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Applied Ocean Resem-ch 28 (2006) 1^8www.elsevier.coni/locate/apor
Numerical methods for calculating the crossing rate of high and extreme
response levels of compliant offshore structures subjected to random waves
A. Naess^''^'*, H.C. Kaiisen'^, P.S. Teigen^
''' Centre for Ships and Ocean Structures, Nonvegian Universit)' of Science and Teciinology, A. Gelz vei 1, NO-7491, Trondlieim, Norway ^ Department of Mathematical Sciences, Norwegian University of Science and Tedmology, A. Getz vei J, NO-7491, Trondheim, Norway
Statoil Research Centre, RotvoU, Trondheim, Norway
Received 16 December 2005; received i n revised form 3 April 2006; accepted 8 April 2006 Available online 5 June 2006
Abstract
The focus of the paper is on methods for calculating the mean upcrossing rate of stationary stochastic processes that can be represented as second order stochastic VolteiTa series. This is the cuiTent state-of-the-art representation of the horizontal motion response of e.g. a tension leg platform i n random seas. Until recently, there has been no method available for accurately calculating the mean level upcrossing rate of such response processes. Since the mean upcrossing rate is a key parameter for estimating the large and extreme responses it is clearly of importance to develop methods for its calculation. The paper describes in some detail a numerical method for calculating the mean level upcrossing rate of a stochastic response process of the type considered. Since no approximations are made, the only source of inaccuracy is m the numerical calculation, which can be controlled. In addition to this exact method, two approximate methods are also discussed.
© 2006 Elsevier Ltd. A l l rights reserved.
Keywords: Second order stochastic VolteiTa model; Mean crossing rate; Extreme response; Slow drift response; Method of steepest descent
1. Introduction
The problem of calculating the extreme response of compliant offshore structures like tension leg platforms or moored spar buoys in random seas, has been a challenge for many years, and, in fact, it still represents a challenge. Starting with the state-of-the-art representation of the horizontal excursions of moored, floating offshore structures in random seas as a second order stochastic VolteiTa series, we shah in this paper develop a general method for estimating the extreme response of such stnictures. Even i f the VolteiTa series model was formulated more than 30 years ago, it is not until quite recently that general numerical methods have become available that allow accurate calculation of the probability distribution and, perhaps more importantly, the mean upcrossing rate of the total response process. This last quantity is the cmcial parameter for estimating extreme responses.
During the 1980s significant efforts were directed towards developing methods for calculating the response statistics of
* Con-esponding author. Tel.: 4-47 73 59 70 53; fax: -f47 73 59 35 24.
E-mail address: ai-vidn@math.ntnu.no (A. Naess).
0141-1187/$ - see front matter © 2006 Elsevier Ltd. A l l rights reserved. doi:10.1016/j.apor.2006.04.001
compliant offshore structures subjected to random waves. The list of contributions is long. To mention but a few, which also contain references to other work focussing on response statistics, see [1-6]. However, none of these works succeeded in developing a general method that made it possible to calculate the exact statistical distribution of the total response process, not to mention the much harder problem of calculating the mean upcrossing rate.
2. The response process
The response process Z ( f ) that is considered here is assumed to be represented as a second order stochastic VolteiTa series. This would apply to the state of the art representation of e.g. the surge response of a large volume, compliant offshore stmcture in random waves. This response would consist of a combination of the wave frequency component Z i (f) and the slow-drift component ZiO)- that is, Z{t) = Z ^ O + Ziit). Naess [6] describes the standard representation of the two response components leading to a second order Volterra series model for the total response. To alleviate the statistical analysis of the response process, it has been shown [2,6] that the slow-drift response Z2(0 can be expressed as
N
Z 2 ( 0 = 5] {1^2^-1(0^ + W2j{t)^}. (1)
; = i
Here Wjit), j = l,...,2N are real stationary Gaussian iV(0, l)-processes. The coefficients iij are obtained by solving the eigenvalue problem (assumed nonsingular)
Quj = iijUj (2)
to find the eigenvalues [tj and orthonormal eigenvectors iij,
j = l,...,N,ofÜieN X A?-matrix Q = (Qij), where
Qij = H2(oJi, -Wj) • ^[Sx(oJi)Sx{coj)]^^^Aoj. (3)
Here H2{_a>, co') denotes the quadratic transfer function between the waves and the surge response, cf. [2,6], denotes the one-sided spectral density of the waves, and 0 < oji < • • • < (W/v is a suitable discretization of the frequency axis. The stochastic processes Wj (t) can be represented as follows (i = ^ / ^ )
N
W2j-dt) + i W2j{t) = V2j2i^ji(^k)Bké'"'' (4) k=l
where Ujiojk) denotes the klh component of Uj and [Bk]
is a set of independent, complex Gaussian A''(0, l)-variables with independent and identically distributed real and imaginary parts. The representation can be arranged so that W2j{t) becomes the Hilbert transform of Wzj-iiO, cf. [6]. For each fixed f, {Wj(t)] becomes a set of independent Gaussian variables.
