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Prediction of safe sea-state using finite element method and artificial neural networks
ARTICLE IN PRESS Ocean Engineering 37 (2010) 200–207
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Prediction of safe sea-state using finite element method and artificial neural networks S.F. Yasseri a, H. Bahai b, H. Bazargan c,n, A. Aminzadeh d a
Kellogg Brown and Root (KBR), UK School of Engineering & Design, Brunel University, London, UK c Shahid Bahonar University of Kerman, College of Engineering, Kerman, Iran d University of California, Berkeley, USA b
a r t i c l e in f o
a b s t r a c t
Article history: Received 29 October 2007 Accepted 22 November 2009 Available online 4 December 2009
This article proposes a predictive method for identifying the range of sea-states considered safe for the installation of offshore structures. A finite element dynamic analysis of the system for various sea-states characterized by significant wave heights and mean zero-up-crossing wave periods and modeled as a combination of several wave components has been performed. Using this procedure a table of safe and unsafe sea-states is generated. The significant wave height (Hs) and mean zero-up-crossing wave period (Tz) of a future sea-state in a location in the north east Pacific were predicted from the distributions whose parameters were estimated using the artificial neural networks (ANNs) trained for this purpose. The location of US National Oceanographic Data Center (NODC) Buoy 46005 is used in this study. The Hs and Tz of some future sea-states were predicted from their corresponding conditional 7-parameter distribution given some information including a number of previously measured Hs’s and Tz’s. This gives a predicted sea-state for a specific time in future. The parameters of the distributions have been estimated from the outputs of two different 7-network sets of trained ANNs. A pile-driving operation is used as a case study in which the pile configuration, including the non-linear foundation and the gap between the pile and the pile sleeve shims, has been modeled by the finite elements method and the range of sea-states suitable for safe pile-driving operation was identified. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Safe pile-driving Prediction Neural networks Finite elements method Simulated annealing Pierson–Moskowitz spectrum Hepta-parameter spline distribution
1. Introduction In the recent years artificial neural networks (ANNs) have been used to predict the sea-states characteristics. Some of the applications of the ANNs to predicting wave characteristics are reviewed here. Deo et al. (2001) used a 3-layered feed-forward ANN to obtain wave height and period (output) from wind speeds (input). They concluded that an appropriately trained network could provide satisfactory results for certain types of predictions and wind fetch. It is noted in their study that, unlike deterministic models, wind duration does not seem to be a necessary input to be given to ANNs. Agrawal and Deo (2002) dealt with the on-line prediction of wave heights by ANNs as well as by first-order autoregressive moving average (ARMA) and auto-regressive integrated moving average (ARIMA) models. They reported that the ANNs resulted in more accurate prediction of wave heights than the two time-series models when shorter prediction intervals were involved; however, for long-range predictions both approaches showed similar performance. Makarynskyy et al. (2002) developed and tested an ANN-based methodology to predict the
n
Corresponding author. Tel./fax: +98 341 2112861. E-mail address: [email protected] (H. Bazargan).
0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.11.006
significant wave height (Hs), the mean zero-up-crossing wave period (Tz) and the peak wave period of sea-states. They dealt with forecasting of subsequent 3-hourly intervals by training a set of ANNs with one hidden layer given the 24-h Hs history as input. This study reports that the accuracy of the forecasts improved with decreasing time intervals. Makarynskyy (2004, 2005) presents two more articles on the prediction of wave characteristics using ANNs. In the latter study, two different neural network strategies have been employed to forecast the sea-state characteristics for 3, 6, 12, and 24 h in advance. In the first approach, eight separate ANNs were implemented to simulate every wave parameter over each of the above four prediction intervals. In the second approach, only two networks provided simultaneous forecasts of the wave characteristics for the four prediction intervals. The suitability of ANNs has been demonstrated through verifying the short-term forecasts by the observed data. However the results of the simultaneous forecasts exhibited less accuracy than those obtained in the separate predictions. The main objective of ANN-based part of this research is to develop an ANN model for the statistical behavior of the stochastic process of the Hs’s i.e to estimate the parameters of the conditional distribution of future Hs or Tz given some past seastate observations; not to forecast the Hs and Tz directly as it is usually done (see Aminzadeh et al., 2009; Bazargan et al., 2007b).
ARTICLE IN PRESS S.F. Yasseri et al. / Ocean Engineering 37 (2010) 200–207
Nomenclature
S
ANNs ai fi FE Hs h pdf PM
SA Se S(fi) T
artificial neural networks the amplitude of ith wave component the frequency of ith wave component finite elements significant wave height time (h) probability density function Pierson–Moskowitz
Pile operation is a typical offshore installation procedure for which a predictive sea-state model is a highly valuable tool. Vertical piles are now normally used for the installation of fixed jacket structures, offering the potential for cost saving. Vertical piling has in most instances been used for the piles for which the pile-driving operation takes place well below the water surface. However, due to poor foundation conditions, the piles may extend well into the wave zone during the installation phase. Consequently, the dynamic response of these long unsupported vertical piles during installation becomes an important consideration. Djahansouzi and Yasseri (1994) analyze the dynamic response of the finite element model of a proposed vertical pile in its installation configuration to identify the possible safe sea-states for undertaking the pile operation and estimating the associated extreme value response parameters. In this study using a finite element dynamic analysis a table of some safe and unsafe seastates has been prepared based on the pile allowable stress. This table is used to determine whether the predicted sea-state by ANNs is safe or not.
2. Data The data used for this study were the 3-hourly Hs’s and Tz’s observed during 1978–1999 at the US National Oceanographic Data Center Buoy 46005 in the north east Pacific near 461N and 1311W. The significant wave height has been calculated as the average of the highest one-third of all of the wave heights observed during the 20-minute sampling period. The buoy performs the sampling mostly every 1 h and sometimes every 3 h. The resolution is one-tenth of a meter. There are some gaps (approximately 15 months) in the record. The details of the missing Hs’s and Tz’s are reported by Anderson et al. (2001). This set of data had been observed for 21 years and 84 days (7754.375 days) and was expected to consist of 7754.375 8= 62,035 values of 3-hourly Hs’s and Tz’s. The data for some periods including the following were missing: February to April 1985, November to December 1985, October to November 1986, June 1987. December 1990, April to June 1993. For the Hs simulation, those data were extracted whose immediate 8 preceding consecutive 3-hourly values of Hs were available. This led to a vector of 49,736 Hs’s. For the simulation of Tz the missing Hs’s and Tz’s whose immediate preceding and immediate next 3-hourly observed data were available were interpolated linearly, resulting in 55,922 values of data.
3. The specifications of the Neural Networks Employed In this work one of the most common neural network modeling techniques, i.e. feed-forward multi-layer networks has been used to simulate the Hs and Tz of a future 3-hourly sea-state. For this purpose two conditional distributions were used, one for
Tz
s
201
the mean of the stress time history of the element (MPa) simulated annealing most likely extreme stress occurred in the element energy density of ith wave component mean zero-crossing period for the stress time history of an element in the pile FE model mean zero-up-crossing wave period of a sea-state standard deviation of the stress time history of an element
the Hs and one for the Tz. The parameters of the distributions were estimated from the outputs of trained ANNs. Simulated annealing (SA), a global optimization technique of operations research, is the training algorithm. The details have been described by Bazargan et al. (2007b) and Aminzadeh et al. (2009). However a brief description is followed:
3.1. Networks used for generating Hs To simulate the Hs of a sea-state, more than fifteen different feed-forward multi-layer network architectures were tested and finally seven networks of size 13 24 13 1 were chosen (Fig. 1). Each of these seven networks are used for estimating the seven parameters of a proposed distribution called Heptaparameter spline distribution (see Appendix A). The transfer functions of the two hidden layers and the output layer were of ‘logsig’, ‘logsig’ and ‘purelin’ type, respectively. For each of the 7 networks, the size of the target vector is 1 49,736. The vector includes the observed Hs’s whose immediate eight preceding consecutive 3-hourly values of Hs were reported by the Buoy. The input for training the networks is a matrix of 13 49,736. For every Hs in the target vector, there are 13 values in the input matrix i.e. its immediate 8 preceding successive 3-hourly observed Hs’s, the time and 4 fuzzy values related to the season of the Hs. The time of an Hs in the target vector refers to the time it occurs. The actual time values lie in /0–24S, but to increase the efficiency of training, the range was scaled to /0–1S . The season is usually considered as spring, summer, fall and winter (1, 2, 3, and 4); however, in this work it was expressed as a fuzzy value, such that a particular day of the year belongs to all the four seasons with 4 membership values each lying in the interval /0–1S. Therefore 4 different membership values were assigned to each day of a year. For example the vector (1, 0, 0, 0) is assigned to half past 7 am on the 127th day of the year or 5th May (day= 127.3125) and (0.0246, 0, 0, 0.9754) is assigned to 6 am on 7th February or 6 am on the 38th day of the calendar (day =38.25). The 7 networks produce 7 real numbers as outputs. These numbers are used to estimate the 7 parameters of the Heptaparameter spline distribution (described in Appendix A) of the desired Hs given its 8 immediate successive preceding 3-hourly Hs’s. In this study, two training algorithms were tested: the gradient descent algorithm and simulated annealing. The stochastic algorithm of simulated annealing (SA) proved to be a much faster training algorithm for our purpose than the deterministic algorithm of gradient descent. For using SA as the training algorithm, a performance or cost function based on the maximum likelihood estimation (MLE) method was chosen. The cost function was the natural logarithm of the product of the conditional pdf’s of all observed Hs’s available in the target vector given their eight preceding
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Fig. 1. The ANN architecture used for Hs simulation.
