Author's Accepted Manuscript

Finite element solution of Fokker-Planckequation of nonlinear oscillators subjected tocolored non-Gaussian noise

Pankaj Kumar, S. Narayanan, Sayan Gupta

PII: S0266-8920(14)00049-6DOI: http://dx.doi.org/10.1016/j.probengmech.2014.07.002Reference: PREM2801

To appear in: Probabilistic Engineering Mechanics

Cite this article as: Pankaj Kumar, S. Narayanan, Sayan Gupta, Finite elementsolution of Fokker-Planck equation of nonlinear oscillators subjected tocolored non-Gaussian noise, Probabilistic Engineering Mechanics, http://dx.doi.org/10.1016/j.probengmech.2014.07.002

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mech

Finite Element solution of Fokker-Planck

equation of nonlinear oscillators subjected to

colored non-Gaussian noise

Pankaj Kumar a S Narayanan a Sayan Gupta b,∗

aDepartment of Mechanical Engineering, Indian Institute of Technology Madras,

Chennai, India

bDepartment of Applied Mechanics, Indian Institute of Technology Madras,

Chennai 600036, India

Abstract

Nonlinear oscillators subjected to coloured Gaussian/non-Gaussian excitations are

modelled through a set of three coupled first-order stochastic differential equations

by representing the excitation as a first-order filtered white noise. A 3-D finite ele-

ment (FE) formulation is developed to solve the corresponding 3-D Fokker Planck

(FP) equations. The joint probability density functions of the state variables, ob-

tained as a solution of the FP equation, are typically non-Gaussian and are used

for computing the crossing statistics of the response - an essential metric for time

variant reliability analysis. The method is illustrated through a noisy Lorenz at-

tractor and a Duffing oscillator subjected to additive coloured noise. The increase

in state-space dimension when the Duffing oscillator is additionally excited with a

parametric Gaussian noise is effectively handled by using stochastic averaging to

reduce the state-space dimension. Investigations are carried out to examine the ac-

curacy of the FE method vis-a-vis Monte Carlo simulations. The proposed method

Preprint submitted to Probabilistic Engineering Mechanics

is observed to be computationally significantly cheaper for these three problems.

Key words: Fokker-Planck equation, Colored noise, Stochastic averaging, Crossing

statistics, Lorenz attractor, Duffing oscillator, Finite Element, Non Gaussian

process, Kullback-Leibler entropy.

1 Introduction

A key step in the time variant reliability analysis of dynamical systems sub-

jected to random excitations involve estimating the expected up-crossings of

the response using Rice’s formula [1], given by

E[N+(α, T )] =∫ T

0

∫ ∞

0x pXX(α, x, t) dx dt. (1)

Here, pXX(α, x, t) is the joint probability density function (pdf) of the re-

sponse process X(t) and its instantaneous time derivative X(t), N+(α, T ) is

the number of up-crossings of X(t) across level α in duration T and E[·] is theexpectation operator. When X(t) is stationary, Eq. (1) can be simplified to

E[N+(α, T )] = Tμ+(α) = T∫ ∞

0x pXX(α, x) dx, (2)

where, μ+(α) is the mean up-crossing intensity across level α. The crux in us-

ing Rice’s formula lies in estimating the joint pdf pXX(x, x). It is well known

that for linear systems subjected to Gaussian white noise excitations or fil-

tered Gaussian excitations, the response X(t) is Gaussian for which closed

form expressions for the joint pdf pXX(x, x) is available. However, for non-

linear systems subjected to Gaussian excitations, closed form expressions for

∗ Corresponding author: gupta.sayan@gmail.com, Phone: +91 44 2257 4055, FAX:

+91 44 2257 4052

2

the joint pdf are available only for a limited class of oscillators [2]. Early

studies in the field therefore have focused attention on the development of

analytical/numerical methods for obtaining approximations for the joint pdf

of the response for a wider class of problems. Recent studies have focussed

on approximating the joint pdf of the response of dynamical systems which

have different nonlinear characteristics as well as when the excitations are

non-Gaussian.

An alternative approach to estimating the joint pdf pXX(x, x) has been to solve

the Fokker-Planck (FP) equation corresponding to the dynamical system. The

FP equation is a partial differential equation for the transition probability den-

sity function (tpdf) of the state variables associated with a dynamical system

[3,4] when the equations of motion are expressed in the form of first-order

stochastic differential equations (SDEs). The equations of motion can be ex-

pressed in the form of SDEs if the excitations are white noise processes and

the response is Markovian. As white noise processes have infinite energy, in

practice they cannot exist. However, in dynamical systems where the system

relaxation time is much greater than the correlation times of excitations, the

response can still be approximated to have Markovian characteristics. Alter-

natively, the non-white excitations can be expressed as filtered white noise

processes. This mathematical artefact however increases the state-space di-

mension of the problem.

Finding solutions for the Fokker-Planck equations is however not easy. Exact

analytical solutions for FP equations are available only for a set of limited

class of oscillators [5,6]. Approximate analytical solutions for the FP equation

have been developed for certain nonlinear dynamical systems [7] using meth-

ods, such as, equivalent linearization [8], Gaussian-closure techniques [9,10],

3

stochastic averaging [11,12], equivalent nonlinearization [13], maximum en-

tropy based methods [14] and non-Gaussian closure based methods [15–18].

In recent years, with the emergence of cheap computing facilities, the use of

numerical methods for the solution of the FP equation has gained wider accep-

tance. This has led to the development of numerical methods based on finite

elements (FE) [19–21], finite difference (FD) [21], cell mapping techniques [22]

and path integrals (PI) [23–25] for solving the FP equation.

The numerical techniques discussed in the literature have been primarily used

to solve the FP equation for oscillators with state vectors limited to dimension

of 2. Efforts have been expended to obtain solutions of FP equations for higher

dimensional problems; see [17,18,26]. However, these methods are applicable

only for special class of problems and extending these methods to higher or-

der systems poses formidable numerical challenges as the number of unknown

variables are of the order of several millions [27]. This places severe constraints

on the scalability and usefulness of solving the FP equations for general large

order systems. Nevertheless, for systems which can be approximated as single

degree of freedom systems, solutions of the corresponding FP equations pro-

vide an efficient methodology for qualitative and quantitative assessment of

their dynamical behaviour.

The focus of this paper is on the development of an efficient FE solution of

a three dimensional FP equation using the FE method and analyzing the

results. First, the FE method is used to solve the 3−dimensional FP equa-

tion for a 3−D Lorenz attractor subjected to additive white noise excitations.

