Local Maxima in Quadratic LMA Processes

Local Maxima in Quadratic LMA Processes

Jithin Jith1 Sayan Gupta 1 Igor Rychlik2

1Department of Applied MechanicsIndian Institute of Technology Madras,

India

2Mathematical Sciences,Chalmers University of Technology,

Sweden

11th ASCE Joint Speciality Conference on Probabilistic Mechanics

and Structural Reliability

Jithin Jith, Sayan Gupta , Igor Rychlik Local Maxima in Quadratic LMA Processes

Introduction

Motivation: Estimating failure probability of an offshore structure.

Difficulties:

1 Wave loadings arenon-Gaussian processes.

2 Nonlinear structurebehavior.

3 Structure response:filtered process; structurebeing the filter.

Z (t) = g [X (t)]


Rice’s formula

Failures can be due to

exceedance of safe threshold levels (overloading)

fatigue failures (gradual damage accumulation)

µ+(u), intensity of upcrossingof level u of the response

µ+(u) =

∫ ∞0

zpZZ (z , z)dz

µ(u), intensity of localmaxima above level u

µ(u) = −∫ 0

−∞

∫ ∞u

zpZZZ (z , 0, z)dzdz


Problem 1: Loadings X (t) are non-Gaussian

Example 1:

40 minutes measurements of sea-surface elevation at a fixedlocation.

Kurtosis = 3.17, Skewness = 0.25; obviously non-Gaussian.

Dotted line: mean crossing intensity assuming process to beGaussian; underestimates at high levels


Problem 2: Non-Gaussian response (due to nonlinearities)

Example 2:

30 minutes of measured stress in a ship under stationary seaconditions.

Kurtosis = 7.66. Skewness = 1.12

900 950 1000 1050 1100150

100

50

0

50

100

150

200

250

300

t seconds

Y(t)

MPa

Dotted line: mean crossing intensity assuming process to beGaussian; significant underestimation at high levels


Objectives

The focus of this study are:

1 Modeling the non-Gaussian load X (t) as a LMA (LaplaceMoving Average) process.

2 Modeling the response Z (t) as a quadratic transformation ofthe load X (t).

3 Developing a methodology to estimate the intensity of localmaxima for Z (t).


Laplace Moving Average (LMA) Process

A LMA process is defined as

X (t) =

∫ ∞−∞

f (t − s)dΛ(s), (1)

Λ(t) = ζ · t + µΓ(t) + σB(Γ(t)) (2)

Λ(s) is a stochastic process (referred as Laplace Motion)

f (·) is a kernel (can be chosen to correspond to the spectrum).

Γ(t) is a Gamma process and B(t) is Brownian motion.

Parameters: drift (ζ), asymmetry (µ), scaling (σ), ν governs theshape of Γ(·)


Properties of LMA

The increments of LMA, dΛ(x), follow generalized Laplacedistribution. Thus, LMA will have a non-symmetricaldistribution function.

The characteristic function of Laplace distribution is describedby four parameters, related to the first four moments.


Advantages of LMA

S(ω) =1

2π

σ2 + µ2

ν|F [f (t)]|2.

Advantages

Captures mean, variance, skewness and kurtosis (asymmetryand heavy tails in the pdf)

Conditioned on the Gamma process, Γ(·) = γ(·), becomesGaussian

λ(s) = ζs + µγ(s) + σB(γ(s)) (3)


Gaussian MA vs LMA (skewness=0.9,kurtosis=7.5)

(e) Time history (GMA) (f) Histogram (GMA)

(g) Time history (LMA) (h) Histogram(LMA)


Second Order Response of Weakly Nonlinear Systems

Volterra series representation

Z (t) = Z1(t) + Z2(t),

Z1(t) =

∫ ∞−∞

k1(t − s)dΛ(s)

Z2(t) =1

2

∫ ∞−∞

∫ ∞−∞

k2(t − s1, t − s2)dΛ(s1)dΛ(s2).

Here,

k1(·) is the linear transfer function, andk2(·, ·) is the quadratic transfer function.


Kac Siegert Representation (1947)

Rewrite the response in terms of its eigenbasis functions

Z (t) ≈n∑

j=1

{aiWi (t) +λi2W 2

i (t)} (4)

where,

Wi (t) =

∫ T

−Tφi (t − s)dΛ(s) (5)

Eigenvalue equation:∫k2(t, s)φ(s)ds = λφ(t). (6)

Response is a quadratic transformation of vector of LMAprocess {Wi (t)}ni=1.


Intensity of Local Maxima

Intensity of Local Maxima above threshold u

µ(u) = −∫ 0

−∞

∫ ∞u

z fZZZ (z , 0, z)dzdz . (7)

Joint pdf fZZZ (z , 0, z) not easy to determine for quadratictransformations of LMA.


Proposed Hybrid Method

Conditionally on Γ(t) = γ(t), Wi (t), Wi (t) & Wi (t) areGaussian processes.

Note:

as Λ(t)|Γ(t) = c · t + µγ(t) + σB(γ(t))

Estimate the crossings of the conditional process Z (t)|γ.

Monte Carlo simulations used to estimate unconditionalcrossing intenisty

µ(u) = E [NZ (u)] = E [E [NZ (u)|Γ(·) = γ(·)]].


Example: Offshore Jacket Platform

120 m water depth

Subjected to small amplitude waves that follow P-Mspectrum, with Hs = 10 m, ωp = 0.5 rad/s

Sηη(ω) =5

16H2s

ω4p

ω5exp

(−

5ω4p

4ω4

). (8)

Loading:

F (t) = kM X (t) + kD |X (t)|X (t). (9)

X (t): skewness = -0.2, kurtosis = 4.5



Figure: Lumped mass model of the jacket platform [Luo, Zhu (2006)]



0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Frequency (rad/s)

Po

we

r sp

ectr

al d

en

sity (

W/r

ad

/s)

(a) Spectrum of X (t)

−40 −30 −20 −10 0 10 20 30 40−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t (s)

f(t)

(b) LMA kernel f (t)



−2 −1.5 −1 −0.5 0 0.5 1 1.5 210

−15

10−10

10−5

100

105

Response level(m)

Inte

nsity o

f m

axim

a(s

−1

)

(c) Intensity of maxima at z = u

−2 −1.5 −1 −0.5 0 0.5 1 1.5 210

−15

10−10

10−5

100

105

Response level(m)

Inte

nsity o

f m

axim

a(s

−1

)

(d) Intensity of maxima at z ≥ u

Figure: Solid line - LMA; Asterisks - LMA MCS; Dashed line - GMA;Circles - GMA MCS


Concluding Remarks

Estimates using proposed method are in agreement with thosefrom MCS

Gaussian models underestimates intensity at high levelsunderlying the importance of modeling the non-Gaussianfeatures of the process.

LMA-process has flexibility to model the skewness andkurtosis of the marginal distribution.

In contrast to pure simulations, the hybrid method canestimate crossings of very high levels.


Local Maxima in Quadratic LMA Processes

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