Nonlinear sea wave crest height exceedance probability prediction system and method based on transformed gaussian process
Patent Information
- Application Number
- CN202310999140.3
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-08-09
- Publication Date
- 2026-09-15
- Estimated Expiration
- 2043-08-09
AI Technical Summary
然而，这些研究者的非线性模拟耗时过长，这一缺点必然会影响非线性模拟方法在一些时间紧迫的海洋工程项目中的应用
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Figure CN117010201B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a technology in the field of nonlinear wave crest height exceedance probability prediction, specifically a nonlinear wave crest height exceedance probability prediction system and method based on a transformed Gaussian process. Background Technology
[0002] Determining the probability distribution of wave crests is crucial for the design and risk analysis of various ships and marine structures. The probability distribution of wave crest elevation (i.e., the highest elevation of a single wave) must be carefully determined as it is used to calculate wave loads on ships or offshore platforms. Extreme wave crest heights are also key input parameters for determining bow height to avoid deck submersion. Furthermore, accurate statistical knowledge of extreme wave crest heights is essential for determining sufficient air clearance below deck on offshore platforms to ensure that wave crests do not compromise platform integrity. Simultaneously, understanding wave crest distributions is also crucial for the successful design of various coastal protection structures.
[0003] In practical marine engineering projects, the known information about environmental conditions is typically the wave spectrum corresponding to short-term sea states. Based on this specific wave spectrum, several methods exist for calculating the exceedance probability of wave crest height. The first and most straightforward method is to use empirical (or theoretical) models, such as the Rayleigh model of the wave crest height distribution of random ocean waves. However, previous research has shown that the Rayleigh model systematically underestimates the wave crest height distribution of nonlinear random ocean waves.
[0004] Depending on the specific spectrum, there is another method to calculate the exceedance probability of wave crest height. For example, based on the spectrum of the GullfaksC platform, some researchers have used linear simulation methods to calculate the wave crest height distribution. However, their calculations show that under actual nonlinear sea conditions, linear simulation methods predict overly nonconservative wave crest height probability distributions, which will lead to the design of unsafe marine engineering structures.
[0005] Some researchers have used nonlinear simulation methods to calculate the crest height distribution of nonlinear irregular waves, and they have verified that the crest height distribution calculated by their nonlinear simulations is more accurate than that calculated by linear simulation methods. However, these nonlinear simulations are too time-consuming, which inevitably affects the application of nonlinear simulation methods in some time-sensitive marine engineering projects. Summary of the Invention
[0006] To address the aforementioned shortcomings of existing technologies, this invention proposes a nonlinear wave crest height exceedance probability prediction system and method based on a transformed Gaussian process. This system can significantly improve the prediction accuracy of nonlinear wave crest height exceedance probability and greatly reduce the computer time required for predicting the exceedance probability of nonlinear wave crest height, thereby greatly improving the accuracy and efficiency of marine engineering structure design.
[0007] This invention is achieved through the following steps:
[0008] This invention relates to a nonlinear wave crest height exceedance probability prediction system based on a transformed Gaussian process, comprising: a non-Gaussian process unit, a saddle point approximation unit, an inverse transform unit, and an output unit. The non-Gaussian process unit performs calculations using Rice's formula based on the joint probability density information to obtain the horizontal upward leap rate of the non-Gaussian wave surface elevation. The saddle point approximation unit performs saddle point approximation based on the relevant information of the cumulant generating function to obtain the joint probability density. The inverse transform unit performs mathematical calculations based on the obtained horizontal upward leap rate of the non-Gaussian wave surface elevation to obtain the inverse transform expression. The output unit performs mathematical calculations based on the obtained inverse transform expression to obtain the nonlinear wave crest height probability distribution.
