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TECHNICAL PAPERS

# Newmark’s Time Integration Method From the Discretization of Extended Functionals

[+] Author and Article Information
Lorenzo Bardella

Department of Civil Engineering, University of Brescia, Via Branze, 38, 25123 Brescia, Italy

Francesco Genna

Department of Civil Engineering, University of Brescia, Via Branze, 38, 25123 Brescia, Italygenna@bscivgen.ing.unibs.it

The amplification matrix $A$ of a time integration algorithm relates quantities at the end of one step to quantities at the beginning of the next one, under initial conditions only (i.e., without considering forcing terms), as follows:

$[un+1υn+1]=A[unυn]=[A11A12A21A22][unυn]$

This phenomenon, which we were unaware of, before finding it in our calculations, can be also obtained in the numerical solution of linear problems, when the physical damping and the spurious introduction or removal of energy by numerical schemes can influence each other in such a way as to produce stable but meaningless results.

J. Appl. Mech 72(4), 527-537 (Feb 15, 2005) (11 pages) doi:10.1115/1.1934648 History: Received September 22, 2003; Revised February 15, 2005

## Abstract

In this note we illustrate how to obtain the full family of Newmark’s time integration algorithms within a rigorous variational framework, i.e., by discretizing suitably defined extended functionals, rather than by starting from a weak form (for instance, of the Galerkin type), as done in the past. The availability of functionals as a starting point is useful both as a tool to obtain new families of time integration methods, and as a theoretical basis for error estimates. To illustrate the first issue, here we provide some examples of how to obtain modified algorithms, in some cases significantly more accurate than the basic Newmark one despite having a comparable computational cost.

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Copyright © 2005 by American Society of Mechanical Engineers
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## Figures

Figure 1

Integration of undamped, unforced linear motion with m=1,k=1. Comparison of standard Newmark, “augmented” Newmark (Sec. 3.1), and “improved” Newmark (Sec. 3.2) solutions, all with β=1/6,γ=1/2,Δt=0.5.

Figure 2

Integration of damped, unforced linear motion with m=1,k=1,c=0.04. Comparison of standard Newmark, “augmented” Newmark (Sec. 3.1), and “improved” Newmark (Sec. 3.2) solutions, all with β=1/6,γ=1/2,Δt=0.5.

Figure 3

Effect of the inclusion of a constant kernel on the algorithm based on functional 36 with discretization as in Eqs. 24,26,212: stability and phase shift

Figure 4

Time histories computed numerically for the modified Duffing equation of Eq. 37. Curve (a): standard Newmark method, β=1/6,γ=1/2; curve (b): variationally derived Newmark method, β=1/6,γ=1/2; curve (c): same as in (b), with analytical integration of the forcing term; curve (d): kernel-modified variational method (functional 312), discretization as in Eqs. 24,26,212, β=1/6,γ=1/2;l1=−1.35221;l2=l3=0. Δt=0.5 in all curves.

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