A Unified Family of High‐Order Energy‐Conserving Time Integrators for Nonlinear Dynamical Problems

ABSTRACT This paper develops a family of energy‐conserving time integration methods for nonlinear conservative dynamical problems. Constructed within a unified, self‐starting, and single‐solve integration framework, the proposed methods avoid multi‐stage or sub‐step procedures, leading to improved computational efficiency while maintaining robust accuracy. By using a nonlinear conservative single‐degree‐of‐freedom system as a benchmark, the proposed framework systematically derives the conditions under which higher‐order consistency in conserving the total energy can be achieved. Three implicit integrators, called IA1‐L, IA1‐O, and IA1‐E, are designed to minimize the local truncation error in displacement, overshoot at the first time step, and total energy error at the first time step, respectively. These three schemes are unconditionally stable and controllably dissipative, and exhibit third‐order consistency in total energy conservation. To further enhance the accuracy in acceleration and total energy conservation, two identically second‐order accurate implicit and explicit schemes, IA2 and EA2, are constructed to achieve fourth‐order consistency in conserving the total energy. A range of numerical examples, including two linear multi‐degree‐of‐freedom problems, a two‐dimensional scalar wave propagation problem, the simple pendulum with distinct initial conditions, hardening/softening elastic springs, and the swinging wire rope with large‐deformation and large‐rotation, are solved and compared. The results demonstrate that the proposed methods significantly outperform the existing schemes, such as TPO/G‐ and EG‐, especially in terms of total energy conservation. Moreover, for spatially discretized nonlinear problems, introducing moderate numerical dissipation further improves energy conservation and numerical stability. Overall, the proposed methods provide some efficient implicit and explicit procedures for simulating nonlinear dynamics with energy‐conserving requirements.