The excitations are modelled as Gaussian white noise; consequently, the sys-

tem response state vector is Markovian enabling the direct formulation of the

3-D FP equations. Next, a Duffing oscillator subjected to additive coloured

4

noise is considered. Expressing the excitation as a first-order filtered white

noise process and augmenting the state-space with the additional state of the

excitation, the equations of motion are expressed in terms of three coupled

first-order SDE. A generic choice for the first-order filter is considered such

that changing a particular parameter in the filter enables modelling the ex-

citation as Gaussian or non-Gaussian. The corresponding 3D FP equation is

solved using the FE method and the mean crossing statistics of the stationary

response are computed. Questions related to the accuracy and convergence

of the solutions are investigated using the Kullback-Leibler entropy measure.

Finally, issues related to dimension reduction of the state-space are examined

when the dimension of the state-space exceeds three. Thus, when the Duffing

oscillator is additionally excited with a parametric Gaussian noise, the state-

space dimension increases to four. Using stochastic averaging, the state-space

is first reduced and the corresponding FP equation in the reduced space is

subsequently solved using the proposed FE method.

The paper is divided into the following sections. First, a brief review of the FP

equation for general nonlinear oscillators subjected to band limited random

excitations is provided. This is followed by a brief discussion on the choice of

a general 1-D filter used in this study to model the band limited excitations

as filtered white noise processes. Next, details of the FE method used in this

study to discretize the 3D FP equations are provided along with methods of

solution and convergence checks. The developments are illustrated through a

set of numerical examples which are presented next. The following section is

devoted to dimension reduction using stochastic averaging and the use of the

FE for solving the associated FP equation in the reduced space. The salient

features arising out of this study are summarized in the concluding section.

5

2 The Fokker-Planck equation

We first review the developments of the FP equation for a general nonlinear

oscillator subjected to random excitations. The governing equations of motion

can be written in the general form as

x(t) + g1(x(t), x(t)) = f(t), (3)

where, g1(·) is a nonlinear function of x(t) and x(t) and f(t) is the random

excitation having a band limited power spectral density function. The band

limited random excitation can be further expressed as a filtered white noise,

f(t) + g2(f(t),W (t)) = 0, (4)

where, g2(·) is in general a nonlinear function and W (t) represents Gaussian

white noise, such that, E[W (t)] = 0, E[W (t)W (t + τ)] = 2Dδ(τ). Here, δ(τ)

is the Dirac-delta function, and D is the spectral density of the white noise.

If the function g2(·) is linear, f(t) is Gaussian, else it is non-Gaussian.

Equations (3-4) when written concisely in the first-order form, constitute a

set of three coupled first-order SDEs. In the Ito form, in general, a set of N

dimensional SDEs can be expressed as

dX(t) = m[X, t]dt+ h[X, t]dB, (5)

where, X = [X1(t), . . . , XN(t)] is the N × 1 dimensional system state vector,

m[X, t] is the Wong-Zakai corrected [28] drift vector of dimensions N × 1,

h[X, t] is the diffusion matrix of dimensions N ×M and dB are Wiener incre-

ments of dimensions M×1. The response X(t) forms a Markov vector process

in �N , whose tpdf p ≡ p(X, t|X0, t0), is governed by the FP equation

6

∂p

∂t=

[−

N∑i=1

∂[mi(X , t)]

∂Xi+

1

2

N∑i=1

N∑j=1

∂2[hij(X , t)]

∂Xi∂Xj

]p. (6)

In the case of the general nonlinear oscillator of Eq. (3), X = [X(t), X(t), f(t)]

is the 3-dimensional state-space vector leading to a 3D FP equation. Equation

(6) can be written in the concise form

∂p(X, t|X0, t0)

∂t= LFP [p(X, t|X0, t0)], (7)

where, LFP [ · ] is the FP operator.The solution of Eq. (6) must satisfy the

initial conditions, boundary conditions and the normalization condition given

as

lim

t → 0p(X, t|X0, t0) = δ(X−X0),

p(X, t|X0, t0)|Xi→±∞ = 0, (i = 1, . . . , n), (8)∫ ∞

−∞p(X, t|X0, t0) dX = 1.

The stationary solution of the FP equation is obtained when ∂p/∂t = 0.

3 Gaussian and non-Gaussian colored noise

In developing the FP equations, the elements of the drift and the diffusion

matrices depend on the form of the functions g1(·), g2(·) and the parameters

associated with these functions. More specifically, the form of the function

g2(·) depends on the pdf of the excitation f(t). As is well known, pure white

noise processes have zero correlation length while physical processes are char-

acterized by correlation functions with finite correlation lengths. Exponential

functions are the most commonly used forms of the correlation functions.

One of the simplest example of a finite time correlated noise is the Ornstein-

7

Uhlenbeck process, ξ(t), with an exponential correlation function of the form

E[ξ(t)ξ(s)] =2D

τexp[−|t− s|

τ], (9)

where, τ denotes the correlation time of the colored noise ξ(t). Such exponen-

tially correlated Gaussian processes, ξ(t), can be obtained by passing Gaussian

white noise W1(t) through a first-order low pass filter of the following form

ξ(t) = − 1

τ1ξ +

1

τ1W1(t). (10)

Due to its Gaussian nature, only the second order cumulant survives.

Equation (10) is a linear filter and hence the filtered white noise is also Gaus-

sian. Non-Gaussian processes can be expressed as filtered white noise pro-

cesses, where the filter is nonlinear. An example of a first-order nonlinear

filter is given by

η(t) = − 1

τ2

d

dηVq(η) +

1

τ2W2(t). (11)

Here, η(t) is a non-Gaussian process having a finite correlation length, W2(t)

is a Gaussian white noise and Vq(η) is of the form [29]

Vq(η) =1

μ(q − 1)ln[1 + β(q − 1)

η2

2], (12)

where, μ = τ2/D2 and q is related to the Tsallis entropy [30]. It can be shown

that when q = 1, Eq. (11) becomes identical to Eq. (10) leading to Gaussian

colored noise. Obviously, q �= 1 leads to non Gaussian noise. In the numerical

examples considered later in this paper, the general form of the filter in Eq.

(12) has been used for various values of the parameter q.

8

4 The Finite Element Method

The FP equation shown in Eq. (6) is a partial differential equation which in

general is not amenable for analytical solutions. A numerical solution of the

FP equation can be obtained by seeking a weak form for Eq. (6). Using the

FE method, the weak form of Eq. (6) is obtained by transforming the set of

coupled partial differential equations to a set of coupled ordinary differential

equations through discretization along the spatial domain.

