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A novel approximate technique based on Wiener path integrals (WPIs) is developed for determining, in a computationally efficient manner, the non-stationary joint response probability density function (PDF) of nonlinear multi-degree-of-freedom dynamical systems. Specifically, appropriate multi-dimensional (time-dependent) global bases are constructed for approximating the non-stationary joint response PDF. In this regard, two distinct approaches are pursued. The first employs expansions based on Kronecker products of bases (e.g., wavelets), while the second utilizes representations based on positive definite functions. Next, the localization capabilities of the WPI technique are exploited for determining PDF points in the joint space-time domain to be used for evaluating the expansion coefficients at a relatively low computational cost. In contrast to earlier implementations of the WPI technique, the herein developed generalization and enhancement circumvents computationally daunting brute-force discretizations of the time domain in cases where the objective is to determine the complete time-dependent non-stationary response PDF. Several numerical examples pertaining to diverse structural systems are considered, including both single-and multi-degree-of-freedom nonlinear dynamical systems subject to non-stationary excitations, as well as a bending beam with a non-Gaussian and non-homogeneous Young's modulus. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the technique.
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Wiener path integrals and multi-dimensional global
bases for non-stationary stochastic response
determination of structural systems
Apostolos F. Psarosa, Ioannis Petromichelakisa, Ioannis A. Kougioumtzogloua,
aDepartment of Civil Engineering and Engineering Mechanics,
Columbia University, 500 W 120th St, New York, NY 10027, United States
Abstract
A novel approximate technique based on Wiener path integrals (WPIs) is devel-
oped for determining, in a computationally efficient manner, the non-stationary
joint response probability density function (PDF) of nonlinear multi-degree-of-
freedom dynamical systems. Specifically, appropriate multi-dimensional (time-
dependent) global bases are constructed for approximating the non-stationary
joint response PDF. In this regard, two distinct approaches are pursued. The
first employs expansions based on Kronecker products of bases (e.g., wavelets),
while the second utilizes representations based on positive definite functions.
Next, the localization capabilities of the WPI technique are exploited for de-
termining PDF points in the joint space-time domain to be used for evaluating
the expansion coefficients at a relatively low computational cost. In contrast
to earlier implementations of the WPI technique, the herein developed gener-
alization and enhancement circumvents computationally daunting brute-force
discretizations of the time domain in cases where the objective is to determine
the complete time-dependent non-stationary response PDF. Several numerical
examples pertaining to diverse structural systems are considered, including both
single- and multi-degree-of-freedom nonlinear dynamical systems subject to non-
stationary excitations, as well as a bending beam with a non-Gaussian and
non-homogeneous Young’s modulus. Comparisons with pertinent Monte Carlo
simulation data demonstrate the accuracy of the technique.
Keywords: path integral, nonlinear systems, stochastic dynamics, global
approximations, multi-dimensional basis, positive definite functions
1. Introduction1
Problems involving a probabilistic description of excitation and media prop-2
erties occur in abundance in engineering (e.g., [1,2]). Clearly, this increased3
Corresponding Author
Email address: ikougioum@columbia.edu (Ioannis A. Kougioumtzoglou)
Preprint submitted to Mechanical Systems and Signal Processing April 1, 2019
sophistication in modeling, in conjunction with complex nonlinear behaviors,4
renders the response analysis of many engineering systems a challenging task.5
In this regard, there is a wide range of solution techniques developed over the6
past few decades in the field of stochastic engineering dynamics and mechan-7
ics (e.g., [39]). Nevertheless, solving in a computationally efficient manner8
high-dimensional nonlinear stochastic differential equations (SDEs) modeling9
the system dynamics remains a persistent challenge.10
Recently, a promising technique has been developed for determining the11
stochastic response of nonlinear multi-degree-of-freedom (MDOF) systems, which12
relates to the concept of the Wiener path integral (WPI) [10,11]. The notion of13
path integral, which generalizes integral calculus to functionals, was introduced14
by Wiener [12] and by Feynman [13], independently, and has been an instrumen-15
tal mathematical tool in the field of theoretical physics [14]. It is noted that the16
WPI based technique developed by Kougioumtzoglou and co-workers exhibits17
significant versatility and can account even for systems endowed with fractional18
derivative terms [15]. Furthermore, it has been extended for addressing certain19
one-dimensional mechanics problems with random material/media properties20
[16], systems subject to non-white and non-Gaussian stochastic processes [17],21
as well as a class of nonlinear electromechanical energy harvesters [18].22
From a computational efficiency perspective, recent work by Kougioumt-23
zoglou et al. [19] and by Psaros et al. [20] has reduced drastically the computa-24
tional cost of the standard numerical implementation of the technique by resort-25
ing to sparse representations of the joint response probability density function26
(PDF) in conjunction with compressive sampling schemes. Nevertheless, the27
above developments and implementations rely on time-invariant joint response28
PDF expansions, or in other words, the objective is to determine the joint re-29
sponse PDF at a specific time instant. Although this localization capability30
can be considered as an advantage of the technique (for instance, the stationary31
response PDF can be determined directly at a reduced cost, without necessar-32
ily obtaining the global solution first), in many cases the determination of the33
complete time-dependent non-stationary joint response PDF is required (e.g.,34
cases of evolutionary excitations [21]). Addressing this challenge by discretizing35
the temporal dimension in a brute-force manner and applying the technique36
for each and every time instant would render the numerical implementation37
computationally daunting. In this regard, the objective of this paper relates to38
generalizing the WPI technique and further enhancing its computational effi-39
ciency by constructing time-dependent bases for determining the non-stationary40
response PDF directly, based on knowledge of relatively few PDF points in the41
joint space-time domain.42
Evaluating the PDF of a stochastic process given partial information is a43
typical problem in a wide range of research fields [22,23]. In stochastic dynamics44
PDF expansions have been utilized for solving the Fokker-Planck (FPE) and the45
Backward Kolmogorov (BKE) equations [24,25], or other alternative equations46
governing the stochastic response of a dynamical system [26]. Indicatively, PDF47
expansions have been coupled with weighted residual methodologies [2730],48
where the approximate PDF is substituted into the FPE and the residual error49
2
is minimized; with moment closure schemes [3135], which yield a finite set of50
moment equations to be solved for approximating the response PDF; with finite51
element method direct numerical solution schemes [36,37]; with discretized52
Chapman-Kolmogorov equation schemes propagating the response PDF in short53
time steps [3841]; and with solution schemes based on the maximum entropy54
principle [42,43]. Typical PDF expansions and approximation schemes utilize55
truncated Gram-Charlier or Edgeworth series [25,27,3032], Hermite or other56
polynomials [29,33,35,36,41,44], Gaussian distributions with varying mean57
and variance [45], kernel density functions [26] and B-splines or piecewise linear58
functions [38,40]. Further, McWilliam et al. [28] employed Shannon wavelets59
for approximating the PDF within the context of the weighted residual method.60
Moreover, radial basis functions (RBFs) demonstrated accurate results in the61
context of numerically solving the FPE [46,47].62
In this paper an approximation scheme based on the WPI technique is de-63
veloped for efficiently determining the non-stationary joint response PDF of64
stochastically excited MDOF dynamical systems. To this aim, two distinct ex-65
pansions are proposed for the PDF; the first is based on Kronecker products of66
bases such as wavelets, and the second is based on positive definite functions,67
which is a more general class of functions than RBFs. As a result, the WPI68
technique is generalized herein to account explicitly for the time dimension in its69
formulation and implementation. The paper is organized as follows: in Section 270
the mathematical formulation of the WPI technique is outlined, in Section 3var-71
ious approximation schemes are developed, and in Section 4 three illustrative72
numerical examples demonstrate the reliability of the proposed technique.73
2. Wiener Path Integral technique74
In this section, the main theoretical and numerical aspects of the WPI tech-75
nique are delineated for completeness. A more detailed presentation can be76
found in [17].77
2.1. Wiener Path Integral formalism78
In general, a wide range of problems in engineering mechanics and dynamics79
can be described by stochastic equations of the form80
F[x] = f(1)
where F[.] represents an arbitrary nonlinear differential operator; fdenotes81
the external excitation; and xis the system response to be determined. In82
the following, an MDOF nonlinear dynamical system with stochastic external83
excitation is considered in the form84
M¨x+C˙x+Kx +g(x,˙x) = f(t) (2)
where xis the displacement vector process (x= [x1, . . . , xm]T); M,C,K85
correspond to the m×mmass, damping and stiffness matrices, respectively;86
3
g(x,˙x) denotes an arbitrary nonlinear vector function; and f(t) is, in general, a87
non-stationary, non-white, and non-Gaussian vector process expressed as [4,5]88
f(t) = D(t)ξ(t) (3)
where D(t) is a diagonal matrix of deterministic time-modulating functions89
dj(t), j∈ {1, . . . , m}, and ξ(t) is given as the response of the nonlinear filter90
equation91
P¨
ξ+Q˙
ξ++u(ξ,˙
ξ) = w(t) (4)
where P,Q,Rdenote coefficient matrices; u(ξ,˙
ξ) is an arbitrary nonlinear92
vector function; and w(t)=[w1, . . . , wm]Tis a white noise stochastic vector93
process with the power spectrum matrix94
Sw=
S0. . . 0
.
