Under the Hamel’s formulism of Hamiltonian mechanics, a fast discrete geometric numerical integration algorithm is proposed for the simulation of infinitedimensional mechanical systems. First, a dual frame operator is introduced, based on which the reduced Poisson bracket is derived. The resulting Hamiltonian equations recover the Hamel field equations and their compatibility conditions. By combining the discrete Poisson bracket with the symplectic Euler scheme and the implicit midpoint scheme, a Poisson integrator is constructed. Next, using the geometrically exact beam’s kinematic model as an example, the reduced Poisson bracket and Hamiltonian equations are derived for both continuous and discrete cases, leading to the Poisson integrator for the geometrically exact beam. Finally, numerical simulations demonstrate that the proposed Poisson integrator preserves energy and momentum while significantly improving computational efficiency compared to the Hamel field integrator.