Fig. 2. The ANNs architecture used for Tz simulation.
next sea-states of the desired period have been generated by the code from their conditional statistical distribution given their eight immediate preceding observed or generated Hs’s in a similar way. For more details see Aminzadeh et al. (2009). 3.2. Networks used for generating Tz Several different feed-forward multi-layer network architectures having 2 hidden layers were tested and finally the size 10 1 15 1 (Fig. 2) was used. The transfer functions of the 2 hidden layers and the output layer were of ‘logsig’, ‘logsig’ and ‘purelin’ type, respectively. A vector of 1 25,000 of 3-hourly Tz’s as described before was used as the target vector, although the code used had the capability of using a target vector of larger size. To increase the efficiency of training, the target vector was scaled using Fig. 3. Pile installation configuration (after Djahansouzi and Yasseri, 1994).
Tzscaled ¼ 0:8
immediate 3-hourly Hs’s (Bazargan et al., 2007a, Aminzadeh, et al., 2009). Since this function is a function of the pdf’s parameters which are estimated from the networks outputs and on the other hand the networks outputs depend on the networks weights and biases, the objective during the process of training was to change the networks weights and biases continually and randomly in order to maximize the cost function. After the training process, to simulate the 3-hourly significant wave heights for a future time period using the trained networks, the starting day of the period (a real number in the interval /1–365.25S) and the eight 3-hourly consecutive observed Hs’s of 24 h previous to the period are used as the input data to the neural networks coded in MATLAB. The code calculates the 4 fuzzy membership values and the time from the day value (see Appendix B) and also obtains the outputs of the trained networks for each series of the input. The estimates of the 7 parameters of the corresponding conditional distribution related to the Hs of the first sea-state of the desired period are then calculated from the 7 outputs using the relationships given in Appendix C. Then the code generates a variate from the distribution by the inversion method (Naylor et al.,1968). This variate is taken as the predicted Hs of the first 3-hourly sea-state of the desired future period. However, a histogram could be created from several of such generated variates and the most frequent variate of the histogram or its mean can be taken as the predicted value. The Hs’s of the
Tz minðTz Þ þ0:1; maxðTz ÞminðTz Þ
ð2Þ
where min(Tz)= 3.7, i.e. the minimum of the Tz data and max(Tz) =17.5, i.e. the maximum of the Tz data. The input data for training the networks was a matrix of 10 25000. For any 3-hourly Tz in the target vector there are 10 pieces of data in the input vector as follows: its two immediate preceding 3-hourly Tz’s, its immediate preceding 3-hourly Hs, the Hs of the corresponding sea-state, the Hs of next immediate sea-state, the occurrence time of the desired sea-state which is a value in the interval /0–24S divided by 24, and finally 4 fuzzy values representing the season of the sea-state (see Appendix B). The data belonging to the future is not given as input; because the code will generate them. To increase the efficiency, the Hs’s of the input matrix were scaled using Hsscaled ¼ 0:8
HS minðHS Þ þ0:1; maxðHS ÞminðHS Þ
ð3Þ
where min(Hs)= 0.2 i.e. the minimum of the Hs data and max(Hs)= 13.6 i.e. the maximum of the Hs data. The networks were trained by SA algorithm. To simulate the Tz’s of future sea-states using the trained networks the starting day of the period, a number in the interval /1–365.25S, and the necessary input Tz’s and Hs’s (see Bazargan
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et al., 2007b) for predicting the Tz of the first sea-state of the future period are given as input to a MATLAB code. The code then calculates the other necessary inputs. In a similar way stated for the prediction of Hs’s, the Tz’s of the desired sea-states are generated by the code from the corresponding conditional statistical distribution of the Tz given 10 inputs. The details are described by Bazargan et al. (2007b).
4. Case study In the case study conducted, the behavior of a vertical pile stick-up just after stabbing but before the driving operation is investigated. Fig. 3 shows the configuration of such a pile. The lower end of the pile is embedded in soil and restrained from lateral movement by the soil and pile sleeves. The pile supports the weight of the pile-driving hammer at the top and is subject to wave and current loads as well as self-weight and buoyancy effects along its length. The objective of this work is to determine the probability that a forecasted sea-state is safe for the operation.
5. Finite element modeling and analysis The pile is a pipe of 72 m long with 8 cm wall thickness and outer radius of 91.45 cm. The model of the pile used for analysis consisted of 30 two-node PIPE-21 beam elements (ABAQUS, 2004). The foundation was represented by non-linear spring elements. GAP elements were used to represent the gap between the pile and its sleeve. The analysis was conducted in a time domain covering 300 s with time increments of 0.05 s to ensure that a converged pile response was computed. To facilitate the derivation of spectral
203
properties from the time series, the response was computed for at least 2048 sampling points. Pile is initially leaning against the pile sleeve. In the first step of a multi-step analysis the pile self-weight was applied and a non-linear static analysis was performed to determine the equilibrium position. In the next step the weight of hammer was applied. The third step was a non-linear time domain analysis in which the pile set up was exposed to the effect of wave loading calculated by Morrison equation in ABAQUS. Airy wave theory was used for this analysis, as it is considered to be accurate for low amplitude waves in deep water. For each sea-state, the sea surface was presented using a large number of Airy waves. ABAQUS/AQUA software was used to analyze the pile for more than 70 sea-states (see Table 1).
6. Decomposition of sea-states ABAQUS/AQUA requires the input sea-states to be given in the form of waves. Thus a sea-state needs to be decomposed into several sinusoidal wave components identified by 3 parameters, i.e. amplitude, period and phase angle. It was assumed the Pierson–Moskowitz spectral formulation (described in Appendix D) represents the sea-state. Each sea-state was decomposed into 30 sinusoidal components having its own amplitude, frequency and phase angle. The phase angles were generated randomly between 01 and 3601. Then for each sea-state the range of frequency (bandwidth), i.e. zero to where the spectrum approaches zero, was determined and divided into 30 subintervals. The frequency of each subinterval end point was selected as each component frequency. Then, given the frequency of the component, its amplitude was calculated using the following formula
Table 1 The results of pile analysis. Sea-state
von Mises predicted
1 = Safe; 0 =Uns.
Sea-state
von Mises predicted
from Eq. (5) for 3 h (MPa) Hs (m)
Tz (s)
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.2 1.2
2 3 4 6 7 8 10 12 2 3.5 4 6 8 10 12 4 10 2 4 6 8 10 12 2 4
19.62 26.70 29.96 23.55 26.82 22.36 21.12 20.29 28.33 55.05 43.94 29.71 26.03 22.89 22.60 55.26 30.12 20.48 45.57 39.25 36.31 31.65 26.74 150.58 59.68
1=S 0 =U
Sea-state
von Mises predicted
from Eq. (5) for 3 h (MPa)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
Hs (m)
Tz (s)
1.2 1.2 1.2 1.2 1.2 1.4 1.4 1.4 1.4 1.4 1.4 1.6 1.6 1.6 1.6 1.6 1.6 1.8 1.8 1.8 1.8 2 2 2 2
4.3 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 6 8 10 12 2 4 6 6.5
79.50 53.19 48.52 34.44 29.71 34.78 52.66 43.29 35.62 31.76 34.00 28.70 120.86 67.55 52.70 35.83 30.68 56.20 41.36 51.77 32.44 26.06 77.63 50.96 64.49
1=S 0 =U
from Eq. (5) for 3 h (MPa)
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1
Hs (m)
Tz (s)
2 2 2 2.2 2.2 2.2 2.2 2.2 2.2 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.8 2.8 2.8 3.2 3.2 3.2 3.5 3.5 3.5
8 10 12 2 4 6 8 10 12 2 3 4 6 8 10 12 3 7 8 4 8 12 4 8 12
59.41 39.31 42.24 24.47 69.64 95.71 51.46 41.50 42.21 24.09 77.03 75.83 78.36 29.71 54.03 39.54 66.87 112.25 79.36 227.90 69.1 57.5 129.97 116.93 53.8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1
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zero-crossing period for the stress time history of the element; in our case T=300/N where N= total number of zero-up-crossings in the stress time history of the element. The most probable maximum von Mises stress predicted to occur in the pile after 3 h is the value obtained from Eq. (5) for the critical element of the pile. This value is the maximum of the values obtained from Eq. (5) for all elements. For more than 70 sea-states this maximum has been calculated. Table 1 shows the pile responses. In this analysis, a sea-state is considered safe for pile driving if the maximum predicted von Mises stress is less than 55% of the yield stress of the material of the pile i.e. steel M10. The basic safety margin in API is 67% of the material yield for normal loading moderated by buckling. Due to unsupported large length, the allowable axial load is very low. The paper uses the von Mises stresses to judge if the operation is safe. The value of 55% allows a safety factor of nearly 2, which the operating company and third party certifier agree to use as an equivalent threshold allowable stress. The case study comes from an actual offshore pile driving during special months of the year. This value within the context of the paper must be taken as an example of how one can make the necessary decision. The sea-states characterized with Hs’s greater than 3.5 m were considered unacceptable(unsafe); because the maximum allowable equivalent stress was agreed to be 55% of the material yield and the stresses that they would generate exceed the agreed acceptable level.
