A surrogate model generation method for dynamic effects of mechanical structures with non-uniform horizontal factors

Abstract

The present invention provides a method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform level factors, and belongs to the field of mechanical structure dynamics analysis. First, according to the non-uniform level of the factors considered by the structure, a test design table is constructed by using dynamic evolution sampling and discrete rounding methods. Secondly, a structural finite element model is constructed by using the finite element method, and the data in the test design table is brought into the structural finite element model for calculation to obtain the structural model response data. Finally, a conditional quotient method is used to establish an efficient proxy model of the response of the structure at different time points. The present invention constructs a test scheme through dynamic evolution and discrete rounding technology. The test scheme not only has good distribution quality in the test space, but also can be applied to the situation where the horizontal distribution of the considered factors is non-uniform; the use of the present invention for proxy model construction has fast calculation speed, high calculation accuracy, and strong versatility, which can provide guarantees for structural design and reliability analysis.

Description

Technical Field

[0001] The invention belongs to the field of mechanical structure dynamics analysis and relates to a proxy model generation method for the dynamic effects of a mechanical structure containing non-uniform horizontal factors. Background Art

[0002] Dynamic effects are factors that must be considered in the safety and reliability analysis of complex mechanical equipment structures. In order to ensure the safety and reliability of the structure, it is usually necessary to conduct multiple rounds of analysis on the dynamic response of the designed structure to ensure that the structure can operate safely and reliably, resulting in a long R&D cycle and high cost for structural equipment. Currently, designers often use experimental design schemes combined with Kriging methods, radial basis function methods, response surface methods, neural networks, etc. to help construct the corresponding surrogate model of the structure, and then select and optimize the key parameters of the structure based on the surrogate model to ensure the safety and reliability of the structure. However, this approach faces two practical problems:

[0003] 1) Non-uniform level problem. Commonly used experimental design methods include orthogonal test method, Latin hypercube sampling method, uniform test method, etc. However, most of these methods are aimed at uniform level. For example, the patent "Adaptive Optimization Design Method for Machine Tool Column Structure for High-Dimensional Mixed Variables" (application number 202410801102.7) uses the Latin hypercube sampling method to sample in the design space, and then constructs an agent model of the machine tool column structure according to the Gaussian process method. The patent "Prediction and Optimization Method for CNC Machine Tool Cutting Stability Considering Parameter Uncertainty" (application number 202210177100.6) uses the orthogonal experimental design method. However, in many actual mechanical structures, due to manufacturing processes or specification constraints, non-uniform levels are often faced, such as machine tool cutting speed level, heat treatment process parameter level, bolt preload level, etc. The non-uniform level problem involved in the structure of complex mechanical equipment reduces the accuracy and efficiency of existing experimental design methods.

[0004] 2) The problem of low computational efficiency. Considering the dynamic effect, it is necessary to establish proxy models at different times based on the sample response. Most of the existing proxy models, such as the Kriging method and the radial basis function method, require optimization methods to determine the parameters in the model. Establishing proxy models at multiple times means that multiple multi-parameter optimization problems need to be solved, which is computationally intensive and computationally inefficient. For example, the proxy models constructed in the patent "Adaptive Optimization Design Method for Machine Tool Column Structure for High-Dimensional Mixed Variables" (application number 202410801102.7) and the patent "A Method for Optimizing the Internal Structure of a Machine Tool Bed" (application number 201410452765.9) are all for structural static problems. When the patent "Bridge Digital Twin Construction Method Based on Kriging Proxy Model" (application number 202311505732.1) uses the Kriging proxy model for analysis, it is necessary to solve an optimization problem to determine the parameters. Summary of the invention

[0005] In view of the problems existing in the prior art, the present invention provides a method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors. The present invention adopts dynamic evolution sampling and discrete rounding technology to establish a high-quality test scheme suitable for non-uniform horizontal factors, and then based on these test schemes, a conditional quotient method is used to establish a proxy model that does not require an optimization process. The present invention can achieve efficient establishment of a proxy model for the dynamic response of a mechanical structure, providing a guarantee for structural design and reliability analysis. The present invention is used to construct a proxy model with fast calculation speed and high calculation accuracy.

