Jacket platform global sensitivity analysis method based on PCE-HDMR proxy model

Abstract

The invention discloses a PCE-HDMR proxy model-based jacket platform global sensitivity analysis method. The method comprises the following steps of S1, constructing a first-order nonlinear component function by adopting a polynomial chaos expansion model; s2, selecting a new input sample by adopting an adaptive sampling algorithm, and judging whether a first-order convergence condition is met or not; s3, judging whether a second-order nonlinear component function exists or not; s4, constructing a second-order nonlinear component function by adopting a polynomial chaos expansion model; s5, selecting a new input sample by adopting an adaptive sampling algorithm, and judging whether a second-order convergence condition is met or not; s6, synthesizing a polynomial chaos expansion-high-dimensional model to represent the proxy model; and S7, deriving a calculation formula for calculating a Sobol global sensitivity index based on the PCE-HDMR proxy model, and carrying out global sensitivity analysis. According to the method, the PCE-HDMR proxy model can be accurately and efficiently established according to the finite element response data of the jacket platform, and global sensitivity analysis of the jacket platform is realized.

Description

Technical Field

[0001] The present invention belongs to the intersection of marine engineering, proxy model and artificial intelligence, and relates to the application of machine learning methods in the uncertainty quantification of jacket platform, and in particular to a global sensitivity analysis method of jacket platform based on PCE-HDMR proxy model. Background Art

[0002] With the increasing demand for oil and gas resources, land resources can no longer meet the existing demand, and the vast ocean resources are being continuously developed. In this process, different types of offshore platforms are designed, built and applied, and the jacket platform is the most common and most widely used fixed offshore platform.

[0003] The calculation of wave-current coupling forces acting on marine structures is the most basic task in structural design, and also one of the most difficult tasks. Therefore, it has always been a research focus in the field of marine engineering. Currently, many marine platform classification societies recommend using Stokes third-order wave or Stokes fifth-order wave theory for structural design and strength verification.

[0004] The calculation of wave-current coupling force is the most basic task in the design of jacket platform structure, and it is also one of the most difficult tasks. It has always been the research focus in the field of marine engineering. For large-scale marine jacket platform structures, many use Stokes third-order wave theory or Stokes fifth-order wave theory as the standard for strength verification and structural design of marine structures. However, it is a deterministic calculation theory that does not take uncertainty into account.

[0005] The marine environment is complex, and with the increase of service life, offshore platform failures occur frequently. Jacket platforms face uncertainties caused by many factors, including but not limited to fluctuations in boundary conditions and initial states, uncertainty in material properties, and external excitations.

[0006] At present, since the surrogate model method allows the use of a small number of samples to build a surrogate model to replace complex physical systems, combining surrogate models to perform quantitative uncertainty analysis of complex structures has become the mainstream method in this field.

[0007] For the jacket platform, methods such as digital twin, Kriging model, and support vector machine have been adopted for structural damage identification and reliability analysis, but global sensitivity analysis has not yet been carried out.

[0008] Traditional local sensitivity analysis requires gradient or derivative information, which is usually used to describe the change of model response due to local changes in input. However, it may be difficult or non-existent to obtain gradient or derivative information based on model output. In contrast, global sensitivity analysis is becoming more and more mainstream, which describes how global changes in input due to input uncertainty affect the overall uncertainty behavior of the model.

[0009] In the context of global sensitivity analysis, the Sobol index based on variance decomposition is widely adopted because it can provide accurate information for most models.

[0010] The "curse of dimensionality" problem faced by high-dimensional input and output systems makes the proxy model method computationally expensive and difficult. However, the high-dimensional model representation method has a good improvement on high-dimensional input problems such as the jacket platform. Previously, radial basis functions, kriging, support vector machines and high-dimensional model representation have been proposed to effectively solve the "curse of dimensionality" problem and improve the accuracy and computational efficiency of high-dimensional input models.

[0011] However, the current methods combining proxy models with high-dimensional model representations all have non-orthogonal component functions and cannot perform global sensitivity calculations. The basis functions of polynomial chaos expansion use multivariate orthogonal bases, which can be combined with the decomposition ability of the high-dimensional model representation structure to achieve global sensitivity analysis of high-dimensional input catheter rack platforms.

[0012] In summary, the prior art has the following technical problems:

[0013] 1. The marine environment is complex, and many working conditions of the jacket platform are difficult to predict. The design and structure are often updated passively after an accident occurs, causing huge losses in manpower and financial resources during the exploration process;

[0014] 2. There are many uncertain factors in the jacket platform, and there is no analytical relationship between the variables. The polynomial chaos expansion agent model will produce "dimensionality disaster" when establishing a high-dimensional model, with high calculation cost and low efficiency. Therefore, it is difficult to achieve jacket platform modeling;

[0015] 3. Traditional high-dimensional model representation methods all have non-orthogonal components, and it is impossible to use the Sobol sensitivity index method to efficiently calculate global sensitivity. Summary of the invention

[0016] In view of the above technical problems, the present invention provides a global sensitivity analysis method of a catheter rack platform based on a PCE-HDMR (polynomial chaos expansion-high dimensional model representation) proxy model, which uses a proxy model represented by a polynomial chaos expansion-high dimensional model to accurately establish a numerical model of the catheter rack platform, thereby improving the modeling efficiency. A calculation formula for calculating the Sobol global sensitivity index is derived based on the PCE-HDMR proxy model, and applied. The global sensitivity analysis of the catheter rack platform is realized based on the orthogonality of the polynomial basis and the Sobol sensitivity index calculation method, thereby alleviating the "dimensionality curse" problem of the proxy model and solving the problem that a variety of high dimensional model representation methods have non-orthogonal components and cannot efficiently calculate the global sensitivity.

[0017] The method of the invention can accurately and efficiently establish a polynomial chaos expansion-high dimensional model representation proxy model according to the finite element response data of the jacket platform, and concisely perform global sensitivity analysis based on the orthogonality of the polynomial basis, providing guidance and reference for the jacket platform structure design, operation and maintenance.