Fig. 4. Stress time history at an integration point on the critical element.
(Goda, 2000): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ai ¼ 2Sðfi Þ Dfi ;
ð4Þ
where ai is the amplitude of ith component, fi the frequency of ith component, S(fi) the energy density given by Eq. (D1) and Dfi = (fi + 1 fi 1)/2.
7. Pile response The von Mises stress was used as a measure of pile strength (API, 1993). Since the pile response is non-linear, the position of the maximum stress depends on the wave excitation. The time history of stresses at a subset of the pile elements were obtained and examined to ensure that the member with maximum response is picked up. The analyses were done by ABAQUS/AQUA in time domain covering 300 s with time increments of 0.05 s to ensure that the pile response has settled down. The von Mises stress has been calculated for nearly 300/0.05 = 6000 time increments for all of the elements in the subset. Fig. 4 shows the stress history occurred in one of the integration points of an element proved to be critical due to a sample sea-state for a time of 300 s. Some 6000-von-Mises-stress values were obtained for each integration point (both the top and bottom integration points). due to a sea-state loading as well as the self-weight and buoyancy effects. Therefore for each element of the subset 2 series of 6000von-Mises-stress values were produced by ABAQUS/AQUA. The mean and standard deviation of these values were computed. Then the most likely extreme amplitude of stress cycles after time h, using a long-term extreme prediction due to Barltrop and Adams (1991), were calculated from the following formula for all elements in the subset: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3600h ; ð5Þ Se ¼ S þ s 2ln T where Se is the most likely extreme stress of the element after h hours (MPa), S= the mean of the stress time history of the element (MPa), s the standard deviation of the stress time history of the element (MPa), h the time (h), in our case h=3, T the mean
8. Safe sea-state for pile driving Given the necessary inputs as described earlier , the Hs and Tz of a desired future 3-hourly sea-state are generated as random variates from their corresponding conditional distribution. To determine the level at which a future sea-state is safe to drive a pile, the 3-hourly sea-state is first predicted K times in this way and then, using Table 1 it is determined whether each of these is predicted to be safe. Suppose the sea-state proves to be safe n times and unsafe K n times; it could therefore be concluded that the sea-state is safe with 100 n/K percent of certainty.
9. Illustration As an illustration, assuming a reference date of 1st January we would like to simulate and forecast the sea-states of a period starting 00 am on 2nd January to 12 pm on 30th December of the same year for the region in the Pacific where Buoy 46005 operates. Table 2 shows the observed 3-hourly Hs’s of 24 h before the above starting time and the two necessary Tz’s. This data is necessary as input for simulation.
Table 2 Inputs for forecasting. Time
Date
Hs (m)
Tz (s)
9 pm 6 pm 3 pm 12 noon 9 am 6 am 3 am 00 am
1st 1st 1st 1st 1st 1st 1st 1st
3.1 3.2 3.0 2.6 2.7 2.0 1.7 1.8
6.7 6.8 n.r. (not required) n.r n.r n.r n.r n.r
January January January January January January January January
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For practical purposes the number of simulation runs should be increased to get more accurate predictions.
Table 3 Ten times prediction of sea-state for 9 a.m. on 2nd January. Run no. 1 2 3 4 5 6 7 8 9 10
Hs (m)
Tz (s)
1 = Safe 0= Unsafe
2.5 3.1 3.7 4.3 4.1 3.8 5.4 2.8 3.9 5.5
7.6 7.9 7.4 7.9 7.0 8.3 7.3 6.6 8.3 7.4
1 1 0 0 0 0 0 1 0 0
Table 4 Ten times prediction of sea-state for 24 p.m. on 30th December. Run no. 1 2 3 4 5 6 7 8 9 10
205
Hs (m)
Tz (s)
1 = Safe 0= Unsafe
5.25 2.25 6.0 4.4 3.3 8.9 3.4 2.6 1.5 8.5
9.0 5.4 9.7 9.5 7.3 11.0 10.5 6.4 6.5 9.8
0 1 0 0 1 0 0 1 1 0
After inputting the data of Table 2 to a MATLAB code, the 3-hourly sea-states for the period were simulated one after another by running the code which calculates the parameters of the 2 required conditional distributions for the Hs and Tz of each 3-hourly sea-state in the period from the outputs of the trained neural networks(see Appendix C) and generates the Hs and Tz from the corresponding distribution. For each time of simulating the 3-hourly sea-state of period, the code generates nearly (365.25–1) 8= 2914 sea-states. Suppose we are interested in the sea-state of 9 am on 2nd January and 12 pm on 30th December. The results of 10-time simulation for 9 am on 2nd January are shown in Table 3. The code has generated the Hs’s and the Tz’s of the sea-states (columns 2 and 3) in 10 runs. Column 4, which indicates whether the generated sea-states are safe or otherwise, was extracted visually from Table 1. Using this prediction it can, therefore be concluded from column 4 that the sea-state of 9 am on 2nd January is safe with a probability of 3 out of 10 i.e. 30 percent (30%). Note that the mean of the 10 predicted values are Hs = 3.91 m and Tz =7.57 s, and the mean of the observed values for 9 am on 2nd January during 1978–1999 were 3.89 m and 7.67 s, respectively. Using this procedure the sea-states of any future time period could be predicted. However, the shorter the time, the more accurate the prediction. The results of 10-time simulation for 24 pm on 30th December are shown in Table 4. The code has generated the Hs’s and the Tz’s of the sea-states. It is worth noting that the mean of the 10 simulated values are Hs =4.61 m, Tz = 8.51 s and the mean of the corresponding observed values during 1978– 1999 for 24 pm on 30th December are 3.46 m and 7.52 s, respectively. Column 4 was extracted visually from Table 1. Therefore it can be concluded that the sea-state of 24 pm on 30th December is safe with a probability of 4 out of 10 i.e. 40 percent (40%).
10. Conclusions From the simulated and observed values it is concluded that: (1) In this study multi-layer feed-forward ANNs proved to be an efficient tool in estimating the parameters of the proposed distribution used to simulate the characteristics of the seastates of a desired time and thereby in the simulation of seastate sequences. (2) The predicted 3-hourly significant wave heights and mean zero-up-crossing wave periods of the sea-state of a time in the near future obtained by the proposed method compare well with their corresponding observed values. (3) The proposed hepta-parameter spline distribution approximating the conditional probability distributions used to simulate the Hs and the Tz of a 3-hourly sea-state in the north east Pacific proved to be a good choice for the approximation. (4) The mean of the conditional distribution or the mean of the simulated values obtained in several runs of simulation provides a good forecast for the Hs and the Tz of a desired time sea-state. (5) The finite element and neural network modeling resulted in an easy-to-use procedure to predict the degree of safety of the sea-states at which the pile-driving operation could be conducted in a future period given the eight successive 3-hourly sea-state history preceding that period. The method introduced for the simulation of sea-state characteristics is applicable to other time-dependent processes which have instinct periodical variations. The application could cover a variety of things such as simulating a related random variable, forecasting its most probable and mean values and also doing an extreme value analysis related to the process.
Acknowlegement The authors would like to thank Mr. Robert Sears who had prepared the ABAQUS model formulation for the pile stick-up analysis initially.
Appendix A. Hepta-parameter spline distribution In this appendix, a parametric distribution named ‘‘heptaparameter spline’’ is proposed. This very flexible distribution was used in this research to approximate the conditional distribution of [Hs(tn)9Hs(tn 1) =hn 1, y,Hs(tn 8)= hn 8], whose pdf is denoted by f(x). The density function f(x) should satisfy the following two properties: (1) limx- 7 1 f ðxÞ ¼ 0, (2) smoothness. These two properties imply that there are two points, say a and e (a oe) where f(x) approaches zero for x oa and x 4e and also there is a point c in the interval /a–eS for which f 0 ðcÞ ¼ 0 (e.g. the maximum of the function). Since there were no a priori information available on the shape of the distribution of Hs, the proposed density function had to be designed in such a way that it could closely approximate a number of possible distributions. The cubic interpolations on the four subintervals of Fig. A1 were assumed to be a good approximation for f(x). Note that a, b, c, d, e, m1, and m2 were chosen such that: b¼
aþc ; 2
d¼
c þe ; 2
m1 ¼ f 0 ðbÞ;
and
m2 ¼ f 0 ðdÞ:
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Fig. A1. The intervals on which the conditional distribution is defined, and the parameters of the distribution.
