Numerical solutions of energy-conserving time integration methods

Two energy-conserving time integration methods developed by Simo and Hughes, respectively, were studied due to their excellent unconditional stability for general nonlinear structures to obtain the optimal method. Firstly, the ways of conserving system energy, hence ensuring unconditional stability, were analyzed by comparing the balance equations of the two methods. Then, numerical solutions of the two methods applied to solving dynamical balance equations were studied theoretically. The theoretical analyses showed that the balance equation of Simo method has the only one solution, while Hughes method may induce multiple solutions which may lead to unreasonable solution when solving balance equations. The results of numerical examples demonstrated the correctness of the theoretical analyses and showed higher computation efficiency of Simo method than those of Hughes one and the typical average acceleration one. The results of theoretical analyses and numerical examples exhibited advantages of Simo method over Hughes one and the average acceleration one.