gram-charlier coefficients
TRANSCRIPT
A.1 Introduction
APPENDIX A
GRAM-CHARLIER COEFFICIENTS
In this section, the relations between the Gram-Charlier coefficients and the
moments and cumulants of joint random variables are given. These relations have been
obtained from Beaman and Hedrick(1981) and Nigam(1983).
A.2 Gram-Charlier Coefficients
A multivariate Gram-Charlier expansion of a probability distribution p(x) for the n
dimensional vector X is written as
().iT cp(x) (A.I)
where
(A.2)
The symbol cP(x) represents a multivariate gaussian distribution which can be written as
(A.3)
where the matrix S is the covariance matrix for the vector x. The Gram-Charlier coefficients, rjlj2'''jn' are related to the cumulants, Kjd2'''jn ' of
the joint random variables by the equation
1 jT= 0
0 0< jT< 3 rjd2'''jn =
(_l)h Kjlh'''jn . , . i ., h· J2' ... In' jT~ 3
The joint moments and cumulants are related to each other through the joint
characteristic function, MOO, which is defmed as
(A.4)
152
M(e) !E < exp(i e . x.) > = J .... J exp(i e . 3) p(3) d.x (A.5)
n
In this equation, the symbol (.) represents the scalar product of two vectors and i = {:l.
The joint moments of the vector ~ are related to the derivatives of the characteristic function
by the equation
(A.6)
The above moments are algebraically related to the joint central moments, J.l.hh'''jn ' which ~
are defmed as n .
lLIJ'Z"'J'n = < n (x·_<X·>)Ji > r-J i=1 1 1 (A.7)
The joint cumulants of the vector ~ are related to derivatives of the natural logarithm
of the characteristic function by the equation
(A.8)
B.1 Introduction
APPENDIX B
EVALUATION OF EXPECTATIONS
In. this appendix, details of the evaluation of two types of expectations are given.
The first kind involves the quadratic nonlinearity used throughout this study. Fonns of this
expectation are given by equations (2.66), (2.71), and (5.20). The second kind of
expectation involves central moments of random variables whose probability distribution is
approximated by a Gram-Charlier expansion. Recall that due to computational limitations,
only moments up to third order are included in the expansion. Thus, fourth and higher
order cumulants of the approximate probability distribution are zero. The fourth and higher
order central moments, however, are not zero and may need to be computed. In. particular,
the fourth order central moment is needed to compute the quadratization coefficients.
Higher order moments may also be needed depending on the nonlinearity. A simple
method for approximating higher order central moments based on the truncated probability
distribution is described.
B.2 Expectations Involving Quadratic Nonlinearity
Consider the following expectation involving a general nonlinearity g(x)
k = 0,1,2 (B.1)
The non-gaussian random variable x is a represented by its mean, J.I.x, and a zero-mean
component x by the equation
(B.2)
The expectation in equation (B.1) needs to be evaluated to detennine the mean response and
the quadfatization coefficients as described in Chapter 2. Using the defmition of
expectation, Ek is written as the following integral
k=O, 1,2 (B.3)
154
where p(i) is the probability distribution of i. This non-gaussian distribution is
approximated by the truncated Gram-Charlier expansion
(B.4)
where cp(i) is a gaussian distribution defmed by the equation
(B.5)
and ~im is the mth central moment ofx. Substituting equation (B.4) into equation (B.3)
leads to
(B.6)
(B.7)
(B.8)
where
n=O, ... ,5 (B.9)
For the quadratic nonlinearity used throughout this study, equation (B.9) can be
written in the following general form
In = J -yla + ~x + il(a + ~x + i) in cp(i) di n=O, ... ,5
This integral is evaluated by the following procedure. A new variable I; = i/{iiii is
introduced and substituted into equation (B.IO). This leads to
n+2 00
In = )'(~2)T J I it + I; I( it + I; )I;n cp!;(I;) dl; """
where
(B.IO)
(B.ll)
_ a+J.l.x a ::;:--..fIii2
155
The integration in equation (B.ll) is split over several regions as
n+2 -li
In ::;: y(~2r2 J -( a + ~ )2 ~n cp~(~) d~ """
n+2 0 + y(J.LX2)"'2 J ( a + ~ )2 ~n cp~(~) d~
-a n+2 co
+ y(J.Lx2f2 J ( a + ~ )2 ~n cp~(~) d~
Making several changes of variables, equation (B.13) is rewritten as
(B.12)
(B.13)
(B.14)
These integrals can be evaluated using standard integral tables and integration by parts.
