An integral differentiation procedure for dynamic time-history response analysis of structures

Traditionally, the response of displacement is selected to be the basic unknown and the responses of velocity and acceleration are usually expressed by linear weighted sum of the displacement when the differential quadrature (DQ) method is applied to the solution of dynamic problems. Either the linear equations or matrix equations (Sylvester equation) has to be processed in such procedure for dynamic solutions and the derived algorithm is conditionally stable in general. In this paper, the DQ principle is used in the inverse way to implement a high-accuracy explicit algorithm for the operation of convolution, and the algorithm is applied to dynamic analysis via the solution of Duhamel's integral. The dynamic response can be solved over a finite time interval according to this procedure, so that the total time-history of response could be obtained step by step. The inverse of Vandermonde matrix is required only once if the distribution of DQ nodes are completely consistent in each time interval and the response at several time instants during the interval can be obtained simultaneously. Hence, the procedure for dynamic solutions numerically achieves a high computational efficiency. It is proved that the spectral radius of the transfer matrix in the dynamic algorithm is always equal to 1, so the algorithm has unconditional stability and no numerical dissipation occurs during the calculation. The numerical accuracy of this algorithm depends on N, the number of DQ nodes within the analyzing time interval, and an algebraic accuracy with the order of N-1 can be achieved. In practice, 10 and even more DQ nodes of N are suggested in order to gain high accuracy for dynamic problems. © 2019, Chinese Journal of Theoretical and Applied Mechanics Press. All right reserved.