Method and system for generating mechanical response proxy model of mechanical structure under multi-source random variable

Abstract

The invention provides a method and system for generating a mechanical response proxy model of a mechanical structure under a multi-source random variable, and relates to the technical field of mechanical response analysis of the mechanical structure, and the method comprises the steps: S1, generating a structural point set of mechanical structure parameters based on a moment integral; s2, assembling a moment integral structure point set of the mechanical structure parameters; s3, constructing a mechanical response agent model of the mechanical structure under the influence of the multi-source random variables; and S4, constructing a sparse mechanical structure mechanical response proxy model for mechanical structure mechanical response analysis. According to the method, structural points used for mechanical response agent model construction are generated through a unified mechanical structure moment integral structural point generation method; on the basis of a moment integral structure point MQDP framework, the algebraic accuracy of Gaussian integral is utilized, so that the constructed agent model can accurately restore a response quantity statistical moment, and the engineering decision is supported; a sparse rule structure is introduced to cope with a high-dimensional problem, and the method is used for efficient modeling of a multi-parameter and high-dimensional practical engineering problem.

Description

Technical Field

[0001] This invention relates to the field of mechanical structure mechanical response analysis technology, specifically to a method and system for generating surrogate models of mechanical structure mechanical response under multi-source random variables. Background Technology

[0002] With the increasing demand for complex system modeling and predictive maintenance, surrogate models are becoming increasingly important in the fields of risk and reliability. As engineering models, such as digital twins and large-scale simulation models, become increasingly complex, the quantification and analysis of model uncertainties from classical to quantum domains pose significant computational challenges. Therefore, constructing optimal surrogate models has become crucial.

[0003] Existing surrogate modeling methods still face the following challenges when dealing with complex black-box computational models in reality: First, although the rules for generating structural points are crucial for all subsequent surrogate modeling, regardless of the basis function chosen, there seems to be a lack of universal generation rules based on reasonable metrics to evaluate the quality of structural point sets; second, for some basis functions, such as the GP / Kriging model, the process of constructing and simplifying high-dimensional real-world surrogate models based on generated structural points can be extremely complex due to the dependence on specific case-specific parameter tuning. Current research on moment methods includes, but is not limited to, point-mapped sparse grid integral methods to improve the efficiency of moment estimation in structural reliability analysis, moment basis frameworks with Bayesian inference assistance for reliability assessment of robot systems under uncertain conditions, and Bayesian update methods combining dimensionality reduction integrals to efficiently evaluate the failure probability of multi-mode systems. These studies mainly focus on moment estimation and reliability inference, while this invention emphasizes deterministic surrogate construction, integrating surrogate modeling, uncertainty propagation, and sensitivity analysis into a unified framework based on moment quadrature design points (MQDP).

[0004] Driven by the need for technological improvements to address the aforementioned challenges, this invention aims to develop a deterministic and explicit method for constructing surrogate models. Determinism means that it eliminates the need for case-specific parameter tuning, as is common in many adaptive methods; explicitness means that the entire process is systematic and can establish a standardized procedure for any black-box computational model. To achieve the first objective, a structure point generation scheme based on the method of moments (moments integral) is proposed. Mathematically, it can be proven that when using polynomial basis functions, regardless of orthogonality, this structure point is optimal in terms of approximate accuracy. To achieve the second objective, ensuring that a surrogate model can always be established, this scheme automatically enumerates conventional polynomial terms based on the tensor product of the generated structure points. These terms with unknown coefficients are then linearly superimposed and combined, and the input-response data evaluated at the structure points is used to solve them as a system of linear equations. The simplification process can be easily achieved by eliminating terms with extremely small coefficients. In this context, the main contribution of this invention lies not only in using moments integrals for numerical computation but also in utilizing the method of moments integrals as a deterministic and uniformly distributed structure point generator to construct surrogate models. The resulting moment integral structure points form a unified computational foundation, simultaneously supporting surrogate fitting, statistical moment evaluation, and global sensitivity analysis. Compared to Gaussian integral-based polynomial chaotic expansion methods, whose integration rules are primarily used for coefficient projection, this framework treats the moment integral structure point (MQDP) as a general structure point for surrogate modeling. Furthermore, by incorporating sparse grid construction techniques, this method can be extended to high-dimensional real-world scenarios while maintaining a fully deterministic, non-adaptive formal expression. More importantly, this method establishes a standardized construction and simplification process for any general computable model with finite-dimensional random inputs and scalar objective outputs, enabling its application to any general scenario, including those that cannot be handled by special parameter tuning. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention aims to provide a method and system for generating surrogate models of mechanical response under multi-source random variables. Through a unified method for generating mechanical structure points using moment integrals (MIBs), it can deterministically generate structure points for constructing surrogate models of mechanical response based on a given input distribution. By establishing a MIB-based structure point MQDP framework, it directly utilizes the algebraic accuracy of Gaussian integrals, enabling the constructed surrogate model to accurately reproduce the key statistical moments of the response, achieving high-precision reproduction of response statistical characteristics and supporting engineering decision-making. Furthermore, through tensor product and covariance transformation, the method is extended to multivariate and correlated input scenarios, and sparse rules are introduced to address high-dimensional problems, making it suitable for efficient modeling of multi-parameter, high-dimensional practical engineering problems.