Evaluating In for n up to 5 yields
where
10 ::;: 2y (J.Lx2){ (a2 + l)ql + it q2)
11 ::;: 4Y(J.Lx2)3/2{a ql + q2}
12 ::;: 2y (~2)2{ (a2 + 3)ql + a q2)
13 ::;: 4y (J.LX2)5/2{ 3a ql + 4 q2)
14 ::;: 2Y(J.Lx2)3{3(a2 + 5)ql + aq2}
Is ::;: 4y (J.Lx2)7/2{ 15a ql + (a2 + 24)q2)
q2 ::;: _1_ exp( - ! a2) {2rt 2
(B.15)
(B.16)
(B.17)
(B.18)
(B.19)
(B.20)
(B.2l)
(B.22)
156
B.3 High Order Central Moments By defmition, the mth central moment, ~im , of a random variable, x, is computed
by the equation
~im = J (x - ~x)m p(x) dx
where p(x) is the probability distribution of x. It is assumed that the probability
distribution is approximated by a Gram-Charlier expansion of the form
00 aj",(x) p(x) = 1 r· ~
j=O J axJ
(B.23)
(B.24)
where C\l(x) is a gaussian distribution and rj is related to the jth moment of the random
variable x as described in Appendix A. If the distribution is truncated at order j=jv the jt+ 1
and higher order cumulants of the approximate probability distribution are zero. The jt+ 1
and higher order central moments, however, are not zero and may need to be computed. In
this study, the expansion is truncated at the third order. However, the fourth order central
moment is needed to compute the quadratization coefficients. A simple method for
approximating these higher order central moments is described.
If p(x) is given by the truncated Gram-Charlier expansion, evaluation of the integral
in equation (B.23) yields integrals of the following form
(B.25)
To evaluate this integral the relations
(B.26)
(B.27)
(B.28)
are used. The first equation reflects a property of gaussian distributions. The second
equation defmes the Hermite polynomial, Hi~). The third relation states simply that the
157
quantity ~m Hj(~) can be written as an expansion of Hennite polynomials where ~'s are the
coefficients. To initiate the evaluation of I.nj. equation (B.26) is substituted into equation
(B.25) and the variable change ~ = (x - 11x)/~ is made. Then. applying equation (B.27)
yields
.!!!i 00 •
Imj = (~2) 2 J ~m (-l)l H/~) q,t;(~) d~ (B.29) """
where q,t;(~) is defined in equation (B.12). Substituting equation (B.28) into (B.29) and
recalling the variable change gives
.!!!i 00 • l+j ~akq,E.(~) I.nj = (~2)2 J (-I)l~ek(~X2)2 a k d~
""" k=O ~x (B.30)
Interchanging the order of summation and integration yields
(B.31)
As an example. the fourth order and fIfth order central moments of a random
variable whose distribution is approximated by a third order Gram-Charlier expansion can
be computed. Specifically. it is found that
(B.32)
(B.33)
APPENDIX C PIERSON-MOSKOWITZ WAVE SPECTRUM
In offshore engineering, the wave elevation spectrum is frequently defmed by the
Pierson-Moskowitz spectrum. The analytical expression for the two-sided fonn of this
spectrum, which is used throughout this investigation, is given by the equation
(C.I)
where Uw is the wind speed, g is the acceleration of gravity, and Ol is the frequency. The
constant parameters (X and p are commonly assumed to have the following values
(X = .0081
P = .74
(C.2)
(C.3)
For given (X and p, the wave spectrum given by equation (C.l) is completely
defmed by the wind speed. In offshore practice, however, it is more common to specify
the wave spectrum in tenns of the significant wave height, Hs' and the mean period, Tavg.
The significant wave height is the average height of the highest one third of the waves in a
particular sea state. The mean period is related to the spectral moments. These two
parameters are related to the wind speed by the following equations
(C.4)
Olavg = 21t = .920 (1tP) 1/4 LU Tavg w
(C.5)
More details on these relationships can be found in references such as Chakrabarti(1987).
Using these equations, an alternative fonn of the Pierson-Moskowitz spectrum can be
159
produced in tenns of the significant wave height and the mean frequency, coavg•
Specifically,
4
Sllll(CO) = .111H; ~:I~ eXP[_.444(CO!Vgj] (C.6)
As an example, Figure C.1 shows a one-sided fonn of the P-M spectrum for
Uw = 30 m/sec. Further, the significant wave height, mean period, and mean frequency
for this and other wind speeds based on the values of a and ~ given in equations (C.2) and
(C.3) are given in Table C.l.
