A New Family of Higher-Order Time Integration Algorithms for the Analysis of Structural Dynamics

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Summary In this article, a two‐stage explicit time integration method is presented for the numerical analysis of various dynamic problems described by second‐order ordinary differential equations in time. The newly proposed method have negligible damping and period errors in the important low‐frequency range and can introduce moderate numerical damping into the spurious high‐frequency range. The computational effort per step required in the new method is the same as the effort needed in the existing two‐stage explicit methods. The proposed method can provide fourth‐order accurate solutions for velocity independent problems and second‐order accurate solutions for velocity dependent problems, which is not possible in the existing methods. The proposed new method is expected to be an excellent tool for numerical analyses of various dynamic problems including structural dynamics.

“…In the past studies, the initial conditions have been chosen as θ 0 =0 and

{\overset{θ}{true}}_{0} = 1.99999923845

Section: Illustrative Linear and Nonlinear Examplesmentioning

confidence: 99%

{\overset{θ}{true}}_{0} = 2.00000076154

to synthesize another highly nonlinear pendulum that rotates completely about the point O .…”

Section: Illustrative Linear and Nonlinear Examplesmentioning

confidence: 99%

“…Comparison of absolute error

(||, {\overset{r}{true}}_{numerical} - {\overset{r}{true}}_{exact})

at t =0.1.

{\overset{r}{true}}_{exact}

at t =0.1 is computed by using the 10th‐order accurate method of Kim and Reddy [Colour figure can be viewed at wileyonlinelibrary.com]…”

Section: Illustrative Linear and Nonlinear Examplesmentioning

confidence: 99%

Section: Illustrative Linear and Nonlinear Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Kim

2019

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A simple explicit single step time integration algorithm for structural dynamics

Kim

2019

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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.

Highly precise and efficient solution strategy for linear heat conduction and structural dynamics

Xing

2021

Time-dependent parabolic and hyperbolic equations are widely encountered in practical engineering. Accurate and fast solution methods for such equations have always attracted attention. To reach this aim, this article proposes a strategy to construct highly precise and efficient time integration methods (TIMs) for linear heat conduction and structural dynamic systems. In the proposed strategy, an integrated amplification matrix of a TIM is created to precisely transfer the free responses of the previous step, and the Gauss-Legendre quadrature is employed to precisely compute the forced responses of the current step. To reduce computational cost and rounding error, a 2 m algorithm and a method of storing incremental matrix are applied in the construction of the integrated matrix.Moreover, the L-stable backward difference formula for heat conduction and the generalized trapezoidal rule for structural dynamics, which is A-stable and has controllable dissipation, are utilized to conduct this strategy. Numerical experiments validate that compared with some existing TIMs, the TIMs generated by the proposed strategy enjoy advantages both in precision and efficiency.