[0018] The present invention achieves the purpose through the following technical solutions.

[0019] A global sensitivity analysis method for a jacket platform based on a PCE-HDMR proxy model comprises the following steps:

[0020] Step S1: Constructing a first-order nonlinear component function using a polynomial chaos expansion model

[0021] Step S2, select new input samples using an adaptive sampling algorithm to determine whether the first-order convergence condition is met. If not, add the new input samples to the existing sample library and update the polynomial chaos expansion model in step S1; if satisfied, proceed to step S3;

[0022] Step S3, determine whether there is a second-order nonlinear component function, if yes, proceed to step S4, if no, proceed to step S6;

[0023] Step S4: Constructing a second-order nonlinear component function using a polynomial chaos expansion model

[0024] Step S5, select new input samples using an adaptive sampling algorithm to determine whether the second-order convergence condition is met. If not, add the new input samples to the existing samples and update the polynomial chaos expansion model in step S4; if satisfied, proceed to step S6;

[0025] Step S6, synthesizing a polynomial chaos expansion-high dimensional model representation (PCE-HDMR) proxy model and performing precision analysis;

[0026] Step S7: Global sensitivity analysis, calculating the Sobol first-order sensitivity index and the total sensitivity index.

[0027] In the above scheme, in step S1, a first-order nonlinear component function is constructed according to the following formula:

[0028]

[0029] x i represents the i-th variable;

[0030] Indicates that except x i Except for , the rest of the variables are taken as mean values;

[0031] is the approximate response value of the i-th first-order component;

[0032] ψ α is a multivariate orthogonal polynomial basis;

[0033] β α are the unknown coefficients corresponding to the polynomial basis.

[0034] In the above solution, in step S2, whether the first-order convergence condition is met is determined by the following formula:

[0035]

[0036] The new sample of the i-th variable selected by the DIRECT sampling method;

[0037] is the approximate value of the new sample;

[0038] is the exact value of the new sample;

[0039] ε1 is the convergence threshold;

[0040] If the convergence condition is not met, add To existing sample library, update The polynomial chaos expansion model of ; if the convergence condition is met, the first-order nonlinear component function is obtained

[0041] In the above scheme, in step S3, it is determined whether there is a second-order nonlinear component function according to the following formula:

[0042]

[0043] i and j are the number of variables, representing 1, 2, 3, ..., n respectively, and i is not equal to j;

[0044] New sample points are formed by random combination of different variables in the existing sample library;

[0045] Enter the corresponding response value for the mean;

[0046] is the exact value of the new sample point;

[0047] and is the approximation calculated by the first-order component function;

[0048] If it is satisfied, the polynomial chaos expansion-high-dimensional model indicates that the construction is complete;

[0049] If not satisfied, continue to construct the second-order nonlinear component function.

[0050] In the above scheme, the second-order nonlinear component function in step S4 is in the form of the following formula:

[0051]

[0052] x j is the jth variable value;

[0053] x i represents the i-th variable;

[0054] Indicates that except x i 、x j Except for , the rest of the variables are taken as mean values;

[0055] is the approximate value of the model response under the i-th and j-th components;

[0056] ψ α is a multivariate orthogonal polynomial basis;

[0057] β α are the unknown coefficients corresponding to the polynomial basis;

[0058] p is the highest order of the given polynomial.

[0059] In the above solution, in step S5, whether the second-order convergence condition is met is determined according to the following formula:

[0060]

[0061] The new sample of the i-th variable selected by the DIRECT sampling method;

[0062] The new sample of the j-th variable selected by the DIRECT sampling method;

[0063] is the approximate value of the new sample;

[0064] is the exact value of the new sample;

[0065] If the convergence condition is not met, add To existing sample library, update Polynomial chaos expansion model of ;

[0066] Until the convergence condition is met, the second-order nonlinear component function is obtained In the above scheme, the synthetic polynomial chaos expansion-high dimensional model representation proxy model in step S6 is as follows:

[0067]

[0068] Enter the corresponding response value for the mean;

[0069] and is the coefficient of the univariate polynomial chaos expansion model;

[0070] and is the coefficient of the bivariate polynomial chaos expansion model;

[0071] ψ α is a multivariate orthogonal polynomial basis;

[0072] When i=0, x i =0,ψ α (x i ,x j )=0;

[0073] is a multidimensional indicator defined as:

[0074]

[0075] represents the multidimensional index when s=1;

[0076] It represents the multidimensional index when s=2;

[0077] In the above scheme, the precision analysis in step S6 includes calculating the model mean, total variance, and partial variance;

[0078] The model mean is calculated by the following formula:

[0079]

[0080] are the coefficients of the bivariate polynomial chaos expansion model, Enter the corresponding response value for the mean,

[0081]

[0082] x0 is the mean vector of the input variables;

[0083] The total variance is calculated by the following formula:

[0084]

[0085] The set of subscripts representing the non-zero coefficients of the univariate PCE,

[0086] The set of subscripts representing the non-zero coefficients of the bivariate PCE,

[0087] E[·] represents data expectation,

[0088] Var[·] represents the variance,

[0089] K n represents the unit hypercube space;

[0090] The partial variance is calculated by the following formula:

[0091]

[0092] 1≤i1<… s ≤n,s=1,…,n

[0093] represents the collection of PCE-HDMR coefficient subscripts,

[0094] are the coefficients of the bivariate polynomial chaos expansion model,

[0095] β α are the unknown coefficients corresponding to the polynomial basis,

[0096] ψ α is a multivariate orthogonal polynomial basis,

[0097] x i represents the i-th variable,

[0098] x j represents the jth variable,

[0099] i and j are the numbers of variables, representing 1, 2, 3, ... respectively, and i is not equal to j.