Table C.1
Jlw mlsec Hli: m Tru sec rom radlsec
20 8.5 11.3 .557
22 10.3 12.4 .506
24 12.3 13.5 .464
26 14.4 14.7 .429
28 16.7 15.8 .398
30 19.2 16.9 .371
32 21.8 18.0 .348
34 24.6 19.2 .328
36 27.6 20.3 .310
120
100
~ U)
80 . N
E
60
:§:
cl 40
20
0
0.0
160
0.2 0.4 0.6 0.8 CD radlsec
Figure C.l Pierson-Moskowitz Wave Spectrum (Uw = 3OmJsec. H. = 19.2m. T""II = 16.9sec)
1.0
D.1 Introduction
APPENDIX D
SIMULATION METHODS
The simulation methods used to generate random wave force time histories are
reviewed in this section. The force time histories are used in a Newmark numerical
integration procedure to obtain simulated responses of an idealized TLP.
D.2 Linear Wave Simulation
In offshore engineering, random wave force time histories are typically computed
by transforming random wave elevation time histories based on suitable physical relations.
The sum of harmonics method described by Borgman( 1969) is a common method used to
simulate linear waves. Wave elevation time histories are generated from a two-sided target
spectrum, Sl1l1(OO), by the following summation
N
11 (x,t) = ~ 11j COS(lCjX - OOjt + £j) )=1
where
(D.I)
(D.2)
(D.3)
The harmonic component with frequency OOj has an amplitude, 11j' which is related to the
area of a slice of the target wave spectrum at OOj. In this study, the frequency width of each
slice, L\oo, is the same, although this is not necessary. In addition, the phase angle, £j' is a
random variable with a uniform probability distribution from 0 to 2n. For a constant x
value, it follows that the random process, l1(X,t), is zero mean. Also by virtue of the
central limit theorem it approaches a gaussian probability distribution as the number of
harmonic components tends to infmity.
D.3 Linear Wave Force Simulation
A linear force, rl)(t), is obtained from the wave elevation by the following linear
transform written in the frequency domain as
162
00
fl>(t) = I H~>(ro) 'i1(X,ro) exp(irot) dro (0.4)
where H~I>(ro) is the linear force transfer function and 'i1(x,ro) is the Fourier transfonn of the
wave elevation. Force time histories are generated by taking the Fourier transfonn of the
hannonic summation in equation (D. 1) and substituting into equation (0.4) which yields
the following hannonic summation for the linear force
It is noted that the linear force depends on the horizontal position x. If x is time dependent,
as is the case in a compliant platfonn, the linear force actually has higher order behavior.
This is the so called body motion effect. By accounting for the time dependent position,
the force is no longer zero mean and is no longer gaussian, even though it is obtained by a
linear transfonnation. This equation is used to simulate the linear force in the numerical
integration computer program. As an option, the displaced position effect can be neglected
to isolate its contribution to the total response.
D.4 Drag Force Simulation
The nonlinear Morison equation with relative velocity effects is used to model the
drag force. The total drag force on a vertical cylinder is obtained by the following
integration over the submerged length of the cylinder
ll(X,t)
fD(t) = tpocCD I I U(z) + U(X,z,t) - xl(t)1 (U(z) + U(X,z,t) - Xl(t») dz (0.6) -ds
The variables have been defmed previously in Chapter 5. The wave field water velocity,
u(x,z,t), is a random process which is simulated by an equation similar to equation (D.5).
Accounting for the displaced position of the TI..P and the integration to the free surface as
opposed to the mean free surface can readily be accomplished in the simulation. These
effects, however, are not readily modeled in the analytical procedure presented in this
study. They are neglected in the simul~tion to prevent obscuring the validity of the
163
analytical procedure. Thus, the simulated drag force is computed for a TLP in the
undisplaced position, and the integration is carried out only to the mean free surface. The
integration in equation (D.6) is performed by the trapezoidal rule.
D.S Quadratic Potential Force Simulation
The wave elevation and velocity head drift forces are simulated directly from the
second order Volterra series functional. The force is written in the frequency domain as
00
£<2)(t) = If H~)(COl,C02) fi(x,col) fi(x,co2) exp(ico1t) exp(ico2t) dcoldco2 (D.7)
where Hi2)(COl,C02) is the quadratic force transfer function.
