1. Stochastic Analysis of Nonlinear Wave Effects
on 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, 2006
Thanks for this opportunity to speak to you!
Zheng/CeSOS-NTNU/2006

2. Stochastic Analysis of Nonlinear Wave Effects
on Offshore Platform Responses
The structural responses of fixed offshore platforms tend to be non-Gaussian because of:
Morison drag term u|u|
Inundation effects (wave fluctuation induced)
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)
a) 3 nonlinearities b) Stochastic study (time- & frequency-domain) of 1 &2, refer to Li et al 1995 & Tognarelli et al. 1997, Kareem et al. 1998, Zheng & Liaw 2003
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3. MAIN TOPICS OF PRESENTATION
Statistics of non-Gaussian wave kinematics under second-order wave
Frequency-domain analyses of offshore structural responses (to obtain first 4 moments)
outline
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4. Wave elevation (η) & kinematics (u & a)
plane-Cartesian coordinate system (x-z); unidirectional wave
(1) Linear random wave theory: (at time t)
(1)
1) Central limit theorem for N → ∞ leads to Gaussianity 2) Linear transform of a Gaussian process is still Gaussian (Papoulis 1991)
φn = knx-ωnt+θn, θn: uniformly distributed
An: amplitude component
ωnRn(z): transfer function for u1(x,z,t)
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5. for the general case of broad-band wave spectrum &
(2) 2nd-order nonlinear random wave theory:
for the general case of broad-band wave spectrum &
finite-water depth (Sharma and Dean, 1979)
(2)
Coefficients p, q, r, s, l, h could be symmetric, e.g., pnm = pmn, or skew-symmetric lnm= - lmn
2n order wave results from interactions between any 2 components
producing frequency difference and sum
double summations make simulation rather time-consuming
2D-FFT most efficient – N=2048 in t < 10 s per realization
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6. 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:
1) P, Q, D is real & symmetric 2) Also applicable to wave elevation, because of interaction matrix is symmetric about m & n
D = P1 Λ1 P1T
Λ1 is a diagonal eigenvalue matrix; P1 the orthonormal eigenvector matrix
Note - D is symmetric and real
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7. Thus:
(5)
that is a quadratic summation of 2N standard Gaussian variables Xn
The first four cumulants are (5th & higher also obtainable):
mean
variance
1. Xn is also orthonormal 2. Also applicable to wave elevation, because of interaction matrix is symmetric about m & n
Skewness & kurtosis excess (normalized):
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8. 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:
1. New matrix along the secondary diagonal 2. Skewness of a is always 0
Now A is real & symmetric. Hence, the first four cumulants can be derived, similar to the velocity.
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9. B. Frequency-domain analyses of offshore structural responses
B.1. Approximation of Morison force by Gaussian u1 & a1
B.1.1. Inertia force: no longer Gaussian as in the linear random wave case
Since a has 0 mean & skewness:
(7)
B.1.2. Drag force: involves even-degree polynomials due to non-Gaussian u
(8)
1) Wave-structure interaction taken into account by adjusting the damping ratio 2) Kareem et al. (1998) missed the u13(z,t) term
b1 & b3 solved by equalizations of variance & kurtosis
B1, B2 , B3 & B4 solved by equalizations of mean, variance, skewness & kurtosis
Solving nonlinear functions
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10. B.2. Third-order Volterra model
Total Morison force on an idealized monopod platform:
(9)
Ψ(z): mode shape
F is composed of:
Wave-structure interaction taken into account by adjusting the damping ratio
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11. Third-order Volterra modely η1 u1 a1 u2 a2 I1 I3 D1 D2 D3 F I II
III IV
Figure 1:
Third-order Volterra model
Input-output relationship: four phases
linear transformations from Gaussian wave elevation to Gaussian kinematics, single-input to multi-output
nonlinear transformations from Gaussian kinematics to non-Gaussian kinematics & associated wave forces, multi-input to multi-output
assemblage of these forces into F, multi-input to single-output
linear transformation from F to deck response y, single-input to single-output
It is a finite-memory model, with linear transformations before & after the zero-memory processes
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12. B.2.1. Power spectrum of F (Volterra-series approach)
(10)
1) Forces I & D uncorrelated ) odd- & even-degree terms uncorrelated
Evaluation of (10) involves bilinear & trilinear transfer functions:
1. Details of force F spectrum not given here 2. Inundation force P spectrum not given here (Zheng & Liaw 2003) 3. P correlated with F
Then the spectrum of structural modal displacement is:
Linear transfer function
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13. B.2.2. Power spectrum of F (Correlation function based)
(11)
where:
(12)
Rff(z,z’,τ) is the cross-correlation of 2 Morison forces at z & z’
e.g.
Inverse Fourier transform for cross-correlation functions of kinematics, straightforward to be obtained
involves the cross-correlation of Gaussian accelerations Ra1a1(z,z’,τ)
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14. B.3. Tri-spectrum of F (13) (14) (15)