[0100] ​In the above scheme, the calculation method of the Sobol first-order sensitivity index in step S7 is as follows:

[0101]

[0102] S i Represents variable x i The first-order sensitivity index is

[0103] D i Represents variable x i The first-order partial variance of

[0104] D PCE-HDMR represents the total variance;

[0105] The calculation method of Sobol total sensitivity index is as follows:

[0106]

[0107] α:α i >0 means x is included i All α i The set of subscripts greater than zero,

[0108] Represents variable x i The overall sensitivity index is

[0109] S α Indicates that the variable x is included i sensitivity index.

[0110] In the above solution, the adaptive sampling algorithm adopts the DIRECT sampling method, but is not limited to this algorithm.

[0111] In the present invention, both the first-order component function and the second-order component function are obtained by and can only be obtained by polynomial chaos expansion.

[0112] Compared with the prior art, the present invention has the following beneficial effects:

[0113] The present invention is oriented to data of multiple dimensions of the jacket platform, combines polynomial chaos expansion with high-dimensional model representation, adopts a proxy model represented by polynomial chaos expansion-high-dimensional model to accurately establish a numerical model of the jacket platform, balances the relationship between the high-dimensional model and the precise modeling, and derives a calculation formula for calculating the Sobol global sensitivity index based on the PCE-HDMR proxy model. The calculation method based on the orthogonality of the polynomial basis and the Sobol sensitivity index is used to realize the global sensitivity analysis of the jacket platform through the sensitivity index calculation formula, thereby feeding back the influence of uncertainty factors on the safety performance of the jacket, which is helpful for the structural design and optimization of the jacket platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0114] The drawings described herein are used to provide a further understanding of the present invention and constitute a part of the present invention. The exemplary embodiments of the present invention and their descriptions are used to explain the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0115] Figure 1 It is a schematic flow chart of the global sensitivity analysis method of the catheter platform based on the PCE-HDMR proxy model of the present invention;

[0116] Figure 2 is the PCE-HDMR component function diagram of the present invention, wherein Figure 2 (a) is the component function graph of variable x1; Figure 2 (b) Component function graph of variable x2; Figure 2 (c) Component function graph of variable x3; Figure 2 (d) Component function diagram of the coupling effect of variables x1 and x3;

[0117] Figure 3 A flow chart of the DIRECT adaptive sampling method provided by the present invention;

[0118] Figure 4 A schematic diagram of the working condition of the jacket platform provided by the present invention working in the ocean;

[0119] Figure 5 The schematic diagram of the first to sixth order modal finite element analysis of the jacket platform provided in the embodiment of the present invention, wherein: Figure 5 (a) is the schematic diagram of the first-order modal finite element analysis. Figure 5 (b) is the schematic diagram of the second-order modal finite element analysis. Figure 5 (c) is the schematic diagram of the third-order modal finite element analysis. Figure 5 (d) is the schematic diagram of the fourth-order modal finite element analysis. Figure 5 (e) is the schematic diagram of the fifth-order modal finite element analysis. Figure 5 (f) is the schematic diagram of the sixth-order modal finite element analysis;

[0120] Figure 6 A comparison diagram of probability density curves of the first to sixth order modes provided by the embodiment of the present invention and the conventional method, wherein: Figure 6 (a) is a comparison of the probability density curve of the first-order mode with that of the conventional method. Figure 6 (b) is a comparison of the probability density curves of the second-order mode and the conventional method. Figure 6 (c) is a comparison of the probability density curves of the third-order mode and the conventional method. Figure 6 (d) is a comparison of the probability density curves of the fourth-order mode and the conventional method. Figure 6 (e) is a comparison of the probability density curves of the fifth-order mode and the conventional method. Figure 6 (f) is a comparison chart of the probability density curve under the sixth-order mode with that of the conventional method;

[0121] Figure 7 A correlation heat map of the first to sixth order modal global sensitivity indicators provided by an embodiment of the present invention, wherein: Figure 7 (a) is the correlation heat map of the first-order modal global sensitivity index, Figure 7 (b) is the correlation heat map of the second-order modal global sensitivity index. Figure 7 (c) is the correlation heat map of the third-order modal global sensitivity index. Figure 7 (d) is the correlation heat map of the fourth-order modal global sensitivity index. Figure 7 (e) is the correlation heat map of the fifth-order mode global sensitivity index. Figure 7 (f) is the correlation heat map of the sixth-order mode global sensitivity index. DETAILED DESCRIPTION

[0122] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution will be clearly and completely described below through the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0123] The present invention relates to a method for performing global sensitivity analysis on a catheter platform based on a polynomial chaos expansion-high dimensional model representation proxy model, aiming to construct a framework for analytically calculating the Sobol sensitivity index based on the orthogonality of basis functions, combining the simple structure of the polynomial chaos expansion with the decomposability of the high dimensional model representation, realizing the establishment of the PCE-HDMR proxy model and global sensitivity analysis, improving the modeling accuracy and efficiency of the proxy model, and providing guidance for the structural design and optimization of the catheter platform.

[0124] like Figure 1 As shown, a global sensitivity analysis method of a jacket platform based on a PCE-HDMR (polynomial chaos expansion-high dimensional model representation) proxy model includes the following steps.

[0125] Step S1: Constructing a first-order nonlinear component function using a polynomial chaos expansion model

[0126] Step S2, select new input samples using an adaptive sampling algorithm to determine whether the first-order convergence condition is met. If not, add the new input samples to the existing sample library and update the polynomial chaos expansion model in step S1; if satisfied, proceed to step S3;

[0127] Step S3, determine whether there is a second-order nonlinear component function, if yes, proceed to step S4, if no, proceed to step S6;

[0128] Step S4: Constructing a second-order nonlinear component function using a polynomial chaos expansion model

[0129] Step S5, select new input samples using an adaptive sampling algorithm to determine whether the second-order convergence condition is met. If not, add the new input samples to the existing samples and update the polynomial chaos expansion model in step S4; if satisfied, proceed to step S6;

[0130] Step S6, synthesizing the polynomial chaos expansion-high dimensional model to represent the proxy model, and performing precision analysis;

[0131] Step S7: Global sensitivity analysis, calculating the Sobol first-order sensitivity index and the total sensitivity index.