Having achieved the desired representation of the quadratic response Z2(0> it can then be shown that the linear response can be expressed as
2^
Ziit) = ^ l 3 j W j { t ) . (5)
The (real) parameters Pj are given by the relations
N
Pij^l + i Plj ^Y^Hi(cok)* ./Sx{(Ok)Aa) Uj{(Dk) (6)
/f=l
where Hi (co) denotes the linear transfer function between the waves and the surge response. Based on the representations given by Eqs. (1) and (5), [11] describes how to calculate the statistical moments of the response process Z{t), while a general and accurate numerical method for calculating the PDF of Zit) is given in [7]. However, for important prediction purposes, like extreme response estimation, the ciucial quantity is the mean rate of level upcrossings by the response process. 3. The mean crossing rate
Let N^{^) denote the rate of upcrossings of the level
t; by Z ( f ) , cf. [12], and let = E[N+iO], that is, v^iO denotes the mean rate of upcrossings of the level f . As
discussed in [9], under suitable regularity conditions on the response process, which can be adopted here, the following formula can be used
4 ( 0 - / sf^^{^,s)ds (7)
Jo
where f^^i-, •) denotes the joint PDF of Z(0) and Z(0) = dZ(f)/df|,=o- Eq. (7) is often refeiTed to as the Rice formula [13]. v j ( 0 is assumed throughout to be finite.
Calculating the mean crossing rate of a stochastic process represented as a second order stochastic VolteiTa series directiy from Eq. (7) has turned out to be very difficult due to the difficulties of calculating the joint PDF fzz(-> O- However, this can be circumvented by involdng the concept of characteristic function.
Denote the characteristic function of the joint variable (Z, Z) by M^zi-, •). or, for simphcity of notation, by M(-, •). Then
M ( H , V) = M^ziii, v) = £[exp(i!(Z -|- i i ; Z ) ] . (8) Assuming that M(-, •) is an integrable function, that is,
M ( - , •) e L \ R ^ ) , it follows that
X exp ( - i ! ( f - ivs) dudv. (9) By substituting from Eq. (9) back into Eq. (7), the mean
crossing rate is formally expressed in terms of the characteristic function, but this is not a very practical expression.
The solution to this is obtained by considering the characteristic function as a function of two complex variables. It can then often be shown that this new function becomes holomorphic in suitable regions of C^, where C denotes the complex plane. As shown in detail in [14], under suitable conditions, the use of complex function theory allows the derivation of two alternative expressions for the crossing rate. Here we shall focus on one of these alternatives, viz.
1 roo—ia rco—ib i
v + ( f ) = - / / — M ( z , u;)e-i^fdzdu) (10) where 0 < A < ai for some positive constant a i , and bo < b <
3
To actually cany out the calculations, the joint characteristic function needs to be known. It has been shown [8] that for the second order stochastic Volterra series, it can be given in closed form. To this end, consider the multidimensional Gaussian vectors W = (W\, ..., Wn)' (' denotes transposition) and W ( W i , . . . , W„y, where u = It is obtained that the covariance matrix of (W', W')' is given by
-£21 ^22 (11)
where E\\ = I = the /; x n identity matrix, Sn =
( n j ) = (ElWiWj]), S21 = (ElWiWj]) and E22 = (sij) = (ElWiWj])- i,./ = 1, . . . , n. = - r j , and E12 = S^^. I t
follows from Eq. (4), that the entries of the covariance matrix
S can be expressed in terms of the eigenvectors 11 j, cf. [2]. Let N
^•J = ^ ( - i « A ' ) « , - ( ( W A ' ) " ; ('"<:)*•
A'=l
Then it can be shown that
' • 2 , - l , 2 y - l = '•2r,2j = ^ ( R i j ) while ' • 2 ; - l , 2 ; = - ' • 2 / , 2 ; - I = ^(Rij) (12) (13) (14) where ^(z) denotes the real part and ü(z) the imaginary part of
z. Similarly, let Sij = N 'Y2^k"i(-^k)llj(0Jk)* k=l Then •S2r-1,2;-1 = ^2!,2j = ^ ( S j j ) while •S'2(-l,2j = - • y 2 ! , 2 j ^ l =
SiSjj).
