Global Format for Conservative Time Integration in Nonlinear Dynamics

The widely used classic collocation-based time integration procedures like Newmark, Generalized-alpha etc. generally work well within a framework of linear problems, but typically may encounter problems, when used in connection with essentially nonlinear structures. These problems are overcome in the new generation of energy conserving algorithms developed over the last two decades. However, the conservative algorithms typically rely on the special structure of the problem to be solved and require intermediate calculations using a mean state. This seems to have limited their use outside academia. In the present paper a conservative time integration algorithm is developed in a format using only the internal forces and the associated tangent stiffness at the specific time integration points. Thus, the procedure is computationally very similar to a collocation method, consisting of a series of nonlinear equivalent static load steps, easily implemented in existing computer codes. The paper considers two aspects: representation of nonlinear internal forces in a form that implies energy conservation, and the option of an algorithmic damping with the purpose of extracting energy from undesirable high-frequency parts of the response. The energy conservation property is developed in two steps. First a fourth-order representation of the internal energy increment is obtained in terms of the mean value of the associated internal forces and an additional term containing the increment of the tangent stiffness matrix over the time step. This explicit formula is exact for structures with internal energy in the form of a polynomial in the displacement components of degree four. A fully general form follows by introducing an additional term based on a secant representation of the internal energy. The option of a simple monotonic algorithmic damping is included by introducing a slight shift in the weighting of the displacement and velocity components at the forward and the current time, and the magnitude of the corresponding algorithmic damping is therefore controlled by a single parameter.