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Improving the accuracy of the Newmark method through backward error analysis

Donát M. Takács, Tamás Fülöp

Abstract

We use backward error analysis for differential equations to obtain modified or distorted equations describing the behaviour of the Newmark scheme applied to the transient structural dynamics equation. Based on the newly derived distorted equations, we give expressions for the numerically or algorithmically distorted stiffness and damping matrices of a system simulated using the Newmark scheme. Using these results, we show how to construct compensation terms from the original parameters of the system, which improve the performance of Newmark simulations. The required compensation terms turn out to be slight modifications to the original system parameters (e.g. the damping or stiffness matrices), and can be applied without changing the time step or modifying the scheme itself. Two such compensations are given: one eliminates numerical damping, while the other achieves fourth-order accurate calculations using the traditionally second-order Newmark method. The performance of both compensation methods is evaluated numerically to demonstrate their validity, and they are compared to the uncompensated Newmark method, the generalized-$α$ method and the 4th-order Runge--Kutta scheme.

Improving the accuracy of the Newmark method through backward error analysis

Abstract

We use backward error analysis for differential equations to obtain modified or distorted equations describing the behaviour of the Newmark scheme applied to the transient structural dynamics equation. Based on the newly derived distorted equations, we give expressions for the numerically or algorithmically distorted stiffness and damping matrices of a system simulated using the Newmark scheme. Using these results, we show how to construct compensation terms from the original parameters of the system, which improve the performance of Newmark simulations. The required compensation terms turn out to be slight modifications to the original system parameters (e.g. the damping or stiffness matrices), and can be applied without changing the time step or modifying the scheme itself. Two such compensations are given: one eliminates numerical damping, while the other achieves fourth-order accurate calculations using the traditionally second-order Newmark method. The performance of both compensation methods is evaluated numerically to demonstrate their validity, and they are compared to the uncompensated Newmark method, the generalized- method and the 4th-order Runge--Kutta scheme.
Paper Structure (20 sections, 50 equations, 11 figures)

This paper contains 20 sections, 50 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Backward error analysis for numerical methods
  3. Asymptotic expansion of the distorted vector field
  4. Demonstration of BEA for the explicit Euler method
  5. Solution of the truncated distorted equation
  6. Distorted equations of the Newmark method
  7. The Newmark method
  8. Deriving the distorted equation
  9. Distorted vector field of the Newmark method
  10. Distorted second-order equation corresponding to the Newmark method
  11. Numerical verification
  12. Numerical demonstration
  13. Applications of the distorted equations for improving the Newmark method
  14. Eliminating numerical damping from the Newmark method using a compensated damping matrix
  15. Numerical demonstration (undamped case)
  16. ...and 5 more sections

Figures (11)

  • Figure 1: Exact solution, numerical explicit Euler solution and analytical solution of the truncated distorted equation for the position of a mass-spring system as the function of time. Simulation parameters: $\Delta t=0.1$, $\omega=0$, $x(0)=1$, $v(0)=0$.
  • Figure 2: The position and velocity of the results in Fig. \ref{['fig:EE_solution_time']} shown in phase space.
  • Figure 3: Convergence of the norm of deviation between the Newmark simulation results and the distorted vector field or the second-order equation, for the position (left) and velocity (right) solutions.
  • Figure 4: Solution for the first component of the position vector $\mathbf{\bm{q}}$ as the function of time, using the Newmark method, compared to solutions of the distorted vector field \ref{['eq:newmark_mvf']} and distorted second-order equation \ref{['eq:newmark_mod_secondorder']}, as well as a reference solution of the original system. (System parameters per \ref{['eq:sys11']}--\ref{['eq:sys12']}, $\Delta t=0.7$.)
  • Figure 5: Solution for the first component of the position vector $\mathbf{\bm{q}}$ as the function of time, from simulating the original, undamped system using the Newmark method, compared to the Newmark damping compensated (NDC) system also simulated with the Newmark method, against a Runge--Kutta (RK4), generalized-$\alpha$ and a reference solution. (Time step for the former four was identically $\Delta t=0.7$.)
  • ...and 6 more figures