A family of second-order fully explicit time integration schemes
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A family of second-order fully explicit time integration schemes Mohammad Rezaiee-Pajand1
· Mahdi Karimi-Rad1
Received: 2 May 2016 / Revised: 26 September 2017 / Accepted: 28 September 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract A new family of fully explicit time integration methods is proposed, which has second-order accuracy for structures with and without damping. Using a diagonal mass matrix, the suggested scheme remains fully explicit not only for structures with a non-diagonal damping matrix, but also when the internal force vector is a nonlinear function of velocity. The present algorithm has an acceptable domain of stability, and it is self-starting. This technique introduces effectively numerical dissipation to suppress the high-frequency spurious modes, while at the same time the lower modes are not affected too much. In addition, numerical dispersion error of the scheme is considerably smaller than that of the central difference method. Solving several linear and nonlinear problems highlights the superior performance of the authors’ approach. Findings demonstrate that solution time for the suggested scheme is much less than that of the central difference technique. The related algorithm can be easily implemented into programs, which already contain the central difference method. Keywords Explicit time integration · Second-order accuracy · Numerical dissipation · Nonlinear structural dynamics · Numerical dispersion
1 Introduction Finite element methods are widely used in structural dynamics in order to obtain numerical solutions. After spatial discretization of the weak form of the momentum equation in standard finite element methods, a system of coupled, second-order ordinary differential equations in time is derived. The related equations are solved numerically using direct time integration methods (Bathe 1996). These techniques can be classified as explicit or implicit. Explicit schemes use a difference expression which gives explicitly the unknown values at the end
Communicated by Jose Alberto Cuminato.
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Mohammad Rezaiee-Pajand [email protected] Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
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M. Rezaiee-Pajand, M. Karimi-Rad
of a time step in terms of preceding solutions. On the other hand, it is necessary to solve a system of equations in each time step by implicit methods (Wood 1990). Therefore, implicit algorithms need higher computational costs per time step in comparison with the explicit strategies. Conditional stability is a significant drawback of the explicit methods. As a result, the time step size must be selected based on the stability limit. In implicit unconditionally stable methods, the time step size can be selected independent of the stability considerations and thus can result in decreasing the number of time steps in time history analysis (Subbaraj and Dokanish 1989). Generally, explicit techniques are suitable for highly nonlinear problems. Using implicit solutions for these problems leads
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