stochastic analysis of nonlinear wave effects on offshore platform responses xiang yuan zheng,...

Stochastic Analysis of Nonlinear Wave EffectsStochastic Analysis of Nonlinear Wave Effects

on Offshore Platform Responseson Offshore Platform Responses

Xiang Yuan ZHENG, Torgeir MOAN

Centre for Ships and Ocean Structures (CeSOS)

Norwegian University of Science and Technology

Ser Tong QUEK

Centre for Offshore Research & Engineering (CORE)

National University of Singapore

March 23, 2006Zheng/CeSOS-NTNU/2006

The structural responses of fixed offshore platforms tend to be non-Gaussian because of:

1. Morison drag term u|u|

2. Inundation effects (wave fluctuation induced)

3. Wave nonlinearity (Harsh sea states)

Deterministic study of 1, 2, 3 √

Stochastic study (time- & frequency-domain) of 1, 2 √

Stochastic study of 3 ? (higher-order moments)

Zheng/CeSOS-NTNU/2006

Stochastic Analysis of Nonlinear Wave EffectsStochastic Analysis of Nonlinear Wave Effects

on Offshore Platform Responseson Offshore Platform Responses

MAIN TOPICS OF PRESENTATION

A. Statistics of non-Gaussian wave kinematics under second-order wave

B. Frequency-domain analyses of offshore structural responses (to obtain first 4 moments)


Wave elevation (η) & kinematics (u & a)plane-Cartesian coordinate system (x-z); unidirectional wave

(1) Linear random wave theory: (at time t)

11

( , ) cosN

n nn

x t A

11

( , , ) ( ) cosN

n n n nn

u x z t R z A

21

1

( , , ) ( ) sinN

n n n nn

a x z t R z A

φn = knx-ωnt+θn, , θn: uniformly distributed

An: amplitude componentcomponent

ωnRRnn((zz)): transfer function : transfer function for for uu11((x,z,tx,z,t))

(1)


21 1

( , ) cos( ) cos( )N N

n m nm n m nm n mn m

u z t A A p q

(2) 2nd-order nonlinear random wave theory:

for the general case of broad-band wave spectrum &

finite-water depth (Sharma and Dean, 1979)

2n order wave results from interactions between any 2 components 2n order wave results from interactions between any 2 components producing frequency difference and sum producing frequency difference and sum double summations make simulation rather time-consuming double summations make simulation rather time-consuming 2D-FFT most efficient – N=2048 in t < 10 s per realization2D-FFT most efficient – N=2048 in t < 10 s per realization

21 1

( ) cos( ) cos( )N N


t A A r s

21 1

( , ) sin( ) sin( )N N


a z t A A l h

(2)



A. Statistics of non-Gaussian wave/kinematics

A.1. Second-order velocity

(3)

in matrix notation (Langley 1987)::

u(z,t) = M xT + x [Q + P] xT + y [Q - P] yT

xn and yn are standard Gaussian variables, mutually orthogonal

u(z,t) = [M 0] [x y]T + [x y] [D] [x y]T (4)

Where:

D = P1 Λ1 P1T

Λ1 is a diagonal eigenvalue matrix; P1 the orthonormal eigenvector matrixNote - D is symmetric and real


Thus:

(5)

that is a quadratic summation of 2N standard Gaussian variables Xn

The first four cumulants are (5th & higher also obtainable):

Skewness & kurtosis excess (normalized):

meanmean variancevariance


A.2. Second-order acceleration

(6)

in matrix notation:

a(z,t) = G yT + x [H + L] yT + y [H - L] xT

Note H is symmetric while L is skew-symmetric.

In order to follow the procedures for u, a modification is made:

a(z,t) = [0 G] [x y]T + [x y][A] [x y]T

where:

Now A is real & symmetric. Hence, the first four cumulants can be derived, similar to the velocity.


B. Frequency-domain analyses of offshore structural responses

B.1. Approximation of Morison force by Gaussian u1 & a1

(7)

B.1.1. Inertia force: no longer Gaussian as in the linear random wave caseB.1.1. Inertia force: no longer Gaussian as in the linear random wave case

Since Since aa has 0 mean & skewness: has 0 mean & skewness:

B.1.2. Drag force: involves even-degree polynomials due to non-Gaussian B.1.2. Drag force: involves even-degree polynomials due to non-Gaussian uu

(8)

b1 & b3 solved by equalizations of variance & kurtosis

B1, B2 , , B3 & & B4 solved by equalizations of mean, variance, skewness & kurtosis

Solving nonlinear functions


B.2. Third-order Volterra model

Total Morison force on an idealized monopod platform:

(9)

Ψ(z):: mode shape

F is composed of:


yη1

u1

a1

u2

a2

I1

I3

D1

D2

D3

F

I II III IV

Figure 1:

Third-order Volterra model

(I) linear transformations from Gaussian wave elevation to Gaussian kinematics, single-input to multi-output

(II) nonlinear transformations from Gaussian kinematics to non-Gaussian kinematics & associated wave forces, multi-input to multi-output