[0009] This invention relates to a nonlinear wave crest prediction method based on the above-mentioned system, comprising:
[0010] Step 1: Waves in an idealized linear Gaussian random sea state exhibit crest-trough symmetry. However, it is well known that sea level rise in the real world deviates from a Gaussian distribution; that is, wave crests become steeper and higher than expected under a Gaussian distribution, and wave troughs become flatter and shallower. If the vertical axis is represented by x and time by t, then the free sea surface elevation under a nonlinear random sea state is... Where: η (1) (x,t) is a first-order linear component, η (2) (x,t) represents the second-order nonlinear correction component. Typically, a sufficiently large positive integer N is chosen, and c... n Represents a random complex-valued amplitude, which can be determined based on a specific spectrum S. η (ω) and angular frequency ω n It was calculated. Furthermore, k n For a specific wavenumber, it is related to ω through the dispersion relation. n Related, and ε n It is a uniformly distributed phase angle between 0 and 2π. and The term is a second-order transfer function, which can be calculated using the following formula (when the water depth d remains constant):
[0011]
[0012]
[0013] When executing the equation η(x,t), η is omitted. (2)The (x,t) term will yield a linearly simulated wave elevation time series; this is the so-called linear simulation method. The complete execution of equation η(x,t) without omitting η... (2) The (x,t) term will calculate the wave elevation time series from the nonlinear simulation; this is the so-called nonlinear simulation method. However, due to r mn and q mn Contains N 2 The sum of frequency components and N 2 With multiple difference frequency components, nonlinear simulations become extremely time-consuming when N is large. Therefore, predicting peak height distribution and exceedance probability based on wave height time series from nonlinear simulations is not an effective method.
[0014] Step 2: Without loss of generality, assume x = 0 in η(x,t) and simplify it to η(t). For a specific nonlinear sea state, η(t) will be a non-Gaussian stochastic process. Here, a widely valid non-Gaussian sea area model will be used to represent η(t) as a static zero-mean Gaussian process with variance 1. The function, i.e. It is important to note that once the peak is calculated... If the distribution of the peaks in η(t) is such that a simple inverse transform of G is sufficient to obtain the distribution of the peaks in η(t), then... As is well known, for a zero-mean Gaussian process... The upward crossing rate of the horizontal line u in: and They are and its derivative The variance.
[0015] Step 3: Now let's look at the upward level leap rate μ(u) of the non-Gaussian process η(t). For all G functions that satisfy the properties in Step 2, the following relationship exists:
[0016] Step 4: The following equation then holds true: If the upward leveling rate μ(u) of the non-Gaussian process η(t) is known, then the inverse transform can be obtained according to the above formula.
[0017] Step 5: Discuss the upward leveling rate μ(u) and the high probability distribution of the peak. The relationship between M * Represents the peak height, which satisfies
[0018] Step 6: Combining Step 4 and Step 5, we arrive at the following conclusion:
[0019] Step 7: To calculate the peak distribution of the non-Gaussian process η(t) (i.e. The key task is to calculate the upward level leap rate μ(u) of the non-Gaussian process η(t), and the principle of the numerical calculation process will be explained below.
[0020] When η(t) is non-Gaussian distributed, for a fixed horizontal line u, let μ(u) be the expected number of times the process η(t) crosses the horizontal line u upwards within the interval [0,1]. Then Rice's formula is extended as follows: That is, when calculating μ(u), it is necessary to estimate η(0) and The joint probability density. For the current problem, there is no η(0) and The explicit closed-form formula for the joint probability density is given. The saddle point approximation method will be used to calculate...
[0021] Step 8: To use the saddle point approximation method, we need to obtain η(0) and The explicit formula for the cumulant generating function is derived in step 1 using the following matrix notation: η(t) = s T X+X T [Q+R]X+Y T [QR]Y, where Q and R are real symmetric matrices, and their nm components are respectively... and ; s is a vector, and its nm component is X and Y are vectors, and their nm components are x and y respectively. n and y n ;x n and y n It is a Gaussian random variable. Then we can calculate... That is, the derivative of η(0). When t = 0, we obtain η(0) and... The value of . Now we get the vector. The explicit formula for the moment generating function: Where: I is a (2N, 2N) dimensional identifier matrix, the matrix vector W = [w mn ], where: w mm =-ω m When m≠n, w mn =0, S = 2(Q+R)W-2W(QR).
