A high-order implicit time integration method for linear and nonlinear dynamics with efficient computation of accelerations

Abstract

An algorithm for a family of self-starting high-order implicit time integration schemes with controllable numerical dissipation is proposed for both linear and nonlinear transient problems. This work builds on the previous works of the authors on elastodynamics by presenting a new algorithm that eliminates the need for factorization of the mass matrix providing benefit for the solution of nonlinear problems. The improved algorithm directly obtains the acceleration at the same order of accuracy of the displacement and velocity using vector operations (without additional equation solutions). The nonlinearity is handled by numerical integration within a time step to achieve the desired order of accuracy. The new algorithm fully retains the desirable features of the previous works: 1. The order of accuracy is not affected by the presence of external forces and physical damping; 2. numerical dissipation in the algorithm is controlled by a user-specified parameter, leading to schemes ranging from perfectly nondissipative A-stable to L-stable; 3. The effective stiffness matrix is a linear combination of the mass, damping, and stiffness matrices as in the trapezoidal rule. The proposed algorithm is shown to replicate the numerical results demonstrated on linear problems in previous works. Additional numerical examples of linear and nonlinear vibration and wave propagation are presented herein. Notably, the proposed algorithms show the same convergence rates for nonlinear problems as linear problems, and very high accuracy. Second-order time integration methods commonly used in commercial software produce significantly polluted acceleration responses for a common class of wave propagation problems. The high-order time integration schemes presented here perform noticably better at suppressing spurious high-frequency oscillations and producing reliable and useable acceleration responses.

Figures (38)

• Figure 1: MATLAB function to determine the coefficients of the polynomial $p_{\mathrm{r}}(x_{\mathrm{r}})$ resulting from shifting the roots of polynomial $p(x)$ with $x_{\mathrm{r}}=r-x$. The input includes a row vector of size $M+1$ whose components are the coefficients of $p(x)$.
• Figure 2: Acceleration error in the $L_{2}$-norm for the single degree of freedom system obtained by the algorithm for mixed-order Padé-based schemes in Song with distinct roots. Left: $\rho_{\infty}=1$ ; Right: $\rho_{\infty}=0$. The dash-dotted lines indicate the optimal rates of convergence corresponding to slopes of 3 (right-pointing triangle), 4 (square), 5 (diamond), 6 (pentagram), 7 (hexagram), and 8 (upward-pointing triangle), respectively.
• Figure 3: Acceleration error in the $L_{2}$-norm for the single degree of freedom system obtained by the algorithm for $M-$schemes in Song2024 with single multiple roots. Left: $\rho_{\infty}=1$ ; Right: $\rho_{\infty}=0$. The dash-dotted lines indicate the optimal rates of convergence corresponding to slopes of 2 (circle), 3 (right-pointing triangle), 4 (square) and 5 (diamond), respectively.
• Figure 4: Nonlinear simple pendulum
• Figure 5: Acceleration error in the $L_{2}$-norm for the nonlinear oscillating simple pendulum obtained by the algorithm for mixed-order Padé-based schemes with distinct roots. Left: $\rho_{\infty}=1$ ; Right: $\rho_{\infty}=0$.