(15) (16) (17) By this, the covariance matrix E is completely specified.It is convenient to introduce a new set of eigenvalues Xj,
j = 1, . . . , n defined by A2,--i — X2i — l^i, ' = 1, . . . , A^.
Let / i = diag(Ai, . . . , A„) be the diagonal matrix with the parameters Xj on the diagonal, and let = (Pi,..., ji,,)'. It can now be shown that [8]
M(w, v) = exp where l n ( d e t ( A ) ) - ^ i ; 2 / ) ' y ^ + \ t ' t / I = / - 2 i H / I - 2 i ri (/11721 + ^12 /I) + 4 v^j\ V / l V = S22- S21 S\2 t = {iul + ivSi2-2v^ / l Y ^ p . (18) (19) (20) (21) 4. Numerical calculation
Previous efforts to carry out numerical calculation of the mean crossing rate using Eq. (10) have been reported in [9]. These initial iiwestigations indicated that the method had the potential to provide very accurate numeiical results. We shall rewrite Eq. (10) as follows
Y rcc—ia | (2n)^ J - c o - i a (22) where / = I{?,w) = ƒ M(z,w) e-'^^dz co-ib CO—ii CO—ii> e x p { - i z f + l n M ( z , w)}dz. (23) A numerical calculation of the mean crossing rate can start by calculating the function / ( ^ , w) for specified values of f and
w. However, a direct numerical integration of Eq. (23) is made
difficult by the oscillatory term exp{-iSH(z)f}. This problem can be avoided by involdng the method of steepest descent, also called the saddle point method. For this purpose, we write
g(z) = g(z; w) = - i z f +\nM{z, w)
= ( P ( x , y ) + i f i x , y ) (24)
where z = x + iy. (j)(x, y) and ^(x, y) become real harmonic functions when g(z) is holomorphic. The idea is to identify the saddle point of the suiface (x, y) (j>(x, y) closest to the integration line from - 0 0 - i& to 00 - \b. By shifting this integration line to a new integration contour that passes through the saddle point, and then follows the path of steepest descent away from the saddle point, it can be shown that the function i/f(x, }') stays constant, and therefore the oscillatory teim in the integral degenerates to a constant. This is a main advantage of the method of steepest descent for numeiical calculations. I t can be shown that the integral does not change its value as long as the function g{z) is a holomoiphic function in the region bounded by the two integration contours and i f the integrals vanish along the contour segments required to close the region.
I f Zs denotes the identified saddle point, where g'{zs) —
0, the steepest descent path away from the saddle point will follow the direction given by -g'{z)*, for z Zs, cL [15]. Typically, the singular points of the function g w i l l be around the imaginary axis, which indicates that the direction of the paths of steepest descent emanating from the saddle point will typically not deviate substantially from a direction orthogonal to the imaginary axis. This provides a guide for setting up a numerical integration procedure based on the path of steepest descent. First the saddle point Zs is identified. Then the path of steepest descent starting at Zs and going 'right' is approximated by the sequence of points {Z; } y l o calculated as follows:
ZO = Zj zi=Zs+ h (25)
Azj = h, j = l , 2 .
\g'{Zj)\
^j+'^=Zj + Azj, ; = 1 , 2 , . . .
where h is a small positive constant.
Similarly, the path of steepest descent going 'left' is approximated by the sequence {zj calculated by
Z-\ ^ Z s - h \g'iZj)\ Zj-i=^Zj + Azj, ; = - l , - 2 . (28) (29) (30) A numerical estimate / of / can be obtained as follows.