Adopting the standard Bubnov-Galerkin finite element formulation, the first

step involves identifying a subset of the domain Ω spanned by the state-space

vectors within which a solution of the FP equation would be sought. Theoreti-

cally, the domain Ω associated with the dynamics of a noisy oscillator typically

extends up to infinity. However, in the numerical procedure, one needs to con-

sider Ω ≈ Ωe to be finite. In general, the spatial extent of Ωe is decided such

that the pdf of the state vector asymptotically approaches ε at the boundaries

of Ωe, where ε is a small number close to zero, and the pdf in the subspace

outside Ωe is assumed to be negligible and is approximated as zero. This is

a valid numerical approximation that has been adopted in the literature to

reduce computational costs without significantly affecting the accuracy of the

solutions [19]. For a nonlinear oscillator subjected to Gaussian excitations, the

extent of Ωe along a state dimension is usually taken to span k×σl, where, σl

is the standard deviation of the state response of the linearized oscillator and k

is an integer usually greater than 4. The underlying principle behind adopting

this procedure is that the response of the linearized oscillator is Gaussian and

P [X ≥ kσ] ≈ 0, when k ≥ 4.

9

4.1 Discretization

Next, the finite domain Ωe is discretized into a collection of NE elements,

each element hosting n interpolating nodes and spanning the domain Ωe. Let

{ψ1(X), . . . , ψn(X)} represent a set of global cardinal interpolation functions

defined on the solution domain. Using the Galerkin method, time varying

solution p(X, t) of the FP equation can be written as a linear combination in

terms of the prior unknown values pr of the probability densities at the global

interpolation nodes, and the interpolating functions {ψr(X)}nr=1, such that,

p(X, t) ≈NG∑r=1

pr(t)ψr(X), (13)

where, NG is the number of global nodes. Though Eq. (6) consists of second

order terms implying the need for the interpolating functions to be at least C1

continuous, it has been shown [19] that C0 continuity of the shape functions

is sufficient.

Substitution of Eq. (13) into Eq. (6) leads to the following expression for the

residual error;

�(X, t) =NG∑r=1

[∂pr(t)ψr(X)

∂t− LFP [pr(t)ψr(X)]

]. (14)

Requiring the error due to the FE approximation to be orthogonal to the basis

functions, the residual error �(X, t) is projected onto a set of independent

weighting functions for the pdf. Taking the inner product of residual error on

L2(Ω), the space of Lebesgue integral functions, leads to

⟨�(X, t), ψs(X)

⟩= 0, (15)

where, < · > is the L2 inner product. The weight function {ψs(X)} represents

the space over which the projection of the residual error is minimal. Hence,

10

Eq. (6) can now be represented as a finite set of ordinary differential equations

in the following form

NG∑r=s

pt(t) < ψr(X), ψs(X) > −NG∑r=s

pr(t) < LFP{ψr(X)}, ψs(X) >= 0.(16)

Taking the inner product of the weighting function {ψi(X)}, with the FP

equation and applying the divergence theorem, leads to the following weak

form of the FP equation

Mp+Kp = 0, (17)

subject to the initial condition p(0) = p, where, p is a vector of the joint pdf

at the nodal points,

Mrs =< ψr, ψs >, (18)

Krs=∫Ω

[ N∑i=1

ψr(X)∂[mi(X)ψs(X)]

∂Xi

]dX

+∫Ω

[ N∑i=1

N∑j=1

∂[ψr(X)]

∂Xi

∂[hijψs(X)]

∂Xj

]dX (19)

It is clear that though the matrix M is symmetric, K need not be symmetric.

4.2 Elements,shape functions and non-uniform grids

Since the FP equation for oscillators under colored noise is limited to dimen-

sion three, i.e., Ωe ∈ �3, the discretization for Ωe is carried out with three

dimensional finite elements. In this study, the discretisation has been carried

out with a three dimensional cuboid isoparametric element such that each edge

of the cuboid has three nodes with nodes at each corner and at the midpoint.

Thus, the number of nodal points in a single element is m = 20; see Fig.1.

Each node has one degree of freedom only. The shape functions are selected

11

Fig. 1. Quadratic element with 20 nodes.

such that the pdf, pe(X), within the element can be represented as

pe(X) =m∑i=1

piψi(X), (20)

where, pi denotes the pdf at the i-th node. ψi(X) are the corresponding shape

functions and have the property ψj(Xk) = δjk, where δjk is the Kronecker-

delta. In terms of the normalized co-ordinates ξ, η, and τ , the shape functions

for the corner nodes are of the form

ψi =1

8(1 + ξξi)(1 + ηηi)(ξξi + ηηi + ττi − 2), (21)

while for a typical mid-side node with ξi = 0, ηi = ±1, τi = ±1, the shape

functions are of the form

ψi =1

8(1− ξ2)(1 + ηηi)(1 + ττi). (22)

Following the formulation in the previous section, the elemental matrices Ke

and Me are constructed and are of typically dimensions 20 × 20. The global

matrices K andM are assembled following standard FE techniques. The num-

ber of unknowns associated with a single element is equal to the number of

nodes in an element.

If each dimension of Ωe is discretized into N elements, the total number of

12

elements would be N3, and the total number of nodes and hence the size

of the systems of equations shown in Eq. (17) would be N3 × 20. Typically,

even if only 50 elements are used for discretizing each dimension of Ωe, the

number of coupled equations that need to be solved is 50× 50× 50× 20 and

is about 2.5 million. It is therefore obvious that the solution of the discretized

FP equations is not a trivial task and requires extensive computing facilities.

4.3 Transient solution

Equation (17) is a first-order ordinary differential equation in time, the so-

lution of which gives the evolution of the joint pdf at the nodal points in

Ωe, with time. A solution of Eq. (17) is obtained using the Crank-Nicholson

method, which is an implicit time integration scheme with second order accu-

racy and unconditional stability and allows large time steps to be used. The

time discretized equation takes the form

[M−Δt(1− θ)K]p(t +Δt) = [M+ΔtθK]p(t), (23)

where, the parameter 0 ≤ θ ≤ 1 and Δt is the time step. For θ = 1/2, Eq.

