-
ADAPTIVE GENERALIZED POLYNOMIAL CHAOS FORNONLINEAR RANDOM
OSCILLATORS∗
D. LUCOR† AND G. E. KARNIADAKIS†
SIAM J. SCI. COMPUT. c© 2004 Society for Industrial and Applied
MathematicsVol. 26, No. 2, pp. 720–735
Abstract. The solution of nonlinear random oscillators subject
to stochastic forcing is investi-gated numerically. In particular,
solutions to the random Duffing oscillator with random Gaussianand
non-Gaussian excitations are obtained by means of the generalized
polynomial chaos (GPC).Adaptive procedures are proposed to lower
the increased computational cost of the GPC approachin
large-dimensional spaces. Adaptive schemes combined with the use of
an enriched representationof the system improve the accuracy of the
GPC approach by reordering the random modes accordingto their
magnification by the system.
Key words. uncertainty, Duffing oscillator, polynomial chaos,
stochastic modeling
AMS subject classifications. 65C20, 60H35, 65C30
DOI. 10.1137/S1064827503427984
1. Introduction. Fully nonlinear oscillators subject to mild or
extreme noisyforces are of great interest for multiple disciplinary
engineering communities (e.g.,ocean structures [1]). Many
mechanical systems involving flow-structure interactioncan be
modeled by the Duffing oscillator equation; see [2, 3]. In the
present work, wedetermine the response of nonlinear
single-degree-of-freedom mechanical systems sub-ject to random
excitations (Gaussian or non-Gaussian). We are particularly
interestedin the determination of second-moment characteristics of
the response of stochasticDuffing oscillators.
The method we adopt in this work is an extension of the
classical polynomial chaosapproach [4]. This representation is an
infinite sum of multidimensional orthogonalpolynomials of standard
random variables with deterministic coefficients. Practically,only
a finite number of terms in the expansion can be retained, as the
sum has to betruncated. Consequently, the multidimensional random
space has a finite number ofdimensions n, and the highest order of
the orthogonal polynomial is finite, denotedhere by p. The
Hermite-chaos expansion, which is the basis of the classical
polynomialchaos, is effective in solving stochastic differential
equations with Gaussian inputs aswell as certain types of
non-Gaussian inputs [5, 6, 7]. Its theoretical justificationis
based on the Cameron–Martin theorem [8]. However, it has been found
that forgeneral non-Gaussian random inputs, optimal exponential
convergence rate is notachieved, and in some cases the convergence
rate is in fact severely deteriorated; see[9, 10]. Another issue
with the polynomial chaos decomposition is the fast growth ofthe
dimensionality of the problem with respect to the number of random
dimensionsand the highest order of the retained polynomial; see
Table 1.1. This issue becomescritical if one deals with a very
noisy input (white noise) or a strongly nonlinearproblem or both.
Indeed, an accurate representation of a noisy input requires usinga
large number of random dimensions, while strong nonlinear dynamics
can only becaptured accurately with the use of a high polynomial
order.
∗Received by the editors May 15, 2003; accepted for publication
(in revised form) March 22,2004; published electronically December
22, 2004. This work was supported by ONR and NSF, andcomputations
were performed at the DoD HPCM centers.
http://www.siam.org/journals/sisc/26-2/42798.html†Division of
Applied Mathematics, Brown University, Providence, RI 02912
([email protected].
edu, [email protected]).
720
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ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 721
Table 1.1Number of unknown deterministic coefficients in the
polynomial chaos representation as a func-
tion of the number of random dimensions n and the highest
polynomial order p.
p = 3 p = 5 p = 7 p = 9n = 2 10 21 36 55n = 4 35 126 330 715n =
8 165 1,287 6,435 24,310n = 16 969 20,349 245,157 2,042,975
In this paper, we consider the case of the random response of a
Duffing oscil-lator subject to nonstationary additive noise, where
the forcing is represented by adeterministic time-dependent
periodic function multiplied by a random variable withdifferent
distributions. We also study the case of the random response of a
Duff-ing oscillator subject to a stationary additive noise
represented by a random processwith different distributions. The
objective is twofold: First, we investigate what typeof stochastic
solutions we obtain in comparison with the well-studied
deterministicDuffing oscillator. Second, we obtain the stochastic
solutions at reduced cost usingadaptive procedures first pioneered
by Li and Ghanem in [11].
The paper is organized as follows. In section 2 we give a brief
overview of thegeneralized polynomial chaos, and subsequently we
apply it to the stochastic Duffingoscillator. In section 3 we first
consider the case of a periodic excitation with randomamplitude and
then focus on the adaptive procedure and present an extension
toGhanem’s original proposal. We conclude in section 4 with a brief
summary.
