Nonlinear Stochastic Response of Offshore Structures: With Focus on Spectral Analysis

CESOS Highlights Conference & AMOS Visions - 2013

Deepwater Research Centre

Dalian University of Technology

Dalian, China

Xiang Yuan ZHENG

27/05/2013

Outline:

1. Background2. Response Extreme Value 2. Response Extreme Value 3. Frequency-domain Spectral Analysis4. Case Study5. Concluding Remarks

No. of platforms in

waves of 10m or larger

No. of platforms

Destroyed or Damaged

Andrew 1992 970 87

Katrina 2005

(August)

697 67

Rita 2005 1055 98

1. BackgroundGOM - Lily 2002

Statistics: Platforms survived hurricanes in GOM.(Hurricanes went across both shallow and deep water.)

33

Rita 2005

(September)

1055 98

Ike 2008 1135 91

(Source: Weather Research Center 2009)

I. More than 90% of platforms can safely survive consecutive

hurricanes.

II. This is much different from land structures attacked by

earthquakes.

Christchurch in New Zealand remained almost intact during ML-7.1

quake in Sep 2010, but was entirely destroyed by ML-6.3 quake in

Feb 2011.

Katrina track

August 23, 2005

44

Rita track

September 28, 2005

In a harsh environment, the major structure can remain linear, but the responseusually is nonlinear and this jeopardizes structural strength.usually is nonlinear and this jeopardizes structural strength.

Sources of nonlinearities:

• Environmental loads (Forcing functions: Morison, 2nd-order drift...)• Waves• Fluid-structure interaction• Damping (viscous) • Soil & Soil-structure interaction• Coupling among loads (wind & waves)

5

QuasiQuasiQuasiQuasi----static static static static P/K * DAF,

Dynamic Dynamic Dynamic Dynamic

Deterministic Analysis(Design wave)

• Airy wave• Stokes waves: 2nd, …, 5th• Cnoidal waves• Stream function• NewWave• Fourier series waves

(Fenton 1999)

Time-domain, Frequency-domain

Approaches for Response Analysis (ULS)

Dynamic Dynamic Dynamic Dynamic

Stochastic analysis(Wave spectrum)

Random wave models:

• Linear random waves*• 2nd, 3rd, 4th-order

nonlinear random waves• Focused waves

Time-domain, Frequency-domain

Fully nonlinear model(Boussinesq for shallow-waterwind turbines)

Time-domain

CFD Time-domain

6

Industrial design often uses Linear random waves + Linearization of wave loads which leads to response EV underestimation.

2. Response Extreme Value (EV)

Linear FRF H(ω)Input: Gaussian Output: Gaussian

Linear systems (LTI):

Input: Non-Gaussian Output: Non-Gaussian

2( ) ( ) ( )yy xxS H Sω ω ω= σ2

Linear FRF H(ω)

7

50 100 150 200 250 300 350 400 450 500

-30

-20

-10

0

10

20

30

t (s)

F(t) (103 ton)

-28.55

29.78 Rayleigh extreme for a narrow-banded Gaussian

process based on independence assumption:

defined by mean m and standard deviation σ.

( )max

0.57722ln 1

2ln

nwY NN

σ

= +

Wide-banded:

( ) ( )max max 1wd nwY Y ρ= −

ρ: bandwidth parameter

H(ω)

Nonlinear FRFH(ω1, ω2,…, ωn)Input: Gaussian or

Non-GaussianOutput:

Non-Gaussian

Nonlinear systems:

� Type I, II, III (Gumbel, Fréchet and Weibull)� Gram-Charlier & Edgeworth Series� Moment-neglect closure method� Maximum entropy distribution

Classical Probability distribution of EV (i.i.d):

8

� Maximum entropy distribution

I. Rare occurrence of EVs requires a large number of

sample functions to be generated in analysis.

II. Monte-Carlo simulations are time-consuming when

nonlinearities are involved, due to iterations in solving

equations and stochastic uncertainty.

Recent approaches for EV predictionLimited response data available from experiments, simulations, observations:

1) Naess-Gaidai method (2007, 2008)(Mean up-crossing rates & Gumbel distribution)

2) First-order Reliability Method (Jensen 2006, 2011)3) Winterstein Hermite polynomial method (1985, 1988, 1994; Bruce 1985)

(Moment-based: mean, standard deviation, skewness and kurtosis)Moments can be also evaluated from spectra.

9

Moments can be also evaluated from spectra.

Adopted in EV prediction of structural response subject to wind, wave, seismic loads.