gðxÞ ¼
8 0 > > > > > > > Cubic-Int: > > > > > > > > > > > > Cubic-Int: > > <
a " "
> > > > Cubic-Int: > > > > > > > > > > > > Cubic-Int: > > > > > : 0
"
0 b a2 c a3 d a4
" with slope ¼ 0
&
with slopem1 40
&
#
"
#
" with slope ¼ 0
&
with slope m2 o 0
&
#
b a2 c a3 d
with slope m1 40
aoxrb
with slope ¼ 0
b ox r c
with slope m2 o0
c o x rd
with slope ¼ 0
doxre
# #
a4 e 0
xra
#
xZe
Fig. A1 also shows a1, a2, a3, a4, l, D, m1, and m2 the parameters of the function used to approximate of the conditional distribution. The density function g(x), the proposed pdf for approximating f(x), is defined as follows: where a, b, c, d, e, a2, a3, a4, m1, m2 are illustrated in Fig. A1, and Cubic-Int. stands for the cubic interpolation of each pair of the points mentioned above according to spline method. Having two points x1 and x2 and the values of a function f(x) and its derivative at x1 and x2, the formula for approximating f(x) according to the spline method is given by gðxÞ ¼ f ðx2 Þ
xx1 x2 x1
þ f ðx1 Þ
3
xx2 x1 x2
3f ðx2 Þ ðxx1 Þ2 ðxx2 Þ þ f 0 ðx2 Þ x2 x1 ðx2 x1 Þ2
3
3f ðx1 Þ ðxx2 Þ2 ðxx1 Þ þ f 0 ðx1 Þ x1 x2 ðx1 x2 Þ2
and 6a4 rm2 r 0: ð1lÞD
ðA4Þ
The distribution has the capability to approximate a number of the conventional distributions including chi-square, positive normal, 3-parameter Weibull. However, an attempt to use the 3-parameter Weibull distribution instead of the hepta-parameter distribution in the study was not found to be satisfactory.
Appendix B. Calculation of the time and four season values from day values ðA1Þ
The time at which the sea-state being predicted occurs is calculated from the following MATLAB command: time ¼ floorðdayÞ;
Since
R1
gðxÞ dx ¼ 1; it could be proved that:
1
lDa2 2
þ
Da3 4
þ
ð1lÞDa4 ¼ 1: 2
ðA2Þ
Hence one of the parameters a2, a3, a4, l, and D could be calculated from the other four and the number of independent parameters reduce to seven. Therefore, g(x) has seven independent parameters and hence the name hepta-parameter spline. Arbitrarily a3 was determined from a2, a4, l, and D. Finally, since all the interpolated values of g(x) have to be positive, it could be shown that the following two inequalities must also be satisfied: 0 r m1 r
6a2
Dl
where day is a number in the interval /0–365.25S with step 0.125 representing the day on which 3-hourly sea-states occur. The four fuzzy membership values related to the season in which the sea-state occurs are calculated using the following four MATLAB instructions: (1) (2) (3) (4)
springvalue= Trianglefun(abs(day-midspring)/(365.25/4)) summervalue= Trianglefun(abs(day-midsummer)/(365.25/4)) fallvalue= Trianglelfun(abs(day-midfall)/(365.25/4)) wintervalue=Trianglefun(abs (day-midwinter)/(365.25/4))
where Trianglefun is the following MATLAB function: ;
ðA3Þ
function v ¼ TrianglefunðdayÞ;
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207
v ¼ ð1absðdayÞÞ ðabsðdayÞ o 1Þ þ ð1absðday4ÞÞ ðabsðday4Þ o 1Þ and since the 36th day of the year (4th February) is the midwinter, midwinter= 36; and also: midspring= 36+ (365.25/4)= 127.3125 (5th April) midsummer=36 +2 (365.25/4)= 218.6250 (6th August) midfall= 36+ 3 (365.25/4)= 309.9375 (6th November)
Appendix C. Calculation of the distribution parameters from the networks outputs Let xj denotes the output of the jth network for j= 1, 2, y,7. Since the range of the networks outputs is ( N to N) due to the transfer function of purelin in the output layers, we cannot directly use the outputs of any of the 7 networks for calculating positive a1. Hence a transformation was applied on x1 to arrive at a positive number: a1 = x21. The relationships used to calculate the parameters from the outputs of the networks are given as follows:
Fig. D1. A typical PM energy density spectrum.
where S(fi)= energy density of ith wave component (m2/Hz), A ¼ Hs2 =4pTz4 , B ¼ 1=pTz4 , 0 ofi oN, fi = the frequency of ith wave component (Hz), As an illustration, the spectrum has been plotted for a sea-state with Hs = 1 m and Tz = 3 s in Fig. D1:
Constraint a1 o0
Relationship a1 ¼ x21
ðC1Þ
0 oa2 oa3
a2 ¼ a3 logsig x2
ðC2Þ
D 40
D ¼ x23
ðC3Þ
References
0 oa4 oa3
a4 ¼ a3 logsig x4
ðC4Þ
ABAQUS 6.4 User’s Manual, 2004. Hibbit, Karlsson & Sorensen Inc. Agrawal, J.D., Deo, M.C., 2002. On-line wave prediction. Marine Structures 15 (1), 57–74. Aminzadeh, A., Bahai, H., Bazargan., H., 2009. Simulation of significant wave height by neural networks and its application to extreme wave analysis. Journal of Atmospheric and Oceanic Technology 26 (4). Anderson C.W., Carter D, Cotton P.D., 2001. Wave climate variability and impact on offshore design extreme. Report for Shell International, and the Organisation of Oil and Gas Producers /http://info.ogp.org.uk/metocean/JIPweek/WCERe port_2sided.pdfS. API, 1993. Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms—Load and Resistance Factor Design. API RP 2A-LRFD, Washington, DC. Barltrop, N.D.P., Adams, A.J., 1991. In: Dynamics of Fixed Marine Structures. Butterworth-Heinemann, Oxford (Chapters 3 and 6). Bazargan, H., Bahai, H., Aminzadeh, A., Bazargan, A., 2007a. Neural network simulation of significant wave height. In: 26th International Conference on Offshore, Mechanic & Arctic Engineering, June 10–15, 2007, San Diego, USA. Bazargan, H., Bahai, H., Aryana, F., 2007b. Simulation of the mean zero-up-crossing wave period using artificial neural networks trained with a simulated annealing algorithm. Journal of Marine Science and Technology 12 (1), 22–34. Buoy Data—46005 WASHINGTON 315NM West of Aberdeen, WA /http://www. nodc.noaa.gov/BUOY/46005.htmlS. Deo, M.C., Jha, A., Chaphekar, A.S., Ravikant, K., 2001. Neural networks for wave forecasting. Ocean Engineering 28 (7), 889–898. Djahansouzi, B, Yasseri, F., 1994. Study of offshore pile stick-up. In: Proceedings of the Ninth UK ABAQUS User Group Conference, Oxford. Goda, Y., 2000. In: Random Seas and Design of Maritime Structures 2nd ed. World Scientific, Singapore (Chapter 10). Makarynskyy, O., Pires-Silva, A.A., Makarynska, D., Ventura-Soares, C., 2002. Artificial neural networks in the forecasting of wave parameter. In: The Seventh International Workshop on Wave Hindcasting & Forecasting, Banff, Alberta Canada 21–25 October 2002, pp. 514–522 /http://www.wavework shop.org/7thwaves/papers/makarynskyy_etal.pdfS. Makarynskyy, O., 2004. Improving wave prediction with artificial neural nets. Ocean Engineering 31 (5–6), 709–724. Makarynskyy, O., 2005. Artificial neural networks for wave tracking, retrieval & prediction. Pacific Oceanography 3 (1), 21–30. Naylor, H.N., Balintfy, J.L., Burdick, D.S., Chu, K, 1968. In: Computer Simulation Techniques. J W & Sons, New York (Chapter 4). Pierson, W.J., Moskowitz, L., 1964. A proposed spectral form for fully developed wind seas based on similarity theory of S.A. Kitaigorodskii. Journal of Geophysical Research 69 (24), 5181–5190.
0 rm1 r
6a2
Dl
m1 ¼
6a2
Dl
logsigx5
6a4 logsigx6 ð1lÞD
6a4 r m2 r 0 ð1lÞD
m2 ¼
0 o l o1
l ¼ logsig x7
ðC5Þ
ðC6Þ ðC7Þ
From Eqs. (A2), (C2), (C3), (C4), and (C7) we have a3 ¼
2x2 3 ðlogsigx7 Þðlogsig x2 Þ þ 0:5 þð1logsig x7 Þðlogsigx4 Þ
ðC8Þ
where logsig xj ¼
1 ; 1þ exj
j ¼ 1; . . .; 7:
ðC9Þ
In practice, in the first step, a3 is calculated by Eq. (C8); then a2 and a4 are calculated from Eqs. (C2) and (C4). Other parameters are calculated simply using the appropriate relationship.
