G. Papazafeiropoulos, V. Plevris and M. Papadrakakis, "A generalized algorithm framework for non-linear structural dynamics", Bulletin of Earthquake Engineering, 15(1), pp. 411-441, 2017.

Abstract:
The performance of a family of nonlinear generalized single step-single solve (GSSSS) time integration schemes is assessed by comparison of their results in terms of total energy and the agreement with respective results published in the literature. The nonlinear algorithms have been developed by their linear counterparts using a Newton–Raphson iterative procedure to ensure dynamic equilibrium inside each time step. A literature review of the available time integration schemes used for nonlinear problems and the family of linear GSSSS algorithms are presented along with several commonly used time integration algorithms as special cases. Afterwards, the nonlinear schemes are formulated, and outlined in an explicit flowchart, which describes the nonlinear integration procedure in detail. The nonlinear family of algorithms is applied to six benchmark problems involving the dynamic response of SDOF systems with various stiffness and damping properties, as well as to a 3dof structure representing finite element systems containing rigid connections, penalty factors and other such types of constraints. It is shown that the schemes with Continuous Acceleration formulation (such as the HHT-a method) perform in general better than the others, even with a large time step, which leads to reduced computational effort for the estimation of the nonlinear dynamic response with relatively little loss of accuracy.

Keywords:
Time integration scheme; Newton–Raphson; Single step; Single solve; Energy conservation