2. Generalized polynomial chaos.
2.1. The Wiener–Askey representation. The Wiener–Askey
polynomialchaos or generalized polynomial chaos (GPC) expansion is
an extension of the orig-inal polynomial chaos. It is well suited
to represent more general (Gaussian andnon-Gaussian) random inputs.
The expansion basis in this case consists of polyno-mial
functionals from the Askey family [9, 12]. Since each type of
polynomial fromthe Askey scheme forms a complete basis in the
Hilbert space, each correspondingWiener–Askey expansion converges
to an L2 functional in the L2 sense in the appro-priate Hilbert
functional space; this is a generalized result of the
Cameron–Martintheorem; see [8, 13].
A general second-order random process X(θ) is represented by
X(θ) = a0I0
+
∞∑i1=1
ci1I1(ζi1(θ))
+
∞∑i1=1
i1∑i2=1
ci1i2I2(ζi1(θ), ζi2(θ))
+
∞∑i1=1
i1∑i2=1
i2∑i3=1
ci1i2i3I3(ζi1(θ), ζi2(θ), ζi3(θ))
+ · · · ,(2.1)
where In(ζi1 , . . . , ζin) denotes the GPC of order n in terms
of the random vectorζ = (ζi1 , . . . , ζin).
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722 D. LUCOR AND G. E. KARNIADAKIS
For example, one possible choice for In are the Hermite
polynomials which cor-respond to the original Wiener–Hermite
polynomial chaos Hn.
In the GPC expansion, the polynomials In are not restricted to
Hermite poly-nomials but rather can be all types of the orthogonal
polynomials from the Askey
scheme. For example, the expression of the Jacobi polynomials
P(α,β)n is given by
In(ζi1 , . . . , ζin) ≡ P(α,β)n (ζi1 , . . . , ζin)
=(1 − ζ)−α(1 + ζ)−β
2nn!(−1)−n∂n
∂ζi1 · · · ∂ζin[(1 − ζ)n+α(1 + ζ)n+β
],
where ζ denotes the vector consisting of n Beta random variables
(ζi1 , . . . , ζin).For notational convenience, we rewrite (2.1)
as
X(θ) =
∞∑j=0
ĉjΦj(ζ),(2.2)
where there is a one-to-one correspondence between the functions
In(ζi1 , . . . , ζin) andΦj(ζ).
The orthogonality relation of the GPC takes the form
〈ΦiΦj〉 = 〈Φ2i 〉δij ,(2.3)
where δij is the Kronecker delta and 〈·, ·〉 denotes the ensemble
average which is theinner product in the Hilbert space of the
variables ζ. We also have
〈f(ζ)g(ζ)〉 =∫
f(ζ)g(ζ)W (ζ)dζ(2.4)
or
〈f(ζ)g(ζ)〉 =∑
ζ
f(ζ)g(ζ)W (ζ)(2.5)
in the discrete case. Here W (ζ) is the weighting function
corresponding to the GPCbasis {Φi}.
Most of the orthogonal polynomials from the Askey scheme have
weighting func-tions that take the form of probability function of
certain types of random distribu-tions. We then choose the type of
independent variables ζ in the polynomials {Φi(ζ)}according to the
type of random distributions as shown in Table 2.1. Legendre
polyno-mials, which are a special case of the Jacobi polynomials
with parameters α = β = 0,correspond to an important
distribution—the Uniform distribution.
2.2. Duffing oscillator and its GPC representation. We consider
the Duff-ing oscillator subject to external forcing, i.e.,
ẍ(t, θ) + cẋ(t, θ) + k[x(t, θ) + �x3(t, θ)
]= f(t, θ).(2.6)
This equation has been normalized with respect to the mass, so
the forcing f(t) hasunits of acceleration. The damping factor c and
spring factor k are defined as follows:
c = 2ζω0 and k = ω20 ,(2.7)
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ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 723
Table 2.1Correspondence between the type of GPC and the type of
random inputs (N ≥ 0 is a finite
integer).
Random inputs GPC Support
Continuous Gaussian Hermite-chaos (−∞,∞)Gamma Laguerre-chaos
[0,∞)
Beta Jacobi-chaos [a, b]Uniform Legendre-chaos [a, b]
Discrete Poisson Charlier-chaos {0, 1, 2, . . . }Binomial
Krawtchouk-chaos {0, 1, . . . , N}
Negative binomial Meixner-chaos {0, 1, 2, . . . }Hypergeometric
Hahn-chaos {0, 1, . . . , N}
where ζ and ω0 are, respectively, the damping ratio and the
natural frequency ofthe system. This system can become stochastic
if the external forcing or the inputparameters or both are some
random quantities. Those random quantities can evolvein time
(random process) or not (random variable).