[0132] The step S1 of constructing a first-order nonlinear component function by polynomial chaos expansion specifically includes the following steps:

[0133] The approximate function of the first-order component response is constructed based on the one-dimensional form of the polynomial chaotic expansion. The steps are as follows.

[0134] Step (1) Calculate the response value when the input variable is the mean according to the following formula:

[0135]

[0136] x0 is the mean vector of the input variables;

[0137] Enter the corresponding response value for the mean;

[0138] Step (2) constructs a first-order polynomial chaos expansion according to the following formula:

[0139]

[0140] x is the input variable sample;

[0141] α=(α i ,…,α n ) is an n-dimensional index, i = 1,...,n, and α i ≥0;

[0142] n is the model dimension;

[0143] ψ α is a multivariate orthogonal polynomial basis;

[0144] ψ αUsually it is selected according to different data distribution types. If the data is uniformly distributed, the Legendre polynomial basis is used. If the data is Gaussian distributed, the Hermite polynomial basis is used.

[0145] β α are the unknown coefficients corresponding to the polynomial basis;

[0146] Step (3) truncate the complete polynomial chaos expansion to keep the total order not exceeding the given maximum order;

[0147] Since there is very little information about the over-coupled terms and over-high-order terms, truncation can both ensure the calculation accuracy and improve the calculation efficiency, so truncation is performed according to the following formula:

[0148]

[0149] p is the highest order of the given polynomial;

[0150] The truncated polynomial chaos expansion is shown as follows:

[0151]

[0152] Step (4) calculating unknown coefficients of polynomial chaos expansion;

[0153] Solve the l1 norm minimization problem according to the following formula:

[0154]

[0155] y is the exact response value of the input sample;

[0156] β is the polynomial basis coefficient;

[0157] ψ is the multivariate orthogonal polynomial basis vector;

[0158] ∈ is the error;

[0159] P represents the spatial dimension;

[0160] It represents the P-dimensional real number space.

[0161] ‖·‖1 represents the 1-norm

[0162] ‖·‖2 represents the 2-norm

[0163] According to the prior art, the PCE-HDMR polynomial basis coefficient β can be solved by the minimum angle regression algorithm, the orthogonal matching pursuit algorithm, the greedy coordinate descent algorithm, etc.;

[0164] Step (5) constructs a first-order nonlinear component function according to the following formula:

[0165]

[0166] x i represents the i-th variable;

[0167] Indicates that except x i Except for , the rest of the variables are taken as mean values;

[0168] is the approximate response value of the i-th first-order component;

[0169] The step S2 determines whether the first-order convergence condition is met:

[0170]

[0171] The new sample of the i-th variable selected by the DIRECT sampling method;

[0172] is the approximate value of the new sample;

[0173] is the exact value of the new sample;

[0174] ε1 is the convergence threshold;

[0175] If the convergence condition is not met, add To existing sample library, update The polynomial chaos expansion model of ; if the convergence condition is met, the first-order nonlinear component function is obtained

[0176] The second-order component function is generally determined by a convergence threshold ε1=0.01. If the threshold is exceeded, a second-order component function will be generated.

[0177] The criterion for judging whether there is a second-order nonlinear component function in step S3 is:

[0178] This part is mainly based on the first-order component function and the existing sample library, and determines whether there is a second-order nonlinear component function according to the following formula;

[0179]

[0180] i and j are the number of variables, representing 1, 2, 3, ..., n respectively, and i is not equal to j;

[0181] New sample points are formed by random combination of different variables in the existing sample library;

[0182] Enter the corresponding response value for the mean;

[0183] is the exact value of the new sample point;

[0184] and is the approximation calculated by the first-order component function;

[0185] If it is satisfied, the polynomial chaos expansion-high-dimensional model indicates that the construction is complete;

[0186] If it is not satisfied, continue to construct the second-order nonlinear component function;

[0187] The step S4 uses the polynomial chaos expansion to construct a second-order nonlinear component function, and the second-order nonlinear component function is in the form of the following formula:

[0188]

[0189] x j is the jth variable value,

[0190] x i represents the i-th variable,

[0191] Indicates that except x i 、x j Except for , all other variables are taken as mean values.

[0192]

[0193] is the approximate model response under the i-th and j-th components,

[0194] The construction method is similar to that of the first-order nonlinear component function;

[0195] In step S5, whether the second-order convergence condition is met is determined as follows:

[0196]

[0197] is the new sample of the i-th variable selected by the DIRECT sampling method,

[0198] is the new sample of the j-th variable selected by the DIRECT sampling method,

[0199] is the approximate value of the new sample,

[0200] is the exact value of the new sample;

[0201] If the convergence condition is not met, add To existing sample library, update Polynomial chaos expansion model of ;

[0202] Until the convergence condition is met, the second-order nonlinear component function is obtained

[0203] The step S6 synthesizes the polynomial chaos expansion-high dimensional model to represent the proxy model:

[0204] The final polynomial chaos expansion-high dimensional model represents the proxy model, which is composed of the first-order component model and the second-order component model, and the accuracy analysis is performed;

[0205] By constructing all variables in sequence, the synthesized model can be expressed as follows:

[0206]

[0207] The precision analysis adopts four criteria, namely, the coefficient of determination (R 2 ), relative average absolute error (RAAE), relative maximum absolute error (RMAE) and root mean square error (RMSE);

[0208]

[0209]

[0210] m is the number of test points,

[0211] is the average response of the test points,

[0212] std is the standard deviation of the function response of the test point,

[0213] Compute an approximation for the function at the test point,

[0214] f(x i ) is the exact value of the test point,

[0215] R 2 , RAAE, RMAE are the global errors between the predicted response and the true response,

[0216] RMSE is the local error between the predicted response and the true response.