[0006] In order to achieve the above object, the technical solution adopted by the present invention is:

[0007] A method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform level factors. The proxy model generation method first constructs an experimental design table based on the non-uniform level of the factors considered by the structure by using dynamic evolution sampling and discrete rounding methods. Secondly, a finite element method is used to construct a structural finite element model, and the data in the experimental design table is brought into the structural finite element model for multiple calculations to obtain the structural model response data. Finally, a conditional quotient method is used to establish an efficient proxy model for obtaining the response of the structure at different time points. The method includes the following steps:

[0008] The first step is to construct an experimental design table based on the symplectic kinetic evolution method, as follows:

[0009] Step 1.1: Determine the number and level of factors to be considered in structural analysis based on actual working conditions. Assume that the number of factors is , remember The factors are Assume that Factors The number of levels is , remember the factors The level is ,in Indicates The first level of a factor, Indicates The second level of the factor, Indicates The first factor By analogy, the first factor is recorded as , the corresponding level number is , the corresponding level is ; The second factor is recorded as , the corresponding level number is , the corresponding level is .

[0010] Compute upper bounds for different dimensions of the test space and lower bound for:

[0011]

[0012] in, Represents the test space The upper bound of the dimension, Represents the test space The lower bound of the dimension, Indicates The first factor level.

[0013] According to the above calculations, we get all the experimental space The upper and lower bounds of the dimensions are given, and the test space is recorded as .in, represents the upper bound of the first dimension of the test space, represents the lower bound of the first dimension of the test space; represents the upper bound of the second dimension of the test space, Represents the lower bound of the second dimension of the test space; Represents the test space The upper bound of the dimension, Represents the test space The lower bound of a dimension.

[0014] Step 1.2, it is known that the number of trials required is , in the test space In, generated by random sampling initial experimental design points, each of which is recorded as a dimensional column vector, representing an experimental scheme, with specific coordinates as:

[0015]

[0016] in, Indicates Initial experimental design points, Indicates The first component of the initial experimental design points, Indicates The second component of the initial experimental design points, Indicates The first point of the initial experimental design A quantity.

[0017] but The set of initial experimental design points Recorded as the initial experimental design table.

[0018] Step 1.3, use the symplectic dynamics evolution method to iterate the initial experimental design table to obtain an experimental design table with a more uniform distribution of experimental design points. During the iteration process, the generalized potential energy of the current experimental design table in the experimental space is calculated at each iteration step, and defined as To indicate the The generalized potential energy of the test point in the test space calculated by the iteration step is When , the iteration ends. Specifically, the iterative steps and formulas are as follows:

[0019] Step 1.3.1, according to The experimental design table obtained by iterative steps ,calculate :

[0020]

[0021] in, It is a quantity to be used in subsequent calculations and has no practical meaning; represents the computational symbolic function; express and The absolute value of the difference, express Middle Experimental design points No. Quantity, express Middle Experimental design points No. Quantity; represents the generalized distance coefficient.

[0022] Step 1.3.2, according to The experimental design table obtained by iterative steps , calculate the generalized moment , specifically:

[0023]

[0024] Step 1.3.3, based on the calculated generalized distance , calculate the generalized potential energy and generalized force , specifically:

[0025]

[0026]

[0027] in, , which is a parameter that controls the convergence of the algorithm.

[0028] Step 1.3.4, calculate the Unrounded experimental design points for the iteration step , the formula is as follows:

[0029]

[0030] in, express No. Quantity; , , They are all intermediate variables used in the calculation process and have no actual meaning; , is a parameter that controls the convergence of the algorithm; and It is defined by the following mathematical expression:

[0031]

[0032] in, , is a parameter that controls the convergence of the algorithm; is the generalized potential energy of the initial experimental design table, and the calculation formula is consistent with (1); The points in the initial experimental design table are Made of Matrix of Indicates calculation of F norm.

[0033] Step 1.3.5, Round off to get the final Experimental design points for iteration steps , the set of these experimental design points is the Experimental design table for iteration steps

[0034] The rounding method is as follows:

[0035]

[0036] The symbols involved in this formula are consistent with those in step 1.1. For details, see the introduction in step 1.1.

