Yi Ji

An implicit truly self-starting time integration method for nonlinear structural dynamical systems is developed in this paper. The proposed method possesses unconditional stability, second-order accuracy, and controllable dissipation, and it has no overshoots. The well-known BN-stability theory is employed in the design of algorithmic parameters, ensuring that the proposed method can stably solve nonlinear structural dynamical systems without restricting the time step size. The spectral analysis shows that compared to existing second-order accurate time integration methods, the proposed method enjoys a considerable advantage in low-frequency accuracy. For nonlinear problems where the currently popular Generalized-[ ] method and [ ]-Bathe method fail, the proposed method shows strong stability and accuracy. Further, for nonlinear problems in which all methods’ results are convergent, the proposed method has greater accuracy, efficiency, and energy-conservation capability.

This paper constructs highly accurate and efficient time integration methods for the solution of transient problems. The motion equations of transient problems can be described by the first-order ordinary differential equations, in which the right-hand side is decomposed into two parts, a linear part and a nonlinear part. In the proposed methods of different orders, the responses of the linear part at the previous step are transferred by the generalized Padé approximations, and the nonlinear part’s responses of the previous step are approximated by the Gauss–Legendre quadrature together with the explicit Runge–Kutta method, where the explicit Runge–Kutta method is used to calculate function values at quadrature points. For reducing computations and rounding errors, the 2^m algorithm and the method of storing an incremental matrix are employed in the calculation of the generalized Padé approximations. The proposed methods can achieve higher-order accuracy, unconditional stability, flexible dissipation, and zero-order overshoots. For linear transient problems, the accuracy of the proposed methods can reach 10⁻¹⁶ (computer precision), and they enjoy advantages both in accuracy and efficiency compared with some well-known explicit Runge–Kutta methods, linear multi-step methods, and composite methods in solving nonlinear problems.

This paper develops a new implicit solution procedure for multibody systems based on a three-sub-step composite method, named TTBIF (trapezoidal–trapezoidal backward interpolation formula). The TTBIF is second-order accurate, and the effective stiffness matrices of the first two sub-steps are the same. In this work, the algorithmic parameters of the TTBIF are further optimized to minimize its local truncation error. Theoretical analysis shows that for both undamped and damped systems, this optimized TTBIF is unconditionally stable, controllably dissipative, third-order accurate, and has no overshoots. Additionally, the effective stiffness matrices of all three sub-steps are the same, leading to the effective stiffness matrix being factorized only once in a step for linear systems. Then, the implementation procedure of the present optimized TTBIF for multibody systems is presented, in which the position constraint equation is strictly satisfied. The advantages in accuracy, stability, and energy conservation of the optimized TTBIF are validated by some benchmark multibody dynamic problems.

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.