I =
I++ r
(31)where
/ + = ^ exp{g(z,)} 4- exp{g(z_,-)} (32)
and
ƒ - = - exp{g(z,)} - J2 ^^J exp{g(zj)} (33)
for a suitably large integer K.
A numencal estimate v|"(0 of the mean crossing rate can
now be obtained by the sum
4 ( 0
-
(2 7r)2 • 9 Ï I ( ^ , W j ) A w jwhere the discretization points wj are chosen to follow the negative real axis from a suitably large negative number up to a point at - e , where 0 < e < a, then follow a semi-circle in the lower half plane to s on the positive real axis, and finally follow this axis to a suitably large positive number. Since the numerical estimate does not necessarily have an imaginary part that is exactly equal to zero, the real part operator has been applied.
Generally, the CPU time required to cany out the computations above can be quite long, depending on the size of the problem, which is related to the number N of eigenvalues. It is therefore of interest to see i f approximating formulas are accurate enough. The first such approximation we shall have a look at is the Laplace approximation for the inner integral over the saddle point [15]. The simplest version of this approximation, adapted to the situation at hand, leads to the result
Ii^,w) 2TT
3.v2
exp{g(zs; w)} (35)
which can be substituted directly into Eq. (34), leading to an approximation of (0. which is denoted by
v^iO-This approximation can be exploited in the following way; (1) The f u l l method is used for an inner interval of w-values, which contribute significantiy to the integral in Eq. (22). (2) A Laplace approximation is then used in an outer interval of w-values where the contribution is less than significant. Of course, the level of significance is chosen according to some suitable
criterion. By this procedure, the CPU time was reduced by a factor of about 3. This method w i l l be refened to as the hybrid method, and the corresponding approximation of (O is denoted by
4(0-A simple approximation proposed in [16,17] is worth a closer scrutiny. It is based on the widely adopted simpUfying assumption that the displacement process is independent of the velocity process. This leads to an alternative approximation of v j ( f ) , which we shall denote by 4(0- It is given by the formula
where f z denotes the marginal PDF of the surge response, and fref denotes a suitable reference level, typically the mean response. Here, fref has been chosen as the point where f z assumes its maximum, which corresponds well with the mean response level. A general approximation for v^i^ief) is given in [17]. I f only slow-drift response is considered, a good approximation is obtained by putting v|"(Cref) ~ l / ^ b , where To = 2n/cüo is the slow-drift period. The advantage of Eq. (36) is that the rhs is much faster to calculate than the exact formula.
An approximation developed by Langley and McWiUiam [18] expresses the joint PDF of Z(/) and' Z(t) as a series i n the following way
(34) f z z i z , ^ ) = f z i z ) f z i s )
oo oo
+ ^ ^ ( _ i ) " + " ' A „ „ , / r ( z ) / f \ . )
/(=1 m=l
(37)
where (z) is the ;?th derivative of / z ( z ) , and f^"\s) is the
mth derivative of fzis). fz(z) and fz{s) are given by infinite
series expressions, which are truncated in practice, cf. [18, 10]. The coefficients A,,,,, are defined in terms of cumulants for the displacement and velocity. Expressions for calculating die mean crossing rate are provided in [19]. Since explicit expressions are given, the coiuputational burden incurred by adopting this approximation is practically negligible. For an example structure studied in [10], it is shown that the approximation expressed in Eq. (37) gives results very similar to Eq. (36).
5. Distribution of extreme response
To provide estimates of the extreme values of the response process, it is necessary to know the probability law of the extreme value of Z{t) over a specified period of time T, that is, of Mz{T) = max{Z(0; 0 < t < T}. An exact expression for this probability law is in general unknown, but a good approximation is usually obtained by assuiuing that upcrossings of high response levels are statistically independent events. Under this assumption, the probabihty distribution of Mz{T) can be written as
Prob(Mz(r) <0 =
exp(-4(0
T} (38)5
As already pointed out, Eq. (38) is only an approximation to the true extreme value distribution. I f this is so, why is it of practical significance to be able to perform an accurate calculation of the upcrossing rate? It is a well-known fact that the exact extreme value distribution is not completely determined by the upcrossing rate alone. That would be true only when the upcrossing events of the high response levels can be assumed to be statistically independent. Usually that is a good approximation except when the total damping is small. For such cases, Naess [20] has developed an effective method to account for the effect of low damping on the extreme value distribution, and the key to an accurate estimation of the extreme value distribution is good estimates of the luean upcrossing rate at high response levels.