[0037] Step 1.3.6, calculate the Experimental design table for iteration steps The generalized potential energy , as shown in formula (1). If , end the iteration, otherwise continue the iteration.

[0038] Step 1.4, based on the iterative results in step 1.3, obtain the final experimental design table:

[0039]

[0040] The design table contains points (vectors), corresponding to different experimental designs, each point (vector) contains Components, corresponding to The values ​​of different factors.

[0041] The second step is to conduct structural dynamics finite element analysis based on the experimental design table obtained in the first step. Specifically:

[0042] Step 2.1, using the finite element method to obtain a finite element model of the structure;

[0043] Step 2.2: Experimental Program As input, we sequentially substitute it into the finite element model obtained in step 2.1 for dynamic calculation and obtain Structural response of interest , Indicates a time node.

[0044] Furthermore, in step 2.2, the structural response can also be obtained by a meshless method, theoretical analysis or actual measurement.

[0045] The third step is to use the conditional quotient method to establish an efficient proxy model of the structural response at different times. Specifically:

[0046] Step 3.1, assume that you need to know the values ​​of different factors and time The structural response . Give Factor Assign a priori error probability density function, specifically:

[0047]

[0048] in, , indicating factors The a priori standard deviation of the values; , and The definition of is as detailed in step 1.1; represents the error vector, which contains Components, corresponding to factors; represents the error vector No. Quantity; The error vector A quantity.

[0049] Step 3.2: Establish the structural response based on the conditional quotient principle , and calculate the variance of the proxy model output. Specifically:

[0050] Structural response The proxy model calculation formula is:

[0051]

[0052] The variance calculation formula of the proxy model output is:

[0053]

[0054] in, This is the variance of the proxy model output, which can be used to determine the accuracy of the proxy model estimate. At, the surrogate model results The error with the real result is large, so the group Substitute into the structural finite element model for dynamic solution and add to the result obtained in step 2.2 Structural response Go in and get Then based on this Repeat steps 1 to 3 for each structural response.

[0055] Furthermore, in step 3.2, the proxy model result The error with the true result is large, which means that the relative error between the proxy model result and the true result is greater than 5%.

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] (1) The present invention constructs a test scheme through dynamic evolution and discrete rounding technology. The constructed scheme not only has good distribution quality in the test space, but also can be applied to the situation where the horizontal distribution of the considered factors is non-uniform. It solves the problem of decreased accuracy of existing test design schemes when dealing with non-uniform horizontal problems involved in complex mechanical equipment structures such as machine tool cutting speed level, heat treatment process parameter level, and bolt preload level.

[0058] (2) The present invention constructs a proxy model based on the key condition quotient, and its basic principle is to use the conditional probability quotient theory in probability theory. According to the law of large numbers, as the number of experimental points increases, the accuracy of the proxy model will converge with a probability of 1. The proxy model constructed based on the conditional probability quotient does not require optimization calculation, has high computational efficiency, and is particularly suitable for constructing a proxy model of dynamic response.

[0059] (3) The proxy model method provided by the present invention can also estimate the accuracy of the results and has strong versatility. It is not only suitable for the construction of proxy models for the dynamic response of complex mechanical structures, but also suitable for the construction of proxy models for other problems involving time variations and the construction of proxy models for non-time-varying problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0060] Figure 1 is a flow chart of the present invention;

[0061] Figure 2 It is a flow chart of the experimental design steps in the process of the present invention. DETAILED DESCRIPTION

[0062] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. This embodiment is implemented based on the technical solution of the present invention, and provides a detailed implementation method and specific operation process, but the protection scope of the present invention is not limited to the following embodiments.

[0063] The specific embodiment considered is an agricultural machine frame, whose vibration response Affected by the side lengths of the four angle steels, they are Each factor has four levels, namely The proxy model is constructed for the dynamic response of the frame. The specific implementation steps are as follows:

[0064] The first step is to construct an experimental design table based on the symplectic kinetic evolution method, as follows:

[0065] Step 1.1: Determine the number and level of factors to be considered in structural analysis based on actual working conditions. Assume that the number of factors is , remember The factors are Assume that Factors The number of levels is , remember the factors The level is ,in Indicates The first level of a factor, Indicates The second level of the factor, Indicates The first factor By analogy, the first factor is recorded as , the corresponding level number is , the corresponding level is ; The second factor is recorded as , the corresponding level number is , the corresponding level is .