Appendix D. Pierson–Moskowitz energy density spectrum Pierson–Moskowitz (PM) energy density spectrum (Pierson and Moskowitz, 1964) with its following formula is suitable for decomposing a sea-state into several sinusoidal wave components. The formula depends on the sea-state representing parameters, i.e. Hs and Tz (Barltrop and Adams, 1991): ! B4 ; A Sðfi Þ ¼ 5 e fi ðD1Þ fi
Contents lists available at ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Prediction of safe sea-state using finite element method and artificial neural networks S.F. Yasseri a, H. Bahai b, H. Bazargan c,n, A. Aminzadeh d a
Kellogg Brown and Root (KBR), UK School of Engineering & Design, Brunel University, London, UK c Shahid Bahonar University of Kerman, College of Engineering, Kerman, Iran d University of California, Berkeley, USA b
a r t i c l e in f o
a b s t r a c t
Article history: Received 29 October 2007 Accepted 22 November 2009 Available online 4 December 2009
This article proposes a predictive method for identifying the range of sea-states considered safe for the installation of offshore structures. A finite element dynamic analysis of the system for various sea-states characterized by significant wave heights and mean zero-up-crossing wave periods and modeled as a combination of several wave components has been performed. Using this procedure a table of safe and unsafe sea-states is generated. The significant wave height (Hs) and mean zero-up-crossing wave period (Tz) of a future sea-state in a location in the north east Pacific were predicted from the distributions whose parameters were estimated using the artificial neural networks (ANNs) trained for this purpose. The location of US National Oceanographic Data Center (NODC) Buoy 46005 is used in this study. The Hs and Tz of some future sea-states were predicted from their corresponding conditional 7-parameter distribution given some information including a number of previously measured Hs’s and Tz’s. This gives a predicted sea-state for a specific time in future. The parameters of the distributions have been estimated from the outputs of two different 7-network sets of trained ANNs. A pile-driving operation is used as a case study in which the pile configuration, including the non-linear foundation and the gap between the pile and the pile sleeve shims, has been modeled by the finite elements method and the range of sea-states suitable for safe pile-driving operation was identified. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Safe pile-driving Prediction Neural networks Finite elements method Simulated annealing Pierson–Moskowitz spectrum Hepta-parameter spline distribution
1. Introduction In the recent years artificial neural networks (ANNs) have been used to predict the sea-states characteristics. Some of the applications of the ANNs to predicting wave characteristics are reviewed here. Deo et al. (2001) used a 3-layered feed-forward ANN to obtain wave height and period (output) from wind speeds (input). They concluded that an appropriately trained network could provide satisfactory results for certain types of predictions and wind fetch. It is noted in their study that, unlike deterministic models, wind duration does not seem to be a necessary input to be given to ANNs. Agrawal and Deo (2002) dealt with the on-line prediction of wave heights by ANNs as well as by first-order autoregressive moving average (ARMA) and auto-regressive integrated moving average (ARIMA) models. They reported that the ANNs resulted in more accurate prediction of wave heights than the two time-series models when shorter prediction intervals were involved; however, for long-range predictions both approaches showed similar performance. Makarynskyy et al. (2002) developed and tested an ANN-based methodology to predict the
n
Corresponding author. Tel./fax: +98 341 2112861. E-mail address: [email protected] (H. Bazargan).
0029-8018/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2009.11.006
significant wave height (Hs), the mean zero-up-crossing wave period (Tz) and the peak wave period of sea-states. They dealt with forecasting of subsequent 3-hourly intervals by training a set of ANNs with one hidden layer given the 24-h Hs history as input. This study reports that the accuracy of the forecasts improved with decreasing time intervals. Makarynskyy (2004, 2005) presents two more articles on the prediction of wave characteristics using ANNs. In the latter study, two different neural network strategies have been employed to forecast the sea-state characteristics for 3, 6, 12, and 24 h in advance. In the first approach, eight separate ANNs were implemented to simulate every wave parameter over each of the above four prediction intervals. In the second approach, only two networks provided simultaneous forecasts of the wave characteristics for the four prediction intervals. The suitability of ANNs has been demonstrated through verifying the short-term forecasts by the observed data. However the results of the simultaneous forecasts exhibited less accuracy than those obtained in the separate predictions. The main objective of ANN-based part of this research is to develop an ANN model for the statistical behavior of the stochastic process of the Hs’s i.e to estimate the parameters of the conditional distribution of future Hs or Tz given some past seastate observations; not to forecast the Hs and Tz directly as it is usually done (see Aminzadeh et al., 2009; Bazargan et al., 2007b).
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Nomenclature
S
ANNs ai fi FE Hs h pdf PM
SA Se S(fi) T
artificial neural networks the amplitude of ith wave component the frequency of ith wave component finite elements significant wave height time (h) probability density function Pierson–Moskowitz
Pile operation is a typical offshore installation procedure for which a predictive sea-state model is a highly valuable tool. Vertical piles are now normally used for the installation of fixed jacket structures, offering the potential for cost saving. Vertical piling has in most instances been used for the piles for which the pile-driving operation takes place well below the water surface. However, due to poor foundation conditions, the piles may extend well into the wave zone during the installation phase. Consequently, the dynamic response of these long unsupported vertical piles during installation becomes an important consideration. Djahansouzi and Yasseri (1994) analyze the dynamic response of the finite element model of a proposed vertical pile in its installation configuration to identify the possible safe sea-states for undertaking the pile operation and estimating the associated extreme value response parameters. In this study using a finite element dynamic analysis a table of some safe and unsafe seastates has been prepared based on the pile allowable stress. This table is used to determine whether the predicted sea-state by ANNs is safe or not.
2. Data The data used for this study were the 3-hourly Hs’s and Tz’s observed during 1978–1999 at the US National Oceanographic Data Center Buoy 46005 in the north east Pacific near 461N and 1311W. The significant wave height has been calculated as the average of the highest one-third of all of the wave heights observed during the 20-minute sampling period. The buoy performs the sampling mostly every 1 h and sometimes every 3 h. The resolution is one-tenth of a meter. There are some gaps (approximately 15 months) in the record. The details of the missing Hs’s and Tz’s are reported by Anderson et al. (2001). This set of data had been observed for 21 years and 84 days (7754.375 days) and was expected to consist of 7754.375 8= 62,035 values of 3-hourly Hs’s and Tz’s. The data for some periods including the following were missing: February to April 1985, November to December 1985, October to November 1986, June 1987. December 1990, April to June 1993. For the Hs simulation, those data were extracted whose immediate 8 preceding consecutive 3-hourly values of Hs were available. This led to a vector of 49,736 Hs’s. For the simulation of Tz the missing Hs’s and Tz’s whose immediate preceding and immediate next 3-hourly observed data were available were interpolated linearly, resulting in 55,922 values of data.
3. The specifications of the Neural Networks Employed In this work one of the most common neural network modeling techniques, i.e. feed-forward multi-layer networks has been used to simulate the Hs and Tz of a future 3-hourly sea-state. For this purpose two conditional distributions were used, one for
Tz
s
201
the mean of the stress time history of the element (MPa) simulated annealing most likely extreme stress occurred in the element energy density of ith wave component mean zero-crossing period for the stress time history of an element in the pile FE model mean zero-up-crossing wave period of a sea-state standard deviation of the stress time history of an element
the Hs and one for the Tz. The parameters of the distributions were estimated from the outputs of trained ANNs. Simulated annealing (SA), a global optimization technique of operations research, is the training algorithm. The details have been described by Bazargan et al. (2007b) and Aminzadeh et al. (2009). However a brief description is followed:
3.1. Networks used for generating Hs To simulate the Hs of a sea-state, more than fifteen different feed-forward multi-layer network architectures were tested and finally seven networks of size 13 24 13 1 were chosen (Fig. 1). Each of these seven networks are used for estimating the seven parameters of a proposed distribution called Heptaparameter spline distribution (see Appendix A). The transfer functions of the two hidden layers and the output layer were of ‘logsig’, ‘logsig’ and ‘purelin’ type, respectively. For each of the 7 networks, the size of the target vector is 1 49,736. The vector includes the observed Hs’s whose immediate eight preceding consecutive 3-hourly values of Hs were reported by the Buoy. The input for training the networks is a matrix of 13 49,736. For every Hs in the target vector, there are 13 values in the input matrix i.e. its immediate 8 preceding successive 3-hourly observed Hs’s, the time and 4 fuzzy values related to the season of the Hs. The time of an Hs in the target vector refers to the time it occurs. The actual time values lie in /0–24S, but to increase the efficiency of training, the range was scaled to /0–1S . The season is usually considered as spring, summer, fall and winter (1, 2, 3, and 4); however, in this work it was expressed as a fuzzy value, such that a particular day of the year belongs to all the four seasons with 4 membership values each lying in the interval /0–1S. Therefore 4 different membership values were assigned to each day of a year. For example the vector (1, 0, 0, 0) is assigned to half past 7 am on the 127th day of the year or 5th May (day= 127.3125) and (0.0246, 0, 0, 0.9754) is assigned to 6 am on 7th February or 6 am on the 38th day of the calendar (day =38.25). The 7 networks produce 7 real numbers as outputs. These numbers are used to estimate the 7 parameters of the Heptaparameter spline distribution (described in Appendix A) of the desired Hs given its 8 immediate successive preceding 3-hourly Hs’s. In this study, two training algorithms were tested: the gradient descent algorithm and simulated annealing. The stochastic algorithm of simulated annealing (SA) proved to be a much faster training algorithm for our purpose than the deterministic algorithm of gradient descent. For using SA as the training algorithm, a performance or cost function based on the maximum likelihood estimation (MLE) method was chosen. The cost function was the natural logarithm of the product of the conditional pdf’s of all observed Hs’s available in the target vector given their eight preceding
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Fig. 1. The ANN architecture used for Hs simulation.