TLP

2nd-order

heave motion

Combined

wind & wave

actions

Wide application of Winterstein’s method in offshore engineering since 2000.

SNAME, 2002/2011

10

Some issues:

Y

Y: Non-Gaussian response; U: Standard Gaussian process.

1) Monotonic assumption

EV of Y can be expressed in terms of EV of U :2 3

max max max 3 max 4 max max( ) ( 1) ( 3 )Y g U m U c U c U Uβσ ≈ = + + − + −

11

2) Empirical fitted coefficients β, c3, c4 depend on skewness k3 and kurtosis excess k4 (Winterstein 1994):

Applicable only to nonlinear systems with

3) Difficult to obtain stable k3 and k4, so it necessitates spectral analysis.

Applicability extended to:

Polynomial coefficients c0, c1, c2, c3 can be accurately solved by moment equalizations:

3 3 4 4, , ,G Y G Y G Y G Ym m σ σ κ κ κ κ= = = =

Y2 3

0 1 2 3( )Y g U C C U C U C U≈ = + + +

1) strongly nonlinear systems of K4>12 (Zheng & Liaw 2003):

2) bivariate nonlinear systems (non-monotonic):

2 30 1 2 3 4

( )

( , )

Y u c u c

g x y C C x C xy C xy C xy

η= + +

≈ = + + + +

xη

η

σ= ，

u

uy

σ=

2) bivariate nonlinear systems (non-monotonic):

Polynomial coefficients c1, c3 are solved using least square method to keep non-monotonicity, c0, c2, c4 solved by moment equalizations.

3. Frequency-domain spectral analysis

Moment functions and cumulants of a random process

[ ]1 ( )x

xR m E X k= =3

11222212121321 )(2)]()()([),(),( xxxxxxx mmmmmmR +−++−= ττττττττ

1st-order:

2nd-order:

Mean3rd-order:

33

(0,0)xRκ

σ= Skewness

13

)(

)()(

cov)()()(

1

2112

21121

τ

τ

ττ

−=

−−=

−=

xx

xx

xxx

R

mm

sequencemmR

41122232132

3222122

1

213423323131231

122321322223212

3214321

)(6)]()()(

)()()([)(2

)],(),(),(),([

)()()()()()(

),,(),,(

xxxx

xxxx

xxxxx

xxxxxx

xx

mmmm

mmmm

mmmmm

mmmmmm

mR

−−+−+−+

+++

+++−−−

−−−−−−

=

ττττττ

τττ

ττττττττττ

τττττττττ

ττττττ

2nd-order:

2(0)xR σ= Variance

4th-order:

Kurtosis excess44

(0,0,0)xRκ

σ=

Cross cumulant spectrum of n random processes (Xi, i=1,2,..,n):

)](,),(),([),,,( 112112121 −− ++= nnnxxx kXkXkXcumRn

τττττ ���

1 2 1 1 2 1

1 1 2 2 1 1 1 2 1

( , , , ) ... ( , , , )

exp{ ( )} ...

x n x n

n n n

S R

j d d d

ω ω ω τ τ τ

ω τ ω τ ω τ τ τ τ

∞

− −

−∞

− − −

= •

− + + +

∫ ∫� �

�

nth-order cross cumulant

n-dimensional

Fourier transformn-dimensional inverse

Fourier transform

nth-order

cross cumulant

spectrum

14

Spatial & time correlations of distributed wave forces on a structure.

Cross-correlation and cross-spectrum

of input x(t) and output y(t):

( ) ( )xy xyR Sτ ω⇔

f1(t)

f2(t)

f3(t)

f4(t)

f5(t)

fn(t)

Spectra of a random process:

∫ −=∞

∞−

τωττω djRS xx )}(exp{)()(Auto-spectrum ( ) ( )x xS Sω ω= −

∫ ∫ +−=∞

∞−2122112121 )(exp{),(),( τττωτωττωω ddjRS xxBi-spectrum:

1 2 2 1

2 1 1 2 2

( , ) ( , )

( , ) ( , )

x x

x x

S S

S S

ω ω ω ω

ω ω ω ω ω∗

=

= − − = − −2 1 1 2 2

1 1 2 1 2 1

2 1 2

( , ) ( , )

( , ) ( , )

( , )

x x

x x

x

S S

S S

S

ω ω ω ω ω

ω ω ω ω ω ω

ω ω ω

= − − = − −

= − − = − −

= − −

　