Nonconservative restoring forces tend to correspond to
hysteretic materials whosestructural properties change in time when
subjected to cyclic stresses. A popularrestoring force model used
in random vibration analysis consists of the superpositionof a
linear force αx(t) and a hysteretic force (1 − α)Q(t) (see [14]),
so that we have
ẍ(t) + cẋ(t) + k (αx(t) + (1 − α)Q(t)) = f(t),(2.8)
Q̇(t) = αẋ(t) − βẋ(t)|Q(t)|n − ρ|ẋ(t)|Q(t)|Q(t)|n−1.(2.9)
The coefficients α, β, ρ, and n control the shape of the
hysteretic loop; some of thesecoefficients may vary in time.
Here, we focus on the case of the Duffing oscillator, while
other cases can bededucted from this one. Let us consider the
stochastic differential equation (2.6),where the damping factor c
and the spring constant k are random processes withunknown
correlation functions and the external forcing is a random process
with agiven correlation function. We decompose the random process
representing the forcingterm in its truncated Karhunen–Loève
expansion up to the nth random dimension;see [15]. We have
f(t, θ) = f̄(t) + σf
n∑i=1
√λiφi(t)ξi(θ) =
n∑i=0
fi(t)ξi.(2.10)
Assuming that the correlation functions for the coefficients c
and k are not known,we can decompose the random input parameters in
their GPC expansion [5, 6, 7] asfollows:
c(t, θ) =
P∑j=0
cj(t)Φj(ξ(θ)) and k(t, θ) =
P∑j=0
kj(t)Φj(ξ(θ)).(2.11)
Finally, the solution of the problem is sought in the form given
by its truncated GPCexpansion
x(t, θ) =P∑i=0
xi(t)Φi(ξ(θ)),(2.12)
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724 D. LUCOR AND G. E. KARNIADAKIS
where n is the number of random dimensions and p is the highest
polynomial orderof the expansion.
By substituting all expansions in the governing equation (see
(2.6)), we obtain
(2.13)P∑i=0
ẍi(t)Φi +
P∑j=0
cj(t)Φj
P∑i=0
ẋi(t)Φi
+
P∑j=0
kj(t)Φj
(P∑i=0
xi(t)Φi + �
(P∑i=0
xi(t)Φi
P∑k=0
xk(t)Φk
P∑l=0
xl(t)Φl
))=
n∑i=0
fi(t)ξi.
We project the above equation onto the random space spanned by
our orthogonalpolynomial basis Φm; i.e., we take the inner product
with each basis and then use theorthogonality relation. We obtain a
set of coupled deterministic nonlinear differentialequations
ẍm(t) +1
〈Φ2m〉
P∑i=0
P∑j=0
cj(t)ẋi(t)eijm
= − 1〈Φ2m〉
P∑i=0
P∑j=0
kj(t)xi(t)eijm
− �〈Φ2m〉
⎛⎝ P∑
i=0
P∑j=0
P∑k=0
P∑l=0
kjxi(t)xk(t)xl(t)eijklm
⎞⎠ + fm(t),(2.14)
where m = 0, 1, 2, . . . , P , eijm = 〈ΦiΦjΦm〉, and eijklm =
〈ΦiΦjΦkΦlΦm〉; here 〈·, ·〉denotes an ensemble average. These
coefficients as well as 〈Φ2m〉 can be determined an-alytically or
numerically using multidimensional numerical quadratures. This
systemof equations consists of (P + 1) nonlinear deterministic
equations with each equationcorresponding to one random mode.
Standard solvers can be employed to obtain thenumerical
solutions.
3. Duffing oscillator.
3.1. Periodic excitation with random amplitude. We consider a
viscouslydamped nonlinear Duffing oscillator subject to random
external forcing excitations:
ẍ(t, θ) + cẋ(t, θ) + k[x(t, θ) + �x3(t, θ)
]= f(t, θ),
x(0, θ) = x0 and ẋ(0, θ) = ẋ0, t ∈ [0, T ] .(3.1)
In this case, the random forcing is treated as a nonstationary
random variable andhas the form
f(t, θ) = f̄(t) + σf (t)ξ(θ) + γf (t)ξ2(θ) + δf (t)ξ
3(θ),(3.2)
where ξ is a random variable of known distribution and the
coefficients are given by
f̄ = Ā(α + Ā2β
), σf = σA
(α + 3Ā2β
), γf = 3Āσ
2Aβ, δf = σ
3Aβ,
α =(k − ω2
)cos (ωt) − cω sin (ωt) , β = k� cos3 (ωt) .
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ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 725
An analytical solution can be obtained for this forcing of the
form
x(t) =(Ā + σAξ
)cos(ωt + φ)(3.3)
with φ = 0 and Ā, σA, and ω being some fixed constants. The
random variable ξ canhave different distributions. In this section,
we focus on a Gaussian (Case I) and aUniform distribution (Case
II).
