Large-scale long-term transient topology optimization method based on improved Duhamel integral

While transient topology optimization is essential for dynamic engineering applications, large-scale long-term problems remain computationally prohibitive due to the iterative nature of conventional time-integration schemes like the Newmark-β method. To improve design efficiency and accuracy, we propose an efficient optimization framework based on the Duhamel integral to solve large-scale problems with high degrees of freedom (DOFs) under prolonged loading. By integrating modal reduction to transform large-scale systems into decoupled single-degree-of-freedom problems, we derive an efficient recursive transient responses and utilize the Modal Acceleration Method (MAM) to ensure accuracy against high-order mode truncation. Unlike traditional numerical integration, this analytical approach ensures unconditional stability even with larger time steps and enables direct sensitivity analysis through explicit formulations. Numerical examples involving millions of DOFs and long durations demonstrate that the proposed method significantly reduces computational cost, making large-scale transient topology optimization practically tractable.