Tri-spectrum: ∫ ∫ ∫ ++−=∞

∞−321332211321321 )}(exp{),,(),,( ττττωτωτωτττωωω dddjRS xx

12 symmetric zones

Having 96 symmetric zones

Spectra of a linear system subject to non-Gaussian loads (Brillinger and Rosenblatt 1967)

1 2 1 1 2 1

1 2 1

1 2 1

( , ,..., )= ( ) ( ) ( )

( ... )

( , ,..., )

y n n

n

x n

S H H H

H

S

ω ω ω ω ω ω

ω ω ω

ω ω ω

− −

∗

−

−

⋅ ⋅ ⋅ ⋅ ⋅

+ + + ⋅2

( ) ( ) ( )y xy xS H Sω ω ω=

Input: Non-Gaussian Output: Non-GaussianLinear FRF: H(ω)

Higher-order spectral analysis of a LTI system Higher-order spectral analysis of a LTI system Rx(τ1, τ2,…, τn) Fourier transforms

Ry(τ1, τ2,…, τn) Sy(ω1, ω2,…, ωn)

Sx(ω1, ω2,…, ωn)

Inverse Fourier transforms

LTI systemVariance, skewness,

kurtosisExtreme response

Application to a jack-up platform with cylindrical legs.

( )

0

0

01

( ) ( , ) ( ) ( , ) ( ) ( , )

( ) ( , ) ( ) ( , )

d d

n

i i i zi

Q z f z t dz z f z t dz z f z t dz

z f z t z z f z t

η η

η

− −

==

= Φ Φ + Φ

≈ Φ ∆ + Φ

∫ ∫ ∫

∑ i

＝

　　

Modal wave force on a leg:

PFx

P

n distributed

Morison forces

An inundation

force at surface

1) Inundation is 1 order higher than Morison forces.

2) Spectral analysis for loads on 3 legs for cancellation effects.

3) Time simulation needs to choose a proper stretching method.

∆zif(zi,t)

x

z

∆zjf(zj,t)

Finite-memory Volterra Series model (Bendat 1998)1 2 3 4

1

2 1 2 1 2 1 2

3 1 2 3 1 2 3 1 2 3

4 1 2 3 4 1 2 3 4 1

( ) ( ) ( ) ( ) ( )

( ) ( )

( , ) ( ) ( )

( , , ) ( ) ( ) ( )

( , , , ) ( ) ( ) ( ) ( )

y t y t y t y t y t

h x t d

h x t x t d d

h x t x t x t d d d

h x t x t x t x t d d

τ τ τ

τ τ τ τ τ τ

τ τ τ τ τ τ τ τ τ

τ τ τ τ τ τ τ τ τ

+∞

−∞

+∞ +∞

−∞ −∞

+∞ +∞ +∞

−∞ −∞ −∞

= + + +

= − +

+ − − +

+ − − −

+ − − − −

∫

∫ ∫

∫ ∫ ∫

2 3 4d dτ τ τ+∞ +∞ +∞ +∞

−∞ −∞ −∞ −∞

∫ ∫ ∫ ∫

Fourth-order Volterra model (Moan, Zheng, Quek 2007)

3-phase I/O

I: SI → MOII: MI → SO

Quadratic,Cubic,Quartic forces

III: SI → SO

Frequency response functions of wave forces (Zheng & Liaw 2005)

∫ Φ=−

0

),()()(d

uII dzzHzkH ωω η �

∫−

Φ=0

11 ),()()()(d

uDD dzzHzzckH ωω η

1st-order: (Inertia, linear drag) 2nd-order: (bivariate inundation)

[ ])()(2

1),( 2121 ωωωω ηη ΟΟ

+= uuPI HHH��

[ ])()(2

1),( 21212 ωωωω ηη ΟΟ

+= uuPD HHH

0

3 3 3( , , ) ( ) ( ) ( , , , )D D D

d

H k c z z H z dzα α ω α α ω−

− = Φ −∫

),(),(),(),,,(3 γβαγβα ηηη zHzHzHzH uuuD =

3rd-order: (Morison drag) 4th-order: (bivariate inundation)

)]()()(

)()()(

)()()(

)()()([4

1),,,(

432

431

421

32143214

ωωω

ωωω

ωωω

ωωωωωωω

ηηη

ηηη

ηηη

ηηη

ΟΟΟ

ΟΟΟ

ΟΟΟ

ΟΟΟ

+

+

+=

uuu

uuu

uuu

uuuPD

HHH

HHH

HHH

HHHH

Auto-spectrum of wave loads (Volterra-series based)

( )2

2

1 3

2

36 ( , , ) ( ) ( ) ( )

( ) 3 ( , , ) ( ) ( )