Case I. If ξ is a Gaussian random variable, the forcing can be
represented exactlyby the GPC basis using Hermite-chaos and has the
following form:
f(t, θ) =(f̄(t) + γf (t)
)+ (σf (t) + 3δf (t)) ξ + γf (t)
(ξ2 − 1
)+ δf (t)
(ξ3 − 3ξ
).(3.4)
Case II. If ξ is a random variable with a Uniform distribution
(particular case ofa Beta distribution), the forcing can be
represented exactly by the GPC basis usingLegendre-chaos
(particular case of Jacobi polynomials) and has the following
form:
f(t, θ) = f̄(t) +γf (t)
3+
[σf (t) +
3
5δf (t)
]ξ +
2
3γf (t)
[1
2(3ξ2 − 1)
]
+2
5δf (t)
[1
2(5ξ3 − 3ξ)
].(3.5)
We decompose the random forcing and the sought solution in its
GPC expansion.After substituting in the equation and projecting
onto the random space, we obtaina set of coupled equations similar
to (2.14). This nonlinear system is simplified if wewrite it as a
state equation. We obtain the following discrete system which
consistsof a set of simultaneous nonlinear first-order differential
equations:⎧⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎩
Ẋ1m(t) = X2m(t),
Ẋ2m(t) + c
P∑i=0
X2i (t) = −kP∑i=0
X1i (t),
− k�〈Φ2m〉
(P∑i=0
P∑k=0
P∑l=0
X1i (t)X1k(t)X
1l (t)eiklm
)+ fm(t),
where eiklm = 〈ΦiΦkΦlΦm〉. These coefficients as well as 〈Φ2m〉
can be determinedanalytically or numerically very efficiently using
multidimensional Gauss–Legendrequadratures.
Obviously, when � �= 0, we need at least a third-order GPC
expansion to representthe forcing exactly. Because of the form of
the solution, we expect the energy injectedin the system through
the forcing to concentrate mainly in the mean and the firstmode of
the solution. The energy present in the other random modes should
be zero.
Since the resulting ODEs are deterministic, we use standard
explicit schemes(Euler-forward and Runge–Kutta of second order and
fourth order) to check the con-vergence rate of the solution in
time. The following results are obtained using thestandard
fourth-order Runge–Kutta scheme. The structural parameters in the
system(see (3.1) and (3.3)) and initial conditions are set to
c = 0.05, k = 1.05, (Ā, σA) = (0.6, 0.06), ω = 1.05, φ = 0;x0(t
= 0) = Ā, x1(t = 0) = σA,ẋ0(t = 0) = ẋ1(t = 0) = 0, xi>1(t =
0) = ẋi>1(t = 0) = 0.
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726 D. LUCOR AND G. E. KARNIADAKIS
0 50 100 150–0.6
–0.4
–0.2
0
0.2
0.4
0.6
time
Solu
tion
x0 (mean)
x1
0 50 100 150– 4
– 2
0
2
4x 10
– 12
time
Solu
tion
x2
x3
x4
x5
Fig. 3.1. Time evolution of the random modes solution for Case I
(Gaussian) using a GPCexpansion of six terms (p = 5); � = 1.0.
Figure 3.1 shows the time evolution of the random modes solution
for Case I(Gaussian) with � = 1.0. A fifth-order polynomial is used
to solve the problem. Thetop plot shows the mean and the first mode
of the solution. We notice that theyhave the proper amplitude and
frequency that we imposed by assuming the form ofthe solution. The
lower plot represents the higher modes, which should be
indenticallyzero. They are very small and completely controlled by
the temporal discretizationerror. In this case, for fixed ∆t, an
increase of the polynomial order p does not improvethe solution
error. In fact, we obtain the same results with a cubic order
polynomial,as we know that it is enough to represent exactly the
forcing term. We notice in thelower plot that there exists a
transient state with a burst of energy in the high modeswhich
interact in a nonlinear manner. At longer times the amplitude of
the high modesremains bounded and the system is stable. Similar
observations and conclusions can bemade for the case of the Uniform
input (Case II) for the same values of the parameters.
Different values of the nonlinear parameter � were investigated
for fixed valuesof the other parameters. The magnitude and duration
of the observed transient ofthe high modes mentioned above depends
on the value of � (and σA). As � increases,the transient state
takes place earlier in time with an increased magnitude. Next,
wechoose � = 5 with the same set of parameters and an input with
Uniform distribution(Case II). We perform a long-time integration
for different values of the polynomialorder (from p = 3 to p = 11).
Figure 3.2 shows results for p = 3. We present the timeevolution of
the four random modes (x0 (mean), x1, x2, and x3); see Figure 3.2.