(23) can be written in the simplified form

[M− Δt

2K]p(t+Δt) = [M+

Δ

2K]p(t). (24)

It is worth noting that Eq. (24) constitutes a set of m coupled ODEs, where m

is of the order of 2.5 million. The solution of Eq. (17) gives the time evolution

of the transitional pdf of the state-space vector for all time and clearly requires

supercomputing resources and extensive computer memory.

13

4.4 Steady state solution

The long time evolution of the response of weakly nonlinear dynamical systems

to stationary excitations can be approximated as stationary processes. This

implies that the time evolution of the pdf of the response can be assumed to

be negligible, i.e., p ≈ 0. The discretized form of the FP equation in Eq. (17)

can now be written as

Kp = 0. (25)

Equation (25) represents a set of m coupled homogenous linear equations,

where, m is typically of the order of 2.5 million. The solution of Eq. (25)

admits a trivial solution where the vector approximating the pdf at all the

nodal points, p = 0. However, this solution does not satisfy the normalization

condition∫Ωe

p(x) dx = 1.

For the existence of a non-trivial solution, the matrix K must be singular.

Additionally, the normalization condition must be satisfied. A solution for Eq.

(25) can be obtained following the procedure outlined in [19]. Here, the value

p(X) at a particular node is arbitrarily assigned a constant c. Subsequently,

Eq. (25) is repartitioned in terms of known and unknown nodal points and

can be expressed as

⎡⎢⎢⎢⎢⎢⎢⎣Kuu Kku

Kuk Kkk

⎤⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

pu

pk

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

=

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

0

0

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭, (26)

where, the subscripts u and k refer to the unknown and known nodes respec-

tively. Considering the first row from the above equation leads to the reduced

form

Kuupu = −cKuk. (27)

14

Here, the dimension ofKuu is (m−1)×(m−1), and the vectors pu andKuk are

of dimensions (m− 1)× 1 respectively. Solving Eq. (27) leads to the solution

for pu in terms of the constant c, which can be evaluated by imposing the

normalization condition∫Ωe

p(x)dx = 1. Numerically, this is implemented by

the simple technique of first assuming c = 1, solving the set of corresponding

linear algebraic equations using standard decomposition methods and finally,

applying the normalization condition.

5 Numerical solution of 3D FP equations

The accuracy of the FE method in solving 3−D FP equation is first demon-

strated through two numerical examples. The first example considers a Lorenz

attractor subjected to additive Gaussian white noise excitation. The second

example considers a Duffing oscillator subjected to additive colored noise.

Both these examples require the solution of 3−D FP equations. However, at

first, discussions on methods for enhancing the computational efficiency in the

numerical solutions of the FE discretised FP equations and questions related

to assessing their convergence and accuracy are presented.

5.1 Enhancing computational efficiency

The solution of the coupled ODEs represented in Eq. (24) requires extensive

computing resources and memory storage requirements as the matrices M

and K are sparse and have very large bandwidth. The numerical efficiency

can be made efficient by proper vectorisation of the matrices and working

only with the non-zero elements. Thus, in the numerical calculations, it has

15

been shown that a proper vectorisation in the calculation of the elemental

stiffness matrices reduce a loop from 25,600 to just 400, thereby significantly

reducing the computational costs. Further reductions in the computational

costs can be effected by exploiting the conditions of symmetry in M and K as

approximately 36% of the computational costs is expended in the generation of

these matrices. This is because while the elemental mass matrices are identical

for all elements, the same is not true for the elemental stiffness matrices.

Hence, then use of parallelisation techniques in forming the K matrix would

significantly reduce the costs. This however has not been undertaken in this

study. However, in certain cases when K is symmetric about the origin, it

is possible to reduce the number of unknowns by half thereby significantly

enhancing the computational efficiency.

5.2 Accuracy and Convergence

A crucial aspect in the proposed numerical solution is to determine the num-

ber of nodal points to be introduced along each component of vector X. Too

coarse discretization would lead to estimates which are not accurate while too

fine discretization would increase the computational cost. To investigate the

accuracy of the investigation, a convergence study is carried out as a func-

tion of the number of nodes introduced into each component of X. For the

sake of simplicity, each of the three components have been discretized into

identical number of nodal points. For 1-D systems, the accuracy of the dis-

cretisation can be visually assessed by comparing the pdf of the state variable

obtained from FE with those obtained from Monte Carlo simulations (MCS),

which is treated as the benchmark. For 2-D systems, one can compare the

16

corresponding contour plots of the joint pdf to assess the accuracy of the FE

based approach. However, the state-space vector of the systems considered in

this paper span 3-dimensions and therefore visual comparisons are difficult.

Instead, the accuracy of the FE based method is assessed by comparing the

Kullback-Leibler entropy measure, D(N), given by [31]

D(N) = D[pNX(x), p

EX(x)

]=

∫ ∞

−∞pNX(x) log

[pNX(x)

pEX(x)

]dX. (28)

Here, pNX(x) is the j-pdf of the state variables obtained from the FE solution

with N nodes along each component of X, and pEX(x) is the corresponding pdf

obtained from MCS, which is treated as the benchmark. It will be shown later

in the numerical examples that D(N) reduces as N increases. However, the

question of what constitutes an acceptable numerical value of D(N) is more

difficult to address and is discussed later in the paper.

In evaluating D(N) in Eq.(28), one needs estimates of the pdf of the state vari-

ables obtained from MCS. This involves the numerical simulation of Wiener

increments and numerically integrating the SDEs, of the general form as

in Eq.(5), using the Euler-Maruyama (EM) integration scheme [32]. This is

termed as direct MCS or D-MCS in this paper. However, the major drawback

in using D-MCS is the requirement of prohibitively large number of samples re-

quired in low probability regions. The computational efficiency associated with

MCS can be improved by the use of various variance reduction strategies. Most

variance reduction MCS either require knowledge on the evolution in time of

the specific nonlinear system or depends on some other prior knowledge. In

this study, we have used one of several controlled MCS schemes (termed as C-

MCS later in this paper) discussed in the literature; see for example, [33–35].

Here, we have used the method suggested in [33].