( ) ( ) ( )FF

D

D

I

D

S

H H S d S

H S

H S S S d d

η

ηη ηη η

ηη

η

η

η

η

α

ω α α ω α α ω

β α ω β α β α ω

ω ω

β

ω

α β+ − −

+ −

=

− −∫

∫

∫

　　　　＋

Total Morison force (F)：

Inertia term

Linear drag, coupling between

linear and cubic drag terms

Cubic drag terms

Inundation force (P)：

( )

( ) ( )

( )

2 2

22

22 4 2 2

2

2

4

2

2

2

2

2

2 ( ) 2 (0) ( , ) ( ) ( )

3 (0) ( ) 2 (

( ) 2 (0) ( , ) ( ) (

, ) ( ) ( )

2 ( ) ( ) ( ) (

)

D

D

u

PP I P

P

D

u D

u u

I

P

b b k H S

PD b k H

S k

S d

H

S

H S S

S

d

d

S

H S

ηη ηη

η η ηη η

ηη

η

η

η η

η

η

η

µ σ δ ω σ

δ ω α ω α ω α α α

α ω α α ω α α

µ α ω

ω α ω α ω α α α

α α ω

Ο Ο

Ο Ο

Φ + − − +

−

+ Φ

= −

−

− −

Φ −

∫

∫

∫

∫

＋

　　　　

　　　

　

　

　

＋

　　　　

( )22 4 4 2 2

4 2

2

4

(0) [9 ( ) 18 ( , ) ( ) ( ) ( ) ( )

24 ( , , , ) ( ) ( ) ( ) ( )

)

]

D u u PD u u

PD

b k H H H S S d

H S S S S d d

d

d

η η ηη ηη

ηη ηη ηη ηη

µ σ δ ω σ α ω α µ α ω α α ω α α

α β α γ β ω γ α β α γ β ω

α

γ

α

α β γ

Ο Ο Ο ΟΦ ⋅ + − + − − +

− − − − − −

∫

∫∫∫

＋

　　　　　　

　　　　

Inertia term

Quadratic drag term

Coupling between quadratic

and quartic drag terms

Quartic drag term

Auto-spectrum analysis using recursive Price’s Theorem

1) Faster in numerical evaluation: no convolutions

2) Symbolic calculation to avoid tedious derivation in terms of Volterra series.

[ ] 2314241334124321 RRRRRRxxxxE ++=

( ) ( )ij i j

R E x t x t τ = +

[ ]][][

][][][

543216643215

653214654213654312654321

xxxxERxxxxER

xxxxERxxxxERxxxxERxxxxxxE

++

++=

Auto-correlation function of non-Gaussian wave loads

0 0 0 0

( ) ( ) ( )

( , ', ( ) ( ') ' ( ) ( ')( , ', ') )D DI If

FF II DD

d d

ff f

d d

R R R

Rz z d R z zzdz z z dzdzz z τ τ

τ τ τ

− − − −

= +

= Φ Φ + Φ Φ∫ ∫ ∫ ∫

2( , ' ) ( , ', ),I If f I uuR z zR z z k ττ =

� �

2 2 2 2

1 3 1 1 3 3

2

3

3

3

3 ( ) ( ') ( ')+ ( ) ( ')+9 ( ) ( ') ( ) ( ,( ')

+6 ( )

', )

( , '( ) ,' )

D u u u

D

uu

uu

R z zk c z c z z c z c z z z c z c z

k c z z R zc z

σ σ

τ

σ τ

∫= ωωπ

τ ωτ dezzSzzR iuuuu ),',(

2

1),',(

����

∫= ωωπ

τ ωτ dezzSzzR iuuuu ),',(

2

1),',(

Auto-correlation function of non-Gaussian wave loads (Borgman 1967; Zheng & Liaw 2005):

The problem reduces to the evaluation of correlations of Gaussian water particle kinematics only.

Further application to complicated bi- & tri-spectral analysis of wave loads:

1) Correlation (coupling) among Morison forces and inundation;

2) Spatial correlation of forces on different legs of a jack-up unit;

3) Correlation involves 4 time phases;

4) More higher-order moment functions emerge.

3 3 3 3

1 2 312 ( ) ( ) ( ) ( )N N N N

i j k lE E u t u t u t u tτ τ τ= = = =

= + + + ∑∑∑∑ i i i (Not considered in Lutes & Papadimitriou 2000)

Coupling among the strongly nonlinear drag terms.