Inthis case, we notice that both the mean and the first mode
eventually deviate fromthe expected solution. Higher modes also
deviate toward another solution, and theirmagnitude becomes
nonnegligible. The temporal location of the onset of the
bifurca-tion varies as a function of the temporal error introduced
by the scheme. However,the bifurcation always exists, even if the
temporal error introduced is slightly abovemachine precision.
Moreover, we observe very similar asymptotic behavior for
higher
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ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 727
0 50 100 150 200 250 300
−0.5
0
0.5
x0
0 50 100 150 200 250 300
−0.1
0
0.1
x1
0 50 100 150 200 250 300
−0.1
0
0.1
x2
0 50 100 150 200 250 300−0.1
0
0.1
x3
time
Fig. 3.2. Time evolution of the random modes solution for Case
II (Uniform) using a GPCexpansion of four terms (p = 3); � = 5.
values of p even though the transient states are somewhat
different.The critical value of � for Case II is around � ≈ 4.8. No
bifurcation of the solution
is obtained for an � below this threshold value. The critical
value of � for Case I isaround � ≈ 3.7. Slightly above this value,
a long-term instability develops that bringsthe initially regular
(expected) solution to a chaotic state. For both distributions,
fora fixed value of �, a change in the standard deviation of the
input noise can changethe regularity of the solution and bring it
to another state. For instance, for � = 5 forCase II, the
transition in the solution to another state takes place if σA/Ā
> 4%.
Because of the way the forcing term is defined, increasing
values of the nonlinearparameter can be seen as increasing forcing
magnitudes in the equivalent normalizedform of the Duffing equation
[16]. Moreover, multifrequencies are introduced in theforcing for �
within some critical range. For instance, for small values of �,
the forcingis very close to a perfectly single-frequency harmonic
signal. However, in our case,the multifrequencies forcing brings
the oscillator’s mean value to two limit cycles ofdifferent
stability which coexist for certain values of the control
parameters; see plot(c-1) in Figure 3.3. For a limited parameter
range, two stable closed orbits coexist.This kind of jump
phenomenon is observed for the Duffing oscillator for which
wechange slightly the forcing frequency [16]. We verified that once
the oscillator jumpsto the new solution, it does not switch back to
the original one. Concerning thefirst mode x1, a flip
bifurcation-like occurs [16] where the initial limit cycle loses
itsstability, while another closed orbit takes place whose period
is half the period of theoriginal cycle; see plot (c-2) in Figure
3.3.
One fundamental question is whether the bifurcation is intrinsic
to the deter-ministic system or whether it is in fact triggered by
the uncertainty of the randominput. Deterministic computations for
this case are done using the extreme valuesof the random input for
the deterministic forcing. This investigates the response ofthe
deterministic oscillator subject to deterministic forcing whose
amplitude is evalu-ated at the boundary of the density probability
support (here Uniform distribution).This is equivalent to setting
the parameters (Ā, σA) = (0.6 ± 0.06, 0.0). While one
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728 D. LUCOR AND G. E. KARNIADAKIS
−0.5 0 0.5
−0.5
0
0.5
y
y t
Deterministic
(a)
−1 −0.5 0 0.5 1
−0.5
0
0.5
1
Deterministic
y
y t
(b)
−0.2 0 0.2
−0.3
−0.2
−0.1
0
0.1
0.2
Stochastic; First mode
y
y t
(c−2)
−0.5 0 0.5−0.6
−0.4
−0.2
0
0.2
0.4
0.6
y
y t
Stochastic; Mean
(c−1)
Fig. 3.3. Phase projections of deterministic solutions and
stochastic (Uniform distribution)solutions.
case gives a single limit cycle solution (see plot (a) in Figure
3.3), the other case((Ā, σA) = (0.6 + 0.06, 0.0); see plot (b) in
Figure 3.3) exhibits two limit cycles withdifferent amplitudes but
the same frequency. So it seems that in this case the bifur-cation
is intrisic to the system.
In summary, what we have studied in this section shows complex
and differentdynamics for the stochastic Duffing oscillator. A
straightforward implementation ofGPC is possible since for the
problem considered the relatively simple forcing doesnot require
very high order in the GPC expansion. However, in the general case
ofarbitrary stochastic forcing, the computational complexity
increases tremendously asshown in Table 1.1. To this end, we need
to implement adaptive procedures to lowerthe computaional
complexity of stochastic nonlinear oscillators.
