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Research Articles

A Nearly Exact Reformulation of the Girsanov Linearization for Stochastically Driven Nonlinear Oscillators

[+] Author and Article Information
Tara Raveendran, R. M. Vasu

Department of Instrumentation and
Applied Physics,
Indian Institute of Science,
Bangalore, Karnataka 560012 India

D. Roy

Computational Mechanics Lab,
Department of Civil Engineering,
Indian Institute of Science,
Bangalore, Karnataka 560012 India
e-mail: royd@civil.iisc.ernet.in

1Corresponding author.

Manuscript received March 18, 2011; final manuscript received September 10, 2012; accepted manuscript posted October 8, 2012; published online January 25, 2013. Assoc. Editor: Wei-Chau Xie.

J. Appl. Mech 80(2), 021020 (Jan 25, 2013) (11 pages) Paper No: JAM-11-1087; doi: 10.1115/1.4007779 History: Received March 18, 2011; Revised September 10, 2012; Accepted October 08, 2012

The Girsanov linearization method (GLM), proposed earlier in Saha, N., and Roy, D., 2007, “The Girsanov Linearisation Method for Stochastically Driven Nonlinear Oscillators,” J. Appl. Mech.,74, pp. 885–897, is reformulated to arrive at a nearly exact, semianalytical, weak and explicit scheme for nonlinear mechanical oscillators under additive stochastic excitations. At the heart of the reformulated linearization is a temporally localized rejection sampling strategy that, combined with a resampling scheme, enables selecting from and appropriately modifying an ensemble of locally linearized trajectories while weakly applying the Girsanov correction (the Radon–Nikodym derivative) for the linearization errors. The semianalyticity is due to an explicit linearization of the nonlinear drift terms and it plays a crucial role in keeping the Radon–Nikodym derivative “nearly bounded” above by the inverse of the linearization time step (which means that only a subset of linearized trajectories with low, yet finite, probability exceeds this bound). Drift linearization is conveniently accomplished via the first few (lower order) terms in the associated stochastic (Ito) Taylor expansion to exclude (multiple) stochastic integrals from the numerical treatment. Similarly, the Radon–Nikodym derivative, which is a strictly positive, exponential (super-) martingale, is converted to a canonical form and evaluated over each time step without directly computing the stochastic integrals appearing in its argument. Through their numeric implementations for a few low-dimensional nonlinear oscillators, the proposed variants of the scheme, presently referred to as the Girsanov corrected linearization method (GCLM), are shown to exhibit remarkably higher numerical accuracy over a much larger range of the time step size than is possible with the local drift-linearization schemes on their own.

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Figures

Grahic Jump Location
Fig. 1

HD oscillator: plot of Ψmax(k); (a) GCLM1, (b) GCLM2: C=5.0, K=100.0, α=100.0, σ = 5.0, hi = 0.01, X0¯ = {0,0}T, N = 2000

Grahic Jump Location
Fig. 2

HD oscillator: second moment histories through GCLM1 and GCLM2; (a) E [x12], (b) E [x22], and (c) acceptance rate; C = 5.0, K = 100.0, α = 100.0, σ = 5.0, hi = 0.01, X0¯ = {0,0}T, N = 10,000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 3

HD oscillator: second-moment histories of second moments through GCLM1 and GCLM2; (a) E [x12], (b) E [x22]; C = 1.0, K = 10.0, α = 1.0,σ = 0.5, hi = 0.01, X0¯ = {0,0}T, N = 5000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 8

The 2-DOF oscillator: second moment histories via LM1 and GCLM1: E [(x2(1))2]; C1=C2=1.0, K1=K2=K3=10.0,σ1=0.5, α=10.0, σ2=0.5,hi=0.25,X¯0={0,0,0,0}T;N=10,000

Grahic Jump Location
Fig. 7

The 2-DOF oscillator: second moment histories via GCLM1 and GCLM2: (a) E [(x2(1))2] and (b) E [(x1(2))2]; C1=C2=5.0, K1=K2=K3=100.0, α=100.0,σ1=5.0,σ2=5.0, hi=0.01,X¯0={0,0,0,0}T;N=10,000

Grahic Jump Location
Fig. 6

HD oscillator: second-moment history of E [x12] via LM1 and GCLM1; C=1.0,K=10.0,α=10.0,σ=0.5,hi=0.25,X¯0={0,0}T,N=10,000; solid black line indicate exact stationary limits

Grahic Jump Location
Fig. 5

HD oscillator: second-moment histories via GCLM1 and GCLM2: (a) E [x12], (b) E [x22], (c) Ψmax(k) for GCLM1 and (d) Ψmax(k) for GCLM2; C=1.0,K=10.0,α=100.0,σ=5.0,hi=0.005,X¯0={0,0}T,N=1000; solid black lines indicate exact stationary limits

Grahic Jump Location
Fig. 4

HD oscillator: second moment histories via LM1 and GCLM1: (a) E [x12] and (b) E [x22];C=5.0,K=100.0,α=100.0,σ=5.0,hi=0.01,X¯0={0,0}T,N=5000; solid black lines indicate exact stationary limits

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