17

5.3 Example 1: Lorenz attractor subjected to Gaussian white noise

The governing equations for a Lorenz attractor subjected to additive Gaussian

white noise, are given by

X = a(Y −X) + σ1W1(t),

Y =X(r − Z)− ψY + σ2W2(t),

Z =XY − bZ + σ3W3(t), (29)

where, a, r, b and ψ are parameters, {Wi(t)}3i=1, are stationary zero mean

Gaussian white noise processes and {σi}3i=1 represent their intensities. In these

equations, the nonlinearities arise due to the multiplicative terms XZ and

XY present in the second and third equations respectively. The deterministic

system has been extensively studied in the literature and does not require

any further discussion here. The numerical values considered are a = 10, r =

ψ = 1, b = 8/3 and σ1 = σ2 = σ3 = 2. For the deterministic case, i.e., when

{σi}3i=1 = 0, the response of the system for these parameters is non-chaotic

and the corresponding Lyapunov exponents are λ1 = −11.00, λ2 = −2.66, and

λ3 = −0.01. The corresponding FP equation for the stochastic non-chaotic

Lorenz attractor is derived as

∂p

∂t=−a(Y −X)

∂p

∂X− {X(r − Z)− ψY } ∂p

∂Y

− (XY − bZ)∂p

∂Z+

σ21

2

∂2p

∂X2+

σ22

2

∂2p

∂Y 2+

σ23

2

∂2p

∂Z2+ (a+ b+ ψ)p. (30)

The FE method is next used to discretize Eq. (30). An inspection of the tra-

jectories reveal that the states of the system lie within [−5, 5]. The domain Ωe

is therefore chosen as [−5, 5]×[−5, 5]×[−5, 5], which is discretized using a grid

of 20 noded isoparametric elements discussed in the previous section. Thus,

the discretized FE model consists of 1.25 × 105 elements with the number of

18

unknowns (nodes) being 2.5 × 106. The discretized FE equations are numer-

ically solved only for the stationary condition. Figure 2 shows the bivariate

stationary pdf for the state variables taken two at a time.

−50

5

−5

0

50

0.05

0.1

0.15

XY

pX

Y(x

,y)

(a) X-Y

−50

5

−5

0

50

0.05

0.1

XZ

pX

Z(x

,z)

(b) X-Z

−50

5

−5

0

50

0.05

0.1

YZ

pY

Z(y

,z)

(c) Y-Z

Fig. 2. Example 1: 2-dimensional pdf for the state variables.

To compare the accuracy of the solutions, the estimated pdf obtained from

solving the FP equations are compared with those obtained from D-MCS as

well as C-MCS. 4× 104 realisations with a time step of δt = 0.002 have been

used to integrate the SDEs in Eq. (29). using the EM scheme. In C-MCS, for

every δt = 0.02, the sample density and its weight have been updated. Figure

3 compares the contour plots for the bivariate pdf of the state vectors taken

two at a time obtained using FE and MCS. A very good match is observed. A

19

comparison of the marginal pdfs for the state variables is shown in Fig.4 and

a very good match is observed here as well.

Figure 5 shows the convergence characteristics of the proposed FE method in

terms of the Kullback-Leibler entropy measure D(N) computed for the bivari-

ate pdfs as well as the marginals. It is clear that as the mesh size is increased,

D(N) decreases indicating a greater match between the FE predicted pdf and

MCS. It is observed that in both the univariate and the bivariate cases, D(N)

is of the same order, O(10−3). A visual inspection of the match between the

FE predicted pdf and those obtained from MCS indicates that D(N) of order

0.04

0.04

0.04

0.04

0.08

0.08

0.08

0.12

0.12

0.16

X

Y

−5 0 5−5

0

5

(a) pXY (x, y)

0.02

0.02

0.02

0.02

0.04 0.04

0.040.

04 0.06

0.06

0.06

0.080.08

0.1

0.1

X

Z

−5 0 5−5

0

5

(b) pXZ(x, z)

0.02

0.02

0.02

0.02

0.04 0.04

0.04

0.06

0.06

0.06

0.08

0.08 0.1

Y

Z

−5 0 5−5

0

5

(c) pY Z(y, z)

Fig. 3. Example 1: Contour plots for the bivariate pdf for two states; FP solution:

full lines; MCS: dashed lines.

20

−5 0 50

0.05

0.1

0.15

0.2

0.25

X

pX

(x)

(a) pX(x)

−5 0 50

0.05

0.1

0.15

0.2

0.25

Y

pY

(y)

(b) pY (y)

−5 0 50

0.1

0.2

0.3

0.4

Z

pZ(z

)

(c) pZ(z)

Fig. 4. Example 1: Comparison of the marginal pdf for the state variables; FP

solution: full lines; MCS: dashed lines.

25 30 35 40 45 500

0.005

0.01

0.015

0.02

N

D(N

)

X−YX−ZY−Z

(a) Bivariate

25 30 35 40 45 500

0.005

0.01

0.015

0.02

N

D(N

)

XYZ

(b) Marginals

Fig. 5. The Kullback-Leibler cross divergence entropy measure D(N) as a function

of the mesh size N ; (a) bivariate pdfs (b) marginal pdfs.

21

O(10−3) is acceptable. The corresponding D(N) for the 3-D pdf is also of the

same order and indicates a convergent solution for the 3−D FP equation.

A comparison of the computational costs reveals that while D-MCS required

about 19 hours, C-MCS reduces the CPU time to just 7 hours - a reduction

in the CPU time of approximately 63%. In contrast the FE method required

about 90 minutes using the same computing resources. Thus, the CPU costs is

about 8% and 21% when compared with D-MCS and C-MCS. This highlights

the importance of the use of FE based approaches in the solution of the FP

equation for problems which are amenable for such solutions.

5.4 Example 2: Duffing oscillator subjected to additive Gaussian and non-

Gaussian colored noise excitation

Next, a bistable Duffing oscillator subjected to band limited random excita-

tions is considered. The equation of motion is given by

X + 2βX − λX + αX3 = γF (t), (31)

where, β, α, λ and γ are constants and F (t) is the external excitation. F (t)

is first assumed to be mean zero exponentially correlated Gaussian process

that has the characteristics as defined in Eqs. (9) and (10). Introducing the

variables X1 = X, X2 = X and X3 = F (t), Eq. (31) can be written in terms

of a set of first-order SDE. Clearly, the state vector X = [X1 X2 X3]T is

Markovian whose tpdf p(X, t|X0, t0), is governed by the FP equation

∂p

∂t=−X2

∂p

∂X1+

∂(2βX2 − λX1 + αX31 − γX2)p

∂X2

+1

τ1

∂(X3)p

∂X3+

2D1

τ 21

∂2p

∂X23

. (32)

22

The numerical values for the parameters corresponding to Eqs. (9), (10) and

(31) are taken to be β = 0.1, α = 0.2, λ = 1, γ = 1.5, τ1 = 0.5, D1 = 0.2.