[0022] Step 9: Calculate using f(θ1,θ2)=M(iθ1,iθ2) The characteristic function can then be calculated using the inverse Fourier transform. Joint probability density function:
[0023] Step 10: Calculate the upward leveling rate
[0024] Step 11: Since matrix A is usually very large (possibly with dimensions (500, 500) or larger), and the inverse (IA) needs to be calculated each time the eigenfunction is found. -1 Therefore, numerical integration in the above equation becomes extremely slow. To improve computational efficiency, the saddle point method will be used for approximation. The joint probability density is as follows: The cumulative generation function K(p1,p2)=lnM(p1,p2) becomes:
[0025] Step 12: Calculate density Approximate value of saddle point
[0026] Step 13: Solve the equation to obtain the saddle point. Specifically: in: It is the Hessian matrix of the cumulative generating function K, and the upward leveling rate μ(u) of η(t) is calculated through numerical steps 10-13. Attached Figure Description
[0027] Figure 1 This is a flowchart of the present invention;
[0028] Figure 2 This is a time series diagram of measured wave crest heights along a good harbor coastline, used as an example.
[0029] Figure 3 This is a comparison chart of the effects of the examples;
[0030] Figure 4 The diagram shows the drilling platform and the air gap of the platform base plate as an example. Detailed Implementation Example 1
[0031] This embodiment is based on measured sea surface elevation data from the coast of Yura Port, Japan. The measured wave surface elevation data for Yura Port was obtained 3 kilometers from Yura fishing port facing the Sea of Japan. The measurements were conducted by the Ship Research Institute of the Ministry of Transport of Japan from 11:49 AM to 12:49 PM on November 24, 1987. An ultrasonic wave meter was installed at a measuring point at a water depth of 42 meters (limited depth) to measure the temporal elevation of the wave surface. The sampling time interval during the measurement was 1 second. Figure 2 The image shows the first 100 seconds of the time series data of the measured wave crest height of the harbor coastline in this embodiment.
[0032] like Figure 1 As shown, the forecasting method in this embodiment specifically includes:
[0033] Step 1) Construct the Rice formula to calculate the upward leveling rate of the non-Gaussian process η(t). Where: u represents a specific level value, z represents the derivative of that specific level value, and η(0) represents the wave elevation at time 0. The derivative of the wave elevation at time 0;
[0034] Step 2) Approximate the calculation of step 1) using the saddle point method. Specifically:
[0035] 2.1) Obtain the saddle point by taking the partial derivative of the cumulant generating function;
[0036] 2.2) Calculate the density based on the saddle point values obtained above. Approximate value of saddle point
[0037] Step 3) Express the non-Gaussian wave elevation η(t) as a static zero-mean Gaussian process with variance 1. The function, i.e. Based on the upward level leap rate μ(u) of the non-Gaussian process η(t) obtained in step 1), the inverse transformation of G is performed as follows:
[0038] 3.1) First, calculate the ratio of the leveling span ratio μ(u) to the leveling span ratio μ(0);
[0039] 3.2) Take the logarithm of the ratio from the previous step;
[0040] 3.3) Take the square root of the negative number that is twice the logarithm of the previous step.
[0041] Step 4) Based on the inverse function G obtained in step 3), -1 The high probability distribution of the wave peaks was calculated. Where: M * Indicates the height of the wave crest.
[0042] like Figure 3As shown, the wave crest height exceedance probability is calculated based on measured sea surface elevation data from the Yuryo Port coast of Japan. The solid green line in the figure represents the calculated wave crest height exceedance probability directly obtained from measured wave data from the Yuryo Port coast containing 3600 wave elevation points. To obtain this solid green line, the wave crest height time series is first extracted from these 3600 wave elevation points. Then, an accurate Ipanechev kernel density estimation is performed to obtain the probability density function of the wave crest height. Next, the cumulative trapezoidal numerical integral is performed on the above probability density function to obtain the probability distribution (F) of the wave crest height. Finally, the exceedance probability of the wave crest height is obtained using the formula P = 1 - F. This solid green line based on measured wave data from the Yuryo Port coast serves as a benchmark for verifying the accuracy of various numerical simulation methods and existing wave crest height models.