.
.....
.
.
0. . . S0
(5)
In passing, it is noted that the WPI technique, developed originally for Gaus-95
sian white noise excitation only [11], has been recently extended by Psaros et al.96
[17] to address general non-white, non-Gaussian and non-stationary cases de-97
cribed by Eqs. (3)-(4). In particular, differentiating Eq. (2) and substituting98
into Eq. (4) yields the 4th-order SDE99
Λ4x(4) +Λ3x(3) +Λ2¨x+Λ1˙x+Λ0x+h(x,˙x,¨x,x(3)) = w(t) (6)
where100
Λ4=P D1M
Λ3=P D1h2˙
DD1M+Ci+QD1M
Λ2=P D1h2˙
DD1˙
DD1¨
DD1M2˙
DD1C+Ki
+QD1˙
DD1M+C+RD1M
Λ1=P D1h2˙
DD1˙
DD1¨
DD1C2˙
DD1Ki
+QD1˙
DD1C+K+RD1C
Λ0=P D12˙
DD1˙
DD1¨
DD1K
QD1˙
DD1K+RD1K
(7)
and101
h(x,˙x,¨x,x(3)) = P D1¨g(x,˙x)
+2P D1˙
DD1+QD1˙g(x,˙x)
+hP D12˙
DD1˙
DD1¨
DD1
QD1˙
DD1+RD1ig(x,˙x) + u(x,˙x,¨x,x(3))
(8)
4
Notice that the differentiation of Eq. (2) yields time derivatives of order higher102
than 2 in Eqs. (6)-(8); i.e., x(3) and x(4) denoting the 3rd - and 4th-order deriva-103
tives of of x(t), respectively.104
Next, as shown in [17], the transition PDF pxf,˙xf,¨xf,x(3)
f, tf|xi,˙xi,¨xi,x(3)
i, ti
105
with {xi,˙xi,¨xi,x(3)
i, ti}the initial state and {xf,˙xf,¨xf,x(3)
f, tf}the final state106
can be written as107
pxf,˙xf,¨xf,x(3)
f, tf|xi,˙xi,¨xi,x(3)
i, ti
=ZC{xi,˙xi,¨xi,x(3)
i,ti;xf,˙xf,¨xf,x(3)
f,tf}
exp
tf
Z
ti
Lhx,˙x,¨x,x(3),x(4) idt
D[x(t)]
(9)
where the Lagrangian Lx,˙x,¨x,x(3),x(4) is given by108
Lhx,˙x,¨x,x(3),x(4) i=1
2Λ4x(4)+Λ3x(3) +Λ2¨x+Λ1˙x+Λ0x+hx,˙x,¨x,x(3)T
×B1Λ4x(4) +Λ3x(3) +Λ2¨x+Λ1˙x+Λ0x+hx,˙x,¨x,x(3)(10)
where109
B=
2πS0. . . 0
.
.
.....
.
.
0. . . 2πS0
(11)
and D[x(t)] is a functional measure [17]. In Eq. (9) C{xi,˙xi,¨xi,x(3)
i, ti;xf,˙xf,¨xf,x(3)
f, tf}110
denotes the set of all paths with initial state {xi,˙xi,¨xi,x(3)
i}at time tiand final111
state {xf,˙xf,¨xf,x(3)
f}at time tf.112
The formal expression of the path integral in Eq. (9) is of little practical use113
as its analytical evaluation is highly challenging [14]. Therefore, an approximate114
approach is required. In this regard, the “most-probable path” approach (e.g.,115
[10,14]) is employed, according to which the largest contribution to the transi-116
tion PDF of Eq. (9) comes from the path xc(t) that minimizes the integral in117
the exponential. In this regard, calculus of variations (e.g., [48]) dictates that118
the most probable path, xc(t), satisfies the extremality condition119
δ
tf
Z
ti
Lhxc,˙xc,¨xc,x(3)
c,x(4)
cidt= 0 (12)
which yields the system of Euler-Lagrange (E-L) equations120
L
∂xc,j
∂t
L
˙xc,j
+2
∂t2
L
¨xc,j
3
∂t3
L
∂x(3)
c,j
+4
∂t4
L
∂x(4)
c,j
= 0,for j= 1, . . . , m (13)
5
together with 8 ×mboundary conditions121
xc,j (ti) = xj,i
˙xc,j (ti) = ˙xj,i
¨xc,j(ti) = ¨xj,i
x(3)
c,j (ti) = x(3)
j,i
xc,j (tf) = xj,f
˙xc,j (tf) = ˙xj,f
¨xc,j(tf) = ¨xj,f
x(3)
c,j (tf) = x(3)
j,f
for j= 1, . . . , m (14)
Next, solving the boundary value problem (BVP) of Eqs. (13) and (14)122
yields the most probable path xc(t) (m-dimensional), and the transition PDF123
from the initial state {xi,˙xi,¨xi,x(3)
i, ti}to the final state {xf,˙xf,¨xf,x(3)
f, tf}124
is determined as125
pxf,˙xf,¨xf,x(3)
f, tf|xi,˙xi,¨xi,x(3)
i, ti=Cexp
tf
Z
ti
Lhxc,˙xc,¨xc,x(3)
c,x(4)
cidt
(15)
where the normalization constant Cis evaluated based on the condition126
Z
−∞
· · ·
Z
−∞
pxf,˙xf,¨xf,x(3)
f, tf|xi,˙xi,¨xi,x(3)
i, tidxf. . . dx(3)
f= 1 (16)
For the special case of time-modulated Gaussian white noise (i.e., ξ(t) =127
[ξ1, . . . , ξm]Tbeing a white noise stochastic vector process) Eq. (9) degenerates128
to129
p(xf,˙xf, tf|xi,˙xi, ti) = ZC{xi,˙xi,ti;xf,˙xf,tf}
exp
tf
Z
ti
L[x,˙x,¨x] dt
D[x(t)]
(17)
and Eq. (15) becomes130
p(xf,˙xf, tf|xi,˙xi, ti) = Cexp
tf
Z
ti
L[xc,˙xc,¨xc] dt
(18)
where131
L[x,˙x,¨x] = 1
2(M¨x+C˙x+Kx +g(x,˙x))Te
B1
×M¨x+C˙x+Kx +g(x,˙x)(19)
6
132
e
B=
2πS0d2
1(t). . . 0
.
.
.....
.
.
0. . . 2πS0d2
m(t)
(20)
and xc(t) is the solution of the system of E-L equations133
L
∂xc,j
∂t
L
˙xc,j
+2
∂t2
L
¨xc,j
= 0,for j= 1, . . . , m (21)
together with 4 ×mboundary conditions134
xc,j (ti) = xj,i
˙xc,j (ti) = ˙xj,i
xc,j (tf) = xj,f
˙xc,j (tf) = ˙xj,f
for j= 1, . . . , m (22)
Finally, the normalization constant Ccan be determined by utilizing the con-135
dition136
Z
−∞
. . .