[0217] R 2 The closer it is to 1, the closer RAAE, RMAE and RMSE are to 0, and the more accurate the model is; the global sensitivity score of step S7 is:

[0218] First, define a multidimensional indicator:

[0219]

[0220] Polynomial chaos expansion-high dimensional model representation can be rewritten as follows:

[0221]

[0222] and is the coefficient of the univariate polynomial chaos expansion model,

[0223] and are the coefficients of the bivariate polynomial chaos expansion model,

[0224] When i=0, x i =0,ψ α (x i ,x j )=0;

[0225] According to the orthogonality of the polynomial basis, the model mean is calculated as follows:

[0226]

[0227] The total variance is calculated as follows:

[0228]

[0229] The set of subscripts representing the non-zero coefficients of the univariate PCE,

[0230] The set of subscripts representing the non-zero coefficients of the bivariate PCE,

[0231] E[·] represents data expectation,

[0232] Var[·] represents the variance,

[0233] K n represents the unit hypercube space;

[0234] The partial variance is calculated by:

[0235]

[0236] 1≤i1<… s ≤n,s=1,…,n

[0237] Denotes the collection of PCE-HDMR coefficient subscripts.

[0238] The calculation method of Sobol's first-order sensitivity index is as follows:

[0239]

[0240] S​i Represents variable x i The first-order sensitivity index is

[0241] D i Represents variable x i The first-order partial variance of

[0242] The calculation method of Sobol total sensitivity index is as follows:

[0243]

[0244] α:α i >0 means x is included i All α i The set of subscripts greater than zero,

[0245] Represents variable x i The overall sensitivity index is

[0246] S α Indicates that the variable x is included i Sensitivity index (for example: S1, S 12 ,S 13 ,S 123 wait).

[0247] The present invention constructs a high-dimensional model based on polynomial chaos expansion to represent the first-order nonlinear component function, adopts DIRECT adaptive sampling, calls finite element software to obtain the response under a specific input sample, obtains the first-order polynomial model of each component in turn, and finally forms the first-order nonlinear component function represented by the high-dimensional model; judges whether there is a second-order nonlinear component function, and if so, constructs a high-dimensional model based on polynomial chaos expansion to represent the second-order nonlinear component function, so as to achieve more accurate modeling; and utilizes the orthogonality of the polynomial basis to achieve global sensitivity analysis. For the final polynomial chaos expansion-high-dimensional model representation proxy model, the Sobol sensitivity index calculation method is analytically derived to calculate the first-order sensitivity index and the total sensitivity index.

[0248] Next, the accuracy of the PCE-HDMR (polynomial chaos expansion-high dimensional model representation) model is verified by comparing the global sensitivity analysis results of the present invention with those of the PCE-HDMR (polynomial chaos expansion-high dimensional model representation), PCE (polynomial chaos expansion) and RS-HDMR (random sampling-high dimensional model representation) on the classic benchmark function Ishigami function of global sensitivity analysis:

[0249] As shown in Table 1, although the first-order PCE-HDMR uses only 26 sample points, its R 2 The construction accuracy is low, which is 0.6271, and there is a second-order coupling effect of the variables. Figure 2 It can be seen that the second-order PCE-HDMR not only approximates the original model better, but also accurately identifies the influence of variables x1, x2, and x3 on the model when they act alone, such as Figure 2 (a) indicates that when variable x1 acts alone, its influence on the model is in the form of a trigonometric function. Figure 2 (b) indicates that when the variable x2 acts alone, the influence on the model is in the form of a symmetric periodic function. Figure 2 (c) indicates that when variable x3 acts alone, it has no effect on the model; and there is a coupling effect between the two variables x1 and x3, and the output is visualized, such as Figure 2 This is consistent with the conclusion that second-order HDMR is usually used in a large number of engineering applications.

[0250] Table 1 Performance of PCE-HDMR for Ishigami function

[0251]

[0252] Note: Firstorder is the first-order PCE-HDMR component function, Second order is the first-order PCE-HDMR component function, R 2 is the coefficient of determination, RAAE is the relative mean absolute error, RMAE is the relative maximum absolute error, RMSE is the root mean square error, and Noe is the number of model evaluations.

[0253] Table 2 shows the comparison of the results of global sensitivity analysis. It can be seen that for the same number of sample points, the calculation results of PCE-HDMR (polynomial chaos expansion-high dimensional model representation) are closer to the theoretical value of the Ishigami function than those of PCE (polynomial chaos expansion) and RS-HDMR (random sampling-high dimensional model representation), which proves that PCE-HDMR has the ability to accurately identify the coupling interaction between variables and the ability to quantify the size of the coupling interaction through the second-order sensitivity index.

[0254] Table 2 Comparison of results of global sensitivity analysis

[0255]

[0256]

[0257] The calculation methods of Ishigami function theoretical value, PCE, and RS-HDMR are prior art. Preferably, the calculation method of Ishigami function theoretical value can refer to the literature: Ishigami T, Homma T. An importance quantification technique in uncertainty analysis for computer models.

[1990] Proceedings First International Symposium on Uncertainty Modeling and Analysis 1990. p. 398-403. The PCE calculation method can refer to the literature: Zhang J, Yue X, Qiu J, Zhuo L, Zhu J. Sparse polynomial chaos expansion based on Bregman-iterative greedy coordinate descent for global sensitivity analysis. Mechanical Systems and Signal Processing. 2021; 157: 107727. The RS-HDMR calculation method can be found in the literature: Li G, Wang SW, Rabitz H, Wang S, Jaffé P. Global uncertainty assessments by high dimensional model representations (HDMR). Chemical Engineering Science. 2002; 57: 4445-60.