[0043] like Figure 3 As shown, the pink solid line represents the wave crest height exceedance probability obtained using the Rayleigh distribution model. It is clear that the wave crest amplitude exceedance probability obtained using the Rayleigh distribution model deviates significantly from the corresponding baseline result obtained directly from measured wave data from the Yuliang Port coast. This is not surprising, because the Rayleigh distribution is an ideal linear model, while the measured wave crest height sequence from the Yuliang Port coast is obviously nonlinear. Figure 3 As shown, the black "*" indicates that... Figure 2 The spectrum generated from the time series shown is obtained using a linear simulation method, yielding peak height transcendence probabilities. The linear simulation process follows step 1, as follows: Figure 2 The wave spectrum generated by the time series shown, omitting the terms in formula (1), generates a wave crest height time series containing 8,000,000 points. Then, the wave crest height time series is extracted from these 8,000,000 wave crest height points. Then, a precise Ipanechev kernel density estimation is performed to obtain the probability density function of the wave crest height. Then, the cumulative trapezoidal numerical integral of the above probability density function is performed to obtain the probability distribution (F) of the wave crest height. Finally, the wave crest height exceedance probability is obtained using the formula P = 1 - F. The entire linear simulation process takes approximately 9 seconds to complete on a desktop computer. It can be observed that the wave crest height exceedance probability obtained using the linear simulation method deviates significantly from the corresponding benchmark result obtained directly from measured wave data from the coastline of a good harbor. To calculate the wave crest height exceedance probability more accurately, a nonlinear simulation method is used in step 1. For example... Figure 3 As shown, the blue solid line represents... Figure 2 The spectrum generated from the time series shown is obtained using nonlinear simulation methods to determine the peak height transcendence probability. The nonlinear simulation process first utilizes methods such as... Figure 1The wave spectrum generated by the time series shown is used to generate a nonlinear wave crest height time series of 800,000 points by applying equations applicable to nonlinear wave simulation in finite-depth waters. Then, the wave crest height time series is extracted from these 800,000 points. A precise Ipanechev kernel density estimation is then performed to obtain the probability density function of the wave crest height. Finally, the cumulative trapezoidal numerical integral of the probability density function is performed to obtain the probability distribution (F) of the wave crest height. Finally, the probability of wave crest height exceeding the limit is calculated using the formula P = 1 - F. The entire nonlinear simulation process took approximately 364 seconds to complete on a single desktop computer. It can be observed that the wave crest height exceeding probability obtained using the nonlinear simulation method is in good agreement with the corresponding benchmark results obtained directly from measured wave data from a good harbor coastline.
[0044] To further improve computational efficiency, the Gaussian transformation process method (steps 2-13) is used to calculate the peak height transcendence probability more efficiently and accurately. For example... Figure 3 As shown, the red "+" indicates that... Figure 2 The spectrum generated from the time series shown is obtained using the peak height transcendence probability calculated using the transformed Gaussian process method. Figure 1 The spectrum generated from the time series is shown, and then the upward leveling rate μ(u) is calculated according to steps 7-13. During the calculation, joint density is also utilized to improve computational efficiency. Approximate saddle point The specific calculations follow steps 10-S13. After obtaining the upward leveling rate μ(u), the transformation function (G(-) function) is theoretically calculated using step 4. Then, the probability distribution of the wave crest height is obtained using steps 5-6 using this G(-) function. Finally, the exceedance probability of the wave crest height is obtained using the formula P = 1 - F. The entire process of the above-described transformed Gaussian process method takes approximately 7 seconds to complete on an identical desktop computer. It can be observed that the exceedance probability of the wave crest height obtained using the transformed Gaussian process method is in good agreement with the corresponding benchmark results obtained directly from measured wave data from a good harbor coastline. This embodiment demonstrates that this method is more accurate and efficient than various other existing methods. Example 2
[0045] like Figure 4 As shown, the offshore drilling platform involved in this embodiment has a designed vertical height of 5 meters from the bottom plate to the sea level, which means that the air gap of the bottom plate of the offshore drilling platform is 5 meters.
[0046] This embodiment is based on the wave spectrum derived from wave data measured from the coastline of a good harbor.
[0047] like Figure 1 As shown, the forecasting method in this embodiment specifically includes:
[0048] Step 1) Construct the Rice formula to calculate the upward leveling rate of the non-Gaussian process η(t). Where: u represents a specific level value, z represents the derivative of that specific level value, and η(0) represents the wave elevation at time 0. The derivative of the wave elevation at time 0;
[0049] Step 2) Approximate the calculation of step 1) using the saddle point method. Specifically:
[0050] 2.1) Based on the wave spectrum and second-order transfer function obtained from the wave data of the good harbor coastline measured in Example 1, the saddle point is obtained by taking the partial derivative of the cumulant generating function.