3.2. Solutions via adaptive GPC. We consider a nonlinear Duffing
oscillatorsubject to a random process excitation f(t, θ) applied
over a time interval. Theequation governing the motion is given
by
ẍ(t, θ) + 2ζω0ẋ(t, θ) + ω20(x(t, θ) + µx
3(t, θ)) = f(t, θ), x(0) = ẋ(0) = 0.(3.6)
We assume that the input process f(t, θ) is a weakly stationary
random process, withzero mean and correlation function Rff (t1,
t2), given by
Rff (t1, t2) = σ2fe
− |t1−t2|A , A > 0,(3.7)
where A is the correlation length and σf denotes the standard
deviation of the process.If we normalize the equation using
nondimensional time τ = ω0t and nondimensionaldisplacement y =
x/σx, where σx represents the standard deviation of the
linearsystem (µ = 0) with a stationary excitation of infinite
duration (T → ∞), we have
ÿ(t, θ) + 2ζẏ(t, θ) + (y(t, θ) + �y3(t, θ)) =f(t, θ)
σxω20, y(0) = ẏ(0) = 0,(3.8)
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ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 729
where � = µσ2x. Using the above nondimensional time, the
autocorrelation functiontakes the form
Rff (∆τ) = σ2fe
− |τ1−τ2|Aω0 , A > 0,(3.9)
where σx is given by
σx =
(2ζω0 +
1A
2ζω30
)(σ2f
ω20 +(
1A
)2+ 2ζω0A
).(3.10)
We also use (2.10) to represent the stochastic forcing,
i.e.,
f(t, θ) =M∑i=0
fi(t)ξi(θ) = f̄ + σf
M∑i=1
√λiφi(t)ξi(θ),(3.11)
and represent the solution y(t, θ) of the problem by its GPC
expansion
y(t, θ) =P∑i=0
yi(t)Φi(ξ(θ)).(3.12)
The number (P + 1) of terms required in the expansion grows very
rapidly as thenumber of (M + 1) terms in the expansion for the
input process f(t, θ) increases;see Table 1.1. However, some of the
terms in the expansion for y(t, θ) do not con-tribute significantly
to its value. An adaptive procedure, first introduced by Li
andGhanem [11], can be designed in order to keep only the terms
which have the greatestcontribution to the solution.
The expansion for the excitation (3.11) is decomposed into two
summations:
f(t, θ) = f̄ + σf
(K∑i=1
fi(t)ξi(θ) +
M∑i=K+1
fi(t)ξi(θ)
).(3.13)
The first summation contains the terms whose higher-order
(nonlinear) contributionsto the solution y(t, θ) will be kept at a
given step of the iterative process. The secondsummation contains
the terms whose higher-order (nonlinear) contributions will
beneglected in the computation. Correspondingly, the expansion of
the solution becomes
y(t, θ) = ȳ +K∑i=1
yi(t)ξi(θ) +
M∑i=K+1
yi(t)ξi(θ)
+
N∑j=M+1
yj(t)Ψj(ξi(θ) |Ki=1) with N < P.(3.14)
The first two summations represent the linear contributions. The
third summationrepresents higher-order terms, i.e., at least
quadratic polynomials in the random vari-ables {ξi}Ki=1. Another
way to understand the method is to consider that we enrichthe space
of random variables by adding L = (M −K) linear terms to the
standardGPC expansion (see (2.12)). With the expansions for y(t, θ)
and f(t, θ), we now solvethe system for the random modes yi(t) over
the time domain. Once the current com-putation is completed, we
then evaluate the L2 norm of each function yi(t) over the
-
730 D. LUCOR AND G. E. KARNIADAKIS
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Seco
nd−
orde
r m
omen
t res
pons
e
ω0t
I
II
III
IV
V
I: GPC (K=20, L=0, p=1); ε=0II: MC (M=30; 500,000 events)III:
GPC (K=20, L=0, p=1)IV: GPC (K=10, L=0, p=3)V: AGPC (K=10, L=10,
p=3)
Fig. 3.4. Case I (Gaussian): Comparison of second-order moment
response obtained by adaptiveGPC (AGPC ) and Monte Carlo simulation
(MC ) (500,000 events). ω0 = 1.0; ζ = 0.1; A = 1.0;� = 1.0.
time interval. The K linear components yi(t), among i ≤ M , with
the largest norm,are sorted and reordered and then used to produce
the higher-order components inthe next iteration. The iterative
process is repeated. Convergence is reached and theiterative
process stops when the ordering of the K largest contributors to
the solutiondoes not change.
We present numerical results for both Case I (Gaussian) and Case
II (Uniform)for different values of the nonlinear parameter � and
different K,L, and polynomialorder p combinations. The values for
the structural parameters are
ω0 = 1.0, (f̄ , σf ) = (0.0, 1.0), A = 1.0.(3.15)
The time domain extends over 30 nondimensional units (T = 30).
Values of thedamping coefficient ζ will be specified for the
different cases, as we will see that itplays a key role in the
efficiency of the adaptive method.
