Two novel explicit time integration methods based on displacement-velocity relations for structural dynamics

•Two novel explicit methods based on displacement-velocity relations were proposed.•They are truly self-starting without the computation of accelerations.•Their optimal schemes with controllable numerical dissipation were presented.•Linear and nonlinear examples were simulated to show their advantages. Two novel explicit time integration methods are proposed based on displacement-velocity relations in this paper for structural dynamics. They avoid the factorization of damping and stiffness matrices, and are truly self-starting due to the exclusion of acceleration vectors. The first method employs the motion equation of expanded form and a linear relation of the displacement and velocity vectors. The recommended parameters, derived from linear analysis, enable the method to possess first-order accuracy mostly and second-order accuracy in the absence of numerical and physical damping. The second method adopts the idea of composite methods, and employs two motion equations of different expanded forms per step. Theoretical analysis indicates that this method can achieve a maximum stability limit of 4, and provides a single-parameter optimal scheme, controlled by the degree of numerical dissipation, for this method. The resulting scheme is second-order accurate with the stability limit ranged from 3.5708 to 4 for the undamped case, and some numerical experiments show that it has better numerical performance compared with some up-to-date explicit methods.