So far the analysis has been limited to a short-term sea state characterized by a wave spectrum. This means that the extreme value distribution of Eq. (38) applies to the response values within this short-term sea state. In the context of a design situation, this would imply that the sea state has to be chosen as a design sea state. In practice this would usually mean that the spectral parameters would be chosen along a sea state contour having a presciibed return period.
A n alternative approach is to use a long-term analysis. Let us assume that a short-term sea state is characterized by significant wave height H, and a spectral period 7;, which have a j oint PDF
/H^TA'''' 0- Then, for each set of values = h and Ts = t,
the mean upcrossing rate can be calculated as we have described above. Let i t be denoted by v+(f | /;, t). As shown by [21], the distribution of the long-term extreme value over the time period
T is given as follows Prob(Mz(r) < O
= exp
Jo Jo
| ( f I h,t)fH,TAh,r)dhdi (39) Haver and Nyhus [22] have proposed a model for the joint PDF
fH,T,{!ht).
6. Numerical examples
The numerical results presented in this section are based on a specific example structure. It is a moored deep floater (MDF) with main particulars as detailed in Table 1. Fig. 1 shows the submerged part of the floater in the form of a computer mesh, which is used for the calculation of the hydrodynamic transfer functions. The total mass (including added mass) of the MDF is
M = 12.5 X 10^ kg. The damping ratio is set equal to f = 0.06,
and the natural frequency in sway is OJQ = 0.047 rad/s. Note that the second order theory is based on the assumption that the QTF H2((ai, -coj) = L{coi - ojj) # 2 ( f t ) / , -ooj), where # 2 ( ' , •)
is a QTF characterizing the slowly varying surge forces on the MDF, and L(-) is a linear transfer function for the surge motion of the MDF, that is
L ( « ) = 1 (40)
Table 1
Main particulars of the M D F Drauglit (in)
Column diameter (m) Natural period surge/sway (s) Natural period yaw (s)
80.0 10.0 133.5 121 ^ 0 - 2 0 -30¬ -40¬ -50¬ -60¬ 7 0
-Fig. 1. Computer mesh of the submerged part of the moored deep floater
The random stationary sea state is specified by a lONSWAP spectrum, which is given as follows
Sx(co) = "-^
(41) where g = 9.81 ms^ denotes the peak frequency in rad/s and a, y and a are parameters related to the spectral shape, a = 0.07 when co < cOp, and cr = 0.09 when co > 0}p. The parameter y is chosen to be equal to 3.0. The parameter a is determined from the following empuical relationship
5.06 (1 - 0.287 I n y ) (42)
Hs = sigiuficant wave height and Tp = 27T/Wp= spectral peak
wave period. For the subsequent calculations, Hs — 10.0 m and
Tp — 12 s. The natural frequency in surge is 0.047 rad/s, which
I I 1 1 1 1 0 20 40 60 80 100
Eigenvalue number
Fig. 2. The 100 normahzed eigenvalues \.
To get an accurate representation of the response process, there is a specific requirement that must be observed. Since the damping ratio is only 6%, the resonance peak of the linear transfer function for the dynamics is quite narrow. Hence, to capture the dynamics conectly, the frequency resolution must secure a sufficient number of frequency values over the resonance peak. This usually leads to an eigenvalue problem with the g-matrix of size of the order of magnitude 100 x 100. Using the f u l l representation of this size in calculating the mean crossing rate by the general method described here, would lead to very heavy calculations. In order to reduce this, we have investigated the effect of restricting the calculations by retaining only some of the terms in Eq. (1).
For the specific example considered here, where we have used exactly 100 (positive) frequencies, the values of the obtained eigenvalues jxj have been plotted in Fig. 2. It is seen that a substantial portion of the response variance, which is given by Var[Z2(f)] = 4 J2'j=i l-^j' would be lost i f only 10 or 20 eigenvalues were retained. This is also a factor to consider when deciding on the number of terms to retain.