[0006] Specifically, on the one hand, the present invention provides a method for generating a surrogate model of mechanical response of a mechanical structure under multi-source random variables, comprising the following steps: S1: Generate a set of structural points for mechanical structural parameters based on moment integrals; generate the moment integral structural points corresponding to each mechanical structural parameter to obtain a polynomial surrogate model of the mechanical response of the mechanical structure. ; S2: Obtain the one-dimensional moment integral structure point set for each mechanical structure parameter in step S1, and assemble it through tensor product operation to obtain the moment integral structure point set of the mechanical structure parameter as follows: in, The variance-covariance matrix of the mechanical structure parameters is as follows And the total number of moment integral structure points is The point set of the moment integral structure; The total number of moment integral structure points involved in the multidimensional actual situation; Let be the variance and covariance matrix of the mechanical structure parameters; It is the lower triangular decomposition matrix of the variance-covariance matrix; This is the matrix transpose symbol; These are points on the moment integral structure; This represents the number of moment integral nodes corresponding to the mechanical structure parameters; Mechanical structure parameters Random input variables; This is an assignment operator; This represents the total number of mechanical structure parameters. This is the tensor product operator; S3: Construct a surrogate model of the mechanical response of a mechanical structure under the influence of multiple source random variables; input the moment integral structure point set obtained in step S2 to obtain the moment integral structure points. Corresponding mechanical structure mechanical response surrogate model response Solve the system of linear equations to obtain the results used for predicting the input. The polynomial surrogate model for the mechanical response of the structure is as follows: ; in, A polynomial surrogate model for the mechanical response of a mechanical structure; For the polynomial basis set of the surrogate model of mechanical response of mechanical structure; These are the basis coefficients of the surrogate model polynomial; S4: Use the polynomial surrogate model of the mechanical response of the mechanical structure obtained in step S3 to construct a sparse surrogate model of the mechanical response of the mechanical structure, which is used for mechanical response analysis of specific mechanical structures in actual engineering application scenarios.

[0007] On the other hand, the present invention provides an automatic generation system for mechanical structure mechanical response surrogate model of mechanical structure surrogate model generation method under multi-source random variables, including: mechanical structure parameter structural point set generation module, mechanical structure parameter moment integral structural point set assembly module, mechanical structure mechanical response surrogate model construction module and mechanical structure mechanical response analysis module; The mechanical structure parameter point set generation module generates the corresponding moment integral structural points for each mechanical structure parameter and constructs a one-dimensional surrogate model; based on the mechanical mechanical response surrogate model response corresponding to the moment integral structural points of the mechanical structure parameter, a polynomial surrogate model of the mechanical mechanical response is constructed. The mechanical structure parameter moment integral structure point set assembly module assembles the one-dimensional moment integral structure point set of each mechanical structure parameter through tensor product operation to obtain the moment integral structure point set of the mechanical structure parameter. The mechanical structure mechanical response surrogate model construction module constructs a mechanical structure mechanical response surrogate model under the influence of multiple source random variables based on the surrogate model of a single mechanical structure parameter; The mechanical structure mechanical response analysis module constructs a sparse mechanical structure mechanical response proxy model based on moment integral structural point MQDP, thereby obtaining a mechanical structure mechanical response proxy model that meets the actual engineering requirements and is used for mechanical structure mechanical response analysis.

[0008] Compared with the prior art, the beneficial effects of the present invention are as follows: (1) This invention establishes a unified method for generating structural points using moment integrals, which can deterministically generate structural points for constructing mechanical response proxy models based on a given input distribution; it provides a stable and reliable structural point generation scheme for response analysis of complex mechanical structures; compared with traditional random sampling methods, this method completely avoids result fluctuations caused by random sampling, and is particularly suitable for practical engineering scenarios with high requirements for robustness of analysis results.

[0009] (2) Based on the framework of the moment integral structure point MQDP proposed in this invention, the algebraic precision of Gaussian integral is directly utilized to enable the constructed surrogate model to accurately restore the key statistical moments of the response quantity; to achieve high-precision restoration of response statistical characteristics and support engineering decision-making; this feature enables the model to accurately predict the fluctuation range and extreme value tendency of the response in mechanical structure safety assessment, providing a direct basis for reliability design and risk assessment, and achieving the best polynomial approximation effect under the same sample size conditions.

[0010] (3) This invention extends the method to multivariate and correlated input scenarios through tensor product and covariance transformation, and introduces sparse rule construction to deal with high-dimensional problems; it is suitable for efficient modeling of multi-parameter, high-dimensional practical engineering problems; while maintaining accuracy, it significantly reduces the number of samples required, and can be effectively used for complex mechanical systems containing multiple design variables and multiple uncertain parameters, thus alleviating the computational pressure.

