Symplectic two-step algorithms of FE dynamic equations

Two symplectic two-step solution algorithms of finite element dynamics equilibrium equations are presented, one of which has the fourth order accuracy, the phase is forward, and the accuracy of the other is the second order, the phase is backward. The calculation work of the fourth order two-step methods is not larger than that of Newmark symplectic method whose accuracy is of the second order, so there is an advantage of the fourth order two-step symplectic algorithm in the application to the practical calculation. The phase errors of the fourth order algorithm are compared in detail with that of Newmark symplectic method. Numerical results are given to illustrate the accuracy of the newly constructed two-step algorithms by comparing the obtained dynamic responses and energy with that of symplectic Newmark and analytical methods.