Because of the mean forcing f̄ being zero, we have an asymptotic
value of themean of the solution that tends to zero, and only the
random modes associated withpolynomials of odd order are excited
due to the form of the nonlinearity. Therefore,we compare the
second-order moment responses obtained by GPC with and withoutthe
use of the adaptive method and also by Monte Carlo simulation. The
variance ofthe solution includes the square of all random modes
(except mode zero), so we expecta truncated representation of the
solution (without reordering) to always underpredictthe exact
variance of the solution.
Figure 3.4 shows results for Case I (Gaussian case) with � =
1.0. The dampingcoefficient ζ = 0.1 is quite large, so the solution
converges quickly within the imposedtime domain. Because the
problem has been normalized, the asymptotic value of thevariance of
the solution for the linear case (� = 0.0) has to be one. The
linear caseis run first to estimate how many terms are needed to
capture the scale associatedwith the correlation length A of f . We
found that 20 terms (K = 20) are enough forthe variance of the
solution to reach its asymptotic value. Monte Carlo simulation
-
ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 731
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
ω0t
Seco
nd−
orde
r m
omen
t res
pons
eI
II
III
IV
V
I: GPC (K=20, L=0, p=1); ε=0II: MC (M=30; 500,000 events)III:
GPC (K=20, L=0, p=1)IV: GPC (K=10, L=0, p=3)V: AGPC (K=10, L=10,
p=3)
Fig. 3.5. Case II: Comparison of second-order moment response
obtained by adaptive GPCand Monte Carlo simulation (500,000
events). ω0 = 1.0; ζ = 0.1; A = 1.0; � = 1.0.
(MC) with 500,000 realizations for the nonlinear case (� = 1.0)
was performed with 30random dimensions to keep a safety margin. We
notice that cubic order (p = 3) GPCwith only 10 random dimensions
(K = 10) is far from converging to the Monte Carlosimulation.
Linear chaos with 20 random dimensions (K = 20) still
underestimatesthe Monte Carlo simulation. The adaptive GPC of cubic
order with the addition of10 more random dimensions (L = 10) shows
very clear improvement to the standardGPC, and it also improves the
phasing of the solution, but it still underestimates thevalue of
the variance. In this case, cubic polynomials are not enough to
capturethe strong nonlinear behavior of the oscillator. It is worth
mentioning that the useof the incomplete, adaptive third-order GPC
(p = 3, K = 10, L = 10) versus thecomplete standard GPC expansion
(p = 3, K = 20, L = 0) lowers significantly thenumber of unknown
random coefficients from 1,771 to 296.
Figure 3.5 shows very similar results for Case II. Structural
parameters, correla-tion length, and nonlinear parameter are set to
the same values, and only the typeof distribution of the input is
changed to the Uniform distribution. Here again, thenonlinearity is
too large for a cubic polynomial order even with reordering of
themodes.
Figures 3.6 and 3.7 show results for Cases I and II with an � =
0.1 smaller thanthe previous cases. The damping coefficient ζ =
0.02 is kept low. Consequently, thesolution does not converge to
its asymptotic value within the imposed time domain.However, low
damping implies sharper peaks in the energy spectrum of the
oscillator.Therefore, a finite number of random dimensions is more
likely to capture most ofthe energy in the system. Reordering in
this case also helps by sorting out the mostsignificant random
modes corresponding to the resonant frequencies and keeping
theassociated nonlinear components.
Figure 3.6 shows the time evolution of variance of the solution
for Case I. Wesee that the adaptive GPC with reordering is very
close to the Monte Carlo simu-lation. In Figure 3.7, we present
differently the same type of result for Case II by
-
732 D. LUCOR AND G. E. KARNIADAKIS
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
ω0t
Seco
nd−
orde
r m
omen
t res
pons
e
I: GPC(K=20,L=0,p=1), ε=0.0II: MC(K=40), ε=0.1III:
GPC(K=20,L=0,p=1), ε=0.1IV: AGPC(K=10,L=10,p=3), ε =0.1,NO
reorderingV: AGPC(K=10,L=10,p=3), ε=0.1,WITH reordering
ω0=1.0
ζ=0.02A=1
Fig. 3.6. Comparison of second-order moment response obtained by
adaptive GPC and MonteCarlo simulation (1,000,000 events). ω0 =
1.0; ζ = 0.02; A = 1.0; � = 0.1 (Case I: Gaussian).
0 5 10 15 20 25 30
−0.01
0
0.01
0.02
ω0t
Poin
twis
e E
rror
III: APC (K=10, L=0, p=1)IV: APC (K=10, L=10, p=3), NO
reorderingV: APC (K=10, L=10, p=3), WITH reordering
Fig. 3.7. Comparison of second-order moment response obtained by
adaptive GPC and MonteCarlo simulation (1,000,000 events). ω0 =
1.0; ζ = 0.02; A = 1.0; � = 0.1 (Case II: Uniform).
showing the pointwise error of the adaptive GPC solution against
the Monte Carlosimulation.