which is the triple FFT of 4th-order cumulant function of F
Assuming that the modal inertia I and modal drag D are independent:
(14)
where:
(15)
because it has only odd-degree polynomial terms of a1
1) Bi-spectrum of F not presented here, because all procedures can be included by those of tri-spectral analysis 2) in spectral analysis, I & D are uncorrelated, but in tri-spectral analysis, we have to assume their independence
is the 2nd-order moment function of I
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15. B.3.1. Fourth-order moment functions of D
(16)
involves 1st, 2nd, 3rd & 4th-order moment functions;
obviously, the 4th-order is the most complicated:
(17)
The 4th-order moment function of I has the same tri-spectral procedures as those of D under linear random wave theory
without wave nonlinearity, it degenerates to (Zheng & Liaw 2003):
(18)
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16. (19) Comparing Eq. (17 & 18), 112 new joint-moment functions of
D0, D1, D2, & D3 will be found, the most intricate is E10
(19)
which has five other patterns (totally 6/112):
In addition to the symmetries, the page-t-page scheme (Zheng & Liaw 2003) can be used to facilitate computation of E10
The following symmetries among them exist to save computation efforts, e.g.:
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17. B.3.2. Kurtosis excess of structural response
Tri-spectrum of platform modal displacement y is:
(20)
by triple inverse Fourier Transform, the 4th-order cumulant function of y is:
(21)
then the kurtosis excess is:
(22)
1) Triple-FFT is much fast than triple integrals 2) No. of FFT points 64 or 128 can render good accuracy
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18. B.4. Case study Table 1. Wave Conditions
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
1st-mode vibration of structure:
Damping ratio Fundamental frequency: rad/s ≈ 2 ωp
120 time simulations (matrix-vector multiplication for simulation)
Δt=0.5 (s), frequency components N=2048
(3) d = 75 m → a finite water depth
(4) Slope = Hs / Lz = < the wave breaking limit;
Lz : wave length at zero-crossing period
1) The wave conditions are especially for North Sea 2) the double summations for simulating η, u & a can be conducted by vector-matrix multiplications
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19. Figure 2. Response power spectrum
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
Figure 2. Response power spectrum
The frequency-domain results of
The contribution of wave nonlinearity
to super-harmonic response at 2ωp
is comparable to that due to inundation
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20. Table 2 Cumulants of modal wave forces (FWD)
Mean
Variance
Skewness
Kurtosis
excess
(i) F
1.1342e+005
6.0990
(ii) Q
1.2714e+005
2.2100
(iii) F
1.3060e+005
9.9109
(iv) F
( ) DW
1.2899e+005
(8.6734e+004) DW
( )DW
9.8450
(7.5137) DW
Table 3 Cumulants of modal displacements (FWD)
Mean
Variance
Skewness
Kurtosis
excess
(i) yF
8.1998e+005
2.2750
(ii) yQ
1.0277e+006
0.3507
5.2622
(iii) yF
1.0198e+006
3.518
(iv) yF
( ) DW
1.0960e+006
(7.2102e+005) DW
( ) DW
2.8343
(2.0502) DW
1) Inundation leads to positive skewness, while wave nonlinearity negative 2) they could cancel off when acting together
Agreements between time- & frequency-domain results (iii) & (iv)
Stronger non-Gaussianities attributable to wave nonlinearity,
see larger skewness & kurtosis excess, compare (i) with (iii)
Force kurtosis excess even larger than 8.667
Deep-water wave theory results in underestimations of non-Gaussianities
Findings:
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21. Figure 3. Tri-spectrum of wave force F (ω3=0)
Linear random waves vs Nonlinear random waves
Shaper peaks at 2ωp indicates stronger non-Gaussian behavior
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22. Concluding Remarks
(1) A modified eigenvalue/eigenvector approach suggested for wave/kinematics statistics (acceleration)
(2) Cumulant spectral analyses for platform response prediction
(3) Non-negligible nonlinear wave effects on platform response (stronger non-Gaussian behavior of response)
(4) Based on first 4 moments, the extreme value estimation can be performed (Winterstein 1988)
Future works be extended to consider 3rd-order nonlinear wave theories, which is not yet established
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23. Extreme Value Estimation Based on first 4 moments (Winterstein 1988)
Using the obtained mean, variance, skewnes & kurtosis excess (m, σ, к3, к4), the
platform response can be approximated by Hermite transformation (monotonic):
u(t) is a standard Gaussian process, of which the mean extreme is:
Peaks of u(t) is approximately Rayleigh distributed
It follows that the response extreme is (for monotonic case):
(1) Coefficients of Hermite transformation can be exactly estimated by moment equalization (Liaw & Zheng 2004), the empirical fit results were given by Winterstein (1994) (2) N is the number of independent amplitudes of u(t), estimated as T/Tz, T is the record length, Tz is the average zero-crossing period

24. Future work ?
2nd order wave nonlinearity on inundation effects
3rd order wave nonlinearity
Floating structures.
Thank you !
Questions?
Zheng/CeSOS-NTNU/2006

25. 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
, Research Fellow (NUS)
, Teaching Fellow (NUS)
, Pos Doc (NTNU)
Brief resume
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