Z
−∞
p(xf,˙xf, tf|xi,˙xi, ti)dx1,f d ˙x1,f . . . dxm,f d ˙xm,f = 1 (23)
It can be readily seen by comparing Eqs. (17) and (18) that in the approxi-137
mation of Eq. (18) only a single trajectory, i.e., the most probable path xc(t),138
is considered in evaluating the path integral of Eq. (17). Nevertheless, direct139
comparisons of Eq. (18) with pertinent MCS data related to various engineering140
dynamical systems have demonstrated a high degree of accuracy [15,17].141
Further, considering fixed initial conditions (xi,˙xi) typically (i.e., system ini-142
tially at rest), the solution of a functional minimization problem (i.e., Eqs. (21)-143
(22)) for determining a single point of the joint response PDF is required. Uti-144
lizing a data analysis perspective and terminology, this can be construed as145
obtaining joint response PDF measurements at pre-specified sampling locations.146
2.2. Numerical implementation aspects147
In general, a numerical solution scheme needs to be implemented for the148
BVP of Eqs. (21)-(22) (or of Eqs. (13)-(14) for non-white and non-Gaussian149
excitation). Without loss of generality and considering fixed initial conditions,150
the only variables describing the PDF p(xf,˙xf, tf|xi,˙xi, ti) at a time instant151
tfare xfand ˙xf. In this regard, dropping the subscript ffor simplicity and152
adopting a brute-force numerical solution approach, for each time instant tan153
effective domain of values is considered for the joint response PDF p(x,˙x, t).154
Next, discretizing the effective domain using Nspoints in each dimension, the155
joint response PDF values are obtained corresponding to the points of the mesh.156
More specifically, for an m-DOF system corresponding to 2mstochastic dimen-157
sions (mdisplacements and mvelocities) the number of measurements required158
7
is N2m
s. These 2mstochastic dimensions will be referred to hereinafter as159
spatial dimensions. Clearly, the computational cost related to a brute-force160
solution scheme implementation becomes prohibitive eventually, especially for161
high-dimensional systems.162
To address the above computational challenge, Kougioumtzoglou et al. [19]163
employed a polynomial expansion for the logarithm of the joint response PDF;164
thus, the number of required PDF measurements becomes equal to the number165
of the expansion coefficients. The rationale for selecting a polynomial expansion166
relates to the fact that in cases of linear systems (i.e., g(x,˙x) = 0) the joint167
response PDF is Gaussian, or, in other words, the function log (p(x,˙x, t)) can168
be expressed exactly as a second-order polynomial. In the general case, where169
g(x,˙x)6= 0, p(x,˙x, t) can be construed as a “perturbation” (not necessarily170
small) from the Gaussian PDF, and, thus, more monomials are required to171
enhance the approximation accuracy. The resulting polynomial is, consequently,172
of higher order. Further, it was shown that the computational cost follows173
approximately a power-law function of the form (2m)λ! (where λis the174
degree of the polynomial), which can be orders of magnitude smaller than N2m
s.175
Moreover, it has been recently shown by Psaros et al. [20] that a compressive176
sampling treatment in conjunction with an appropriate optimization algorithm177
can further reduce drastically the required number of PDF measurements (i.e.,178
number of deterministic BVPs to be solved numerically).179
Nevertheless, all the above enhancements in terms of computational effi-180
ciency of the WPI technique relate to determining the joint response PDF at a181
specific fixed final time instant t. In other words, in cases where the determina-182
tion of the complete time-dependent non-stationary response PDF is of interest,183
the procedure should be applied for each and every time instant. Indicatively,184
the joint response PDF of a 10-DOF nonlinear dynamical system at a given time185
instant has been obtained with only ns= 3,200 measurements as shown in [20],186
whereas a brute-force PDF domain discretization scheme would require 3020
187
measurements (for Ns= 30). However, even with the efficient implementation188
of [20], a brute-force discretization of the time domain (temporal dimension)189
into Nt= 1,000 points, for instance, would still require Ntns= 3.2×106PDF190
measurements. In this paper, motivated by the aforementioned challenge, the191
computational efficiency of the WPI is further increased by resorting to expan-192
sions based on Kronecker products of basis matrices (Section 3.1) and on positive193
definite functions (Section 3.2). In this regard, the WPI technique is enhanced194
herein and becomes capable of determining directly the global time-dependent195
non-stationary joint response PDF in a computationally efficient manner.196
3. Non-stationary joint response PDF approximation197
As discussed in Section 2.2, the computational cost associated with deter-198
mining the global non-stationary joint response PDF by applying a brute-force199
discretization of the time domain remains significant, even when using the effi-200
cient implementations of [19] and [20]. To address this challenge, the approxima-201
tion schemes developed in this section consider the response PDF as a function202
8
of time explicitly. In this regard, generalizing the formulations of [19] and [20],203
p(x,˙x, t) is written as204
p(x,˙x, t)exp (µ(x,˙x, t)) (24)
or, alternatively, as205
p(x,˙x, t)ν(x,˙x, t) (25)
where µ(x,˙x, t) and ν(x,˙x, t) are approximating functions. Therefore, depend-206
ing on whether Eq. (24) or Eq. (25) is used, a measurement of the response PDF207
at a specific location (x,˙x, t) via the WPI technique refers to either the exponent208
or the exponential function of Eq. (18), respectively.209
First, in Section 3.1, a separable basis is constructed for approximating the210
non-stationary PDF by combining the bases/structures selected for each dimen-211
sion [49]. Such a basis proves, in general, capable of handling the anisotropic212
features of multivariate functions and appears a natural choice for approximat-213
ing the response PDF. Next, an alternative approach is followed in Section 3.2,214
where the approximation takes the form of a scattered data fitting problem [50].215
The non-stationary PDF is sampled at various locations in the spatio-temporal216
domain and a fit to the dataset based on positive definite functions (which can217
be construed as a generalization of the widely used RBFs [50]) is sought for.218
Also, it is noted that positive definite functions have been deliberately selected219
over RBFs for better coping with the potentially anisotropic features of the220
non-stationary PDF [50].221
3.1. Kronecker product approach222
3.1.1. Kronecker product bases223
Various multivariate bases have been developed based on Kronecker prod-224
ucts [51]. Remarkably, the applications of Kronecker structure range from im-225
age/video processing [52] and distributed sensing [49] to pre-conditioning for226
linear system solution [53] and matrix approximation [54].227
Specifically, the Kronecker product HJof two matrices HRH1×H2and228
JRJ1×J2is a matrix of size H1J1×H2J2defined by (e.g., [51])229
HJ=
h11J. . . h1H2J
.
.
.....
.
.