[0011] (4) This invention realizes an automated proxy model construction process from polynomial basis function generation, coefficient regression to model simplification, provides an integrated modeling and analysis process, and improves the efficiency of engineering applications. In addition, thanks to the numerical integration characteristics of moment integral structure points, the same set of model results can be directly used for uncertainty quantification and global sensitivity analysis without additional simulation calculations. This integrated advantage is particularly prominent in engineering optimization and diagnosis scenarios, and meets the actual application analysis process. Attached Figure Description

[0012] Figure 1 A control block diagram for generating a surrogate model of mechanical response of a mechanical structure under multi-source random variables; Figure 2 A two-dimensional tensor product mesh diagram in which the highest polynomial order in each dimension is 5; Figure 3 A three-dimensional tensor product mesh diagram in which the highest polynomial order in each dimension is 2; Figure 4 A flowchart outlining the standard steps for constructing a polynomial surrogate model of the mechanical response of a mechanical structure. Figure 5 This is a tensor product mesh diagram in this embodiment of the invention, where the moment integral node terms are monomials. Figure 6 This is a moment integral node diagram of the mechanical response variables of the mechanical structure in an embodiment of the present invention; Figure 7 This is a sparse tensor product mesh diagram in an embodiment of the present invention; Figure 8 A standard flowchart for constructing a polynomial surrogate model of the mechanical response of sparse mechanical structures in embodiments of the present invention; Figure 9 This is a schematic diagram of a 17-segment stepped cantilever beam in an embodiment of the present invention; Figure 10 This is a comparison chart of the surrogate model prediction and the actual results in an embodiment of the present invention; Figure 11 This is a histogram of the model prediction residuals in an embodiment of the present invention; Figure 12 For the model prediction and in the embodiments of the present invention A comparison graph of the true values ​​of a random sample; Figure 13This is a graph of the empirical probability density function of the model prediction residuals in an embodiment of the present invention; Figure 14 This is a schematic diagram of a disc-shaft rotor in an embodiment of the present invention; Figure 15 For the model prediction and in the embodiments of the present invention A comparison graph of the true values ​​of a random sample; Figure 16 This is a graph of the empirical probability density function of the model prediction residuals in an embodiment of the present invention; Figure 17 For the model prediction and in the embodiments of the present invention A comparison graph of the true values ​​of a random sample; Figure 18 This is a graph of the empirical probability density function of the model prediction residuals in an embodiment of the present invention. Detailed Implementation

[0013] Hereinafter, embodiments of the present invention will be described with reference to the accompanying drawings.

[0014] This invention proposes a method for generating a surrogate model of the mechanical response of a mechanical structure under multi-source random variables, such as... Figure 1 As shown; a set of structural points for generating mechanical structural parameters based on moment integrals is generated; the set of structural points for the moment integrals of mechanical structural parameters is assembled; a surrogate model for the mechanical response of the mechanical structure under the influence of multi-source random variables is constructed; a sparse surrogate model for the mechanical response of the mechanical structure is constructed for mechanical response analysis; the mechanical structural parameters involved in the mechanical response problem include external loads, geometric dimensions, material properties, and mass distribution; this embodiment of the invention takes the deflection at the tip of a cantilever beam, the end position of a robotic arm, the torsional vibration of a disc rotor, and the weight of an airfoil as examples to construct a surrogate model for mechanical response analysis; specifically, the following steps are included: Step S1: Generate a set of structural points for mechanical structural parameters based on moment integrals. This invention is applicable to any man-made physical system whose performance can be characterized by a finite set of parameters and evaluated through physical laws or simulation calculations. For mechanical structures falling within this category, the set of mechanical structural parameters determining structural performance is extracted, the structural performance indicators are defined, and a surrogate model is established based on the moment integral structural point MQDP, with the mechanical structural parameter set as input and the structural performance indicators as output. To demonstrate the versatility of this method, this embodiment analyzes actual conditions such as cantilever beam tip deflection, robotic arm end position, disk rotor torsional vibration, and wing weight.

[0015] Step S11: Generate the one-dimensional moment integral node set corresponding to each mechanical structure parameter; specifically, for the deflection analysis of the stepped cantilever beam tip in this embodiment of the invention, the external loads in all mechanical structure parameters involve the lateral force applied at the tip, the geometric dimensions involve the width, height, and length of each rectangular section segment, and the material properties involve the Young's modulus of the stepped cantilever beam; for the end-position problem of the robotic arm, the external loads in all mechanical structure parameters involve the rotation angle of the robotic arm, and the geometric dimensions involve the length of the robotic arm segment; for the torsional vibration problem of the disc rotor, the geometric dimensions in all mechanical structure parameters involve the diameter and length of the shaft, and the material properties involve the Young's modulus and material density; for the wing weight problem, the external loads in all mechanical structure parameters involve the cruise dynamic pressure and the sparsity of the ultimate load, the geometric dimensions involve the wing area, aspect ratio, quarter-chord sweep angle, aspect ratio, and airfoil thickness chord ratio, and the mass distribution involves the total flight design weight, the fuel weight inside the wing, and the total paint weight. Generate the corresponding one-dimensional moment integral node set for all mechanical structure parameters affected by uncertainties; let a certain mechanical structure parameter The probability density function is Calculate its predecessor First-order origin moment Constructing mechanical structure parameters Hankel matrix for: ; in, Mechanical structure parameters The Hankel matrix is ​​a special type of matrix in which all elements on each subdiagonal are equal. Mechanical structure parameters The First-order origin moment, ; The total number of nodes in the one-dimensional moment integral.

[0016] Mechanical structural parameters The The first-order origin moment is: ; in, Mechanical structure parameters The probability density function; Mechanical structure parameters Random input variables; Mechanical structure parameters The range of values ​​for .

[0017] Mechanical parameters of mechanical structures Hankel matrix Perform the Choleski decomposition, so that Obtain mechanical structure parameters Decomposition matrix for: ; in, Mechanical structure parameters The decomposition matrix, whose inverse matrix elements constitute the mechanical structure parameters. An orthogonal polynomial system; For the matrix of the first Line number Column elements.