Fig. 2. The ANNs architecture used for Tz simulation.
next sea-states of the desired period have been generated by the code from their conditional statistical distribution given their eight immediate preceding observed or generated Hs’s in a similar way. For more details see Aminzadeh et al. (2009). 3.2. Networks used for generating Tz Several different feed-forward multi-layer network architectures having 2 hidden layers were tested and finally the size 10 1 15 1 (Fig. 2) was used. The transfer functions of the 2 hidden layers and the output layer were of ‘logsig’, ‘logsig’ and ‘purelin’ type, respectively. A vector of 1 25,000 of 3-hourly Tz’s as described before was used as the target vector, although the code used had the capability of using a target vector of larger size. To increase the efficiency of training, the target vector was scaled using Fig. 3. Pile installation configuration (after Djahansouzi and Yasseri, 1994).
Tzscaled ¼ 0:8
immediate 3-hourly Hs’s (Bazargan et al., 2007a, Aminzadeh, et al., 2009). Since this function is a function of the pdf’s parameters which are estimated from the networks outputs and on the other hand the networks outputs depend on the networks weights and biases, the objective during the process of training was to change the networks weights and biases continually and randomly in order to maximize the cost function. After the training process, to simulate the 3-hourly significant wave heights for a future time period using the trained networks, the starting day of the period (a real number in the interval /1–365.25S) and the eight 3-hourly consecutive observed Hs’s of 24 h previous to the period are used as the input data to the neural networks coded in MATLAB. The code calculates the 4 fuzzy membership values and the time from the day value (see Appendix B) and also obtains the outputs of the trained networks for each series of the input. The estimates of the 7 parameters of the corresponding conditional distribution related to the Hs of the first sea-state of the desired period are then calculated from the 7 outputs using the relationships given in Appendix C. Then the code generates a variate from the distribution by the inversion method (Naylor et al.,1968). This variate is taken as the predicted Hs of the first 3-hourly sea-state of the desired future period. However, a histogram could be created from several of such generated variates and the most frequent variate of the histogram or its mean can be taken as the predicted value. The Hs’s of the
Tz minðTz Þ þ0:1; maxðTz ÞminðTz Þ
ð2Þ
where min(Tz)= 3.7, i.e. the minimum of the Tz data and max(Tz) =17.5, i.e. the maximum of the Tz data. The input data for training the networks was a matrix of 10 25000. For any 3-hourly Tz in the target vector there are 10 pieces of data in the input vector as follows: its two immediate preceding 3-hourly Tz’s, its immediate preceding 3-hourly Hs, the Hs of the corresponding sea-state, the Hs of next immediate sea-state, the occurrence time of the desired sea-state which is a value in the interval /0–24S divided by 24, and finally 4 fuzzy values representing the season of the sea-state (see Appendix B). The data belonging to the future is not given as input; because the code will generate them. To increase the efficiency, the Hs’s of the input matrix were scaled using Hsscaled ¼ 0:8
HS minðHS Þ þ0:1; maxðHS ÞminðHS Þ
ð3Þ
where min(Hs)= 0.2 i.e. the minimum of the Hs data and max(Hs)= 13.6 i.e. the maximum of the Hs data. The networks were trained by SA algorithm. To simulate the Tz’s of future sea-states using the trained networks the starting day of the period, a number in the interval /1–365.25S, and the necessary input Tz’s and Hs’s (see Bazargan
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et al., 2007b) for predicting the Tz of the first sea-state of the future period are given as input to a MATLAB code. The code then calculates the other necessary inputs. In a similar way stated for the prediction of Hs’s, the Tz’s of the desired sea-states are generated by the code from the corresponding conditional statistical distribution of the Tz given 10 inputs. The details are described by Bazargan et al. (2007b).
4. Case study In the case study conducted, the behavior of a vertical pile stick-up just after stabbing but before the driving operation is investigated. Fig. 3 shows the configuration of such a pile. The lower end of the pile is embedded in soil and restrained from lateral movement by the soil and pile sleeves. The pile supports the weight of the pile-driving hammer at the top and is subject to wave and current loads as well as self-weight and buoyancy effects along its length. The objective of this work is to determine the probability that a forecasted sea-state is safe for the operation.
5. Finite element modeling and analysis The pile is a pipe of 72 m long with 8 cm wall thickness and outer radius of 91.45 cm. The model of the pile used for analysis consisted of 30 two-node PIPE-21 beam elements (ABAQUS, 2004). The foundation was represented by non-linear spring elements. GAP elements were used to represent the gap between the pile and its sleeve. The analysis was conducted in a time domain covering 300 s with time increments of 0.05 s to ensure that a converged pile response was computed. To facilitate the derivation of spectral
203
properties from the time series, the response was computed for at least 2048 sampling points. Pile is initially leaning against the pile sleeve. In the first step of a multi-step analysis the pile self-weight was applied and a non-linear static analysis was performed to determine the equilibrium position. In the next step the weight of hammer was applied. The third step was a non-linear time domain analysis in which the pile set up was exposed to the effect of wave loading calculated by Morrison equation in ABAQUS. Airy wave theory was used for this analysis, as it is considered to be accurate for low amplitude waves in deep water. For each sea-state, the sea surface was presented using a large number of Airy waves. ABAQUS/AQUA software was used to analyze the pile for more than 70 sea-states (see Table 1).
6. Decomposition of sea-states ABAQUS/AQUA requires the input sea-states to be given in the form of waves. Thus a sea-state needs to be decomposed into several sinusoidal wave components identified by 3 parameters, i.e. amplitude, period and phase angle. It was assumed the Pierson–Moskowitz spectral formulation (described in Appendix D) represents the sea-state. Each sea-state was decomposed into 30 sinusoidal components having its own amplitude, frequency and phase angle. The phase angles were generated randomly between 01 and 3601. Then for each sea-state the range of frequency (bandwidth), i.e. zero to where the spectrum approaches zero, was determined and divided into 30 subintervals. The frequency of each subinterval end point was selected as each component frequency. Then, given the frequency of the component, its amplitude was calculated using the following formula
Table 1 The results of pile analysis. Sea-state
von Mises predicted
1 = Safe; 0 =Uns.
Sea-state
von Mises predicted
from Eq. (5) for 3 h (MPa) Hs (m)
Tz (s)
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.8 0.8 1.0 1.0 1.0 1.0 1.0 1.0 1.2 1.2
2 3 4 6 7 8 10 12 2 3.5 4 6 8 10 12 4 10 2 4 6 8 10 12 2 4
19.62 26.70 29.96 23.55 26.82 22.36 21.12 20.29 28.33 55.05 43.94 29.71 26.03 22.89 22.60 55.26 30.12 20.48 45.57 39.25 36.31 31.65 26.74 150.58 59.68
1=S 0 =U
Sea-state
von Mises predicted
from Eq. (5) for 3 h (MPa)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1
Hs (m)
Tz (s)
1.2 1.2 1.2 1.2 1.2 1.4 1.4 1.4 1.4 1.4 1.4 1.6 1.6 1.6 1.6 1.6 1.6 1.8 1.8 1.8 1.8 2 2 2 2
4.3 6 8 10 12 2 4 6 8 10 12 2 4 6 8 10 12 6 8 10 12 2 4 6 6.5
79.50 53.19 48.52 34.44 29.71 34.78 52.66 43.29 35.62 31.76 34.00 28.70 120.86 67.55 52.70 35.83 30.68 56.20 41.36 51.77 32.44 26.06 77.63 50.96 64.49
1=S 0 =U
from Eq. (5) for 3 h (MPa)
1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1
Hs (m)
Tz (s)
2 2 2 2.2 2.2 2.2 2.2 2.2 2.2 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.8 2.8 2.8 3.2 3.2 3.2 3.5 3.5 3.5
8 10 12 2 4 6 8 10 12 2 3 4 6 8 10 12 3 7 8 4 8 12 4 8 12
59.41 39.31 42.24 24.47 69.64 95.71 51.46 41.50 42.21 24.09 77.03 75.83 78.36 29.71 54.03 39.54 66.87 112.25 79.36 227.90 69.1 57.5 129.97 116.93 53.8
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 0 0 1
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zero-crossing period for the stress time history of the element; in our case T=300/N where N= total number of zero-up-crossings in the stress time history of the element. The most probable maximum von Mises stress predicted to occur in the pile after 3 h is the value obtained from Eq. (5) for the critical element of the pile. This value is the maximum of the values obtained from Eq. (5) for all elements. For more than 70 sea-states this maximum has been calculated. Table 1 shows the pile responses. In this analysis, a sea-state is considered safe for pile driving if the maximum predicted von Mises stress is less than 55% of the yield stress of the material of the pile i.e. steel M10. The basic safety margin in API is 67% of the material yield for normal loading moderated by buckling. Due to unsupported large length, the allowable axial load is very low. The paper uses the von Mises stresses to judge if the operation is safe. The value of 55% allows a safety factor of nearly 2, which the operating company and third party certifier agree to use as an equivalent threshold allowable stress. The case study comes from an actual offshore pile driving during special months of the year. This value within the context of the paper must be taken as an example of how one can make the necessary decision. The sea-states characterized with Hs’s greater than 3.5 m were considered unacceptable(unsafe); because the maximum allowable equivalent stress was agreed to be 55% of the material yield and the stresses that they would generate exceed the agreed acceptable level.