hH11J. . . hH1H2J
(26)
Further, given the basis matrices D1Rn1×n1and D2Rn2×n2, consider a230
transform applied to a data matrix YRn1×n2by using the separable basis231
constructed by their Kronecker product. Vectorizing matrix Y, i.e., concate-232
nating its columns vertically, so that y=vec(Y)Rn1n2, the data vector can233
be written as (e.g., [51])234
y= (D2D1)c(27)
where cdenotes the coefficient matrix in vectorized form. Generalizing, consider235
pdimensions in total and n1n2. . . npmeasurements taken from a multivariate236
9
function y(q), where qRp. The measurement tensor YRn1×n2×···×np
237
admits a Kronecker expansion of the form of Eq. (27) expressed as238
y= (Dp⊗ · · · D2D1)c(28)
where y,cRn1n2...np. Therefore, after collecting n=n1n2. . . npmeasure-239
ments from the p-dimensional space and selecting a basis in each dimension240
(D1,...,Dp) the coefficients of the Kronecker expansion can be obtained by241
solving the linear system of Eq. (28). It is noted that the columns of the ba-242
sis matrices D1,...,Dpin Eq. (28) are the basis functions selected for each243
dimension discretized into n1, . . . , nppoints, respectively.244
3.1.2. Multi-dimensional basis construction for approximating the non-stationary245
joint response PDF246
Following the procedure outlined in Section 3.1.1 it is rather straightforward247
to construct a multi-dimensional basis for approximating the non-stationary248
response PDF by employing Eqs. (25) and (28). Specifically, the response PDF249
is a function of (x,˙x, t), which is of size 2m+ 1; that is, 2mspatial dimensions250
and 1 temporal dimension. In this regard, various (potentially different) bases251
can be chosen for the approximation in each dimension in conjunction with the252
numbers nk,k∈ {1,...,2m+ 1}. Next, the expansion coefficients vector c253
is determined by solving the linear system of Eq. (28), where yis the vector254
containing n1. . . n2m+1 measurements of p(x,˙x, t) determined via WPI and255
employing a uniform mesh. In the following, and without loss of generality, two256
distinct approaches are pursued in choosing the bases to be utilized in Eq. (28).257
First, the same one-dimensional wavelet basis is used for each and every258
dimension. In particular, an arbitrary function f(t) can be expressed as259
f(t) =
X
l=−∞
X
r=−∞
clrψlr(t) (29)
where care the expansion coefficients to be determined, rand ldenote the260
different scales and translation levels, respectively, and ψlr(t) = 1
2lψ(t
2lr), with261
ψ(t) the wavelet family to be chosen. Alternatively, Eq. (29) can be expressed262
via the associated scaling function φ(t) as263
fn(t) =
n1
X
r=0
crφLr(t) (30)
where fn(t) denotes the n-term approximation of the function f(t) with only264
n= 2Lscaling functions, given as φLr(t) = 1
2Lφ(t
2Lr), and Ldenotes the265
selected scale, or equivalently the approximation level. A detailed presentation266
of wavelet theory can be found in several books, such as [55]. Obviously, the267
efficacy of the chosen wavelet family is application-dependent. Thus, various268
both discrete and continuous wavelets have been developed over the past decades269
[55], as well as generalizations with additional parameters such as harmonic270
10
wavelets (e.g., [5658]) and chirplets (e.g., [59]); see also the review paper by271
Spanos and Failla [60] for diverse wavelet applications in engineering dynamics.272
In the ensuing analysis, the Meyer wavelet (e.g., [55]) is used in the related273
expansions.274
Second, an alternative approach is pursued, which exploits the flexibility275
of the herein proposed framework to use different bases. In this regard, and276
considering Eq. (24), a multivariate polynomial can be employed for the spatial277
dimensions, as in [19], and a wavelet basis for the temporal dimension. There-278
fore, the linear system of Eq. (27) becomes279
y= (DwP)c(31)
where ycontains the measurements of log (p(x,˙x, t)) determined via the WPI,280
Dwdenotes the one-dimensional wavelet basis, Pthe monomial basis (e.g.,281
[61]) and cthe coefficient vector. Specifically, Pis an ns×nsmatrix, where282
ns=λ+2m
2mfor a polynomial of degree λ, and Dwis an nt×ntmatrix.283
Therefore, n=nsntmeasurements of the joint response PDF via the WPI284
technique are required.285
Overall, it is readily seen that utilizing a Kronecker product formulation is286
a conceptually simple approach for higher-dimensional approximations by com-287
bining several lower-dimensional approximations in a straightforward manner.288
This yields enhanced flexibility in the implementation of the approach as vari-289
ous, potentially different, bases can be used, which have already proven to be290
well-suited for the respective lower-dimensional problems. For instance, the291
monomial basis has exhibited significant accuracy in approximating the spatial292
dimensions of a class of problems in [19] and in [20]. Thus, under the Kronecker293
product formulation, it can be directly used in conjunction with an additional294
basis related to the temporal dimension. Nevertheless, the lower-dimensional295
bases and the respective number of measurements need to be selected a pri-296
ori, while as noted in [20], the monomial basis is prone to ill-conditioning, and,297
hence, the points of the mesh should be selected based on certain optimality298
criteria for enhanced robustness and accuracy of the approximation (see for in-299
stance [62]). If, alternatively, only one-dimensional wavelet bases are used for300
constructing the multi-dimensional basis via Eq. (28), the associated computa-301
tional cost increases exponentially with increasing number of dimensions and302
becomes eventually prohibitive for relatively high-dimensional problems.303
To address the above points, a mesh-free approximation scheme is developed304
in Section 3.2 by utilizing positive definite functions. The advantages of such an305
approach pertain mainly to the fact that the basis functions depend on the mea-306
surement locations, and thus, are not selected a priori. Therefore, as explained307
in detail in Section 3.2, the resulting interpolation matrix is well-conditioned308
yielding a robust and accurate approximation. Overall, positive definite func-309
tions appear more general and suitable for higher-dimensional systems, whereas310
Kronecker product bases perform better for lower-dimensional systems, espe-311
cially when there is some available information regarding the response PDF.312
11
3.2. Positive definite functions approach313
3.2.1. Positive definite functions aspects314
In this section, the multivariate (p-dimensional) approximation problem is315
formulated as a scattered data fitting problem, which is a fundamental problem316
in approximation theory and is summarized in the following [50]: Given a set of317
measurements (qi, yi) from a function y(qi), where i∈ {1, . . . , n},qiRpand318
yiR, determine a function µ(q) such that319
µ(qi) = yi(32)
i∈ {1, . . . , n}. Even though in the univariate case (i.e., p= 1) this meshfree320
problem has a unique solution using ndistinct measurements and a polynomial321
of order n1, the multivariate case is more complex leading to ill-conditioned322
interpolation matrices [63]. According to the Mairhuber-Curtis theorem (e.g.,323
[64]), for the problem to be well-posed, i.e., for a solution to exist and be unique,324
the basis functions cannot be fixed a priori.325
The above challenge has led mathematicians to introduce data-dependent326
bases, which are bases created following the selection of the sampling locations.327
In this direction, positive definite functions (or kernels more generally) have been328
commonly used in approximation theory [50,65]. Following [63], a complex-329
valued continuous function Φ : RpCis called positive definite on Rpif330
n
X
i=1
n
X
j=1
ci¯cjΦ(qiqj)0 (33)
for any npairwise different points q1,...,qnRpand c= [c1, . . . , cn]TCn.331
Among the most widely used positive definite functions is the Gaussian func-332
tion, i.e., Φ(q) = e2kqk2
2, with qRpand a shape parameter  > 0. The333
widespread utilization of positive definite functions in the approximation field334
can be attributed, at least partly, to their connection with the scattered data fit-335
ting problem of Eq. (32), and to the existence of well-behaved (i.e., non-singular)336
interpolation matrices (e.g., [66]). Further, there are constantly new classes of337
positive definite functions being introduced in conjunction with related theoret-338
ical work on error bounds [63]. Finally, it is worth noting that the numerical339
implementation of positive definite functions is amenable to high-performance340
computing [67], while their applications range from meshfree interpolation and341
solution of partial differential equations (PDEs) [68] to simulation of stochastic342
processes [69] and machine learning [70].343
3.2.2. Multi-dimensional basis construction for approximating the non-stationary344
joint response PDF345
As mentioned in Section 3.2.1, given the measurements (qi, yi) the objective346
is to determine an interpolating function µ(q), expressed as347
µ(q) =
n
X
i=1
ciΦi(q) (34)
12
where the basis functions Φi, for i∈ {1, . . . , n}, are positive definite and348
c= [c1, . . . , cn]TRndenotes the expansion coefficient vector. Clearly, the349
choice of the specific basis functions is problem-dependent, with RBFs being350
among the most popular choices [71]. RBFs are rotationally and translationally351
invariant and are commonly used in engineering problems. For RBF interpola-352
tion the basis functions are expressed as a function of kqqik, where qi, for353
i∈ {1, . . . , n}, corresponds to the sampling locations.354
Alternative choices include, but are not limited to, multiscale kernels [72],355
which are defined as linear combinations of shifted and scaled versions of a356
single function and exhibit properties similar to wavelets, and translationally357
invariant functions [50]. The latter are constructed by relaxing the rotational358
invariance property of RBFs and have been found to provide further flexibility359
in the interpolation and to improve the condition number of the interpolation360
matrix [50]. A typical example that is also adopted in the ensuing analysis is361
the anisotropic multivariate Gaussian function362
Φi(q) = exp
p
X
k=1
2
k(qkqik)2!(35)
where k, for k∈ {1, . . . , p}, denotes the shape parameter for the k-th dimension,363
while qkand qik denote the k-th component of qand qi, respectively. The basis364
then becomes a collection of functions of the form of Eq. (35), i.e.,365
{Φ1(q),...,Φn(q)}(36)