[0018] According to Mysovskih's theorem, mechanical structural parameters Decomposition matrix The elements of the inverse matrix correspond to an orthogonal polynomial system. The expression for the orthogonal polynomial is: ; in, Mechanical structure parameters Random input variables; Mechanical structure parameters The corresponding highest order is Orthogonal polynomials; For the first The first recursive coefficient; For the first The second recursive coefficient; Mechanical structure parameters Decomposition matrix The Line number The elements of the column.

[0019] Let the tridiagonal matrix of the first term on the right-hand side of the above equation be denoted as Through the solver Calculate its eigenvalues and eigenvectors Specifically: ; in, Mechanical structure parameters tridiagonal matrix eigenvalues; Mechanical structure parameters tridiagonal matrix eigenvectors; For solver commands; It is a tridiagonal matrix of mechanical structure parameters.

[0020] Obtain mechanical structure parameters Moment integral structure points and weight for: ; in, Mechanical structure parameters The first point in the moment integral structure One-dimensional moment integration node; Mechanical structure parameters The first point in the moment integral structure Each one-dimensional moment integral node corresponds to a weight; A tridiagonal matrix The One eigenvalue; For the first The square of the first component of each eigenvector.

[0021] Traversing mechanical structure parameters Given all external loads, geometric dimensions, material properties, and mass distribution parameters involved in the actual mechanical structure response problem, obtain the one-dimensional moment integral node set for each mechanical structure parameter.

[0022] Step S12: Generate one-dimensional parameters for each mechanical structure. The point set of the integral structure is constructed and a one-dimensional surrogate model is built; in the one-dimensional case... The point set of the integral structure of the element moment is equivalent to The node set of the moment integral is used to obtain the mechanical structure parameters obtained in step S11. One-dimensional moment integral node set Then the mechanical structure parameters Elementary moment integral structure point set for: ; in, One-dimensional mechanical response of a mechanical structure The point set of the matrix integral structure; This is an assignment operator; It is a set of nodes for one-dimensional moment integration.

[0023] Obtain mechanical structure parameters One-dimensional moment integral node set Corresponding mechanical structure mechanical response surrogate model response To construct a one-dimensional polynomial surrogate model of the mechanical response of a mechanical structure for: ; in, A one-dimensional polynomial surrogate model for the mechanical response of a mechanical structure, representing the mechanical structure parameters. The value is a real number The corresponding proxy model response result; For the first The coefficients of the polynomial surrogate model for the mechanical response of a mechanical structure.

[0024] Due to the polynomial surrogate model of the mechanical response of the mechanical structure The highest polynomial order of the variance is Based on the algebraic precision characteristics of Gaussian integrals, their mean and variance can be accurately calculated; the coefficients of the polynomial surrogate model of the mechanical response of the mechanical structure can be obtained by solving the following system of linear equations, specifically: ; in, For one-dimensional moment integration nodes The corresponding mechanical structure mechanical model response results; For the first Moment integral nodes Multiplication Second-rate.

[0025] The first term on the right-hand side of the above system of equations is a one-dimensional moment integral node matrix. The mechanical response of the left-end mechanical structure is denoted as Then the polynomial surrogate model coefficients of the mechanical response of the mechanical structure for: ; in, These are the coefficients of the polynomial surrogate model for the mechanical response of the mechanical structure; These are the nodes for moment integrals; Let be the response vector corresponding to the point of the moment integral structure.

[0026] Based on the above, a one-dimensional polynomial surrogate model of the mechanical response of the mechanical structure is obtained. for: ; Step S2: Assemble the multidimensional moment integral structural point set of the mechanical structure parameters. The multidimensional moment integral structural point set of the mechanical structure parameters is obtained by assembling the one-dimensional moment integral structural point set of each mechanical structure parameter in Step S1 through tensor product operation. All mechanical structure parameters include one or more of external loads, geometric dimensions, material properties, and mass distribution. For specific practical situations, such as the actual deflection at the tip of a stepped cantilever beam, the external load involves the lateral force applied at the tip, the geometric dimensions involve the width, height, and length of each rectangular cross-section, and the material property involves Young's modulus. First, generate the one-dimensional moment integral node set of external loads, geometric dimensions, material properties, and mass distribution according to Step S11. The probability density function and number of nodes for external loads, geometric dimensions, material properties, and mass distribution are set according to the actual situation; the number of integral nodes for different mechanical structure parameters does not need to be the same. Subsequently, assemble the multidimensional moment integral structural point set based on tensor product. A random vector of the mechanical response of a dimensional random mechanical structure The elements are, in order, the external loads, geometric dimensions, material properties, and mechanical structural parameters related to mass distribution involved in the actual situation, namely, the lateral force applied at the tip, the width, height, and length of each rectangular cross-section, and Young's modulus. The total number of mechanical structural parameters is... For the first Mechanical structure parameters The one-dimensional mechanical response of the mechanical structure generated in step S12 The point set of the integral structure of the element moment, composed of The point set of the moment integral structure is: ; in, One-dimensional mechanical response of a mechanical structure The point set of the matrix integral structure; For the first The number of moment integral nodes corresponding to each mechanical structural parameter; This represents the total number of mechanical structure parameters. This represents the total number of moment integral structure points involved in the multidimensional real-world scenario. The tensor product of two vectors, for example

[0027] For linearly correlated mechanical structural parameters, calculate their variance-covariance matrix. The lower triangular matrix is ​​obtained by performing the Cholliski decomposition. Specifically: in, This is the variance-covariance matrix between the mechanical structure parameters; It is the lower triangular decomposition matrix of the variance-covariance matrix; The variance-covariance matrix is ​​the first... Line number Column elements.

