Bi-discontinuous time step integration algorithms––Part 2: second-order equations

Abstract

In this paper, bi-discontinuous time step integration algorithms for Linear second-order equations are investigated. The second-order equations are manipulated directly. Both the initial and final displacement and velocity are only weakly enforced at the beginning and at the end of the time interval. As a result, there may be discontinuous jumps for both the displacement and velocity at the two ends of the time interval. If the displacement is assumed to be a polynomial of degree n+1 within the time interval, there are n+4 unknowns in the formulation. It is found that the order of accuracy would be at least n+1 for displacement and n for velocity in general. It is shown that the relevant algorithmic parameters for the second-order equations can be related to the algorithmic parameters presented in Part 1 of this paper for the first-order equations. Unconditionally stable higher order accurate time step integration algorithms with controllable numerical dissipation equivalent to the generalized Padé approximations with an order of accuracy 2 n+3 can then be constructed systematically. The weighting parameters and the corresponding weighting functions are given explicitly. It is also shown that the accuracy of the particular solutions is compatible with the accuracy of the homogeneous solutions if the proposed weighting functions are employed.