Figure 3.8 shows the energy distribution among the random modes
for Case Ibefore and after reordering. Region I in the figure
represents the linear terms orrandom modes associated with the
linear polynomials. Similarly, region II representsthe quadratic
terms (which are zero as explained previously). Finally, the cubic
termsare all grouped in region III. The linear terms distribution
before reordering clearly
-
ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 733
0 50 100 150 200 250 30010
−8
10−6
10−4
10−2
100
102
P
||yi(t
)||
I II III
NO reorderinglast reordering
Fig. 3.8. L2 norm of the random adaptive GPC modes with no
reordering and with reordering.ω0 = 1.0; ζ = 0.02; A = 1.0; � =
0.1. Relates to V: adaptive GPC (K = 10, M = 10, p = 3).
illustrate the concentration of energy in the system around the
peak of resonance. Wealso represent the last reordering after the
iterative process has converged. We noticethat the most energetic
frequencies have been placed first and the corresponding cubicterms
have increased by as much as four orders of magnitude and about two
ordersof magnitude on average.
3.3. A new adaptive approach to GPC. The concept of truncated
repre-sentation of the solution in the framework of the adaptive
GPC method can be ex-tended further. This time, the solution is
again expanded as in (3.14) but with thedistinction that not all of
the nonlinear terms from the third summation based on theK first
random dimensions are kept in the decomposition. In our case, we
observethat the modal energy is always large for the nonlinear
terms corresponding to crossproducts between random dimensions (see
Figure 3.8). Accordingly, we keep only thecoefficients
corresponding to the nonlinear polynomials of the form
Ψj(ξi(θ) |Ki=1) =∏
Cξ1(θ)···ξi(θ)···ξK(θ)l with l = 1, 2, . . . ,K,(3.16)
where the operator∏
C represents the product of the combination of the K
possiblelinear polynomials ξi(θ) taken l at a time.
The case of Figure 3.4 is repeated using the aforementioned
method with adaptiveseventh-order GPC (p = 7, K = 7, L = 13) which
represents a total number of 141random modes instead of 888,030 for
a standard complete GPC expansion (p= 7,K = 20, L= 0). Results are
shown in Figures 3.9 and 3.10. In this case, the adaptiveGPC
solution does not approach uniformly the Monte Carlo solution over
the entiretime domain, but it is locally very accurate. The error
is very small in some places,and this is an example of local
nonuniform convergence of the method. A finitenumber of modes might
be enough to capture the behavior of the oscillator at someinstants
of time but insufficient at others.
-
734 D. LUCOR AND G. E. KARNIADAKIS
0 5 10 15 20 25 300
0.2
0.4
0.6
ω0t
Seco
nd−
orde
r m
omen
t res
pons
e
I: MC(K=40), ε=0.1II: AGPC(K=7,L=13,p=7), ε =0.1, NO
reorderingIII: AGPC(K=7,L=13,p=7), ε=0.1,WITH reordering
Fig. 3.9. Comparison of second-order moment response obtained by
adaptive GPC and MonteCarlo simulation (1,000,000 events). ω0 =
1.0; ζ = 0.02; A = 1.0; � = 0.1 (Case I: Gaussian).
0 5 10 15 20 25 300
0.01
0.02
0.03
ω0t
|Poi
ntw
ise
Err
or|
II: AGPC(K=7,L=13,p=7), ε =0.1, NO reorderingIII:
AGPC(K=7,L=13,p=7), ε =0.1,WITH reordering
Fig. 3.10. Absolute value of second-order moment pointwise error
obtained by adaptive GPCand Monte Carlo simulation (1,000,000
events). ω0 = 1.0; ζ = 0.02; A = 1.0; � = 0.1 (Case
II:Uniform).
4. Summary. High-order polynomial chaos solutions are
prohibitively expen-sive for strongly nonlinear systems when the
number of dimensions of the stochasticinput is large. Progress can
be made, however, by careful adaptive procedures andselectively
incorporating the nonlinear expansion terms. In this paper, we
demon-strated such a procedure, proposed previously by Li and
Ghanem [11], in the contextof the stochastic Duffing oscillator.
The adaptive scheme improves the accuracy ofthis method by
reordering the random modes according to their magnification by
thesystem. An extension of the originally proposed adaptive
procedure was presented
-
ADAPTIVE GPC FOR NONLINEAR RANDOM OSCILLATORS 735
that uses primarily contributions corresponding to cross
products between randomdimensions.
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