[0258] A specific embodiment of the present invention:

[0259] This embodiment provides a method for performing global sensitivity analysis using a polynomial chaos expansion-high-dimensional model representation proxy model, and the application object is a jacket platform.

[0260] This embodiment uses a method provided by the present invention to represent a proxy model based on polynomial chaos expansion-high dimensional model, and models and performs global sensitivity analysis on the jacket platform based on the jacket platform's geometric parameters, material parameters, and operating parameters such as wave height and period.

[0261] According to the input variable distribution law and coefficient of variation of the jacket platform, the input sample points are adaptively selected, and the finite element analysis software is called to simulate the structural response. Sample data is added according to the convergence conditions in the operation process until the model accuracy reaches the convergence conditions.

[0262] A total of 20 input parameters including structural geometry parameters, material parameters, and load parameters were designed for the jacket platform, and a high-dimensional proxy model was established with six modes from the first to the sixth order as output.

[0263] As attached Figure 4 As shown in FIG. 1 , the natural load conditions on the jacket platform in the ocean are complex.

[0264] For the first-order mode, a single variable first-order model is firstly established using polynomial chaos expansion as a high-dimensional model to represent the first-order component function, and the Legendre polynomial basis is selected.

[0265] Use the mean of the input variable as the initial input point and DIRECT sampling.

[0266] The input sample points are simulated by finite element software to obtain the output frequency.

[0267] As attached Figure 5 As shown in the figure, the simulation results of the first to sixth order modes in the finite element software show that the deformation trends of different modal structures are different. The most unfavorable positions appear in different positions, such as: in the first order mode, the most unfavorable position of the jacket platform is the first layer of vertical rods; in the third order mode, the most unfavorable position of the jacket platform is the platform layer.

[0268] Figure 5 The schematic diagram of the first to sixth order modal finite element analysis of the jacket platform provided in the embodiment of the present invention, wherein: Figure 5 (a) is the schematic diagram of the first-order modal finite element analysis. Figure 5 (b) is the schematic diagram of the second-order modal finite element analysis. Figure 5 (c) is the schematic diagram of the third-order modal finite element analysis. Figure 5 (d) is the schematic diagram of the fourth-order modal finite element analysis. Figure 5 (e) is the schematic diagram of the fifth-order modal finite element analysis. Figure 5 (f) is the schematic diagram of the sixth-order modal finite element analysis.

[0269] As attached Figure 3 As shown, during the modeling process, the value interval is adaptively updated according to the variance.

[0270] Take R 2 , RAAE, RMAE and RMSE evaluation.

[0271] R 2 =0.9988, RAAE=0.0270, RMAE=0.0996, RMSE=0.0022.

[0272] Finally, the first-order nonlinear component function of the first-order modal proxy model of the jacket platform uses a total of 87 sample points.

[0273] Then a second-order polynomial chaos expansion model is established as a high-dimensional model to represent the second-order component function.

[0274] The sampling method and evaluation index are constructed in the same form as the first-order component function.

[0275] The evaluation indicators of the second-order component function are as follows:

[0276] R 2 =1.0000, RAAE=0.0042, RMAE=0.0096, RMSE=3.04×10 -4 .

[0277] A total of 509 sample points are used in the second-order nonlinear component function of the first-order modal proxy model of the jacket platform.

[0278] The first-order modal surrogate model of the jacket platform represented by synthetic polynomial chaos expansion-high dimensional model.

[0279] The second to sixth order modal surrogate models are established in turn. The evaluation index values ​​and the number of samples required for modeling of the first and second order surrogate models represented by the polynomial chaos expansion-high dimensional model are shown in Table 3. 2 The closer it is to 1, the closer RAAE, RMAE, RMAE, and RMSE are to 0, which indicates that the model has a higher accuracy in replacing the actual jacket platform for finite element analysis.

[0280] Table 3 Evaluation indexes and sample numbers of polynomial chaos expansion-high-dimensional model representation surrogate model

[0281] Natural frequency Error indicators <![CDATA[R 2 ]]> RAAE RMAE RMSE Number of samples 1 First order 0.9988 0.0270 0.0996 0.0022 87 Second order 1.0000 0.0042 0.0096 304E-04 509 2 First order 0.9986 0.0277 0.1155 0.0023 88 Second order 10000 0.0029 0.0122 240E-04 518 3 First order 09988 0.0255 0.1246 0.0026 86 Second order 1.0000 0.0029 0.0131 312E-04 480 4 First order 0.9922 0.0442 0.4388 0.0098 88 <![CDATA[second orde r ]]> 0.9953 00421 03150 7.60E-03 540 5 First order 0.9883 0.0551 0.3926 0.0199 88 <![CDATA[Second orde r ]]> 0.9964 0.0428 0.1492 110E-02 361 6 First order 0.9166 01800 1.0846 0.0361 94 Second order 0.9767 0.1125 0.4143 1.91E-02 555

[0282] By comparing the evaluation indicators of the first-order and second-order proxy models, it can be seen that for the modeling of the first five modes of the jacket platform, the first-order modeling accuracy R represented by the polynomial chaos expansion-high-dimensional model is 2 All of them can reach greater than 0.98; RAAE, RMAE, RMAE, and RMSE are close to 0, indicating that the first-order polynomial chaos expansion-high-dimensional model representation model can more accurately replace the time-consuming modeling of the first five modal models of the jacket platform. At the same time, the modeling accuracy of the second-order polynomial chaos expansion-high-dimensional model representation model can also reach a high level. Therefore, the polynomial chaos expansion-high-dimensional model representation of the second-order model is more accurate and more suitable for engineering examples.

[0283] At the same time, the number of samples required for the first-order model is less than 100, and the number of samples required for the second-order model is about 500. In comparison, the second-order model requires more computational cost.

[0284] But in general, for the 20-dimensional catheter rack platform model, the number of samples is about 500, which is a low computing cost to achieve accurate modeling and is acceptable in engineering practice.