[0028] In the variance-covariance matrix, The moment integral structure points of a multidimensional correlated random variable are obtained according to the following formula: in, This is the lower triangular decomposition matrix of the variance-covariance matrix of mechanical structural parameters that exhibit linear correlation. The variance-covariance matrix of the mechanical structure parameters is as follows And the total number of moment integral structure points is The multidimensional moment integral structure point set.

[0029] Step S3: Construct a surrogate model for the mechanical response of a mechanical structure under the influence of multiple random variables. A surrogate model for the mechanical response of a mechanical structure under the influence of a single random variable is a surrogate model for a single mechanical structure parameter, obtained from step S12. At this time, a complete The polynomial basis set of the mechanical structural mechanical response surrogate model of order polynomial is: ; in, For the polynomial basis set of the surrogate model of mechanical response of mechanical structure; For a single random variable, the value space is defined. It is the highest monomial order.

[0030] In this case, the number of MQDP-based structure points required to identify the polynomial coefficients is: For cases involving multiple sources of random variables, i.e., multidimensional situations, the effect is expanded using a singlet tensor product grid. Maintain the status quo; such as Figure 2 The image shows a two-dimensional tensor product grid where the highest monomial order in each dimension is 5, illustrating the two-dimensional polynomial basis sets in two-dimensional space. The resulting two-dimensional tensor product mesh. (Example) Figure 3 The image shows a 3D tensor product mesh where the highest monomial order in each dimension is 2, illustrating the three-dimensional polynomial basis sets in 3D space. This forms a three-dimensional tensor product mesh. The highest monomial order of each dimension can be different, for example, A 24-node three-dimensional tensor product mesh is formed. In this vector space, the set of polynomial bases consists of 24 monomials. In this example, the number of polynomial terms in the three dimensions are 3, 4 and 2, respectively, and the corresponding number of moment integral structure point (MQDPs) are 3, 4 and 2, respectively.

[0031] The polynomial basis set of the above mechanical structure mechanical response surrogate model The elements in the formula form the basis of the polynomial proxy model, which is composed of... Figure 2 The polynomial surrogate model constructed from a two-dimensional tensor product mesh has a basis set with 25 terms and a highest order of 10. The specific polynomial basis set is as follows: ; in, For a surrogate model of the mechanical response of a two-dimensional mechanical structure, there is a set of polynomial bases. Let be the value space of the first random variable; Let be the range of values ​​for the second random variable.

[0032] Multiplying all elements of the polynomial basis set of the surrogate model for the mechanical response of the mechanical structure by their corresponding coefficients and summing them, we obtain the polynomial surrogate model for the mechanical response of the mechanical structure under the influence of multiple random variables as follows: ; in, These are the basis coefficients of the polynomials in the surrogate model for the mechanical response of the mechanical structure. It is the coefficient vector that needs to be identified; The polynomial basis set of the surrogate model for the mechanical response of a mechanical structure The number of elements in it.

[0033] In this embodiment, the number of basis coefficients of each polynomial in the mechanical structure mechanical response surrogate model is consistent with the number of MQDPs based on moment integral structural points. Therefore, the basis coefficients of each polynomial in the mechanical structure mechanical response surrogate model are obtained by solving the following system of linear equations, specifically: ; in, For multidimensional moment integral structure points The corresponding mechanical structure mechanical response surrogate model response; For all elements in the basis set of the surrogate model polynomial at the multidimensional moment integral structure point The response vector at that location; These are the basis coefficients of each polynomial in the surrogate model.

[0034] Solving the above equation, we obtain the basis coefficients of each polynomial in the surrogate model of the mechanical structure's mechanical response: ; in, For multidimensional moment integral structure points The corresponding mechanical structure mechanical response surrogate model response; For all elements in the basis set of the surrogate model polynomial at the multidimensional moment integral structure point The response vector at that point.

[0035] Since the first term on the right-hand side of the above linear equation system is a square matrix, an inverse matrix exists; thus, the matrix used for predicting the input is obtained. The MQDP surrogate model for the mechanical response of the structure at the point of mechanical integration is as follows: ; The standard steps for building an MQDP proxy model based on moment integral structure points are summarized as follows: Figure 4 As shown. This is a mechanical response problem involving three dimensions: external load, geometric dimensions, and material properties. The external load, geometric dimensions, and material properties each involve only one random variable, following a uniform distribution U(0,1), a standard normal distribution N(0,1), and an exponential distribution Exp(1), respectively. Each dimension uses two moment integral nodes; therefore, the basis set of the surrogate model is formed by the tensor product. Composition, specific The corresponding three-dimensional tensor product mesh for this problem is as follows: Figure 5 As shown; in the three-dimensional variable space, the moment integral structural point values ​​of external loads, geometric dimensions, and material properties can be calculated according to S12, and the results are as follows. Figure 6 As shown; the MQDPs based on moment integral structure points thus formed are: ; in, Let be the set of three-dimensional moment integral structure points with a total of 8 point structures.

