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

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 Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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) Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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): Zheng/CeSOS-NTNU/2006

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. Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

Third-order Volterra model y 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 Zheng/CeSOS-NTNU/2006

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: 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 Zheng/CeSOS-NTNU/2006

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) Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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) Zheng/CeSOS-NTNU/2006

(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.: Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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 0.07 Fundamental frequency: 0.848 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 = 0.0602 < 0.0625 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 Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

Table 2 Cumulants of modal wave forces (FWD) Mean Variance Skewness Kurtosis excess (i) F 1.1342e+005 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 3 Cumulants of modal displacements (FWD) Mean Variance Skewness Kurtosis excess (i) yF 8.1998e+005 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) 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: Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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 Zheng/CeSOS-NTNU/2006

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

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

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) Brief resume Zheng/CeSOS-NTNU/2006