Next, considering q= (x,˙x, t) yields p= 2m+ 1 dimensions, while the same366
shape parameter value sis used for all the spatial dimensions and the value t
367
for the temporal dimension. In this regard, by employing anisotropic Gaussian368
functions, Eq. (24) becomes369
p(x,˙x, t)exp "n
X
i=1
ciexp
m
X
k=1
2
s(xkxik)2
m
X
k=1
2
s( ˙xk˙xik )22
t(tti)2!#
(37)
Note that the nsampling locations need to be well-distributed in the (2m+370
1)-space. To this aim, the Halton sequence is used [73], which is also frequently371
employed in quasi-Monte Carlo methods for multi-dimensional integration; see372
also the papers by Bratley et al. [74] and by De Marchi et al. [75] for some373
alternative sampling strategies. Further, it is desirable in many cases to have374
an interpolant that exactly reconstructs a polynomial of a given order; see, for375
instance, the “patch test” in finite elements (e.g., [76]). To this aim, the basis376
of Eq. (36) can be augmented by including monomials to a given order; that is377
{Φ1(q),...,Φn(q),1, q1, q2, . . . , q2
1,2q1q2, . . . }(38)
Therefore, the size of the basis of Eq. (38) becomes n+np, where nis the num-378
ber of measurements, and np=λ+2m+1
2m+1 is the number of monomials Pu(q),379
13
u∈ {1, . . . , np}, for a polynomial of degree λ. In the following, a 4th-degree380
polynomial is considered in the augmented basis of Eq. (38), in accordance with381
the rationale presented in [20]. Concisely, for problems of the form of Eq. (2),382
the joint response PDF is Gaussian for g(x,˙x) = 0; thus, it is represented ex-383
actly by a 2nd-degree polynomial. The nonlinear system joint response PDF384
can be construed as a perturbation (not necessarily small) from the Gaussian,385
and it can be approximated by higher-order polynomials. In several examples,386
including rather challenging cases of bimodal response PDFs [20], it has been387
demonstrated that the choice of a 4th-degree polynomial reflects a reasonable388
compromise between accuracy and efficiency. In this regard, enforcing the n389
interpolation conditions of Eq. (34) and adding npconditions of the form390
n
X
i=1
ciPu(qi) = 0,for u∈ {1, . . . , np}(39)
leads to the augmented linear system of equations [63]391
ΦP
PT0c
cp=y
0(40)
where Φ= [Φ1(qi),...,Φn(qi)]n×n,P= [Pu(qi)]n×np,cdenotes the expansion392
coefficients vector, cpthe polynomial coefficients vector, and ythe measurement393
vector. It is noted that the conditions of Eq. (39) are arbitrary and have been394
added for obtaining a non-singular interpolation matrix [63]. Once the system395
of Eq. (40) is solved, the coefficient vector [c;cp] is determined; see also [77] for396
more details on the conditions to be satisfied for the well-posedness of Eq. (40).397
Note that although the augmented coefficient vector of Eq. (40) is of length n+398
np> n, the number of measurements required for the approximation remains the399
same and equal to n. Further, Eq. (37) is modified to account for the augmented400
basis, and the non-stationary joint response PDF can be approximated as401
p(x,˙x, t)exp n
X
i=1
ciΦi(x,˙x, t) +
np
X
u=1
cpuPu(x,˙x, t)!(41)
3.2.3. Selection of shape parameters402
Positive definite functions have been criticized for producing ill-conditioned403
interpolation matrices Φ, and thus, causing numerical instability issues [78].404
Note, however, that a careful examination of the matter [63] reveals that there405
is a trade-off between accuracy and stability. Theoretical bounds pertaining to406
several positive definite functions indicate that by decreasing the values of the407
shape parameters kin Eq. (35), or the separation distance between the sampling408
locations (i.e., by increasing n), the accuracy of the interpolation is improved.409
Nevertheless, this theoretically attainable accuracy is hard to be reached in410
practice. This is due to numerical stability issues related to the rapid increase411
of the interpolation matrix condition number. This trade-off has led researchers412
14
to seek for “optimal” shape parameters, which provide high accuracy without413
compromising numerical stability [79].414
The approach adopted in the ensuing analysis was developed in [80] and is415
based on leave-one-out cross validation. Specifically, for fixed shape parameters416
sand t, fitting an interpolant of the form of Eqs. (34) and (40) to n1417
measurements ntimes (one is left out each time) yields an interpolation error418
Ei, for i∈ {1, . . . , n}, by comparing the interpolant with the measurement419
left out. The error, E, associated with the pair (s,t) is then selected to420
be the maximum of all the errors Ei. Therefore, the error associated with421
the pair (s,t) becomes the cost function in an optimization algorithm that422
searches for the pair (s,t) with the minimum error E. Finally, as stated in423
Section 3.2.2, introducing the anisotropic Gaussian function of Eq. (35) also424
improves the condition number of the interpolation matrix as a “side-effect”.425
Of course, alternative approaches such as pre-conditioning [63], or exploration426
of other bases [81], can always be used for addressing cases of ill-conditioned427
matrices.428
3.3. Mechanization of the technique429
The mechanization of the Kronecker product approach of Section 3.1 involves430
the following steps:431
(a) Select plower dimensional bases D1,...,Dp.432
(b) Create the interpolation matrix D=Dp⊗ · · · D2D1.433
(c) Obtain n=n1n2. . . npmeasurements of the PDF via the WPI by utilizing434
Eq. (18).435
(d) Determine the coefficient vector cby solving the linear system of Eq. (28).436
(e) The complete non-stationary joint response PDF is evaluated by employing437
the Kronecker product basis and the coefficient vector via Eqs. (25) and438
(28).439
Further, the mechanization of the positive definite functions approach of440
Section 3.2 involves the following steps:441
(a) Select the nbasis functions of Eq. (38), in conjunction with Eq. (35).442
(b) Obtain nmeasurements of the PDF via the WPI by utilizing Eq. (18)443
and by employing the Halton sequence for selecting the locations of the444
measurements [73].445
(c) Determine the coefficient vector cby solving the linear system of Eq. (40).446
(d) The complete non-stationary joint response PDF is approximated via Eq. (41).447
4. Numerical examples448
To assess the performance and demonstrate the efficacy of the developed449
approximation schemes, three examples with distinct features are considered.450
In Section 4.1, two single-degree-of-freedom (SDOF) Duffing nonlinear oscil-451
lators subject to Gaussian white noise are considered: one with a standard452
15
hardening restoring force (Section 4.1.1), and another exhibiting a bimodal re-453
sponse PDF (Section 4.1.2). In Section 4.1.1 the Kronecker product approach454
with a 4th-degree polynomial for the spatial dimensions and an one-dimensional455
wavelet basis for the temporal dimension is used, whereas in Section 4.1.2 a ba-456
sis constructed via a Kronecker product of three one-dimensional wavelet bases457
is employed. Next, in Section 4.2 a 2-DOF nonlinear oscillator subject to non-458
stationary time-modulated Gaussian white noise is considered, and the positive459
definite functions approach of Section 3.2 is employed. Finally, the positive460
definite functions approach is also employed in Section 4.3, where a statically461
determinate Euler-Bernoulli beam is considered with Young’s modulus modeled462
as a non-Gaussian, non-white and non-homogeneous stochastic field.463
4.1. SDOF Duffing nonlinear oscillator464
4.1.1. SDOF Duffing oscillator with a hardening restoring force465
Consider an SDOF Duffing oscillator, whose equation of motion is given by466
Eq. (2) with parameter values (M= 1; C= 0.1; K= 1; g=x3). The stochas-467
tic excitation is given by Eq. (3) with D(t) = 1 and ξ(t) is a white noise process.468
Therefore, f(t) is a Gaussian white noise process, whose power spectrum is given469
by Eq. (5) with S0= 0.0637. Assuming quiescent initial conditions, its transi-470
tion PDF, written as p(x, ˙x, t), is a function of the two spatial dimensions, i.e.,471
xand ˙x, and of the temporal dimension t. In implementing the approximate472
WPI technique developed herein, the monomial basis is used for the two spa-473
tial dimensions, while the wavelet basis is used for the temporal dimension as474
discussed in Section 3.1.2. In particular, utilizing a 4th-degree polynomial, the475
joint response PDF is sampled at n=nsnt= 15 ×32 = 480 locations in the476
spatio-temporal domain and the expansion coefficient vector cis determined by477
solving Eq. (31). Finally, p(x, ˙x, t) is approximated by utilizing the constructed478
basis and the coefficient vector via Eq. (28). The non-stationary marginal PDFs479
of x(t) and ˙x(t) are shown in Fig. 1, where it is seen that the oscillator response480
PDF does not experience any significant changes after about t= 6s; that is, the481
system has reached stationarity effectively. Moreover, the marginal PDFs of482
x(t) and ˙x(t) for two arbitrary time instants are shown in Fig. 2. Although the483
accuracy of the technique depends, in general, on the choice of the polynomial484
degree and the number of points in the temporal dimension, it is shown in this485
example that a 4th-degree polynomial and nt= 32 points are adequate in de-486
termining the non-stationary PDF of this Duffing oscillator with high accuracy487
as compared to pertinent MCS data (50,000 realizations).488
To provide a rough comparison and highlight the gain of the proposed tech-489
nique in terms of computational efficiency, it is worth noting that a brute-force490
numerical implementation of the WPI technique as described in Section 2.2491
would require a number of PDF measurements of the order of 106(assuming492
that the temporal dimension is, indicatively, discretized into 1,000 points). Fur-493
ther, the approximation based on polynomials and wavelets employed in this494
example also requires a smaller number of measurements as compared to the495
efficient implementation of [19]. Specifically, by utilizing the approximate tech-496
nique developed in [19] the response PDF needs to be separately determined at497
16
every time instant, which (for an indicative discretization of the time domain498
into 1,000 points) yields approximately 35,000 required PDF measurements via499
the WPI technique.500
Fig. 1. Non-stationary marginal PDF of x(t) and ˙x(t) for an SDOF hardening Duffing
oscillator under Gaussian white noise excitation, as obtained via the WPI technique (a and
c); comparisons with MCS data - 50,000 realizations (b and d).