[0036] use The points in the model are substituted into the actual mechanical structure's mechanical response model for calculation, and the fitting coefficients are obtained using these eight model responses. Assume the actual mechanical structure's mechanical response model is... The calculated fitting coefficient is The model construction results are consistent with the real model. Although this example is simple, a noteworthy fact is that some coefficients are zero, indicating that elimination... Terms with coefficients of zero or near zero are included. The accuracy of the surrogate model depends on the Weierstrass approximation theorem, which states that any continuous real-valued function set on a closed interval can be approximated by a polynomial function. That is, the mechanical response problem of any continuously varying mechanical structure can be constructed with a certain accuracy using this method. Since the number of moment integral-based structural points (MQDPs) is equal to the number of moment integral nodes, and the number of moment integral nodes determines the polynomial accuracy of the surrogate, the accuracy of the surrogate increases monotonically with the increase of the number of moment integral nodes.

[0037] Step S4: Based on the moment integral structural point MQDP surrogate model obtained in Step S3, construct a sparse moment integral structural point MQDP-based mechanical structure mechanical response surrogate model. This yields a mechanical structure mechanical response surrogate model that meets the actual engineering requirements and is used for mechanical structure mechanical response analysis. As the number of mechanical structure parameters increases, the dimensionality of the mechanical structure mechanical response problem continuously increases. The construction of a multidimensional surrogate model based on tensor product operations will be limited by the curse of dimensionality. Selection of each dimension... points Mechanical response problems of mechanical structures will arise The number of elements in the polynomial basis set of the corresponding surrogate model is MQDPs, which are MQDPs with a moment integral structure. The coefficient solution in its mechanical structure mechanical response surrogate model corresponds to a scale of The matrix inversion operation of dimension 1, this operation when Relatively large, for example, In practice, this is often difficult to handle. To alleviate this situation, a sparse moment integral method is used to generate a sparse set of mechanical structure parameters based on moment integral structure point (MQDP) polynomials to construct a sparse mechanical structure mechanical response surrogate model based on moment integral structure point (MQDP) polynomials.

[0038] Step S41: Sparsification of the MQDP surrogate model based on moment integral structure points; the sparse moment integral method is implemented by introducing the sparse rule Smolyak into the standard moment integral method. The core idea of ​​the sparse rule Smolyak is to ignore higher-order interaction terms in the linear functional approximating the true function. This is achieved by truncating the full tensor integral formula by pre-setting a level parameter. The level parameter and dimension together determine the maximum order in the truncated set. When constructing the surrogate model of the mechanical response of a mechanical structure, the corresponding sparse tensor product reflects a set of sparse monomials used to construct the sparse mechanical response surrogate model. By reducing the full tensor set to a sparse mechanical structure parameter set, the number of terms is reduced from an exponential scale to a polynomial scale, and at most to a linear scale. The level parameter also affects the polynomial accuracy of the linear functional; therefore, increasing the level parameter improves the accuracy of the approximation.

[0039] In the embodiments of the present invention, for In practice, let the indices of the moment integral nodes in each dimension be denoted as... Let the sum of the multidimensional index set be . Set the upper bound of the cutoff as ,in, For horizontal parameters, i.e., retain The terms are used as the sparse mechanical structure parameter set. Again, taking three-dimensional space as an example, considering the mechanical response problem of a mechanical structure involving external loads, geometric dimensions, and material properties, assuming the number of moment integral nodes for each mechanical structure parameter is 3, the moment integral structure point set of the mechanical structure parameters based on full tensor computation is obtained as follows: ; in, For the index of the first-dimensional space; For the index of the second-dimensional space; For the index of the third-dimensional space; This is a label for the full tensor.

[0040] The elements and sum of the multi-index set in the embodiments of the present invention , set The order of the corresponding monomial. For example, correspond If we take the horizontal parameter ,show If higher-order terms can be ignored, then the point set of the moment integral structure based on the full tensor operation of the mechanical structure parameters can be reduced to the sparse mechanical structure parameter set of the mechanical structure as follows: ; in, This is a sparsification marker.

[0041] The polynomial basis set for the mechanical structural response problem corresponding to the sparse mechanical structural parameter set is: ; like Figure 7 The diagram shown is a sparse parametric mesh diagram of the mechanical structure in an embodiment of the present invention. The crosses in the diagram represent polynomial terms that have been eliminated, and the solid circles represent polynomial terms that have been retained. The plane divides the tensor mesh into two subdomains: retained terms and eliminated terms. The polynomial in the expression can be obtained through the following expression: ; in, To obtain the operations of polynomial terms.

[0042] Applicable to mechanical response conditions of mechanical structures, for Based on the actual situation, Specifically; ; in, These are index vectors in each dimensional space; Mechanical structure parameters Random input variables; These are the polynomial terms expressed under the corresponding index vector and variable vector.

[0043] All polynomial terms can be obtained by iterating through all possible polynomial terms. To enumerate, so that .

[0044] Step S42: In step S3, the number of moment integral structure points and the number of fitting coefficients for the mechanical structure parameters are equal. However, in the sparse model, the number of sparse moment integral structure points for the mechanical structure parameters is greater than all possible values. The total number of moment integral structure points involved in the multidimensional real-world implementation of each index in the set is as follows: in, For the first The number of point structures in the moment integral of a dimensional space. For horizontal parameters; index vector The set domain; This is an operation to find the maximum value.

