On the optimization of n -sub-step composite time integration methods
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ORIGINAL PAPER
On the optimization of n-sub-step composite time integration methods Huimin Zhang · Runsen Zhang · Yufeng Xing · Pierangelo Masarati
Received: 12 May 2020 / Accepted: 12 October 2020 © The Author(s) 2020
Abstract A family of n-sub-step composite time integration methods, which employs the trapezoidal rule in the first n − 1 sub-steps and a general formula in the last one, is discussed in this paper. A universal approach to optimize the parameters is provided for any cases of n ≥ 2, and two optimal sub-families of the method are given for different purposes. From linear analysis, the first sub-family can achieve nth-order accuracy and unconditional stability with controllable algorithmic dissipation, so it is recommended for highaccuracy purposes. The second sub-family has secondorder accuracy, unconditional stability with controllable algorithmic dissipation, and it is designed for heuristic energy-conserving purposes, by preserving as much low-frequency content as possible. Finally, some illustrative examples are solved to check the performance in linear and nonlinear systems. Keywords n-Sub-step composite method · Optimization · High-accuracy · Energy-conserving
Huimin Zhang · Runsen Zhang · Yufeng Xing School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China Huimin Zhang (B) · Runsen Zhang · Pierangelo Masarati Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, 20156 Milan, Italy e-mail: [email protected]
1 Introduction Direct time integration methods are frequently used to predict accurate numerical responses for general dynamic problems after spatial discretization. Driven by the pursuit of desirable properties, including higher accuracy and efficiency, robust stability, and many others, a number of excellent methods were proposed in the past decades. In terms of the formulations, existing methods are generally classified into explicit and implicit schemes. Explicit methods are mostly used in wave propagation problems, as their conditional stability limits the allowable time step size to the highest system frequency. Implicit methods have fewer restrictions on the problems to be solved due to the unconditional stability, but they require more computational efforts per step. In another way, the integration methods can also be divided into single-step, multi-sub-step and multistep techniques. The single-step methods only adopt the states of the last step to predict the current one, while the multi-sub-step methods also need the states at the intermediate collocation points, and the multistep methods require the states of more than one previous step. Each of them has specific advantages and disadvantages. From the literature, representative single-step methods include the Newmark method [25], the HHT-α method (by Hilbert, Hughes, and Taylor) [17], the WBZ-α method (by Wood, Bossak, and Zienkiewicz) [29], the generalized-α method [9], the GSSSS (gener-
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alized single-step single-solve) method [34], and many others
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