17
-2 0 2
0
0.5
1
1.5
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
Fig. 2. Marginal PDFs of x(t) (a) and ˙x(t) (b) at time instants t= 1s and t= 12s for an
SDOF hardening Duffing oscillator under Gaussian white noise excitation, as obtained via
the WPI technique; comparisons with MCS data (50,000 realizations).
4.1.2. SDOF Duffing oscillator with a bimodal response PDF501
Although example 4.1.1 has shown that utilizing a Kronecker product of a502
polynomial and a wavelet bases can be adequate for a certain class of prob-503
lems, the resulting interpolation matrix may often be ill-conditioned. Such is504
the case of the SDOF oscillator, whose equation of motion is given by Eq. (2)505
with parameter values (M= 1; C= 1; K=0.3; g=x3) and external506
excitation as in example 4.1.1. In fact, attempting to use the same basis as in507
4.1.1 has led to ill-conditioning. To bypass this limitation, a multi-dimensional508
wavelet basis is instead utilized for this case, as discussed in Section 3.1.2. In509
this regard, a mesh is employed for discretizing the three-dimensional spatio-510
temporal domain characterizing the transition PDF. Specifically, following the511
procedure delineated in Section 3.1.2 the two spatial dimensions are discretized512
into n1=n2= 16 points and the temporal dimension into nt= 32 points;513
thus, yielding n= 8,192 required measurements via the WPI technique. To514
put it into perspective, note that a brute-force implementation of the technique515
would require a number of PDF measurements of the order of 106, while516
applying the efficient implementation of [19] for each and every time instant517
would yield approximately 35,000 required measurements (assuming that the518
temporal dimension is, indicatively, discretized into 1,000 points). Following the519
determination of the expansion coefficient vector cby solving Eq. (28), where520
all basis matrices correspond to the wavelet basis in each dimension, p(x, ˙x, t) is521
approximated based on Eq. (25). In this regard, p(x, ˙x, t) can be approximated522
at any location by utilizing the constructed basis and the coefficient vector. In523
18
Fig. 3 the joint response PDF p(x, ˙x, t) is shown at three arbitrary time instants524
t= 1,2 and 6s. Comparisons with corresponding MCS based results demon-525
strate the relatively high accuracy of the technique for addressing dynamical526
systems even with relatively complex PDF shapes, such as the bimodal. Fi-527
nally, in Figs. 4 and 5the marginal PDFs of x(t) and ˙x(t) are shown for various528
time instants, as obtained by utilizing the herein developed technique. Perti-529
nent MCS based results (50,000 realizations) are included as well for comparison530
purposes. Overall, it is seen that for this specific numerical example the gain of531
the proposed technique in terms of computational efficiency, as compared both532
to the standard [11] and to the enhanced [19] implementations, is drastic. It is533
worth noting that the herein developed technique based on global bases can be534
potentially coupled with sparsity concepts and compressive sampling for further535
reducing the associated computational cost (see [20] for more details on sparse536
representations).537
Fig. 3. Non-stationary joint PDF of x(t) and ˙x(t) at time instants t= 1,2 and 6s for an
SDOF Duffing oscillator with bimodal response PDF under Gaussian white noise excitation,
as obtained via the WPI technique (a); comparisons with MCS data - 50,000 realizations (b).
19
Fig. 4. Non-stationary marginal PDF of x(t) and ˙x(t) for an SDOF Duffing oscillator with
bimodal response PDF under Gaussian white noise excitation, as obtained via the WPI
technique (a and c); comparisons with MCS data - 50,000 realizations (b and d).
20
-2 0 2
0
0.5
1
1.5
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
1
Fig. 5. Marginal PDFs of x(t) (a) and ˙x(t) (b) at time instants t= 1s and t= 12s for an
SDOF Duffing oscillator with bimodal response PDF under Gaussian white noise excitation,
as obtained via the WPI technique; comparisons with MCS data (50,000 realizations).
4.2. MDOF nonlinear oscillator subject to non-stationary time-modulated Gaus-538
sian white noise539
In this section the efficacy of the mesh-free approximate WPI technique540
presented in Section 3.2 is assessed, in conjunction with a 2-DOF nonlinear541
dynamical system whose equation of motion is given by Eq. (2) with542
M=m00
0m0,(42)
543
C=2c0c0
c02c0,(43)
544
K=2k0k0
k02k0,(44)
and545
g(x,˙x) = 1k0x3
1
0(45)
and parameter values (m0= 1; c0= 0.35; k0= 0.5; 1= 0.2; and S0= 0.1).546
The matrix D(t) of Eq. (3) containing the time-modulating functions d1(t) and547
d2(t) is diagonal with548
d1(t) = d2(t) = γ+η(eαt eβt ) (46)
21
and parameters values (α= 0.4; β= 1.6; γ= 103; and η= 5). The non-549
stationary excitation power spectrum is, thus, given as550
Sw=Sw(t) 0
0Sw(t),(47)
where Sw(t) = S0d2
1(t) = S0d2
2(t) is shown in Fig. 6.551
Fig. 6. Non-stationary Gaussian white noise excitation power spectrum, given by Eq. (47),
where Sw(t) = S0d2
1(t) = S0d2
2(t) and d1(t) = d2(t) = γ+η(eαt eβt ) with parameter
values (S0= 0.1; α= 0.4; β= 1.6; γ= 103; and η= 5).