[0045] Therefore, the number of sparse version MQDPs based on moment integral structure points is always greater than that of the sparse version. The number of fitting coefficients. Figure 7 In the example shown, the total number of structural points is 28, and the total number of fitting coefficients is 10. Using the least squares method to identify the fitting coefficients, and based on the model response and polynomial basis set response evaluated at the sparse moment integral structural points of the mechanical structural parameters, the formula to be solved is as follows: ; in, These are the optimal coefficients corresponding to the polynomial basis of the surrogate model; The operation is to find the parameters that make the function reach its minimum value; These are the possible values ​​of the basis coefficients of each polynomial in the surrogate model; The coefficients of the polynomial basis of the surrogate model are The model residual vector at that time; The coefficients of the polynomial basis of the surrogate model are The transpose of the model residual vector at that time.

[0046] The coefficients of each polynomial basis in the surrogate model are Model residual vector at time for: ; Optimal coefficients corresponding to each polynomial basis of the proxy model The transpose of can be expressed as: ; The resulting sparse version of the mechanical structure mechanical response surrogate model based on moment integral structural point MQDP polynomials is as follows: ; The construction process of the sparse version of the MQDP proxy model based on moment integral structure points is as follows: Figure 8 As shown. The main difference between constructing the standard model based on the moment integral structure point MQDP and constructing the sparse version of the model based on the moment integral structure point MQDP lies in the generation of sparse structure points and sparse polynomial basis sets, as well as the solution of fitting coefficients. Furthermore, the sparse model always has fitting residuals, while the standard model does not.

[0047] One of the significant advantages of surrogate model construction based on moment integral structure points (MQDP) is that once the model response evaluation at the moment integral structure points is completed, various important indicators can be directly calculated using the weight combinations corresponding to these moment integral structure points. These include the mean and variance for uncertainty quantification, and the Sobol sensitivity index for sensitivity analysis. "Direct" means that no additional calculations are required by calling the original model or the surrogate model. This advantage stems from the fact that the moment integral structure points are essentially integral nodes, and the model response obtained during the surrogate model construction process can be directly combined with the integral weights to calculate the required statistics—because statistical moments and the Sobol index are essentially integral operations, which is precisely the core computational objective of the moment integral method.

[0048] The first embodiment of this invention applies a surrogate model generation method for mechanical structural mechanical response under multi-source random variables to the analysis of a known stepped cantilever beam problem. The known stepped cantilever beam problem consists of N segments, each involving three variables: width, height, and length of a rectangular cross-section, thus involving a total of d stepped cantilever beam parameters. Now, a surrogate model based on moment integral structural points is constructed to replace the actual model of the tip deflection for this d-dimensional problem. The formula for the tip deflection of the N-segment stepped cantilever beam is: ; in, The deflection at the tip of the stepped cantilever beam; The number of segments of the stepped cantilever beam, i.e. The lateral force applied at the tip of the stepped cantilever beam; The Young's modulus of the stepped cantilever beam; For the first The width of a rectangular cross-section; For the first The height of each rectangular cross-section segment; For the first The length of each rectangular cross-section segment; Number the segments of the stepped cantilever beam; Number the rectangular section segments of the stepped cantilever beam; For the stair-mounted cantilever beam parameter set, the random input variables are due to the lateral force. Young's modulus All are deterministic parameters, therefore It consists only of the width, height, and length of each rectangular cross-section; specifically, Due to random factors such as processing and measurement errors, the width, height, and length of each rectangular cross-section take random values ​​within a given interval. Assuming that the probability of each point taking a value is equal, then... The components are uniformly distributed within a given interval; in this embodiment... .

[0049] A method for automatically generating a surrogate model of the deflection response at the tip of a stepped cantilever beam under the influence of multiple source random variables, specifically including the following steps: Step S51: Generate the structural point set of parameters for the stepped cantilever beam based on moment integrals; according to the tip deflection formula of the N-segment stepped cantilever beam, only consider the vector composed of parameters of the stepped cantilever beam with uncertain influences, due to the lateral force Young's modulus Since these are deterministic parameters, for an N-segment stepped cantilever beam, the parameter set for the stepped cantilever beam affected by uncertainties includes three variables: the width, height, and length of each rectangular section. Therefore, the parameter set for the stepped cantilever beam includes a total of d parameters, where d = 3N. It is necessary to generate the structural point sets for each of these d stepped cantilever beam parameters, specifically: Step S511: Generate the moment integral structural points corresponding to the parameter X of each stepped cantilever beam; stepped cantilever beam parameters The probability density function is Calculate the front of the probability density function First-order origin moment Construct the Hankel matrix of the parameters X of the stepped cantilever beam. Perform the Chollisky decomposition to make ,get Decomposition matrix Obtain parameters of the stepped cantilever beam. Moment integral structure points and weight Number the segments of the stepped cantilever beam.

[0050] Specifically, the external loads in all parameters of the stepped cantilever beam involve the lateral force applied at the tip, the material properties involve the Young's modulus of the stepped cantilever beam, and the geometric dimensions involve the width of each rectangular section. ,high and length The stepped cantilever beam parameter set, influenced by uncertainties, includes three variables: the width, height, and length of each rectangular section. In this embodiment, it is assumed that the width of different rectangular section segments... ,high and length The values ​​of are in the same range, and the geometric dimensions follow a uniform distribution; therefore, the width of each rectangular cross-section segment is... ,high and length The moment integral structure points are the same; stepped cantilever beam parameters The probability density function is .

