEM: Comput. Struct. 24: 611–622
Robinson, J. 1987: Some new distortion measures for quadrilaterals. Finite Elements in Analysis and Design 3: 183–197
Robinson, J. 1988: Distorition measures for quadrilaterals with curved boundaries, Finite Elements in Analysis and Design 4: 115–131
Veblen, O. 1962: Invariants of quadratic differential forms. Cambridge University Press
Yuan, K. Y.; Pian, T. H. H. 1993: The reference coordinates and distortion measures for quadrilateral hybrid stress membrane element. Computational Mechanics, to be published
Yuan, K. Y.; Huang, Y. S.; Pian, T. H. H. 1993: Inverse mapping and distortion measures for quadrilaterals with curved boundaries, Int. J. Numer. Methods Eng., to be published.
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Communicated by S. N. Atluri, 6 September 1993
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Yuan, K.Y., Huang, Y.S., Yang, H.T. et al. The inverse mapping and distortion measures for 8-node hexahedral isoparametric elements. Computational Mechanics 14, 189–199 (1994). https://doi.org/10.1007/BF00350284
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DOI: https://doi.org/10.1007/BF00350284