[0066] For this example, the number of factors to consider is , the number of levels of each factor is 4, that is, , the level of each factor is also the same, .

[0067] Compute upper bounds for different dimensions of the test space and the lower bound for:

[0068]

[0069] in, Represents the test space The upper bound of the dimension, Represents the test space The lower bound of the dimension, Indicates The first factor level.

[0070] According to the above calculation, we can get all the The upper and lower bounds of the dimensions are given, then the experimental space can be written as .in, represents the upper bound of the first dimension of the test space, Represents the lower bound of the first dimension of the test space; represents the upper bound of the second dimension of the test space, Represents the lower bound of the second dimension of the test space; Represents the test space The upper bound of the dimension, Represents the test space For this embodiment, the test space is

[0071] Step 1.2, the number of trials required is known , in the test space In, generated by random sampling initial experimental design points, each of which can be recorded as a dimensional column vector, representing an experimental scheme, the specific coordinates can be recorded as:

[0072]

[0073] in, Indicates Initial experimental design points, Indicates The first component of the initial experimental design points, Indicates The second component of the initial experimental design points, Indicates The first point of the initial experimental design This The set of initial experimental design points Recorded as the initial experimental design table.

[0074] Step 1.3, use the symplectic dynamics evolution method to iterate the initial experimental design table to obtain an experimental design table with a more uniform distribution of experimental design points. During the iteration process, the generalized potential energy ( To indicate the The generalized potential energy of the test point in the test space calculated by the iteration step) When , the iteration ends. Specifically, the iterative steps and formulas are as follows:

[0075] Step 1.3.1, according to The experimental design table obtained by iterative steps ,calculate :

[0076]

[0077] in, It is a quantity to be used in subsequent calculations and has no practical meaning; represents the computational symbolic function; express and The absolute value of the difference, express Middle Experimental design points No. Quantity, express Middle Experimental design points No. Quantity; represents the generalized distance coefficient.

[0078] Step 1.3.2, according to The experimental design table obtained by iterative steps , calculate the generalized moment , specifically:

[0079]

[0080] Step 1.3.3, based on the calculated generalized distance , calculate the generalized potential energy and generalized force , specifically:

[0081]

[0082]

[0083] in, , which is a parameter that controls the convergence of the algorithm.

[0084] Step 1.3.4, calculate the Unrounded experimental design points for the iteration step , the formula is as follows:

[0085]

[0086] in, express No. Quantity; , , They are all intermediate variables used in the calculation process and have no actual meaning; , is a parameter that controls the convergence of the algorithm; and It is defined by the following mathematical expression:

[0087]

[0088] in, , is a parameter that controls the convergence of the algorithm; is the generalized potential energy of the initial experimental design table, and the calculation formula is consistent with (1); The points in the initial experimental design table are Made of Matrix of Indicates calculation of F norm.

[0089] Step 1.3.5, Round off to get the final Experimental design points for iteration steps , the set of these experimental design points is the Experimental design table for iteration steps

[0090] The rounding method is as follows:

[0091]

[0092] The symbols involved in this formula are consistent with those in step 1.1. For details, see the introduction in step 1.1.

[0093] Step 1.3.6, calculate the Experimental design table for iteration steps The generalized potential energy , as shown in formula (1). If , end the iteration, otherwise continue the iteration.

[0094] Step 1.4, based on the iterative results in step 1.3, obtain the final experimental design table:

[0095]

[0096] The design table contains points (vectors), corresponding to different experimental designs, each point (vector) contains Components, corresponding to The values ​​of different factors.

[0097] The second step is to perform structural dynamics finite element analysis based on the experimental design table obtained in the first step. Specifically:

[0098] Step 2.1, using the finite element method to obtain a finite element model of the frame, the unit used is a three-dimensional solid unit;

[0099] Step 2.2: The Experimental Program As input, we sequentially substitute it into the finite element model obtained in step 2.1 for dynamic calculation and obtain Structural response of a rack , Indicates a time node.