[0285] The mean, variance and partial variance are calculated based on the polynomial chaos expansion-high dimensional model representation surrogate model.

[0286] Then, the first-order sensitivity index and the total sensitivity index are obtained according to the Sobol sensitivity index calculation method.

[0287] The input variables and distribution of the jacket platform are shown in Table 4. The input variables of the jacket platform include deck beam height (h / m), deck beam width (b / m), deck thickness (t / m), upper column diameter (d1 / m), bottom column diameter (d2 / m), deck main pipe and horizontal brace diameter (d3 / m), diagonal brace diameter (d4 / m), upper column wall thickness (t1 / m), bottom column wall thickness (t2 / m), deck main pipe and horizontal brace wall thickness (t3 / m), diagonal brace wall thickness (t4 / m), elastic modulus (E / Pa), Poisson's ratio (υ), material density (D / kg / m 3 ), wave height (H / m), wave period (T / s), upper seawater velocity (v1 / m / s 2 ), upper seawater velocity (v1 / m / s 2 ), bottom water velocity (v3 / m / s 2 ), wave phase angle (W / m / s 2 ).

[0288] Table 4 Input variables and distribution of jacket platform

[0289]

[0290]

[0291] As shown in Tables 5 and 6, the calculation results of the first-order sensitivity index and the global sensitivity index of the first to sixth order modes. i Indicates the influence of a single variable on the jacket platform mode, the total sensitivity index It indicates the influence of a single variable and its coupling with other variables on the jacket platform mode. It can be seen that for the first, second, third and fifth order modes of the jacket platform, ∑S i is almost 1, indicating that the coupling between the variables is negligible. For the fourth and sixth order modes of the jacket platform, ∑S i Less than 0.80 indicates that there is a second-order (or higher-order) coupling effect between the variables.

[0292] Table 5 shows the global sensitivity analysis results of the first to third order modes provided by the embodiment of the present invention.

[0293]

[0294] Table 6 shows the global sensitivity analysis results of the fourth to sixth order modes provided by the embodiment of the present invention.

[0295]

[0296] According to the results, the elastic modulus E and steel density D are the factors that have the greatest impact on the natural frequency of the structure.

[0297] As attached Figure 6 As shown, by comparing the method of the present invention with the probability density curves of Monte Carlo simulation and polynomial chaos expansion, it can be seen that the method of the present invention is almost consistent with the probability density curves of Monte Carlo simulation and polynomial chaos expansion (especially near the peak value), thereby verifying the accuracy of the method. Figure 6 A comparison diagram of probability density curves of the first to sixth order modes provided by the embodiment of the present invention and the conventional method, wherein: Figure 6 (a) is a comparison of the probability density curve of the first-order mode with that of the conventional method. Figure 6 (b) is a comparison of the probability density curves of the second-order mode and the conventional method. Figure 6 (c) is a comparison of the probability density curves of the third-order mode and the conventional method. Figure 6 (d) is a comparison of the probability density curves of the fourth-order mode and the conventional method. Figure 6 (e) is a comparison of the probability density curves of the fifth-order mode and the conventional method. Figure 6 (f) is a comparison of the probability density curves of the sixth-order mode and the conventional method.

[0298] Monte Carlo simulation results are generally used as benchmark results for sensitivity analysis.

[0299] It can be found that the prediction result of the method of the present invention at high-order frequencies is closer to that of Monte Carlo simulation, and the effect is better, while the modeling cost is much lower than the modeling cost of Monte Carlo simulation and polynomial chaos expansion.

[0300] As attached Figure 7 As shown in Figure 1, it is a heat map analysis of the first-order and second-order global sensitivity indicators. The darker the color, the greater the sensitivity index. It can be clearly seen that the brace diameter d4 and its coupling with other variables have a greater impact on the jacket platform mode.

[0301] Figure 7 A correlation heat map of the first to sixth order modal global sensitivity indicators provided by an embodiment of the present invention, wherein: Figure 7(a) is the correlation heat map of the first-order modal global sensitivity index, Figure 7 (b) is the correlation heat map of the second-order modal global sensitivity index. Figure 7 (c) is the correlation heat map of the third-order modal global sensitivity index. Figure 7 (d) is the correlation heat map of the fourth-order modal global sensitivity index. Figure 7 (e) is the correlation heat map of the fifth-order mode global sensitivity index. Figure 7 (f) is the correlation heat map of the sixth-order mode global sensitivity index.

[0302] The present invention is based on the basic framework of performing global sensitivity analysis of the jacket platform by using polynomial chaos expansion-high dimensional model proxy model representation, constructs a polynomial chaos expansion-high dimensional model representation proxy model, realizes high dimensional data modeling of the jacket platform, performs global sensitivity analysis by using an analytical method, and improves the modeling accuracy and efficiency of the proxy model.

[0303] The present invention adopts DIRECT adaptive sampling, calls finite element analysis software for simulation, constructs a high-dimensional model to represent the first-order nonlinear component function by polynomial chaos expansion, determines whether there is a second-order nonlinear component function, and if so, constructs a second-order nonlinear component function, synthesizes a polynomial chaos expansion-high-dimensional model to represent the proxy model, and derives a calculation formula for calculating the Sobol global sensitivity index based on the PCE-HDMR proxy model, and applies it to calculate the Sobol sensitivity index based on the orthogonality of the polynomial basis, including the first-order sensitivity index and the total sensitivity index, to complete the global sensitivity analysis. The method of the present invention can accurately and efficiently establish a polynomial chaos expansion-high-dimensional model representation proxy model based on the finite element response data of the jacket platform, and concisely perform global sensitivity analysis based on the orthogonality of the polynomial basis, providing guidance and reference for the jacket platform structure design, operation and maintenance.