(III) assemblage of these forces into F, multi-input to single-output (IV) linear transformation from F to deck response y, single-input to single-output

Input-output relationship: four phases


B.2.1. Power spectrum of F (Volterra-series approach)

(10)

1) Forces I & D uncorrelated 2) 2) odd- & even-degree terms uncorrelated

Evaluation of (10) involves bilinear & trilinear transfer functions:

Then the spectrum of structural modal displacement is:

Linear transfer function


B.2.2. Power spectrum of F (Correlation function based)

(11)

Rff(z,z’,τ) is the cross-correlation of 2 Morison forces at z & z’

where:

(12)

e.g.

involves the cross-correlation of Gaussian accelerations Ra1a1(z,z’,τ)


B.3. Tri-spectrum of F

(13)

Assuming that the modal inertia I and modal drag D are independent::

which is the triple FFT of 4th-order cumulant function of F

1 1 2 2 3 3( )1 2 3 1 2 3 1 2 3( , , ) ( , , ) j

FFFF FFFFS R e d d d

1 2 3 1 2 3 1 2 3( , , ) ( , , ) ( , , )FFFF IIII DDDDR R R (14)

1 2 3

2 1 2 3 2 2 2 2 3 1 2 3 2 24 2 3 11

( , , )

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

IIII

I I I I I II

R

mm m m m m m

where:

(15)

because it has only odd-degree polynomial terms of a1

0 0

2 ( ) ( ) ( ) ( ') ( , ', ) 'I I

III f f

d d

m R z z R z z dzdz

is the 2nd-order moment function of I


B.3.1. Fourth-order moment functions of D

(16)

(17)

1 2 3

2 1 2 3 2 2 2 2 3 1 2 3 2 2 1

1 3 2 1 3 1 3 2 3 3 2 4 3 1 2

21 2 1 2 2 2 3 2 3 1 2 3 2

4 1 2 3

2 2

( , , )

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

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

2( ) ( )

( , , )

( ) ( ) ( ) ( )

(

DDDD

D D D D D D

D D D D D

D D D D D D

D

D

R

m m m m m m

m m m m m

m m m m m

m

m

m

4

1 1) 6( )Dm involves 1st, 2nd, 3rd & 4th-order moment functions;

obviously, the 4th-order is the most complicated:

4 1 2 3

1 2 3

0 1 2 3 0 1 1 2 1 3 1

0 1 2 2 2 3 2 0 1 3 2 3 3 3

( , , )

( ) ( ) ( ) ( )

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

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

Dm

E D t D t D t D t

E D D t D t D t D D t D t D t

D D t D t D t D D t D t D t

without wave nonlinearity, it degenerates to (Zheng & Liaw 2003):

4 1 2 3 1 3 1 1 3 1

1 2 3 2 1 3 3 3

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

( ) ( ) ( ) ( )

Dm E D t D t D t D t

D t D t D t D t

(18)


Comparing Eq. (17 & 18), 112 new joint-moment functions of

D0, D1, D2, & D3 will be found, the most intricate is E10

(19) 1 1 2 3 3 3 1 2 2 2 310 ( , , ) [ ( ) ( ) ( ) ( )]thE E D t D t D t D t

which has five other patterns (totally 6/112):

2 1 2 3 3 2 1 2 2 3 310 ( , , ) [ ( ) ( ) ( ) ( )]ndE E D t D t D t D t

3 1 2 3 3 2 1 3 2 2 310 ( , , ) [ ( ) ( ) ( ) ( )]rdE E D t D t D t D t

4 1 2 3 2 3 1 2 2 3 310 ( , , ) [ ( ) ( ) ( ) ( )]thE E D t D t D t D t

5 1 2 3 2 3 1 3 2 2 310 ( , , ) [ ( ) ( ) ( ) ( )]thE E D t D t D t D t

6 1 2 3 2 2 1 3 2 3 310 ( , , ) [ ( ) ( ) ( ) ( )]thE E D t D t D t D t

2 1 2 3 1 3 2 110 ( , , ) 10 ( , , )nd stE E

The following symmetries among them exist to save computation efforts, e.g.:

4 1 2 3 2 1 2 1 3 110 ( , , ) 10 ( , , )th ndE E


B.3.2. Kurtosis excess of structural response

Tri-spectrum of platform modal displacement y is:

*1 2 3 1 2 3 1 2 3 1 2 3( , , ) ( ) ( ) ( ) ( ) ( , , )yyyy Fy Fy Fy Fy FFFFS H H H H S (20)

by triple inverse Fourier Transform, the 4th-order cumulant function of y is:

1 1 2 2 3 3( )1 2 3 1 2 3 1 2 33

1( , , ) ( , , )

(2 )j

yyyy FFFFR S e d d d

(21)

then the kurtosis excess is:

4 2

(0,0,0)

(0)yyyyy

y

R

R (22)


B.4. Case study

Water depth d 75 m

Significant wave height Hs 12.9 m

Peak frequency ωp 0.417 rad/s

peak enhancement factor γ (JONSWAP Spectrum)

3.3

Table 1. Wave Conditions

(1) 120 time simulations (matrix-vector multiplication for simulation)