Fig. 4. Stress time history at an integration point on the critical element.
(Goda, 2000): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ai ¼ 2Sðfi Þ Dfi ;
ð4Þ
where ai is the amplitude of ith component, fi the frequency of ith component, S(fi) the energy density given by Eq. (D1) and Dfi = (fi + 1 fi 1)/2.
7. Pile response The von Mises stress was used as a measure of pile strength (API, 1993). Since the pile response is non-linear, the position of the maximum stress depends on the wave excitation. The time history of stresses at a subset of the pile elements were obtained and examined to ensure that the member with maximum response is picked up. The analyses were done by ABAQUS/AQUA in time domain covering 300 s with time increments of 0.05 s to ensure that the pile response has settled down. The von Mises stress has been calculated for nearly 300/0.05 = 6000 time increments for all of the elements in the subset. Fig. 4 shows the stress history occurred in one of the integration points of an element proved to be critical due to a sample sea-state for a time of 300 s. Some 6000-von-Mises-stress values were obtained for each integration point (both the top and bottom integration points). due to a sea-state loading as well as the self-weight and buoyancy effects. Therefore for each element of the subset 2 series of 6000von-Mises-stress values were produced by ABAQUS/AQUA. The mean and standard deviation of these values were computed. Then the most likely extreme amplitude of stress cycles after time h, using a long-term extreme prediction due to Barltrop and Adams (1991), were calculated from the following formula for all elements in the subset: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3600h ; ð5Þ Se ¼ S þ s 2ln T where Se is the most likely extreme stress of the element after h hours (MPa), S= the mean of the stress time history of the element (MPa), s the standard deviation of the stress time history of the element (MPa), h the time (h), in our case h=3, T the mean
8. Safe sea-state for pile driving Given the necessary inputs as described earlier , the Hs and Tz of a desired future 3-hourly sea-state are generated as random variates from their corresponding conditional distribution. To determine the level at which a future sea-state is safe to drive a pile, the 3-hourly sea-state is first predicted K times in this way and then, using Table 1 it is determined whether each of these is predicted to be safe. Suppose the sea-state proves to be safe n times and unsafe K n times; it could therefore be concluded that the sea-state is safe with 100 n/K percent of certainty.
9. Illustration As an illustration, assuming a reference date of 1st January we would like to simulate and forecast the sea-states of a period starting 00 am on 2nd January to 12 pm on 30th December of the same year for the region in the Pacific where Buoy 46005 operates. Table 2 shows the observed 3-hourly Hs’s of 24 h before the above starting time and the two necessary Tz’s. This data is necessary as input for simulation.
Table 2 Inputs for forecasting. Time
Date
Hs (m)
Tz (s)
9 pm 6 pm 3 pm 12 noon 9 am 6 am 3 am 00 am
1st 1st 1st 1st 1st 1st 1st 1st
3.1 3.2 3.0 2.6 2.7 2.0 1.7 1.8
6.7 6.8 n.r. (not required) n.r n.r n.r n.r n.r
January January January January January January January January
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For practical purposes the number of simulation runs should be increased to get more accurate predictions.
Table 3 Ten times prediction of sea-state for 9 a.m. on 2nd January. Run no. 1 2 3 4 5 6 7 8 9 10
Hs (m)
Tz (s)
1 = Safe 0= Unsafe
2.5 3.1 3.7 4.3 4.1 3.8 5.4 2.8 3.9 5.5
7.6 7.9 7.4 7.9 7.0 8.3 7.3 6.6 8.3 7.4
1 1 0 0 0 0 0 1 0 0
Table 4 Ten times prediction of sea-state for 24 p.m. on 30th December. Run no. 1 2 3 4 5 6 7 8 9 10
205
Hs (m)
Tz (s)
1 = Safe 0= Unsafe
5.25 2.25 6.0 4.4 3.3 8.9 3.4 2.6 1.5 8.5
9.0 5.4 9.7 9.5 7.3 11.0 10.5 6.4 6.5 9.8
0 1 0 0 1 0 0 1 1 0
After inputting the data of Table 2 to a MATLAB code, the 3-hourly sea-states for the period were simulated one after another by running the code which calculates the parameters of the 2 required conditional distributions for the Hs and Tz of each 3-hourly sea-state in the period from the outputs of the trained neural networks(see Appendix C) and generates the Hs and Tz from the corresponding distribution. For each time of simulating the 3-hourly sea-state of period, the code generates nearly (365.25–1) 8= 2914 sea-states. Suppose we are interested in the sea-state of 9 am on 2nd January and 12 pm on 30th December. The results of 10-time simulation for 9 am on 2nd January are shown in Table 3. The code has generated the Hs’s and the Tz’s of the sea-states (columns 2 and 3) in 10 runs. Column 4, which indicates whether the generated sea-states are safe or otherwise, was extracted visually from Table 1. Using this prediction it can, therefore be concluded from column 4 that the sea-state of 9 am on 2nd January is safe with a probability of 3 out of 10 i.e. 30 percent (30%). Note that the mean of the 10 predicted values are Hs = 3.91 m and Tz =7.57 s, and the mean of the observed values for 9 am on 2nd January during 1978–1999 were 3.89 m and 7.67 s, respectively. Using this procedure the sea-states of any future time period could be predicted. However, the shorter the time, the more accurate the prediction. The results of 10-time simulation for 24 pm on 30th December are shown in Table 4. The code has generated the Hs’s and the Tz’s of the sea-states. It is worth noting that the mean of the 10 simulated values are Hs =4.61 m, Tz = 8.51 s and the mean of the corresponding observed values during 1978– 1999 for 24 pm on 30th December are 3.46 m and 7.52 s, respectively. Column 4 was extracted visually from Table 1. Therefore it can be concluded that the sea-state of 24 pm on 30th December is safe with a probability of 4 out of 10 i.e. 40 percent (40%).
10. Conclusions From the simulated and observed values it is concluded that: (1) In this study multi-layer feed-forward ANNs proved to be an efficient tool in estimating the parameters of the proposed distribution used to simulate the characteristics of the seastates of a desired time and thereby in the simulation of seastate sequences. (2) The predicted 3-hourly significant wave heights and mean zero-up-crossing wave periods of the sea-state of a time in the near future obtained by the proposed method compare well with their corresponding observed values. (3) The proposed hepta-parameter spline distribution approximating the conditional probability distributions used to simulate the Hs and the Tz of a 3-hourly sea-state in the north east Pacific proved to be a good choice for the approximation. (4) The mean of the conditional distribution or the mean of the simulated values obtained in several runs of simulation provides a good forecast for the Hs and the Tz of a desired time sea-state. (5) The finite element and neural network modeling resulted in an easy-to-use procedure to predict the degree of safety of the sea-states at which the pile-driving operation could be conducted in a future period given the eight successive 3-hourly sea-state history preceding that period. The method introduced for the simulation of sea-state characteristics is applicable to other time-dependent processes which have instinct periodical variations. The application could cover a variety of things such as simulating a related random variable, forecasting its most probable and mean values and also doing an extreme value analysis related to the process.
Acknowlegement The authors would like to thank Mr. Robert Sears who had prepared the ABAQUS model formulation for the pile stick-up analysis initially.
Appendix A. Hepta-parameter spline distribution In this appendix, a parametric distribution named ‘‘heptaparameter spline’’ is proposed. This very flexible distribution was used in this research to approximate the conditional distribution of [Hs(tn)9Hs(tn 1) =hn 1, y,Hs(tn 8)= hn 8], whose pdf is denoted by f(x). The density function f(x) should satisfy the following two properties: (1) limx- 7 1 f ðxÞ ¼ 0, (2) smoothness. These two properties imply that there are two points, say a and e (a oe) where f(x) approaches zero for x oa and x 4e and also there is a point c in the interval /a–eS for which f 0 ðcÞ ¼ 0 (e.g. the maximum of the function). Since there were no a priori information available on the shape of the distribution of Hs, the proposed density function had to be designed in such a way that it could closely approximate a number of possible distributions. The cubic interpolations on the four subintervals of Fig. A1 were assumed to be a good approximation for f(x). Note that a, b, c, d, e, m1, and m2 were chosen such that: b¼
aþc ; 2
d¼
c þe ; 2
m1 ¼ f 0 ðbÞ;
and
m2 ¼ f 0 ðdÞ:
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S.F. Yasseri et al. / Ocean Engineering 37 (2010) 200–207
Fig. A1. The intervals on which the conditional distribution is defined, and the parameters of the distribution.