[0100] In the third step, the conditional quotient method is used to establish an efficient proxy model of the structural response at different time points. Specifically:

[0101] Step 3.1, assume that you need to know the values ​​of different factors and time is The structural response . Give Factor Assign a priori error probability density function, specifically:

[0102]

[0103] in, , indicating factors The a priori standard deviation of the values; , and The definition of is as detailed in step 1.1; represents the error vector, which contains Components, corresponding to factors; represents the error vector No. Quantity; The error vector A quantity.

[0104] Step 3.2: Establish the structural response based on the conditional quotient principle , and calculate the variance of the proxy model output. Specifically:

[0105] Structural response The proxy model calculation formula is:

[0106]

[0107] The variance calculation formula of the proxy model output is:

[0108]

[0109] in, This is the variance of the proxy model output, which can be used to determine the accuracy of the proxy model estimate.

[0110] The above-described embodiments merely express the implementation methods of the present invention, but they cannot be understood as limiting the scope of the patent of the present invention. It should be pointed out that for those skilled in the art, several modifications and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors, characterized in that: The proxy model generation method comprises the following steps: The first step is to construct the experimental design table based on the non-uniform level of the factors considered by the structure, using the dynamic evolution sampling and discrete rounding method; including the following steps: Step 1.1, determine the number and level of factors to be considered in structural analysis according to the actual working conditions; assume that the number of factors is s, and the kth factor is X k , k = 1, 2, ..., s; assuming that the kth factor X k The number of levels is L k , factor X k The level is Where m k1 represents the first level of the kth factor, m k2 represents the second level of the kth factor, represents the Lth kth factor k levels; by analogy, the first factor is recorded as X1, the corresponding level number is L1, and the corresponding level is The second factor is recorded as X2, and the corresponding level is L2, and the corresponding level is Calculate the upper bound a of different dimensions of the test space k and the lower bound b k for: Among them, a k represents the upper bound of the kth dimension of the test space, b k represents the lower bound of the kth dimension of the test space, represents the kth factor (L k -1) level; According to the above calculations, we can get the upper and lower bounds of all s dimensions of the test space, which is recorded as [a1,b1]×[a2,b2]×…×[a s ,b s ]; where a1 represents the upper bound of the first dimension of the test space, b1 represents the lower bound of the first dimension of the test space; a2 represents the upper bound of the second dimension of the test space, b2 represents the lower bound of the second dimension of the test space; a s represents the upper bound of the sth dimension of the test space, b s represents the lower bound of the sth dimension of the test space; Step 1.2: It is known that the number of trials required is n. In the test space [a1,b1]×[a2,b2]×…×[a s ,b s ], n initial experimental design points are generated by random sampling. Each initial experimental design point is recorded as an s-dimensional column vector, representing an experimental scheme, and the specific coordinates are as follows: in, represents the jth initial experimental design point, represents the first component of the jth initial experimental design point, represents the second component of the jth initial experimental design point, represents the sth component of the jth initial experimental design point; Then the set of n initial experimental design points is Recorded as the initial test design table; Step 1.3, using the symplectic dynamics evolution method, iterate the initial experimental design table to obtain an experimental design table with a more uniform distribution of experimental design points; during the iteration process, calculate the generalized potential energy of the current experimental design table in the experimental space at each iteration step, and define U (g) is the generalized potential energy of the test point in the test space calculated in the g-th iteration step. (g+1) ≥U (g) When , the iteration ends; Step 1.4, based on the iterative results in step 1.3, obtain the final experimental design table: The design table contains n points, corresponding to n different experimental design schemes, and each point contains s components, corresponding to the values ​​of s different factors; The second step is to construct a structural finite element model using the finite element method according to the test design table obtained in the first step, and bring the data in the test design table into the structural finite element model for multiple calculations to obtain the structural model response data; The third step is to use the conditional entropy method to establish an efficient proxy model for the response of the structure at different time points; including the following steps: Step 3.1, assume that the values ​​of different factors need to be known as X = (X1, X2, ..., X s ) and time t k The structural response u(X,t k ); Give the factor X a priori error probability density function, specifically: in, Representation factor X k The a priori standard deviation of the values; a k represents the upper bound of the kth dimension of the test space, b k represents the lower bound of the kth dimension of the test space, L k represents the kth factor X k The number of levels; V represents the error vector, which contains s components, corresponding to s factors; V k represents the kth component of the error vector V; V s represents the sth component of the error vector; Step 3.2: According to the conditional entropy principle, establish the structural response u(X,t k ) and calculate the variance of the proxy model output; specifically: Structural response u(X,t k ) is calculated as follows: Where n represents the initial experimental design point; x j Represents the experimental design points in the first step of the experimental design table, where the set of n initial experimental design points Recorded as the initial test design table; The variance calculation formula of the proxy model output is: Among them, D(X,t k ) is the variance of the proxy model output, which is used to determine the accuracy of the proxy model estimation.

