Summary: The unsymmetric finite element is based on the virtual work principle with different sets of test and trial functions. In this article, the incompatible four-node quadrilateral element and eight-node hexahedral element originated by Wilson et al. are extended to their unsymmetric forms. The isoparametric shape functions together with E. L. Wilson’s incompatible functions [in: S. T. Fenven (ed.), Numerical and computer methods in structural mechanics. Cambridge, MA: Academic Press (1973; doi:10.1016/B978-0-12-253250-4.50008-7)] are chosen as the test functions, while internal nodes at the middle of element sides/edges are added to generate the trial functions with quadratic completeness in the Cartesian coordinate system. A local area/volume coordinate frame is established so that the trial shape functions can be explicitly obtained. The key idea which avoids the matrix inversion is that the trial nodal shape functions are constructed by standard quadratic triangular/tetrahedral elements and then transformed in consistent with the quadrilateral/hexahedral elements. Numerical examples show that the present elements keep the merits of both incompatible and unsymmetric elements, that is, high numerical accuracy, insensitivity to mesh distortion, free of trapezoidal and volumetric locking, and easy implementation.
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