[0051] 2.2) Calculate the density based on the saddle point values obtained above. Approximate value of saddle point
[0052] Step 3) Express the non-Gaussian wave elevation η(t) as a static zero-mean Gaussian process with variance 1. The function, i.e. Based on the upward level leap rate μ(u) of the non-Gaussian process η(t) obtained in step 1), the inverse transformation of G is performed as follows:
[0053] 3.1) First, calculate the ratio of the leveling span ratio μ(u) to the leveling span ratio μ(0);
[0054] 3.2) Take the logarithm of the ratio from the previous step;
[0055] 3.3) Take the square root of the negative number that is twice the logarithm of the previous step.
[0056] Step 4) Based on the inverse function G obtained in step 3), -1 The high probability distribution of the wave peaks was calculated. Where: M * Indicates the height of the wave crest.
[0057] Table 1 summarizes the exceedance probabilities of wave crest height (h = 5 meters) obtained using various methods. Table 1 shows that the exceedance probability of wave crest height (h = 5 meters) obtained from measured wave data from a good harbor coastline is 0.0047 (47 / 10,000), which is the baseline value. Table 1 also shows that the exceedance probability of wave crest height (h = 5 meters) obtained using the linear simulation method is 0.0013 (13 / 10,000). This indicates that the forecast value obtained using the linear simulation method is overly optimistic, potentially misleading platform designers, builders, and operators. This could lead to the danger of waves potentially striking the platform's bottom plate, causing damage to the platform's bottom plate and other structures, resulting in loss of life and property, and affecting drilling platform operations. Table 1 shows that the probability of exceeding the wave crest height (h = 5 meters) calculated based on the Rayleigh distribution is 0.0014 (14 in 10,000). This indicates that the forecast value derived from the Rayleigh distribution is overly optimistic and could mislead platform designers, builders, and operators, potentially leading to waves striking the platform's bottom plate, damaging the platform's bottom plate and other structures, resulting in loss of life and property, and affecting drilling platform operations. Table 1 also shows that the probability of exceeding the wave crest height (h = 5 meters) calculated using the nonlinear simulation method is 0.0038 (38 in 10,000), which is very close to the benchmark value of 0.0047 derived from measured wave data from a good harbor coastline. As shown in Table 1, the exceedance probability of the wave crest height (h = 5 meters) obtained by this method is 0.0037 (37 per ten thousand). This value is very close to the benchmark value of 0.0047 obtained from wave data measured from good harbor coastlines. Moreover, the prediction time of this method is only about 2% of that of the nonlinear simulation method. It is evident that this method can accurately and efficiently predict the exceedance probability of nonlinear wave crest heights, thereby enabling accurate and efficient design and calculation of the air gap value of the platform floor of offshore drilling platforms. This avoids the danger of waves slamming into the platform floor, ensuring the safety of the platform floor and other structures of offshore drilling platforms, preventing loss of life and property, and ensuring the normal operation of the drilling platform.
[0058] Table 1. Crest height (h = 5 meters) exceedance probability obtained using various methods.
[0059] Compared with existing technologies, this invention can significantly improve the prediction accuracy of the exceedance probability of nonlinear wave crest height and significantly reduce the computer time required to predict the exceedance probability of nonlinear wave crest height, thereby greatly improving the accuracy and efficiency of designing marine engineering structures and enhancing my country's scientific and technological innovation capabilities and international competitiveness in the field of marine engineering.
[0060] The above-described specific implementations can be partially adjusted by those skilled in the art in different ways without departing from the principles and purpose of the present invention. The scope of protection of the present invention is defined by the claims and is not limited to the above-described specific implementations. All implementation schemes within the scope of the claims are bound by the present invention.