The FE method is used to obtain a solution for the FP equation. As in

the previous example, a reduced state-space Ωe is defined to be spanned by

[−5, 5]×[−5, 5]×[−5, 5]. The three dimensional state-space is discretized using

the 20 noded isoparametric element shown in Fig. 1. For obtaining the steady

state solution, only the matrix K is constructed according to Eq. (19). The 2-

dimensional joint pdfs and their corresponding contour plots for pX1X2(x1, x2),

pX1X3(x1, x3) and pX2X3(x2, x3) are shown in Fig. 6. A comparison of the accu-

racy of the pdfs can be seen from the contour plots where the corresponding

plots obtained from MCS are also shown. As can be observed, a very good

match is obtained. The D(N) measures obtained for the bivariate pdf are

calculated to be 8.79 × 10−4, 8.86 × 10−4, and 8.78 × 10−4 for pX1X2(x1, x2),

pX1X3(x1, x3) and pX2X3(x2, x3) respectively when N = 50.

An inspection of the bivariate pdf plots in Fig. 6 reveal the bimodal nature

of pX1X2(x1, x2) and pX1X3(x1, x3), while pX2X3(x2, x3) is observed to be uni-

modal. This can be attributed to the cubic nonlinearity associated with the

state vector X1.

Next, the Duffing oscillator is analyzed for the case when F (t) is non-Gaussian

colored noise. Here, F (t) is assumed to be a filtered white noise process as given

by Eqs. (11-12). Introducing the variables X1 = X, X2 = X and X3 = F (t),

the corresponding FP equation for the first-order SDE can be expressed as

∂p

∂t=−X2

∂p

∂X1+

∂(2βX2 − λX1 + αX31 − γX2)p

∂X2

+∂(AX3)p

∂X3+

2D2

τ 22

∂2p

∂X23

. (33)

23

−5

0

5

−5

0

50

0.02

0.04

0.06

0.08

X1X2

pX

1X

2(x

1,x

2)

(a) pX1X2(x1, x2)

0.020.02

0.02

0.020.02

0.02 0.040.04

0.04

0.04

0.06

0.06

X1

X2

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

(b) Contour plot for pX1X2(x1, x2)

−5

0

5

−5

0

50

0.05

0.1

0.15

X1X3

pX

1X

3(x

1,x

3)

(c) pX3X1(x3, x1)

0.04 0.04

0.040.04

0.04

0.08

0.080.08

0.08

0.12 0.12

X1

X3

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

(d) Contour plot for pX3X1(x3, x1)

−5

0

5

−5

0

50

0.05

0.1

0.15

0.2

0.25

X2X3

pX

2X

3(x

2,x

3)

(e) pX3X2(x3, x2)

0.04

0.04

0.04

0.08

0.080.08

0.12

0.12

0.16

0.2

X2

X3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

(f) Contour plot for pX3X2(x3, x2)

Fig. 6. Duffing oscillator under Gaussian colored noise q = 1, τ1 = 0.5, D1 = 0.2 ;

FP solution: full lines; MCS: dashed lines.

where, A = 1/τ2(1 + μ(q − 1)X23/2). To compare the effects of adopting a

non-Gaussian noise, the same system and filter parameters are considered,

but q is taken to be 1.25 instead of 1.0. Figure 7 shows the bivariate joint pdfs

pX1X2(x1, x2), pX1X3(x1, x3) and pX2X3(x2, x3) and the corresponding contour

24

plots. A comparison of the contour plots for the bivariate pdfs obtained from

MCS once again shows a good match. The Kullback-Leibler entropy measures

are computed to be 8.92 × 10−4, 9.0 × 10−4, and 8.98× 10−4 respectively for

pX1X2(x1, x2), pX1X3(x1, x3) and pX2X3(x2, x3) when N = 50.

A comparison of the corresponding marginal pdfs for pX1(x1), pX2(x2) and

pX3(x3) for the cases when q = 1, i.e., when the excitation is Gaussian and

when q = 1.25, i.e., for non-Gaussian excitation, are shown in Fig. 8. It is

to be noted that both the Gaussian and the non-Gaussian excitations have

identical correlation function. Fig 8 reveals that the marginal pdfs for the

states corresponding to the non-Gaussian excitations have a greater scatter.

This in turn, would imply that the mean crossing rates of the response for the

non-Gaussian excited Duffing oscillator is expected to be higher. Using the

computed joint pdf pX1,X2(x1, x2), the mean upcrossing statistics are numer-

ically computed using Eq. (1) for both the Gaussian and the non-Gaussian

excited Duffing oscillators under stationary conditions. A comparison of the

stationary mean up crossing intensities for the Duffing oscillator under Gaus-

sian and non-Gaussian excitations is shown in Fig. 9. The higher intensity

of the crossings of the response of the Duffing oscillator when excited by a

non-Gaussian noise is clearly visible. A higher crossing intensity implies that

both the first passage failure probability and the expected rain-flow fatigue

damage [36] is higher. This example therefore demonstrates that approximat-

ing non-Gaussian excitations as Gaussian can lead to underestimating the risk

associated with the structure, even though the correlation function is modeled

correctly.

The accuracy of the predicted marginal pdfs shown in Fig. 8 as well as the

crossing intensities shown in Fig. 9 are benchmarked with those obtained from

25

−5

0

5

−5

0

50

0.01

0.02

0.03

0.04

0.05

X1X2

pX

1X

2(x

1,x

2)

(a) pX1X2(x1, x2)

0.010.01 0.01

0.010.010.01

0.01

0.02

0.02

0.02

0.020.02

0.02

0.03

0.03

0.03

0.03

0.04

0.04

X1

X2

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

(b) Contour plot for pX1X2(x1, x2)

−5

0

5

−5

0

50

0.02

0.04

0.06

0.08

0.1

X1X3

pX

1X

3(x

1,x

3)

(c) pX3X1(x3, x1)

0.020.02

0.02

0.020.020.02

0.040.04

0.04

0.040.04

0.060.06

0.060.060.06 0.08

0.08

0.1 0.1

X1

X3

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

(d) Contour plot for pX3X1(x3, x1)

−5

0

5

−5

0

50

0.05

0.1

0.15

X2X3

pX

2X

3(x

2,x

3)

(e) pX3X2(x2, x2)

0.04

0.04

0.040.04

0.08

0.080.08

0.12

0.12

X2

X3

−3 −2 −1 0 1 2 3−3

−2

−1

0

1

2

3

(f) Contour plot for pX3X2(x3, x2)

Fig. 7. Duffing oscillator under non-Gaussian colored noise q = 1.25, τ2 = 0.5,

D2 = 0.2; FP solution: full lines; MCS: dashed lines.