2. The method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors according to claim 1, characterized in that: In step 1.3, the iteration process is as follows: Step 1.3.1, based on the experimental design table obtained in the g-th iteration step calculate Where sgn(·) represents the computational sign function; express and The absolute value of the difference, express The i-th experimental design point The kth component of express The jth experimental design point in The kth component of ; q = 1 represents the generalized distance coefficient; Step 1.3.2, based on the experimental design table obtained in the g-th iteration step Calculate generalized moment Specifically: Step 1.3.3, based on the calculated generalized distance Calculate the generalized potential energy U (g) and generalized force Specifically: Among them, p = 2, which is a parameter that controls the convergence of the algorithm; Step 1.3.4, calculate the unrounded experimental design point of the (g+1)th iteration step j=1,2,…,n, the formula is as follows: in, express The kth component of ; A0, A2, A3 are all intermediate variables used in the calculation process and have no practical meaning; Δt = 1, which is a parameter for controlling the convergence of the algorithm; m and c are defined by the following mathematical expressions: Among them, κ = 10, which is a parameter for controlling the convergence of the algorithm; U (0) is the generalized potential energy of the initial experimental design table; The points in the initial experimental design table are j=1,2,…,n to form an s×n matrix; ||.|| F Indicates calculation of F norm; Step 1.3.5, Round off to get the final experimental design point of the (g+1)th iteration step j=1,2,…,n, the set of experimental design points is the experimental design table of the (g+1)th iteration step Step 1.3.6, calculate the experimental design table for the (g+1)th iteration step The generalized potential energy U (g+1) , as shown in formula (1); if U (g+1) ≥U (g) , end the iteration, otherwise continue the iteration.

3. The method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors according to claim 2, characterized in that: In step 1.3.5, the rounding method is specifically as follows: The symbols involved in the formula are the same as in step 1.

1.

4. The method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors according to claim 1, characterized in that: The second step comprises the following steps: Step 2.1, using the finite element method to obtain a finite element model of the structure; Step 2.2: The n experimental schemes x obtained in the first step j As input, we sequentially substitute into the finite element model obtained in step 2.1 to perform dynamic calculations and obtain n structural responses u of interest. j (x j ,t k ), t k Indicates a time node.

5. The method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors according to claim 4, characterized in that: In step 2.2, the structural response can also be obtained through a meshless method, theoretical analysis or actual measurement method.

6. The method for generating a proxy model for the dynamic effects of a mechanical structure containing non-uniform horizontal factors according to claim 1, characterized in that: In step 3.2, if at a certain set of factors X, the proxy model result u(X,t k ) has a large error with the actual result, then substitute this group X into the structural finite element model for dynamic solution and add it to the n structural response u obtained in step 2.2 j (x j ,t k ), and then (n+1) structural responses are obtained; then based on the (n+1) structural responses, the first to third steps are repeated.

7. The method for generating a proxy model for dynamic effects of mechanical structures containing non-uniform horizontal factors according to claim 6, characterized in that: The proxy model result u(X,t k ) refers to the large error between the proxy model result and the true result, which is greater than 5%.

Images