Considering the system initially at rest, the joint response PDF p(x,˙x, t)552
is sampled at n= 60,000 Halton points (see Section 3.2 for more details).553
Note that this is a rather challenging example from an approximation theory554
perspective due to the fact that the response PDF is non-stationary. As a result,555
the bounds of the effective PDF domain may vary continuously with time in an556
arbitrary manner. In this regard, if the bounds are pre-specified and fixed, there557
are measurements of the PDF whose values are effectively zero; thus, causing558
numerical instabilities in the approximation. This challenge can be addressed559
by considering “adaptive” bounds, whose time-varying values can be estimated,560
for instance, via a preliminary MCS analysis with very few realizations (e.g.,561
of the order of 102). Next, the leave-one-out cross validation follows and the562
set of optimal parameters (s,t) is determined. It is noted that, as discussed563
in Section 3.2.2, a 4th-degree polynomial is also added in the approximation564
scheme, and thus, the joint response PDF is approximated via Eq. (41) with565
the augmented coefficient vector determined via Eq. (40).566
In Figs. 7 and 8the joint PDFs p(x1,˙x1, t) and p( ˙x1, x2, t) obtained by567
the approximate WPI technique based on positive definite functions are plot-568
ted, respectively. Comparisons with pertinent MCS data (50,000 realizations)569
demonstrate a relatively high accuracy degree. Further, as shown in Figs. 9570
and 10 for the non-stationary marginal PDFs of x1(t) and x2(t), respectively,571
and based on comparisons with MCS data, the herein developed technique is572
22
capable of capturing accurately the evolution in time of the PDF shape and573
features. Furthermore, in Fig. 11 the marginal PDFs of x2(t) and ˙x2(t) for two574
arbitrary time instants are shown and compared with respective MCS-based575
results. In passing, note that an alternative brute-force implementation (e.g.,576
[11,17]) and employing an expansion basis for each and every time instant inde-577
pendently [19] would require approximately 109and 70,000 PDF measurements,578
respectively (assuming that the temporal dimension is, indicatively, discretized579
into 1,000 points).580
Fig. 7. Non-stationary joint PDF of x1(t) and ˙x1(t) at time instants t= 1,2 and 6s for a
2-DOF nonlinear system subject to time-modulated Gaussian white noise, as obtained via
the WPI technique (a); comparisons with MCS data - 50,000 realizations (b).
23
Fig. 8. Non-stationary joint PDF of ˙x1(t) and x2(t) at time instants t= 1,4 and 6s for a
2-DOF nonlinear system subject to time-modulated Gaussian white noise, as obtained via
the WPI technique (a); comparisons with MCS data - 50,000 realizations (b).
24
Fig. 9. Non-stationary marginal PDF of x1(t) and ˙x1(t) for a 2-DOF nonlinear system
subject to time-modulated Gaussian white noise, as obtained via the WPI technique (a and
c); comparisons with MCS data - 50,000 realizations (b and d).
25
Fig. 10. Non-stationary marginal PDF of x2(t) and ˙x2(t) for a 2-DOF nonlinear system
subject to time-modulated Gaussian white noise, as obtained via the WPI technique (a and
c); comparisons with MCS data - 50,000 realizations (b and d).
26
-4 -2 0 2 4
0
0.2
0.4
0.6
0.8
-4 -2 0 2 4
0
0.1
0.2
0.3
0.4
0.5
Fig. 11. Marginal PDFs of x2(t) (a) and ˙x2(t) (b) at time instants t= 1s and t= 6s for a
2-DOF nonlinear system subject to time-modulated Gaussian white noise, as obtained via
the WPI technique (a); comparisons with MCS data - 50,000 realizations (b).
4.3. Beam bending problem with a non-Gaussian and non-homogeneous stochas-581
tic Young’s modulus582
The example considered in this section serves to demonstrate that the WPI583
formalism delineated in Section 2 (see [17] for more details) can account not584
only for stochastically excited dynamical systems governed by Eq. (2), but also585
for certain engineering mechanics problems with stochastic media properties. In586
this regard, it has been shown [16,17] that a class of one-dimensional mechan-587
ics problems with stochastic system parameters, such as the herein considered588
Euler-Bernoulli beam with stochastic Young’s modulus, can be cast equivalently589
in the form of Eq. (6). Thus, the left hand-side of Eq. (6) can be used for defin-590
ing an auxiliary Lagrangian function, and the WPI solution technique can be591
applied in a rather straightforward manner.592
In this regard, a statically determinate Euler-Bernoulli beam is considered593
next whose response is governed by the differential equation594
d2
dz2[E(z)I¨q(z)] = l(z) (48)
where E(z) is the Young’s modulus; Iis the constant cross-sectional moment595
of inertia; q(z) is the deflection of the beam; and l(z) denotes a deterministic596
distributed force. In this static problem the dot above a variable denotes differ-597
entiation with respect to the space variable z. Further, as explained in [16] and598
27
[17], Eq. (48) can be integrated twice and cast in the form599
M(z)
I¨q(z)=E(z) (49)
where the Young’s modulus is modeled as a non-Gaussian, non-white and non-600
homogeneous stochastic field as601
˙
E(z)
E(z)=w(z) (50)
with E(0) = EM, and w(z) is the white noise process as defined in Eq. (4).602
It can be readily seen that Eq. (50) represents a standard geometric Brownian603
motion SDE, whose space-dependent response PDF is log-normal (e.g., [82]).604
Combining Eq. (49) and (50) yields an equation in the form of Eq. (6); that is,605
˙
M(z)
M(z)q(3)(z)
¨q(z)=w(z) (51)
Next, the case of a cantilever beam subject to a single point moment at its606
free end is considered (Fig. 12). Thus, taking into account that M(z) is constant607
along the length of the beam, i.e., M(z) = M0, Eq. (51) becomes608
q(3)(z)
¨q(z)=w(z) (52)
while based on Eq. (10) the expression609
Lq, ˙q, ¨q, q(3) =(q(3)(z))2
4πS0(¨q(z))2(53)
can be construed as the corresponding Lagrangian function. In this regard, the610
E-L equation becomes611
L
∂qc
∂z
L
˙qc
+2
∂z2
L
¨qc
3
∂z3
L
∂q(3)
c
= 0 (54)
together with the initial conditions for zi= 0, qc(zi) = qi= 0, ˙qc(zi) = ˙qi= 0612
and ¨qc(zi) = M0
EMI.613
L
zM
Fig. 12. Cantilever beam subject to a single-point moment.
28
Subsequently, the joint response PDF p(q, ˙q, ¨q, z) is sampled at n= 20,000614
Halton points with bounds that vary with z. Specifically, the bounds of the615
response PDF space-varying effective domain are determined via a preliminary616
MCS with only a few realizations (see Example 4.2 for details). Next, follow-617
ing the evaluation of the augmented coefficient vector via Eq. (40), Eq. (41)618
is utilized in conjunction with an 8th-degree polynomial, and p(q, ˙q, ¨q, z) is de-619
termined. In Figs. 13 and 14 the WPI-based non-stationary (space-dependent)620
marginal PDFs of q(z) and ˙q(z) are shown, respectively, while MCS-based data621
(50,000 realizations) are also provided for comparison. Moreover, Fig. 15 shows622
the marginal PDFs of q(z) and ˙q(z) at z= 0.6, z= 0.8 and z= 1, as obtained623
via the herein developed technique, and includes comparisons with pertinent624
MCS data. It is worth mentioning that the considered beam bending problem625
is significantly challenging from a global approximation point of view, since the626
response mean varies considerably in the spatial domain; thus, rendering neces-627
sary the utilization of an adaptive with zeffective PDF domain. Nevertheless,628
it has been shown that the WPI technique in conjunction with positive definite629
functions for approximating the joint response PDF yields accurate results at a630
relatively low computational cost. In this regard, note for comparison purposes631
that alternative implementations, such as the brute-force scheme delineated in632
Section 2.2, would require a several orders of magnitude higher number of PDF633
measurements. Further, a direct comparison in terms of cost with the enhanced634
implementation in [19] is not possible as the 4th-order polynomial employed in635
[19] would be, most likely, an inappropriate choice for approximating the joint636
response p(q, ˙q, ¨q, z).637
29
Fig. 13. Non-stationary (space-dependent) marginal PDF of q(z) for a statically
determinate beam with a non-Gaussian and non-homogeneous stochastic Young’s modulus,
as obtained via the WPI technique (a); comparisons with MCS data - 50,000 realizations (b).
30
Fig. 14. Non-stationary (space-dependent) marginal PDF of ˙q(z) for a statically
determinate beam with a non-Gaussian and non-homogeneous stochastic Young’s modulus,
as obtained via the WPI technique (a); comparisons with MCS data - 50,000 realizations (b).