A straightforward procedure to simulate the quadratic order force is to take the
Fourier transform of the harmonic summation in equation (D.I) and substitute it into
equation (D.7). This yields the following double summation for the quadratic force
N N £<2)(t) = ~1 ~ t11m 11n {Re[Hi)(com,con)] COS[(lCm+lCn)X - (com+con)t + (Em+en)]
At a particular time, t, 4N2 harmonic computations are performed to evaluate the
force. IfNg is the number of steps in the time history, the total number of harmonic
computations is 4Ns N2. Since Ng is in general several times greater than N, the number of
harmonic computations for one time history is of order N3. This proves to be too costly
for practical use. A much more efficient method makes use of digital FFf algorithms. In
order to use this more efficient method, a tradeoff must be made in that the displaced
position cannot be included in the quadratic force calculation. However, as with the drag
force, the displaced position effect in the quadratic force causes higher order effects which
are neglected in the analytical method anyway.
164
The digital procedure makes use of the Fourier series transform which along with
its inverse is defmed by the following equations
T
fk = t J f(t) exp( -ikL\rot) dt (D.9)
-f(t) = L fk exp(ikL\rot) k~
(0.10)
where T is the period over which f(t) repeats itself and fk is the Fourier series transform of
f(t).
The Fourier series transform of the quadratic force is obtained by substituting the
time domain form of the second order Volterra series which is written as
00
£<2}(t) = II h7}('tl,'t2) 11(t-'tl) 11(t-'t2) d'tld't2 (0.11)
into equation (0.9). Changing the order of integration and summation yields
~} = i H7}(m.(\ro,(k-m).(\ro) 11m 11k-m (0.12) m=-oo
where 11m is the Fourier series transform of 11(t). Using the sum of harmonics
representation for the wave elevation and using the same frequency increment for the
harmonic summation and the Fourier series it is readily shown that
r t11m exp( -im) m~O
11m = ~ * m ~ 0
(0.13)
l 11_m
Since the wave elevation is band limited at the frequency NL\CI) the summation in equation
(0.12) need only be carried out over a fmite number of points. The summation can be
rewritten as
k (2) N·k (2) __
L H f (m,k-m) 11m 11k-m + 2 L H f (k+m,-m) 11-m 11k+m m=O m=l
(0.14)
165
where the symmetry properties of the quadratic transfer function have been utilized. Also
the presence of the frequency increment, Aco, in the arguments of the transfer function is
implied. The first tenn on the right hand side of equation (0.14) is the Fourier series of the
high frequency part of the quadratic force, the other tenn is the low frequency part.
The quadratic force time history is obtained from Ji> by applying the inverse Fourier
series transfonn given by equation (0.10). An FFr algorithm can be used to do this
efficiently. The time savings of this method compared to the straightforward method is
obvious. The number of operations in equation (0.13) is of order N, in equation (D.14) of order N2, and for the FFr algorithm of order Ns 10g(Ns). Since the total number of
operations for one time history is obtained by addition of all these operations, this
procedure requires an order N2 number of operations. This is a substantial savings over
the direct method.
This digital procedure has been verified by averaging the quadratic force power
spectral density from 100 realizations and comparing it to the analytical power spectral
density. This has been done for quadratic surge force and pitch moment on the idealized TLP for a P-M spectrum with Hs=1.36 m and Tavg= 4.51 sec. The results are shown in
Figures D.1 and D.2 and show good agreement in both the low frequency and high frequency regions.
&! ~
OJ
Z ~
g g =:
(/)
~ "'.-. ~ Z ~
~ N'''';: -rn-
0.016
0.014
0.012
0.010
0.008
0.006
0.004
0.002
0.0
0
0.20
0.15
0.10
0.05
166
-- Analytical
••• •••• . Simulation
2 3 ro rad/sec
Figure 0.1 Spectral Density of Quadratic Wave ElevationNelocity Head Force on a TLP (llw = 8 mlsec. Ii. = 1.36 m. Ts..g" 4.51 sec)
-- Analytical
•....... Simulation
4
0.0 E::::::~:::"""":==========--~..I...-___ --1. __ -==::J o 2 3
ro rad/sec
Figure 0.2 Spectral Density of Quadratic Wave ElevationNelocity Head Moment on a TLP
(llw .. 8 mlsec . Ii.. 1.36 m. TB"9" 4.51 sec)
4
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Lecture Notes in Engineering Edited by C.A. Brebbia and S.A. Orszag
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