Stochastic Analysis of Nonlinear Wave Effects

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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

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The structural responses of fixed offshore
platforms tend to be non-Gaussian because of
Stochastic Analysis of Nonlinear Wave Effects on
Offshore Platform Responses

• Morison drag term uu
• Inundation effects (wave fluctuation induced)
• Wave nonlinearity (Harsh sea states)
• Deterministic study of 1, 2, 3 v
• Stochastic study (time- frequency-domain) of 1,
  2 v
• Stochastic study of 3 ? (higher-order moments)

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MAIN TOPICS OF PRESENTATION

1. Statistics of non-Gaussian wave kinematics under
   second-order wave
2. Frequency-domain analyses of offshore structural
   responses (to obtain first 4 moments)

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Wave elevation (?) kinematics (u a)

• plane-Cartesian coordinate system (x-z)
  unidirectional wave
• (1) Linear random wave theory (at time t)

(1)
fn knx-?nt?n, ?n uniformly distributed An
amplitude component ?nRn(z) transfer function
for u1(x,z,t)
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• (2) 2nd-order nonlinear random wave theory
• for the general case of broad-band wave
  spectrum
• finite-water depth (Sharma and Dean, 1979)

(2)
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 N2048 in
t lt 10 s per realization
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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 yT x y D x yT
(4)
Where
D P1 ?1 P1T
?1 is a diagonal eigenvalue matrix P1 the
orthonormal eigenvector matrix Note - D is
symmetric and real
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Thus
(5)
that is a quadratic summation of 2N standard
Gaussian variables Xn
The first four cumulants are (5th higher also
obtainable)
mean
variance
Skewness kurtosis excess (normalized)
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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 yT x yA x yT
where
Now A is real symmetric. Hence, the first four
cumulants can be derived, similar to the velocity.
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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)
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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B.2. Third-order Volterra model
Total Morison force on an idealized monopod
platform

(9)
?(z) mode shape
F is composed of


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Figure 1 Third-order Volterra model
Input-output relationship four phases

1. linear transformations from Gaussian wave
   elevation to Gaussian kinematics, single-input to
   multi-output
2. nonlinear transformations from Gaussian
   kinematics to non-Gaussian kinematics
   associated wave forces, multi-input to
   multi-output
3. assemblage of these forces into F, multi-input to
   single-output
4. linear transformation from F to deck response y,
   single-input to single-output

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B.2.1. Power spectrum of F (Volterra-series
approach)
(10)

1) Forces I D uncorrelated 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
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B.2.2. Power spectrum of F (Correlation function
based)
(11)
where
(12)

Rff(z,z,t) is the cross-correlation of 2 Morison
forces at z z

e.g.
involves the cross-correlation of Gaussian
accelerations Ra1a1(z,z,t)

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B.3. Tri-spectrum of F
(13)
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

is the 2nd-order moment function of I
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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)
without wave nonlinearity, it degenerates to
(Zheng Liaw 2003)
(18)
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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)
The following symmetries among them exist to save
computation efforts, e.g.
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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)
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B.4. Case study
Table 1. Wave Conditions
Water depth d 75 m
Significant wave height Hs 12.9 m
Peak frequency ?p 0.417 rad/s
peak enhancement factor ? (JONSWAP Spectrum) 3.3
1st-mode vibration of structure Damping ratio
0.07 Fundamental
frequency 0.848 rad/s 2 ?p

• 120 time simulations (matrix-vector
  multiplication for simulation)
• ?t0.5 (s), frequency components N2048
• (3) d 75 m ? a finite water depth
• (4) Slope Hs / Lz 0.0602 lt 0.0625 the wave
  breaking limit
• Lz wave length at zero-crossing period

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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 contribution of wave nonlinearity to
super-harmonic response at 2?p is comparable to
that due to inundation
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Table 2 Cumulants of modal wave forces (FWD)
Mean Variance Skewness Kurtosis excess
(i) F 0 1.1342e005 0 6.0990
(ii) Q 51.3431 1.2714e005 2.2100 15.9613
(iii) F -13.9402 1.3060e005 -1.2450 9.9109
(iv) F -15.8551 (-11.4725) DW 1.2899e005 (8.6734e004) DW -1.3194 (-0.9954)DW 9.8450 (7.5137) DW
Table 3 Cumulants of modal displacements (FWD)
Mean Variance Skewness Kurtosis excess
(i) yF 0 8.1998e005 0 2.2750
(ii) yQ 71.3992 1.0277e006 0.3507 5.2622
(iii) yF -19.4008 1.0198e006 -0.0347 3.518
(iv) yF -22.0592 (-15.8507) DW 1.0960e006 (7.2102e005) DW -0.1616 (-0.0399) DW 2.8343 (2.0502) DW

• 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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Figure 3. Tri-spectrum of wave force F (?30)
Linear random waves vs.
Nonlinear random waves
Shaper peaks at 2?p indicates stronger
non-Gaussian behavior
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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)

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Extreme Value Estimation Based on first 4
moments (Winterstein 1988)
Using the obtained mean, variance, skewnes
kurtosis excess (m, s, ?3, ?4), 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)
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• Future work ?
• 2nd order wave nonlinearity on inundation effects
• 3rd order wave nonlinearity
• Floating structures.

Thank you !
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• 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)

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