31
-5 -4 -3 -2
10-3
0
2000
4000
6000
8000
MCS
WPI
-10 -8 -6
10-3
0
500
1000
1500
2000
MCS
WPI
Fig. 15. Marginal PDFs of q(z) (a) and ˙q(z) (b) at z= 0.6, z= 0.8 and z= 1 for a
statically determinate beam with a non-Gaussian and non-homogeneous stochastic Young’s
modulus, as obtained via the WPI technique; comparisons with MCS data (50,000
realizations).
5. Concluding remarks638
In this paper, the WPI technique has been generalized and enhanced for639
determining directly, and in a computationally efficient manner, the complete640
time-dependent non-stationary response PDF of stochastically excited nonlinear641
multi-degree-of-freedom dynamical systems. This has been done, first, by con-642
structing multi-dimensional (time-dependent) global bases for approximating643
the non-stationary joint response PDF, and second, by exploiting the localiza-644
tion capabilities of the WPI technique for determining PDF points in the joint645
space-time domain. These points have been used for evaluating the expansion646
coefficients at a relatively low computational cost. Specifically, two distinct ex-647
pansions have been constructed: the first is based on Kronecker products of648
bases (e.g., wavelets), while the second employs positive definite functions. Al-649
though the performance of the expansions in approximating the response PDF is,650
in general, problem-dependent, it can be argued that positive definite functions651
appear more versatile and suitable for handling higher-dimensional problems,652
whereas Kronecker products perform better for lower-dimensional problems, es-653
pecially when some information regarding the PDF is available a priori.654
Several numerical examples pertaining to both single- and multi-degree-of-655
freedom nonlinear dynamical systems subject to non-stationary excitations have656
been considered for assessing the reliability of the technique. Further, to il-657
lustrate that the technique can account also for certain engineering mechanics658
32
problems with stochastic media properties, a bending beam with a non-Gaussian659
and non-homogeneous Youngs modulus has been included in the numerical ex-660
amples as well. The latter example has also been found to be significantly661
challenging from a global approximation perspective, since the response PDF662
effective domain varies considerably along the spatial dimension. Nevertheless,663
this challenge can be addressed by utilizing adaptive PDF domain bounds. Com-664
parisons with pertinent MCS data have demonstrated a relatively high accuracy665
degree for all the considered examples.666
Finally, although the related computational cost is, in general, problem-667
dependent, it has been shown that in most examples considered herein the668
computational efficiency exhibited by the developed technique is significant.669
Specifically, compared both to a brute-force numerical implementation (see Sec-670
tion 2.2) and to employing an expansion basis for each and every time instant671
independently [19], the gain in terms of computational cost of the herein pro-672
posed WPI technique enhancement is notable, measured even at several orders673
of magnitude for some cases. Future work relates to coupling the proposed WPI674
technique with sparse representations and compressive sampling [20] for further675
reducing the computational cost.676
6. Acknowledgment677
I. A. Kougioumtzoglou gratefully acknowledges the support through his678
CAREER award by the CMMI Division of the National Science Foundation,679
USA (Award number: 1748537).680
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39
... Moreover, it was shown in [14] that the computational cost can be reduced drastically by employing sparse representations for the system response PDF in conjunction with compressive sampling schemes and group sparsity concepts. Also, the efficacy of employing global multi-dimensional bases for determining the non-stationary joint response PDF in a direct manner was demonstrated in [15]. ...
... It is noted that the most probable path approximation has exhibited a relatively high degree of accuracy in various diverse applications [10,11,13,14,15]. In fact, as proved in [18], for the case of linear systems the most probable path approximation yields the exact joint response PDF. ...
... Clearly, the computational cost increases exponentially with increasing number of dimensions; see also [13] for a relevant discussion. Thus, alternative more efficient formulations have been developed recently by relying on appropriate PDF expansions [15,19]. In this regard, the problem of evaluating the joint response PDF is recast into determining the PDF expansion coefficient vector. ...
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The computational efficiency of the Wiener path integral (WPI) technique for determining the stoch-astic response of diverse nonlinear dynamical systems is enhanced herein by relying on advanced compressive sampling concepts and tools. Specifically, exploiting the sparsity of appropriately selected expansions for the joint response probability density function (PDF), and leveraging the localization capabilities of the WPI technique for direct evaluation of specific PDF points, yields an underdetermined linear system of equations to be solved for the PDF expansion coefficients. This is done by resorting to L p-norm (0 < p < 1) minimization formulations and algorithms, which exhibit an enhanced sparsity-promoting behavior compared to standard L 1-norm minimization approaches. This translates into a significant reduction of the associated computational cost. In fact, for approximately the same accuracy degree , it is shown that the herein developed technique based on L p-norm (0 < p < 1) minimization requires, in some cases, even up to 40% fewer boundary value problems to be solved as part of the solution scheme than a standard L 1-norm minimization approach. The reliability of the technique is demonstrated by comparing WPI-based response PDF estimates with pertinent Monte Carlo simulation (MCS) data (10,000 re-alizations). In this regard, realizations compatible with the excitation stochastic process are generated, and response time-histories are obtained by integrating numerically the nonlinear system equations of motion. Next, MCS-based PDF estimates are computed based on statistical analysis of the response time-histories. Several numerical examples are considered pertaining to various stochastically excited oscillators exhibiting diverse nonlinear behaviors. These include a Duffing oscillator, an oscillator with asymmetric nonlinearities, and a nonlinear vibro-impact oscillator.
... 20 Further, the computational efficiency of the WPI technique was enhanced recently. This was done by resorting to compressive sampling (CS) concepts and tools in conjunction with appropriate expansions for the joint response probability density function (PDF) [16,17]; see also [18] for a broad perspective. ...
... Solving the problem of Eq. (10) yields the sparse coefficient vector to be substituted into the expansion of Eq. (9) for approxi-mating the system joint response PDF. The interested reader is also directed to [16][17][18] for a more detailed presentation. ...
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The Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear systems is enhanced herein based on a Bayesian compressive sampling (CS) treatment. Specifically, first, sparse expansions for the system response joint probability density function (PDF) are employed. Next, exploiting the localization capabilities of the WPI technique for direct evaluation of specific PDF points leads to an underdetermined linear system of equations for the expansion coefficients. Further, relying on a Bayesian CS solution formulation yields a posterior distribution for the expansion coefficient vector. In this regard, a significant advantage of the herein developed methodology relates to the fact that the uncertainty of the response PDF estimates obtained by the WPI technique is quantified. Furthermore, an adaptive scheme is proposed based on the quantified uncertainty of the estimates for optimal selection of PDF sample points. This yields considerably fewer boundary value problems to be solved as part of the WPI technique, and thus, the associated computational cost is significantly reduced. Two indicative numerical examples pertaining to a Duffing nonlinear oscillator and to an oscillator with asymmetric nonlinearities are considered for demonstrating the capabilities of the developed technique. Comparisons with pertinent Monte Carlo simulation data are included as well.
... Many studies show that the moment equivalence method can only guarantee integral equivalence, but improper results would be given in the prediction of power spectral density (PSD) [9] . Furthermore, the random average approach [10][11] , the Fokker-Planck-Kolmogorov (FPK) equation technique [12][13] , and the Wiener path integral approach [14][15] have been developed. In the field of practical application, the spectral analysis of nonlinear systems in the frequencydomain is very attractive. ...
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A consequence of nonlinearities is a multi-harmonic response via a mono-harmonic excitation. A similar phenomenon also exists in random vibration. The power spectral density (PSD) analysis of random vibration for nonlinear systems is studied in this paper. The analytical formulation of output PSD subject to the zero-mean Gaussian random load is deduced by using the Volterra series expansion and the conception of generalized frequency response function (GFRF). For a class of nonlinear systems, the growing exponential method is used to determine the first 3rd-order GFRFs. The proposed approach is used to achieve the nonlinear system’s output PSD under a narrow-band stationary random input. The relationship between the peak of PSD and the parameters of the nonlinear system is discussed. By using the proposed method, the nonlinear characteristics of multi-band output via single-band input can be well predicted. The results reveal that changing nonlinear system parameters gives a one-of-a-kind change of the system’s output PSD. This paper provides a method for the research of random vibration prediction and control in real-world nonlinear systems.
... The whole convergence process takes 8.1s. Compared with RBF [34] and NN [35][36], it can be seen that both the convergence speed and the calculation accuracy are due to them. ...
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