MCS. In MCS, the governing set of three coupled first-order SDE are solved

using the forward EM numerical integration. A time step of 5×10−4 was used

and the number of time steps required to obtain a stationary state for the re-

sponse was approximately 80×106 for both the Gaussian and the non-Gaussian

26

−5 0 50

0.05

0.1

0.15

0.2

0.25

X1

pX

1(x

1)

FE (q=1)MCS (q=1)FE (q=1.25)MCS (q=1.25)

(a) pX1(x)

−5 0 50

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X2

pX

2(x

2)

FE (q=1)MCS (q=1)FE (q=1.25)MCS (q=1.25)

(b) pX2(x)

−5 0 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

X3

pX

3(x

3)

FE (q=1)MCS (q=1)FE (q=1.25)MCS (q=1.25)

(c) pX3(x)

Fig. 8. Marginal stationary pdf for Duffing oscillator,τ = 0.5; D1 = 0.2.

0 1 2 3 4 510−4

10−3

10−2

10−1

100

X1

E[ N

+(α

,T)]

FE (q=1)MCS (q=1)FE (q=1.25)MCS (q=1.25)

Fig. 9. Mean upcrossing intensity for Duffing oscillator; τ = 0.5; D1 = 0.2.

excited cases. It is observed that there is very good agreement between the

computed marginal densities and those estimated from MCS. A good agree-

ment is also observed in the crossing intensity estimates.

27

The computational time required in D-MCS varied between 16 to 18 hours for

the different cases. Using C-MCS algorithm, the CPU time was approximately

8 hours - a reduction of almost 50%. In contrast, the solution of the FP

equation using the proposed FE method required about 90 minutes using the

same computing system, indicating that the computational costs in solving

the FP equation is just about 10% when compared to D-MCS and about 20%

in comparison to C-MCS.

6 Dimension reduction using stochastic averaging

From the discussions considered so far, it is quite clear that application of

the proposed methodology to systems of FP equations whose dimensions are

greater than three is not possible. In this section, we focus on the use of the

method of stochastic averaging to reduce the dimension of the state-space and

subsequently use FE to solve the corresponding FP equations. With this in

view, the bistable Duffing oscillator considered in the previous example is now

assumed to have an additional parametric band limited Gaussian excitation.

The corresponding equation of motion is given by

X + 2βX − (λ+ ξ2(t))X + αX3 = ξ1(t). (34)

where, β, α, λ are system parameters and ξ1(t) and ξ2(t) are zero mean Gaus-

sian colored noise having the correlation function as shown in Eq. (9). Ex-

pressing the colored noise processes ξ1(t) and ξ2(t) as first-order filtered white

noise processes of the form in Eq. (10) - D1 and D2 being the intensities of the

white noise and τ1 and τ2 being their corresponding correlation lengths, the

FP equation corresponding to Eq. (34) is of 4-dimensions in the state-space.

28

Assuming small intensities of the noise and light damping, it has been shown

that a weakly nonlinear Duffing oscillator exhibits a quasi-harmonic behavior

[37]. In the quasi-harmonic regime, the joint response process (X(t), X(t)) can

be transformed to a pair of slowly varying processes, a(t) and Φ(t), as

X(t) = a(t) cosΦ(t),

X(t) =−a(t) sin Φ(t), Φ(t) = t+ φ(t), (35)

where,

a(t) =√X2(t) + X2(t). (36)

For linear oscillators under Gaussian white noise excitation, the response X(t)

and X(t) are Gaussian and independent. It follows that [38] the transient pdf

of the amplitude process a(t) is a time dependent Rayleigh distributed process.

However, in the presence of nonlinearities as in the Duffing oscillator, even if

the excitations are Gaussian, the response is significantly non-Gaussian and

estimating the pdf in closed form is not easy.

One approach to the problem of determining an approximation for p(a, t) is

based on the FP equation in combination with stochastic averaging. Substi-

tuting for X(t) and X(t), as given in Eq. (35), in Eq. (34) and applying an

averaging procedure based on deterministic and stochastic averaging [6], an

equation for the amplitude process a(t) is obtained, which when written in

the SDE form is given by

da=[− βa+

D1

2a(1 + τ 21 )+

3D2a

8(1 + 4τ 22 )

]dt

+

√√√√ D1

1 + τ 21+

D2a2

4(1 + 4τ 22 )dB1(t). (37)

Here, B1(t) is a Wiener process. Since Eq. (37) is uncoupled from the phase

angle Φ, a(t) can be viewed as a univariate Markov process , and the corre-

29

sponding FP equation in one dimensional space is given by

∂

∂tp(a, t) = − ∂

∂a[Y (a)p(a, t)] +

1

2

∂2

∂a2[Z(a)p(a, t)]. (38)

Here, p(a, t) is the tpdf of the random process a(t), subjected to the restriction

0 ≤ a < ∞, while Y (a) and Z(a) are defined as

Y (a) =−βa+D1

2a(1 + τ 21 )+

3D2a

8(1 + 4τ 22 ), (39)

Z(a) =D1

1 + τ 21+

D2a2

4(1 + 4τ 22 ). (40)

By letting ∂p/∂t = 0, the solution of Eq. (38) for the stationary pdf, ps(a),

can be expressed in closed form as

ps(a) =C

Z(a)exp

[− 2

∫ Y (a)

Z(a)da

], (41)

where, C is a normalizing constant. The solution of Eq. (38) for the non-

stationary pdf p(a, t) is more difficult. Recently, several numerical methods

have been developed for the analysis of the non-stationary random response

of lightly damped nonlinear oscillators using the stochastic averaging method

[39]. Nevertheless, it is essential to estimate both the stationary and the tran-

sient pdf in many situations; for example, see [40] for an application.

The numerical values for the parameters corresponding to Eq. (38) are se-

lected as β = 0.04, τ1 = τ2 = 0.5, D1 = D2 = 0.04. In this study, the FP

equation in Eq. (38) is discretized using the proposed FE approach leading to

a set of discretized equations of the form as in Eq. (17). For this problem of

dimension 1, the FE solution is straightforward and can easily be obtained.

