A generalized class of energy conservation/dissipation [GInO]-σ family of time operators with improved physical attributes for non-linear computational dynamics

Direct extension of linear multistep time integration methods (LMS) when applied to non-linear situations exhibit unbounded growth in energy with both dissipative and non-dissipative framework. It is well known that energy-momentum conserving method (EMM) when applied to non-linear situations, conserves energy. The key difference between the EMM and the Mid-point rule, although they are equivalent in the linear regime is the unique treatment of the stress update. Therefore, in this paper, extension of the generalized integration operator [GInO] framework to represent the non-linear internal force by employing the true algorithmic stress is presented. In addition to modeling of conservation/dissipation of energy correctly, the methodology which is termed here as [GInO]-σ is also inherits optimal properties such as second-order accuracy, minimal numerical dissipation and dispersion, and zero-order displacement and velocity overshoot behavior.