In this paper, we have focussed on the slow-drift response. Hence, only results for Z2(t) w i l l be presented. In the tables, ^z, (0. i^z-j (O. 4, (0- and 4 (O denote the mean upcrossing rate of Z2(f) calculated by the full numerical method, the hybrid method, the Laplace approximation, and the simplified method of Eq. (36), respectively.
To highlight the effect of the increment parameter h, Table 2 compares the results obtained by the f u l l numerical method for two values of h for 10 eigenvalues, that is, for a response representation retaining the first 10 terms. The CPU time differs by a factor of roughly 10 between the two choices of a value for
h. Since the differences between the calculated crossing rates
are fairly small, we have chosen to use the larger value to save CPU time.
In Tables 3-6 we have written down the results obtained for 10, 20, 30 and 50 eigenvalues, respectively. It is apparent that there is some variability of the calculated upcrossing rates depending on the number of eigeiwalues included in the analysis. Ideally, i t would therefore be desirable to carry out
Table 2
Comparison of calculated upcrossing rate 0^ (if) for different step lengths
'J = f / A ' l /i = 1.0 X 10-3 h = 1.0 X 10' -2 2.0 8.38 X 10-3 8.38 X 10-3 5.0 3.93 X 10-3 3.93 X 10-3 10.0 5.53 X lO-'* 5.50 X l O - ' ' 15.0 5.70 X 10-5 5.65 X 10-5 20.0 5.34 X 10-'' 5.26 X 1 0 - ' ' 25.0 4.81 X 10-^ 4.71 X 1 0 - ' Table 3
Calculated upcrossing rates for 10 eigenvalues
'? = f / / - M 2.0 8.38 X 10-3 g 3g ^ -3 7.41 X 10" -3 5.0 3.93 X 10-3 3 93 ^ lO" -3 3.59 X 10--3 10.0 5.50 X IO-"* 5.50 X 10--4 5.23 X 10--4 15.0 5.65 X 10-5 5 g5 X -5 5.59 X 10--5 20.0 5.26 X 10-^ 5.26 x 10" -6 5.36 X 10--6 25.0 4.71 X 10-^ 4.71 X 10" -7 4.92 X 10--7 Table 4
Calculated upcrossing rates for 20 eigenvalues
') = f / / - i l 2.0 8.09 x 10-3 8.10 x 10--3 7.27 X 10--3 5.0 4 . 1 9 x 1 0 - 3 4 . 1 9 x 1 0 --3 3.82 X 10--3 10.0 6.51 X 10-"* 6.51 X 10--4 6.06 X 10--4 15.0 6.95 X 10-5 g 95 iQ--5 6.74 X 10--5 20.0 6.56 X 10-'' 6.56 x IQ--6 6.58 X 10--6 25.0 5.91 X 1 0 - ' 5.91 X 10--7 6.11 X 10--7 Table 5
Calculated upcrossing rates for 30 eigenvalues
0 + ( / ; )
4(")
2.0 7.21 X 10-3 7.21 X 10" -3 6.51 X 10-3 5.0 3.58 X 10-3 3.58 X 10" -3 3.27 X 10-3 10.0 5.48 X 10-'' 5.48 X 10--4 5.11 X 10-'* 15.0 5.85 X 10-5 5.85 X 10--5 5.67 X 10-5 20.0 5.54 X IO-*" 5.54 X 10--6 5.55 X 10-'' 25.0 5.00 X 1 0 - ' 5.00 X 10--7 5.17 X 1 0 - 'the calculations with at least 50 eigenvalues. However, our present implementation of the exact method is too expensive computationally to allow for more than about 50 eigenvalues. Even i f this can be iiuproved significantly, like for the hybrid method, it would still be a method requiring heavy computations.