Claims
1. A nonlinear sea wave crest height exceedance probability forecasting method based on a nonlinear sea wave crest height exceedance probability forecasting system, characterized by, The system includes: a non-Gaussian process unit, a saddle point approximation unit, an inverse transform unit, and an output unit. The non-Gaussian process unit performs calculations using Rice's formula based on the joint probability density information to obtain the horizontal upward stride rate of the non-Gaussian wave surface elevation. The saddle point approximation unit performs saddle point approximation based on the relevant information of the cumulant generation function to obtain the joint probability density. The inverse transform unit performs mathematical calculations based on the horizontal upward stride rate of the non-Gaussian wave surface elevation obtained above to obtain the inverse transform expression. The output unit performs mathematical calculations based on the previously obtained inverse transform expression to obtain the probability distribution of the nonlinear wave crest height. The nonlinear wave crest height exceedance probability prediction method includes: Step 1) Construct the Rice formula for calculating non-Gaussian processes upward level leap rate Where: u represents a specific level value, and z represents the derivative of that specific level value. This represents the wave elevation at time 0. The derivative of the wave elevation at time 0; Step 2) Approximate the calculation of step 1) using the saddle point method. Specifically: 2.1) Obtain the saddle point by taking the partial derivative of the cumulant generating function; 2.2) Calculate the density based on the saddle point values obtained above. Approximate value of saddle point ; Step 3) Elevation of non-Gaussian waves Represented as a static zero-mean Gaussian process with variance of 1 The function, i.e. Based on the non-Gaussian process obtained in step 1), upward level leap rate The inverse transformation of G is as follows: 3.1) First, calculate the leveling span rate. With level crossing rate The ratio; 3.2) Take the logarithm of the ratio from the previous step; 3.3) Take the square root of the negative number that is twice the logarithm of the previous step; Step 4) Based on the inverse function G obtained in step 3), -1 The high probability distribution of the wave peaks was calculated. in: Indicates the height of the wave crest.
2. The nonlinear wave crest height exceedance probability prediction method according to claim 1, characterized in that, specifically... include: Step 1: Construct the free sea surface elevation under nonlinear random sea states ,in: It is a first-order linear component. For second-order nonlinear correction components, positive integers , Represents random complex-valued amplitude, based on a specific spectrum. and angular frequency Calculations show that; furthermore, For a specific wavenumber, it is related to the dispersion relation. Related, and It's 0 and 2 Uniformly distributed phase angles between them, second-order transfer function , ; Step 2: Represented as a static zero-mean Gaussian process with variance of 1 The function, i.e. For zero-mean Gaussian processes Horizontal line upward leap rate ,in: and They are and its variance The derivative; ; Step 3: Now let's look at the non-Gaussian process. upward level leap rate For all G functions that satisfy the properties in step 2, the following relationship exists: ; Step 4: According to If a non-Gaussian process upward level leap rate Given that, the inverse transform can be obtained from the above formula. ; Step 5: Discuss the upward level crossing rate High probability distribution of wave peaks The relationship between them Represents the peak height, which satisfies ; Step 6: Combining Step 4 and Step 5, we arrive at the following conclusion: ; Step 7: Calculate the non-Gaussian process The peak distribution, i.e. The key task is to compute non-Gaussian processes. upward level leap rate Specifically: when It follows a non-Gaussian distribution; for a fixed horizontal line ,set up For the interval [0,1], The process crosses the horizontal line upwards. If the expected number of times is given, then Rice's formula is extended to: That is, calculation It is necessary to estimate at this time and The joint probability density; for the current problem, there is no and The explicit closed-form formula for the joint probability density will be used; the saddle point approximation method will be employed to calculate... ; Step 8: To use the saddle point approximation method, we need to obtain... and The explicit formula for the cumulant generating function is derived in step 1 using the following matrix notation: ,in: and It is a real symmetric matrix, and its nm components are respectively and ; It is a vector, and its nm component is... ; and It is a vector, and its nm components are respectively and ; and It is a Gaussian random variable; then we calculate... ,Right now The derivative; when t=0, we get and The value; now we get the vector ( , The explicit formula for the moment generating function of ) is: ,in: Let the matrix be a (2N, 2N) dimensional identifier matrix. ,vector , ,in: ,when hour, , ; Step 9: Through Calculated ( , The characteristic function of ) is then calculated using the inverse Fourier transform. , The joint probability density function of ) ; Step 10: Calculate the upward leveling rate ; Step 11: Approximate using the saddle point method ( , The joint probability density of ) is as follows: , Cumulative generation function It becomes: ; Step 12: Calculate density Approximate value of saddle point ; Step 13: Solve the equation to obtain the saddle point. Specifically: ,in: It is a cumulative generating function The Hessian matrix is obtained through numerical steps 10-13. upward level leap rate .