1 1 1 1i j k l= = = = ∑∑∑∑

3 3 3 3

1 1 2 2 3 316 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )E E t u t t u t t u t t u tη η τ τ η τ τ η τ τΟ Ο Ο Ο

= + + + + + + i i i(Moan & Zheng 2007)

10395 correlation functions of kinematics.

2027025 correlation functions of kinematics.

But all can be fast evaluated.

4. Case study

Platform natural period (Tn) 7.4 s

(0.85 rad/s)

Water depth 75 m

Peak wave period (Tp) 15.1 s

(0.42 rad/s)

JONSWAP peak

enhancement parameter

3.3

ωn ≈ 2ωp or 3ωp ， the super-harmonics will boost thenonlinearity of wave loads andtherefore the platform deckresponse. Ringing can occur.See also Spidsoe & Karunakaran (1997).

A true platform in Kjeøy et al. 1989 (J. Marine Structures)

23

Significant wave height 12.9 m

Total damping 7%

4000 4050 4100 4150 4200 4250 4300 4350 4400-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t (s)

Deck sway (m)

Auto-spectrum of deck sway

5

10

15

20

25

30

35

40

SYY( ω)/(M/M*)2

Freq. domain: Linearization

Freq. domain: Cubicization

Time domain simulation

1) Time-domain vs.

Frequency-domain;

2) Drag linearization vs.

Drag cubic/quartic

approximation

24240 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

40

ω/ωp

SYY( ω)/(M/M*)2

without inundation

with inundation

Inundation boosts the energy

of wave loads at ωn≈2ωp

and the resultant deck

response there.

0.5 1 1.5 2 2.5 3 3.5 40

ω/ωp

Tri-spectra of wave loads and deck sway (at ω3=0)

Tri-spectrum of wave load4th-order cumulant of wave load

2525

Tri-spectrum of wave load4th-order cumulant of wave load

Tri-spectrum of deck sway4th-order cumulant of deck sway

Statistics of platform deck sway (Time-domain vs. Frequency-domain)

Statistical moments and extreme value

MeanTime-domain 0.018 m

Frequency-domain 0.019 m

VarianceTime-domain 0.0522 m2

Frequency-domain 0.0546 m2Skewness

Time-domain 0.151Frequency-domain 0.351

Time-domain 5.050

26

Kurtosis excessTime-domain 5.050

Frequency-domain 5.262

Extreme valueTime-domain 1.91 m (8.7σ)

Frequency-domain 2.09 m (8.9σ)

1. 200 Time-domain Monte-Carlo simulations, each of duration 3 hour;Δt = 0.33s, very time-consuming.

2. Frequency-domain: CPU Time of only a few hours.3. If the equivalent drag linearization is used, the variance is 0.0467m2, resulting in EV of 0.84m only.

0 20 40 60 80 100 120 140 160 180 2005

5.5

6

6.5

7

7.5

8

8.5

Number of sample functions

Excess of kurtosis of Deck Displacement

Kurtosis excess

Development of averaged statistics of deck sway from time simulations

0 20 40 60 80 100 120 140 160 180 2002

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

Number of sample functions

Mean extreme deck sway (m)

Extreme deck sway

SNAME/2011’s mandatory requirement is farbelow the necessary number of simulations.

28

1) Higher-order spectral analysis is an effective approach to obtain higher-order statistics and the extreme value of offshore structural response. Weextended the research of

Borgman (1965，1967) Lutes & Chen (1991)

Lutes & Papadimitriou (2000) Najafian & Burrows (1994)

Li et al (1995) Tognarelli et al. (1997) Tognarelli & Kareem (1999)

Karunakaran and Spidsoe (1997)

2) Much faster than time-domain simulations when the I/O relationship for

Concluding remarks

2) Much faster than time-domain simulations when the I/O relationship forVolterra series can be explicitly given.

3) But it has difficulty in dealing with nonlinear time variant systems and caseswhose FRFs are not derivable.

[ ]2 1

( ) ( )4 2

m da x u xD

u xf a C C Dπ

ρ ρ= − −+ −+�� � �

Further applications, e.g.:1) Ship motions driven by 2nd-order drift forces.2) Joint wind and wave excitations on offshore wind turbines and induced

structural response: Strength and Fatigue Assessment.

Ongoing projects sponsored by:Ongoing projects sponsored by:Ongoing projects sponsored by:Ongoing projects sponsored by:• The Innovation group DLUT – National Science Foundation (5122-1961)

• State key laboratory of Coastal and Offshore Engineering (SL2012-3)

30

Multi-function Wave Basin (55 x 34 x 0.7) at Dalian University of Technology

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