A Novel Newmark Family of Fourth-Order Accurate Algorithms with Complex Sub-Steps for Structural Dynamics

Abstract

Abstract

A new family of fourth-order accurate time integration schemes is developed by introducing two complex time sub-steps into the classical Newmark family of second-order algorithms. These sub-steps consist of a pair of complex conjugate numbers, enabling the triangularization of a complex-valued effective stiffness matrix. The proposed formulation can be easily implemented in existing codes with only minor modifications to the standard Newmark algorithm. The solution is composed of both real (physical) and imaginary components. The real component provides fourth-order accuracy even in the presence of external loads and physical damping, while the imaginary component offers additional insight into the distribution of numerical errors, an original feature not previously reported for implicit formulations. Compared to the classical Newmark method with a time-step size four times smaller, the proposed scheme exhibits significantly lower numerical dissipation and dispersion errors. Furthermore, the sub-step procedure extends the critical time step of conditionally stable members of the Newmark family by a factor of 3. The numerical analysis performed in the proposed time integration method, along with the results obtained for dynamic structural problems, including a complex three-dimensional (3D) application, clearly demonstrate that the method outperforms both Fung’s fourth-order complex scheme and the classical Newmark approach in terms of accuracy.