[0304] The preferred embodiments of the present invention disclosed above are only used to help explain the present invention. The preferred embodiments do not describe all the details in detail, nor do they limit the specific implementation methods of the invention. This specification selects and specifically describes the embodiments in order to better explain the principles and practical applications of the present invention, so that those skilled in the art can understand and use the present invention well. The present invention is limited only by the claims and their full scope and equivalents.

Claims

1. A global sensitivity analysis method for a catheter platform based on a PCE-HDMR proxy model, characterized in that: The steps include: Step S1: Constructing a first-order nonlinear component function using a polynomial chaos expansion model Step S2, select new input samples using an adaptive sampling algorithm to determine whether the first-order convergence condition is met. If not, add the new input samples to the existing sample library and update the polynomial chaos expansion model in step S1; if satisfied, proceed to step S3; Step S3, determine whether there is a second-order nonlinear component function, if yes, proceed to step S4, if no, proceed to step S6; Step S4: Constructing a second-order nonlinear component function using a polynomial chaos expansion model Step S5, select new input samples using an adaptive sampling algorithm to determine whether the second-order convergence condition is met. If not, add the new input samples to the existing samples and update the polynomial chaos expansion model in step S4; if satisfied, proceed to step S6; Step S6, synthesizing the PCE-HDMR proxy model and performing accuracy analysis; Step S7: Global sensitivity analysis, calculating the Sobol first-order sensitivity index and the total sensitivity index.

2. The method for global sensitivity analysis of a conductor frame platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: In step S1, a first-order nonlinear component function is constructed according to the following formula: x i represents the i-th variable; x0 ～i Indicates that except x i Except for , the remaining variables are taken as mean values; is the approximate response value of the i-th first-order component; ψ α is a multivariate orthogonal polynomial basis; β α are the unknown coefficients corresponding to the polynomial basis.

3. The method for global sensitivity analysis of a catheter platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: In step S2, whether the first-order convergence condition is met is determined by the following formula: The new sample of the i-th variable selected by the DIRECT sampling method; is the approximate value of the new sample; is the exact value of the new sample; ε1 is the convergence threshold; If the convergence condition is not met, add To existing sample library, update Polynomial chaos expansion model of ; If the convergence condition is met, the first-order nonlinear component function is obtained 4. The method for global sensitivity analysis of a conductor frame platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: In the step S3, it is determined whether there is a second-order nonlinear component function according to the following formula: i and j are the numbers of variables, representing 1, 2, 3, ..., respectively, and i is not equal to j; New sample points are formed by random combination of different variables in the existing sample library; Enter the corresponding response value for the mean; is the exact value of the new sample point; and is the approximation calculated by the first-order component function; If it is satisfied, the polynomial chaos expansion-high-dimensional model indicates that the construction is complete; If not satisfied, continue to construct the second-order nonlinear component function.

5. The method for global sensitivity analysis of a jacket platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: The second-order nonlinear component function in step S4 is in the form of the following formula: x j is the jth variable value; x i represents the i-th variable; Indicates that except x i 、x j Except for , the remaining variables are taken as mean values; is the approximate value of the model response under the i-th and j-th components; is a multivariate orthogonal polynomial basis; β α are the unknown coefficients corresponding to the polynomial basis; p is the highest order of the given polynomial.

6. The method for global sensitivity analysis of a jacket platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: In step S5, whether the second-order convergence condition is met is determined according to the following formula: The new sample of the i-th variable selected by the DIRECT sampling method; The new sample of the j-th variable selected by the DIRECT sampling method; is the approximate value of the new sample; is the exact value of the new sample; If the convergence condition is not met, add To existing sample library, update Polynomial chaos expansion model of ; Until the convergence condition is met, the second-order nonlinear component function is obtained 7. The method for global sensitivity analysis of a jacket platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: The PCE-HDMR proxy model synthesized in step S6 is as follows: Enter the corresponding response value for the mean; and is the coefficient of the univariate polynomial chaos expansion model; and is the coefficient of the bivariate polynomial chaos expansion model; ψ α is a multivariate orthogonal polynomial basis; When i=0, x i =0,ψ α (x i ,x j )=0; is a multidimensional indicator, defined as represents the multidimensional index when s=1; Represents the multidimensional index when s=2.

8. The method for global sensitivity analysis of a jacket platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: The precision analysis in step S6 includes calculating the model mean, total variance, and partial variance; The model mean is calculated by the following formula: are the coefficients of the bivariate polynomial chaos expansion model, Enter the corresponding response value for the mean, x0 is the mean vector of the input variables; The total variance is calculated by the following formula: The set of subscripts representing the non-zero coefficients of the univariate PCE, The set of subscripts representing the non-zero coefficients of the bivariate PCE, E[·] represents data expectation, Var[·] represents the variance, K n represents the unit hypercube space; The partial variance is calculated by the following formula: represents the collection of PCE-HDMR coefficient subscripts, are the coefficients of the bivariate polynomial chaos expansion model, β α are the unknown coefficients corresponding to the polynomial basis, ψ α is a multivariate orthogonal polynomial basis, x i represents the i-th variable, x j represents the jth variable, i and j are the numbers of variables, representing 1, 2, 3, ..., n respectively, and i is not equal to j.

9. The method for global sensitivity analysis of a jacket platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: In step S7, a calculation formula for calculating the Sobol global sensitivity index is derived based on the PCE-HDMR proxy model obtained in step S6. The calculation method for the Sobol first-order sensitivity index is shown in the following formula: S i Represents variable x i The first-order sensitivity index is D i Represents variable x i The first-order partial variance of D PCE-HDMR represents the total variance; The calculation method of Sobol total sensitivity index is as follows: α:α i >0 means x is included i All α i The set of subscripts greater than zero, Represents variable x i The overall sensitivity index is S α Indicates that the variable x is included i sensitivity index.

10. The method for global sensitivity analysis of a jacket platform based on a PCE-HDMR proxy model according to claim 1, characterized in that: The adaptive sampling algorithm adopts the DIRECT sampling method.

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