The domain size Ωe is assumed to be[0, 5

]. The solution domain was divided

into 400 elements, using 2 noded line elements with standard shape function

N1(x) = 1 − x/le, N2(x) = x/le, where le = Ωe/N . Estimates of the tpdf for

30

various time instants are obtained by numerically integrating the discretized

equations for the transient solutions using the Crank-Nicholson method de-

scribed earlier. A deterministic initial condition is defined in terms of a pdf

which is a Dirac-delta function at t = 0. This is however, difficult to imple-

ment numerically. For numerical stability, a deterministic initial condition is

assumed as a Gaussian distribution with small variance. The transient solu-

tion is obtained by numerically integrating the equations with a time step

Δt = 0.003 s.

The steady state and transient pdfs p(a, t) computed by the FE method are

shown in Fig. 10(a). As t increases, the transient pdfs show greater variabilities

and finally converge to the stationary pdf obtained by solving the stationary

FP equation. The increase in scatter in the amplitude of the response as t

progresses from zero is reasonable as the uncertainty associated with the re-

sponse gets accumulated over time. It can also be seen that as t becomes

larger, the solution converges correctly to the steady state solution. It took

approximately 100s for the transient pdf to converge to the steady state solu-

tion. For given system parameters, the stationary pdf of amplitude; ps(a) ≈ 0

for a ≥ 2.5. Here, it must be noted that the solution of the steady state solu-

tion is obtained by solving directly the steady state form of the discretized FP

equation. The good agreement between the steady state pdf obtained using

the two approaches indicates the accuracy of the algorithm for solving the

transient FP equation.

We next consider the influence of damping coefficient β on the pdf of ampli-

tude process. Keeping the other parameters as per Figure 10(a), the damping

coefficient β is reduced from 0.04 to 0.02. The steady state and the tran-

sient pdfs p(a, t) computed by the FE method are shown in Figure 10(b).

31

As before, the stationary pdfs are obtained by directly solving the stationary

form of the FE discretized FP equation while the transient solutions are ob-

tained by application of the Crank-Nicholson algorithm. It is observed that

as the damping coefficient is reduced, ps(a) shows a greater variance. This is

expected as damping coefficients decrease, the uncertainty easily propagates

through the filter into the response. At reduced damping coefficient β = 0.02,

the stationary pdf of amplitude; ps(a) ≈ 0 for a ≥ 4. It took approximately

100s for transient PDF to converge to the steady state solution.

The influence of noise correlation time on ps(a) is investigated next. Keep-

ing the other parameters as per Figure 10(a), the correlation time for both

parametric as well as additive noise is increased from 0.5 to 2. The steady

state and transient pdfs p(a, t) computed by the FE method are shown in

Figure 10(c). It is observed from Fig. 10(c) that the variance of the station-

ary pdf ps(a) decreases with an increase in the correlation times τ1 and τ2.

This is expected because a higher correlation time implies a spectrum with

a smaller bandwidth and a smaller energy content. At increased correlation

time τ1 = τ2 = 2, for amplitude a ≥ 1.2, the stationary pdf of amplitude is

ps(a) ≈ 0. It took approximately 60s for the transient pdf to converge to the

steady state solution.

Next, we consider the influence of noise intensity on the amplitude process.

The noise intensity parameters for both parametric as well as additive noise

are increased from 0.04 to 0.08. The steady state and transient pdfs p(a, t)

computed by the FE method are shown in Figure 10(d). At increased noise

intensity D1 = D2 = 0.08 for amplitude a ≥ 4.5, the stationary pdf of the

amplitude ps(a) ≈ 0. It took approximately 120s for the transient pdf to

converge to the steady state solution. Similar to the effect of reducing the

32

damping coefficient, it is observed that as the intensity of the colored Gaussian

excitations is increased, ps(a) shows a greater variance. This is expected as

the scatter associated with the input is larger for higher intensities of the

excitation and this uncertainty propagates through the Duffing oscillator into

the response.

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5t=2 s

t=6 s

t=10 s

t=20 s

t=100 s

a

p(a

,t)

(a)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

t=2 s

t=8 s

t=30 st=60 s

t=200 s

a

p(a

,t)

(b)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t=2 s

t=4 s

t=10 s

t=20 s

t=100 s

a

p(a

,t)

(c)

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

t=2 s

t=8 st=15 s

t=30 s

t=120 s

a

p(a

,t)

(d)

Fig. 10. Example 3: Stationary and transient pdf for the amplitude for the Duffing

oscillator; stationary solution (◦)

Figure 11 shows the variation of D(N) as a function of node of FE mesh

size. The Kullback-Leibler divergence entropy measure indicates that FE dis-

cretization shows convergence upto N = 200 and further refinement does not

improve the solution.

33

0 100 200 300 400 5004

5

6

7

8

9

10 x 10−4

N

D(N

)Fig. 11. The Kullback-Leibler cross divergence entropy measure D(N) as a function

of the mesh size N .

7 Conclusions:

A finite element based methodology for the solution of 3-D Fokker-Planck

equations arising due to oscillators subjected to coloured noise has been pre-

sented. The dimension of the state variables in the oscillators increase by one

due to representing the coloured noise as the output of a white noise pro-

cess when passed through a first-order filter. The solution of the discretised

FE equations were numerically cumbersome due to the large number of un-

known nodal variables. Issues related to the convergence and accuracy of the

solutions were examined and secondary measures for enhancing the compu-

tational efficiency of the solution were investigated. Additionally, questions

related to dimension reduction using the method of stochastic averaging has

been explored.

The solution of the FP equations enabled direct computation of the joint

pdf of the response and its instantaneous time derivative accurately. This

enables computing the crossing statistics and obtaining estimates of the time

variant reliability. Numerical studies have been carried out to illustrate how

the extent of non-Gaussianity in the excitations affects the crossing statistics

34

of the response of the oscillators considered in this study. Further, it has

been shown that for the examples considered, despite the complexities in the

numerical solution of the FE discretised equations, the computational costs are

significantly less than MCS, even with the use of variance reduction techniques.

Further reduction in the computational costs in either method is possible by

employing parallelisation methods. However, investigations on this aspect is

beyond the scope of the present paper.

The applicability of the proposed method to higher order structural systems

is difficult due to the enormous complexities involved in the solution of higher

dimensional FP equations. For such problems, MCS appears to be a more

robust method for dynamical analysis irrespective of the extent of structural

nonlinearity and non-Gaussianity of the excitations. Nevertheless, for dynam-

ical systems which can be appropriately modelled as sdof oscillators, or whose

FP equations are amenable for dimension reduction, the proposed FE based

methodology appears to lead to accurate estimates of the joint pdf of the

response and its time derivative at a fraction of the computational costs.

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