A truly self-starting implicit family of integration algorithms with dissipation control for nonlinear dynamics
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ORIGINAL PAPER
A truly self-starting implicit family of integration algorithms with dissipation control for nonlinear dynamics Jinze Li
· Kaiping Yu
Received: 16 May 2020 / Accepted: 17 November 2020 © Springer Nature B.V. 2020
Abstract In this paper, a novel implicit family of composite two sub-step algorithms with controllable dissipations is developed to effectively solve nonlinear structural dynamic problems. The primary superiority of the present method over other existing integration methods lies that it is truly self-starting and so the computation of initial acceleration vector is avoided, but the second-order accurate acceleration responses can be provided. Besides, the present method also achieves other desired numerical characteristics, such as the second-order accuracy of three primary variables, unconditional stability and no overshoots. Particularly, the novel method achieves adjustable numerical dissipations in the low and high frequency by controlling its two algorithmic parameters (γ and ρ∞ ). The classical dissipative parameter ρ∞ determines numerical dissipations in the high-frequency while γ adjusts numerical dissipations in the low-frequency. Linear and nonlinear numerical examples are given to show the superiority of the novel method over existing integration methods with respect to accuracy and overshoot.
J. Li, · K. Yu (B) Department of Astronautic Science and Mechanics, Harbin Institute of Technology, No.92 West Dazhi Street, Harbin 150001, China e-mail: [email protected] J. Li e-mail: [email protected]
Keywords Implicit integration algorithm · Two sub-step scheme · Controllable dissipation · Truly self-starting · Structural dynamics
1 Introduction In this paper, the main attention is paid on the numerical integration of the second-order ordinary differential equations (ODEs) from the spatial discretization of original partial differential equations (PDEs) describing various engineering problems. For instance, after the original PDEs governing the linear elastic dynamical problems are spatially discretized by using some finite element techniques [1,2], the following secondorder linear ODEs can be obtained, ¨ + Cu(t) ˙ + Ku(t) = F(t) Mu(t)
(1)
with the appropriate initial conditions. In Eq. (1), M, C and K denote the global mass, damping and stiffness ˙ and u(t) ¨ are unknown matrices, respectively; u(t), u(t) global nodal displacement, velocity and acceleration vector, respectively; and F(t) represents the external load force as a known function of time t. The resulting second-order ODEs are often called the semi-discrete equation with respect to time, and further solutions to Eq. (1) generally require some numerical time integration schemes to obtain their numerical responses. In the mathematical point of view, the above system (1) can be converted into the first-order ODEs and some
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well-known numerical methods, such as the RungeKutta methods [3], are then used to solve the resulting first-order systems. However, converting into the first-order ODEs c
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