A simple explicit single step time integration algorithm for structural dynamics

Abstract

In this article, a new single-step explicit time integration method is developed based on the Newmark approximations for the analysis of various dynamic problems. The newly proposed method is second-order accurate and able to control numerical dissipation through the parameters of the Newmark approximations. Explicitness and order of accuracy of the proposed method are not affected in velocity-dependent problems. Illustrative linear and nonlinear examples are used to verify performances of the proposed method. KEYWORDS controllable numerical dissipation, explicit time integration method, impact and wave propagation, linear and nonlinear structural dynamics, pendulums Int J Numer Methods Eng. 2019;119:383-403.wileyonlinelibrary.com/journal/nme 384 KIM highly nonlinear problems, the higher-order methods are more suitable than the second-order methods. However, improved accuracy of the higher-order accurate methods is often accompanied by more complicated computations and increased computational efforts. For example, four stages are required to advance a step in the fourth-order accurate Runge-Kutta method, whereas only one stage is required in many of the second-order methods. Due to this reason, second-order accurate methods are broadly used for practical analyses. Among the classical explicit methods, the central difference (CD) 14 method and the leap-frog method 15 have simple and intuitive computational structures. Both methods do not have undetermined parameters, whereas algorithmic parameters should be properly determined in many of the recently developed explicit methods. The CD method is known as one of the most accurate nondissipative second-order explicit methods with the highest stability limit. Due to this reason, the CD method is often considered as a reference method for the study of accuracy and stability of explicit methods. The leap-frog and Verlet methods are also nondissipative and broadly used in various engineering and science research works. However, the aforementioned methods become only first-order accurate for velocity dependent problems when the velocity vector is explicitly treated.The objective of the current work is to propose a simple second-order explicit time integration algorithm with dissipation control capability. To verify the linear performance of the proposed explicit method, commonly used spectral analysis 16,17 is conducted by using the single-degree-of-freedom (SDOF) problem. Furthermore, linear and nonlinear test problems are solved by using the new and existing methods. The numerical results obtained from the various methods are compared with each other to investigate improved performances of the newly proposed method.

An accurate two‐stage explicit time integration scheme for structural dynamics and various dynamic problems

“…The central difference method can be summarized as When C =0 and f is velocity independent, the method presented in Equation (50) becomes explicit, but it becomes a conditionally stable second‐order accurate implicit method if C ≠ 0 or f is velocity dependent. Related discussions are provided in the work of Kim . In the central difference method, the evaluation of is required to start the procedure at t s =0.…”

“…From the conditions given in Equations (13) and (14), 2 c i 's in Equation (12) can be determined, and they can be substituted back into Equation (12). The displacement vector 2ū (t) can be rearranged in the form of…”

“…Related discussions are provided in the work of Kim. 12 In the central difference method, the evaluation ofü 0 = M −1 f (u 0 , . u 0 , 0) is required to start the procedure at t s = 0.…”

“…Thus, implicit methods require more computational efforts when compared with explicit methods. Many of the recently developed implicit methods [7][8][9][10][11][12][13] have been designed to have dissipation control capabilities. The spurious high-frequency mode can be effectively eliminated from numerical solutions by manipulating numerical dissipations.…”

Higher-order explicit time integration methods for numerical analyses of structural dynamics

“…As more sophisticated spatial finite element models [1][2][3][4] are developed constantly, demands for more accurate time integration methods are also increasing to take full advantage of improved spatial models in transient analyses. Recently, numerous implicit [5][6][7][8][9] and explicit [10][11][12][13][14][15][16] time integration methods have been introduced to effectively analyze challenging dynamic problems.…”