(2) Δt=0.5 (s), frequency components N=2048

(3) d = 75 m → a finite water depth

(4) Slope = Hs / Lz = 0.0602 < 0.0625 the wave breaking limit;

Lz : wave length at zero-crossing period

1st-mode vibration of structure:Damping ratio 0.07 Fundamental frequency: 0.848 rad/s ≈ 2 ωp


A comparative study among 4 cases:

(i) Without inundation (just F), linear random wave, frequency-domain

(ii) With inundation (Q), linear random wave, frequency-domain

(iii) Without inundation (F), nonlinear random wave, frequency-domain

(iv) Without inundation (F), nonlinear random wave, time-domain

0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

25

30

35

40

45

/p

SY

Y(

)/(M

/M*)

2

(i)(ii)

(iii)

(iv)

Figure 2. Response power spectrum

The contribution of wave nonlinearity

to super-harmonic response at 2ωp

is comparable to that due to inundation


Mean Variance Skewness Kurtosis excess

(i) F 0 1.1342e+005 0 6.0990

(ii) Q 51.3431 1.2714e+005 2.2100 15.9613

(iii) F -13.9402 1.3060e+005 -1.2450 9.9109

(iv) F -15.8551

(-11.4725) DW

1.2899e+005

(8.6734e+004) DW

-1.3194

(-0.9954)DW

9.8450

(7.5137) DW

Table 2 Cumulants of modal wave forces (FWD)

Table 3 Cumulants of modal displacements (FWD)Mean Variance Skewness Kurtosis

excess

(i) yF 0 8.1998e+005 0 2.2750

(ii) yQ 71.3992 1.0277e+006 0.3507 5.2622

(iii) yF -19.4008 1.0198e+006 -0.0347 3.518

(iv) yF -22.0592

(-15.8507) DW

1.0960e+006

(7.2102e+005) DW

-0.1616

(-0.0399) DW

2.8343

(2.0502) DW

1) Agreements between time- & frequency-domain results (iii) & (iv)2) Stronger non-Gaussianities attributable to wave nonlinearity, see larger skewness & kurtosis excess, compare (i) with (iii)3) Force kurtosis excess even larger than 8.6674) Deep-water wave theory results in underestimations of non-Gaussianities

Findings:


Figure 3. Tri-spectrum of wave force F (ω3=0)

-20

2

-2

0

2

0

2

4

6

8

10

12

x 109

1/

p

2/

p

SF

FF

F(

1,

2,

3)

(N4s4)

-20

2

-2

0

2

0

2

4

6

8

10

x 1010

1/

p

2/

p

SF

FF

F(

1,

2,

3)

(N4S4)

Linear random waves vs. Nonlinear random waves

Shaper peaks at 2ωp indicates stronger non-Gaussian behavior


Concluding Remarks

(1) A modified eigenvalue/eigenvector approach suggested for (1) A modified eigenvalue/eigenvector approach suggested for wave/kinematics statistics (acceleration)wave/kinematics statistics (acceleration)

(2) Cumulant spectral analyses for platform response prediction(2) Cumulant spectral analyses for platform response prediction

(3) Non-negligible nonlinear wave effects on platform response (3) Non-negligible nonlinear wave effects on platform response (stronger non-Gaussian behavior of response)(stronger non-Gaussian behavior of response)

(4) Based on first 4 moments, the extreme value estimation can be (4) Based on first 4 moments, the extreme value estimation can be performed (Winterstein 1988)performed (Winterstein 1988)

Extreme Value Estimation

Based on first 4 moments (Winterstein 1988)Based on first 4 moments (Winterstein 1988)

Using the obtained mean, variance, skewnes & kurtosis excess (Using the obtained mean, variance, skewnes & kurtosis excess (mm, , σσ, , кк3, , кк4), the), the

platform response can be approximated by Hermite transformation (platform response can be approximated by Hermite transformation (monotonicmonotonic):):

uu((tt) is a standard Gaussian process, of which the mean extreme is:) is a standard Gaussian process, of which the mean extreme is:

Peaks ofPeaks of u u((tt) is approximately Rayleigh distributed) is approximately Rayleigh distributed

It follows that the response extreme is (for monotonic case):It follows that the response extreme is (for monotonic case):

3

1)()(

i iUY y

uufyf

Thank you !


Future work ?Future work ?

•22ndnd order wave nonlinearity on inundation effects order wave nonlinearity on inundation effects

•33rdrd order wave nonlinearity order wave nonlinearity

•Floating structures.Floating structures.

Education

B. E. Offshore Engineering, Tianjin University, China, 1996 M. Sc. Earthquake Engineering, Institute of Engineering Mechanics,

China Seismological Bureau, China, 1999 Ph. D. National University of Singapore (NUS), Singapore, 2003

Research & Teaching

2003-2004, Research Fellow (NUS) 2004-2005, Teaching Fellow (NUS) 2005-2006, Pos Doc (NTNU)


stochastic analysis of nonlinear wave effects on offshore platform responses xiang yuan zheng,...

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