  In this article, a family of $H^2$-nonconforming finite elements on
tetrahedral grids is constructed for solving the biharmonic equation in 3D. In
the family, the $P_\ell$ polynomial space is enriched by some high order
polynomials for all $\ell\ge 3$ and the corresponding finite element solution
converges at the optimal order $\ell-1$ in $H^2$ norm. Moreover, the result is
improved for two low order cases by using $P_6$ and $P_7$ polynomials to enrich
$P_4$ and $P_5$ polynomial spaces, respectively. The optimal order error
estimate is proved. The numerical results are provided to confirm the
theoretical findings.
