Damping perturbation based time integration asymptotic method for structural dynamics

TL;DR

The paper tackles transient analysis of linear structural dynamics with viscous damping by artificially perturbing the damping term and summing the resulting asymptotic series to yield an explicit time-integration scheme. The method (PER) produces an explicit update $\mathbf{U}_{k+1} = \mathbf{a}(\Delta t)\mathbf{U}_k + \mathbf{b}_k(\Delta t)$ after summing all perturbation orders, with convergence guaranteed by $\rho(\boldsymbol{\beta})<1$ and practical time-step bounds derived from a modal, Neumann-series analysis. Detailed algorithms are provided for computing the main matrices $\mathbf{a}(\Delta t)$ and $\mathbf{b}_k(\Delta t)$ using truncated series and a $2^p$-type precise integration, along with stability and accuracy results that quantify error contributions from interpolation, truncation, and nonhomogeneous-term approximation. The approach is validated on a discrete 12-DOF system and a continuous Euler-Bernoulli beam model, showing high accuracy and competitive computational efficiency when compared to implicit methods and MPIM, especially at larger time steps. Overall, the damping-perturbation strategy offers a practical, explicit, and accurate alternative for transient structural dynamics with lightweight damping, expanding the toolbox of time-integration methods for large-scale problems.

Abstract

The light damping hypothesis is usually assumed in structural dynamics since dissipative forces are in general weak with respect to inertial and elastic forces. In this paper a novel numerical method of time integration based on the artificial perturbation of damping is proposed. The asymptotic expansion of the transient response results in an infinite series which can be summed, leading to a well-defined explicit iterative step-by-step scheme. Conditions for convergence are rigorously analyzed, enabling the determination of the methodology boundaries in form of maximum time step. The numerical properties of the iterative scheme, i.e. stability, accuracy and computational effort are also studied in detail. The approach is validated with two numerical examples, showing a high accuracy and computational efficiency relative to other methods.

Figures (12)

• Figure 1: Eigenvalues of $\bm{\sigma}_m(\tau)$ in absolute value: $\left|\mu_{1}(m,\tau) \right|$, $\left| \mu_{2}(m,\tau) \right|$
• Figure 2: Absolute value of eigenvalues of matrix $\mathbf{a}(\Delta t _0) = (\mathbf {I}_{2} - \bm{\beta} )^{-1} \left( \mathbf {T} \, + \bm{\alpha} \right)$ in a single dof system with natural frequency $\omega= \sqrt{K/M} = 2 \pi / T$ and damping ratio $\zeta = c/2M\omega$, for different cases of damping level: (a) $\zeta=0$, (b) $\zeta = 0.005$, (c) $\zeta = 0.05$ and (d) $\zeta = 0.50$. Abscissas represent the reduced time step $\Delta t _0/T$. Curves for different truncation orders $m_a$ are plotted in colors: $m_a = 2$ (red), $m_a = 4$ (blue), $m_a = 6$ (green), $m_a = 8$ (black), $m_a = \infty$ (magenta). The inverse matrix is approximated by $(\mathbf{I} - \bm{\beta})^{-1} \approx \mathbf{I} + \bm{\beta}_a + \bm{\beta}_a^2$ ($r_a=2$). Plots (e) and (f): absolute value of eigenvalues of matrix $\bm{\beta}_a$ for two damping cases: (e) $\zeta = 0.05$, (f) $\zeta = 0.50$
• Figure 3: Discrete system of Example 1. (Top) 12-dof sketch of the lumped-mass structure. External applied force is located at dof # 3. (Bottom) Time function of external force $f(t)$, formed by a superposition of harmonic functions weighted by a Gaussian function.
• Figure 4: (Example 1) Time-domain numerical solutions for the different methods at dof $u_1(t)$. Dimensionless damping parameter $\zeta = 0.10$. Time-step $\Delta t = 0.24$ s = $0.75T$. Newmark--$\beta$ method with parameters $\gamma = 1/2, \ \beta = 1/4$. Wilson--$\theta$ method is calculated for $\theta = 1.4$. Bathe's method is implemented with $\gamma=1/2$. Modified PIM (MPIM) uses $g=4$ Gauss quadrature points in integration of nonhomogeneous term. Order of truncation in proposed method in the non-homogeneous term: $m_b = 8$ and $r_b=4$. The reference solution has been evaluated using RK4 with a time-step $\Delta t /500$. (a) Displacement $u_1(t)$, (b) Local relative errors of displacements. (c) Velocity of dof--1, $\dot{u}_1(t)$, (d) Local relative error of velocity, (e) and (f) Detail plots of displacement and velocity in the range $15 \leq t \leq 20$ s.
• Figure 5: Example 1. Simulation of time-domain response of dof--1. Dimensionless damping parameter $\zeta = 0.10$. Time-step $\Delta t = 0.024 = 0.075T$ s. Newmark--$\beta$ method with parameters $\gamma = 1/2, \ \beta = 1/4$. Wilson--$\theta$ method is calculated for $\theta = 1.4$. Bathe's method is implemented with $\gamma=1/2$. Modified PIM (MPIM) uses $g=4$ Gauss quadrature points in integration of nonhomogeneous term. Order of truncation in proposed method in the non-homogeneous term: $r_b=4$, $m_b = 4$. (a) Displacement, $u_1(t)$. (b) Relative error of displacement response. (c) Velocity, $\dot{u}_1(t)$. (d) Relative error of velocity response.