An efficient generic direct integration method for the generalized damping structure dynamic system

Generalized damping is a time-memory nonlocal model that takes the complete history velocity into account through a convolution. The kernel function of the convolution represents the distribution or the weight of history velocity in the time dimension. In this work, a high-efficiency algorithm is proposed for the calculation of generalized damping convolution. In this algorithm, the kernel function in the convolution is approximated by Fourier series. The relationship between the second derivative of the convolution and the convolution itself has been established. The derivative of the convolution is computed through multi-points difference scheme. This algorithm combined with direct time integration method is a generic method that can be applied for any causal kernel functions. By analyzing the computational complexity of different methods, it can be seen that the computational complexity of the existing methods is not only related to the number of degrees of freedom of the system, but also has a quadratic or cubic relationship with the number of computational steps. The computational efficiency of the proposed method is only related to the number of degrees of freedom of the system, not the number of steps, its efficiency is higher than that of the existing direct integration methods. Numerical examples are provided to illustrate the accuracy and especially the efficiency of the integration.