Precise Time-Integration Method with Dimensional Expanding for Structural Dynamic Equations

Abstract

The precise time-integration method proposed for a linear time-invariant dynamic system can give precise numerical results approaching the exact solution at the integration points. However, dife culties arise when the algorithm is used for nonhomogeneous dynamic systems due to the inverse matrix calculation required. A new algorithm isproposedto convertnonhomogeneousdynamicequationsinto homogeneousequationsbymeansofthe dimensional expanding method. With this conversion, the inverse matrix calculation is not required in the precise time-integration method. The new algorithm has enhanced the precise time-integration method by benee ting the programming implementation and the numerical stability; it has improved the computational efe ciency as well. Numerical examples are given to demonstrate the validity and efe ciency of the algorithm. Nomenclature Ai = coefe cient of the solution of Fourier series ai = coefe cient vector Bi = coefe cient of the solution of Fourier series bi = coefe cient vector C = magnitude of the load on the nodes of the structure D = coefe cient matrix of ordinary differential equations satise ed by the nonhomogeneous vector d = number of terms of one series f = load vector G = damping matrix H = coefe cient matrix of structural dynamic system H ¤ = coefe cient matrix of expanding structural dynamic system I = identity matrix K = number of time steps K = stiffness matrix k = order of time step l = number of dynamic modes M = mass matrix m = number of 2 N N = algorithm parameter of precise time integration n = dimension of the structural dynamic system p = transformed system state variable q = displacement vector r = nonhomogeneous vector of structural dynamic system Ta = small part of the matrix exponential Tc = part of matrix exponential according to matrix C Td = part of matrix exponential according to matrix D t = time v = state variable of structure dynamic system v ¤ = state variable vector of expanding structural