Table 6 Table 7
Calculated upcrossing rates for 50 eigenvalues Calculated upcrossing rates for 100 eigenvalues
1) =
4(")
= f//-<l 4,(7) 4,(7)2.0 _3 6.17 X 10-3 5.59 X 10-3
2.0 6.55 X 10~3 6.55 X 10-3 5.93 X 10" 2.0 6.17 X 10-3 5.59 X 10-3 6.03 X 10"^ 5.0 3.25 X 10-3 3.25 X 10-3 2.98 X 10" -3 5.0 3.03 X 10-3 2,78 X 10-3 2.74 X 10-3 10.0 5.04 X IQ-'* 5.04 X IO-"* 4.70 X 10" -4 10.0 4.71 X lO-'* 4.40 X IQ-'* 3.65 X l O - ' ' 15.0 5.44 X 10-5 5.44 X 10-5 5.28 X 10" -5 15.0 5.11 X 10-5 4.96 X 10-5 3.44 X 10-5 20.0 5.19 X 10-6 5.19 X 10-6 5.20 X 10" -6 20.0 4.88 X 10-6 4.90 X 10-6 2.94 X 10-6 25.0 4.71 X 1 0 - ' 4.71 X 1 0 - ' 4.86 X 10" -7 25.0 4.44 X 1 0 - ' 4.63 X 1 0 - ' 2.44 X 1 0 - ' 1.6 1.4 1.2 & 0.8 X 10 • 0.4 i Original system Updated system i i ' ' 1 '' ' / \ jV: ' , / A > / \ 7 ^' / \^ 20 30 40 50 60 Numbet of eigenvalues 70
Fig. 3. The mean upcrossing rate of the level i; = 20 as a function of the number of retained eigenvalues.
upcrossing rate is also shown in Fig. 3. The figure indicates a couple of interesting conclusions. Updating for variance can lead to inaccurate results for the crossing rate for small to moderate number of eigenvalues retained. Comparing Figs. 2 and 3 it is seen that suiprisingly accurate results are obtained for even a small number of retained eigenvalues when the truncation is done exacdy where negative eigenvalues are followed by positive eigenvalues. This seems to provide the right balance between the terms in the response representation, and it indicates a useful criterion for truncating the response representation for crossing rate calculations.
It is also of great interest to observe that the simple Laplace approximation in fact provides quite accurate estimates of the mean upcrossing rates, and for this method the number of eigenvalues has practically no effect on the computational burden. Hence, from a practical point of view this is an extremely appealing method. In Table 7 we have listed the results obtained by the hybrid method, the Laplace approximation and also the simple approximation of Eq. (36) for 1 0 0 eigenvalues. It is seen that while there is excellent agreement between the hybrid method and the Laplace approximation, the simple approximation leads to significantiy lower values. In terius of extreme value predictions, for the example structure at hand the Laplace approximation is within about 1 % of the hybrid method, while the simple approximations would lead to an underestimation of typically 5 % - 1 0 % compared with the two more accurate methods. While
Fig. 4. The exceedance probabihty Ppiiy, T) as a function of ?; = f o r r = 3, 6, 18 h for 22 and 100 eigenvalues.
0 -0.5 CO R--1.5 Q-o -2.5 - - Hybrid Laplace Approx
Fig. 5. The exceedance probability Pp(iV, T) as function of ï) = f / M l for T • 3 h and 100 eigenvalues for the hybrid, Laplace and simple approximations.
7. Concluding remarks
In this paper we have described a general numerical method for calculating the mean level upcrossing rate of a second order stochastic Volten-a series. The specific model considered was the surge response of a moored deep floater in random seas. The method has been successfully implemented, and results have been calculated. However, it has been pointed out that a numerical calculation for the full problem by the general method is i n many cases too demanding computationally for practical use. Therefore, two approximations have been investigated, and i t has been shown that the approximation obtained by using what is known as a Laplace approximation for integrals, appears to provide estimates that are sufficiently accurate for practical purposes. The second approximation, which was based on a widely used simphfication, did not in general perform that well.
The main accoiuplishment of the present work, is that there is now avaUable a general numerical method for calculating the mean upcrossing rate of any second order stochastic VolteiTa system with a stationary Gaussian process as input. Since this is the state-of-the-art representation of the motion response of any large volume floating offshore stiucture subjected to random seas, this makes it possible to calculate accurately the response statistics of such structures.
Acknowledgements
The financial support from the Statoü Research Centre and the Research Council of Norway (NFR) through the Centre for Ships and Ocean Structures (CeSOS) at the Norwegian University of Science and Technology is gratefully acknowledged.
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