gðxÞ ¼
8 0 > > > > > > > Cubic-Int: > > > > > > > > > > > > Cubic-Int: > > <
a " "
> > > > Cubic-Int: > > > > > > > > > > > > Cubic-Int: > > > > > : 0
"
0 b a2 c a3 d a4
" with slope ¼ 0
&
with slopem1 40
&
#
"
#
" with slope ¼ 0
&
with slope m2 o 0
&
#
b a2 c a3 d
with slope m1 40
aoxrb
with slope ¼ 0
b ox r c
with slope m2 o0
c o x rd
with slope ¼ 0
doxre
# #
a4 e 0
xra
#
xZe
Fig. A1 also shows a1, a2, a3, a4, l, D, m1, and m2 the parameters of the function used to approximate of the conditional distribution. The density function g(x), the proposed pdf for approximating f(x), is defined as follows: where a, b, c, d, e, a2, a3, a4, m1, m2 are illustrated in Fig. A1, and Cubic-Int. stands for the cubic interpolation of each pair of the points mentioned above according to spline method. Having two points x1 and x2 and the values of a function f(x) and its derivative at x1 and x2, the formula for approximating f(x) according to the spline method is given by gðxÞ ¼ f ðx2 Þ
xx1 x2 x1
þ f ðx1 Þ
3
xx2 x1 x2
3f ðx2 Þ ðxx1 Þ2 ðxx2 Þ þ f 0 ðx2 Þ x2 x1 ðx2 x1 Þ2
3
3f ðx1 Þ ðxx2 Þ2 ðxx1 Þ þ f 0 ðx1 Þ x1 x2 ðx1 x2 Þ2
and 6a4 rm2 r 0: ð1lÞD
ðA4Þ
The distribution has the capability to approximate a number of the conventional distributions including chi-square, positive normal, 3-parameter Weibull. However, an attempt to use the 3-parameter Weibull distribution instead of the hepta-parameter distribution in the study was not found to be satisfactory.
Appendix B. Calculation of the time and four season values from day values ðA1Þ
The time at which the sea-state being predicted occurs is calculated from the following MATLAB command: time ¼ floorðdayÞ;
Since
R1
gðxÞ dx ¼ 1; it could be proved that:
1
lDa2 2
þ
Da3 4
þ
ð1lÞDa4 ¼ 1: 2
ðA2Þ
Hence one of the parameters a2, a3, a4, l, and D could be calculated from the other four and the number of independent parameters reduce to seven. Therefore, g(x) has seven independent parameters and hence the name hepta-parameter spline. Arbitrarily a3 was determined from a2, a4, l, and D. Finally, since all the interpolated values of g(x) have to be positive, it could be shown that the following two inequalities must also be satisfied: 0 r m1 r
6a2
Dl
where day is a number in the interval /0–365.25S with step 0.125 representing the day on which 3-hourly sea-states occur. The four fuzzy membership values related to the season in which the sea-state occurs are calculated using the following four MATLAB instructions: (1) (2) (3) (4)
springvalue= Trianglefun(abs(day-midspring)/(365.25/4)) summervalue= Trianglefun(abs(day-midsummer)/(365.25/4)) fallvalue= Trianglelfun(abs(day-midfall)/(365.25/4)) wintervalue=Trianglefun(abs (day-midwinter)/(365.25/4))
where Trianglefun is the following MATLAB function: ;
ðA3Þ
function v ¼ TrianglefunðdayÞ;
ARTICLE IN PRESS S.F. Yasseri et al. / Ocean Engineering 37 (2010) 200–207
207
v ¼ ð1absðdayÞÞ ðabsðdayÞ o 1Þ þ ð1absðday4ÞÞ ðabsðday4Þ o 1Þ and since the 36th day of the year (4th February) is the midwinter, midwinter= 36; and also: midspring= 36+ (365.25/4)= 127.3125 (5th April) midsummer=36 +2 (365.25/4)= 218.6250 (6th August) midfall= 36+ 3 (365.25/4)= 309.9375 (6th November)
Appendix C. Calculation of the distribution parameters from the networks outputs Let xj denotes the output of the jth network for j= 1, 2, y,7. Since the range of the networks outputs is ( N to N) due to the transfer function of purelin in the output layers, we cannot directly use the outputs of any of the 7 networks for calculating positive a1. Hence a transformation was applied on x1 to arrive at a positive number: a1 = x21. The relationships used to calculate the parameters from the outputs of the networks are given as follows:
Fig. D1. A typical PM energy density spectrum.
where S(fi)= energy density of ith wave component (m2/Hz), A ¼ Hs2 =4pTz4 , B ¼ 1=pTz4 , 0 ofi oN, fi = the frequency of ith wave component (Hz), As an illustration, the spectrum has been plotted for a sea-state with Hs = 1 m and Tz = 3 s in Fig. D1:
Constraint a1 o0
Relationship a1 ¼ x21
ðC1Þ
0 oa2 oa3
a2 ¼ a3 logsig x2
ðC2Þ
D 40
D ¼ x23
ðC3Þ
References
0 oa4 oa3
a4 ¼ a3 logsig x4
ðC4Þ
ABAQUS 6.4 User’s Manual, 2004. Hibbit, Karlsson & Sorensen Inc. Agrawal, J.D., Deo, M.C., 2002. On-line wave prediction. Marine Structures 15 (1), 57–74. Aminzadeh, A., Bahai, H., Bazargan., H., 2009. Simulation of significant wave height by neural networks and its application to extreme wave analysis. Journal of Atmospheric and Oceanic Technology 26 (4). Anderson C.W., Carter D, Cotton P.D., 2001. Wave climate variability and impact on offshore design extreme. Report for Shell International, and the Organisation of Oil and Gas Producers /http://info.ogp.org.uk/metocean/JIPweek/WCERe port_2sided.pdfS. API, 1993. Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms—Load and Resistance Factor Design. API RP 2A-LRFD, Washington, DC. Barltrop, N.D.P., Adams, A.J., 1991. In: Dynamics of Fixed Marine Structures. Butterworth-Heinemann, Oxford (Chapters 3 and 6). Bazargan, H., Bahai, H., Aminzadeh, A., Bazargan, A., 2007a. Neural network simulation of significant wave height. In: 26th International Conference on Offshore, Mechanic & Arctic Engineering, June 10–15, 2007, San Diego, USA. Bazargan, H., Bahai, H., Aryana, F., 2007b. Simulation of the mean zero-up-crossing wave period using artificial neural networks trained with a simulated annealing algorithm. Journal of Marine Science and Technology 12 (1), 22–34. Buoy Data—46005 WASHINGTON 315NM West of Aberdeen, WA /http://www. nodc.noaa.gov/BUOY/46005.htmlS. Deo, M.C., Jha, A., Chaphekar, A.S., Ravikant, K., 2001. Neural networks for wave forecasting. Ocean Engineering 28 (7), 889–898. Djahansouzi, B, Yasseri, F., 1994. Study of offshore pile stick-up. In: Proceedings of the Ninth UK ABAQUS User Group Conference, Oxford. Goda, Y., 2000. In: Random Seas and Design of Maritime Structures 2nd ed. World Scientific, Singapore (Chapter 10). Makarynskyy, O., Pires-Silva, A.A., Makarynska, D., Ventura-Soares, C., 2002. Artificial neural networks in the forecasting of wave parameter. In: The Seventh International Workshop on Wave Hindcasting & Forecasting, Banff, Alberta Canada 21–25 October 2002, pp. 514–522 /http://www.wavework shop.org/7thwaves/papers/makarynskyy_etal.pdfS. Makarynskyy, O., 2004. Improving wave prediction with artificial neural nets. Ocean Engineering 31 (5–6), 709–724. Makarynskyy, O., 2005. Artificial neural networks for wave tracking, retrieval & prediction. Pacific Oceanography 3 (1), 21–30. Naylor, H.N., Balintfy, J.L., Burdick, D.S., Chu, K, 1968. In: Computer Simulation Techniques. J W & Sons, New York (Chapter 4). Pierson, W.J., Moskowitz, L., 1964. A proposed spectral form for fully developed wind seas based on similarity theory of S.A. Kitaigorodskii. Journal of Geophysical Research 69 (24), 5181–5190.
0 rm1 r
6a2
Dl
m1 ¼
6a2
Dl
logsigx5
6a4 logsigx6 ð1lÞD
6a4 r m2 r 0 ð1lÞD
m2 ¼
0 o l o1
l ¼ logsig x7
ðC5Þ
ðC6Þ ðC7Þ
From Eqs. (A2), (C2), (C3), (C4), and (C7) we have a3 ¼
2x2 3 ðlogsigx7 Þðlogsig x2 Þ þ 0:5 þð1logsig x7 Þðlogsigx4 Þ
ðC8Þ
where logsig xj ¼
1 ; 1þ exj
j ¼ 1; . . .; 7:
ðC9Þ
In practice, in the first step, a3 is calculated by Eq. (C8); then a2 and a4 are calculated from Eqs. (C2) and (C4). Other parameters are calculated simply using the appropriate relationship.
Appendix D. Pierson–Moskowitz energy density spectrum Pierson–Moskowitz (PM) energy density spectrum (Pierson and Moskowitz, 1964) with its following formula is suitable for decomposing a sea-state into several sinusoidal wave components. The formula depends on the sea-state representing parameters, i.e. Hs and Tz (Barltrop and Adams, 1991): ! B4 ; A Sðfi Þ ¼ 5 e fi ðD1Þ fi