[0051] Obtain parameters for each stepped cantilever beam The front of the probability density function First-order origin moment Constructing the parameters of the stepped cantilever beam Hankel matrix for: ; in, Parameters of stepped cantilever beam The Hankel matrix is ​​a special type of matrix in which all elements on each subdiagonal are equal. Parameters of stepped cantilever beam The First-order origin moment, The total number of nodes in the one-dimensional moment integral.

[0052] Stepped cantilever beam parameters The The first-order origin moment is: ; in, Parameters of stepped cantilever beam The probability density function; Parameters of stepped cantilever beam Random input variables; Parameters of stepped cantilever beam The range of values ​​for .

[0053] Parameters of stepped cantilever beam Hankel matrix Perform the Choleski decomposition, so that Established, the parameters of the stepped cantilever beam are obtained. Decomposition matrix for: ; in, Parameters of stepped cantilever beam The decomposition matrix, whose inverse matrix elements constitute the parameters of the stepped cantilever beam. An orthogonal polynomial system; For the matrix of the first Line number Column elements.

[0054] According to Mysovskih's theorem, the parameters of the stepped cantilever beam Decomposition matrix The elements of the inverse matrix correspond to an orthogonal polynomial system. The expression for the orthogonal polynomial is: ; in, Parameters of stepped cantilever beam Random input variables; Parameters of stepped cantilever beam The corresponding highest order is Orthogonal polynomials; For the first The first recursive coefficient; For the first The second recursive coefficient; Parameters of stepped cantilever beam Decomposition matrix The Line number The elements of the column.

[0055] Let the tridiagonal matrix of the first term on the right-hand side of the above equation be denoted as Through the solver Calculate its eigenvalues and eigenvectors Specifically: ; in, Parameters of stepped cantilever beam tridiagonal matrix eigenvalues; Parameters of stepped cantilever beam tridiagonal matrix eigenvectors; For solver commands; Parameters of stepped cantilever beam A tridiagonal matrix.

[0056] Obtaining parameters of stepped cantilever beam Moment integral structure points and weight for: ; in, Parameters of stepped cantilever beam The first point in the moment integral structure One-dimensional moment integration node; Parameters of stepped cantilever beam The first point in the moment integral structure Each one-dimensional moment integral node corresponds to a weight; A tridiagonal matrix The One eigenvalue; For the first The square of the first component of each eigenvector.

[0057] Setting the width of the stepped cantilever beam parameters ,high and length Then, when the parameters of the stepped cantilever beam... Stepped cantilever beam parameters when the width is [value missing]. The probability density function is for: ; in, Let be the probability density function for the width; To select the judgment condition.

[0058] Obtain the parameters of the stepped cantilever beam representing the width of the rectangular cross-section. Moment integral structure points and weight for: ; When the parameters of the stepped cantilever beam Parameters of stepped cantilever beam at height The probability density function is for: ; Obtain the parameters of the stepped cantilever beam representing the height of the rectangular cross-section segment. Moment integral structure points and weight .

[0059] When the parameters of the stepped cantilever beam Parameters of stepped cantilever beam when the length is given The probability density function is for: ; Obtain the parameters of the stepped cantilever beam representing the length of the rectangular cross-section segment. Moment integral structure points and weight .

[0060] Among them, superscript Indicates the width of the rectangular cross-section; superscript Indicates the height of a rectangular cross-section segment, superscript This indicates the length of the rectangular cross-section segment.

[0061] Step S512: Generate parameters for each stepped cantilever beam One-dimensional The point set of the moment integral structure is used to construct a one-dimensional surrogate model; in the one-dimensional case... The point set of the integral structure of the first moment is equivalent to Step moment integral structural points, obtain the stepped cantilever beam parameters obtained in step S511. Moment integral structure points Then the parameters of the stepped cantilever beam of Step moment integral structure point set for: ; in, One-dimensional deflection response at the tip of a stepped cantilever beam Point set of integral structure of order moment; Parameters of stepped cantilever beam One-dimensional moment integral nodes; This is the assignment operator.

[0062] If we only consider the influence of a single dimensional parameter of a rectangular cross-section on the deflection response at the tip of a stepped cantilever beam, then the deflection response at the tip of the stepped cantilever beam degenerates into a one-dimensional problem. In this case, we can construct the parameters of this one-dimensional stepped cantilever beam. A one-dimensional surrogate model for the tip deflection response problem is presented. The following considers only the parameters of a stepped cantilever beam. The influence of uncertainties on the deflection response at the tip of the stepped cantilever beam is investigated. The width, height, and length of the remaining rectangular cross-sections are taken as fixed values, such as 0.045, 0.4, and 0.9. These values ​​characterize the parameters of a stepped cantilever beam. Moment integral structure points Substitute into the formula for the tip deflection of an N-segment stepped cantilever beam. To obtain the actual model response To construct a polynomial surrogate model of the mechanical response of a mechanical structure for: ; in, A polynomial surrogate model for the deflection response at the tip of a stepped cantilever beam is given, representing the parameters of the stepped cantilever beam. The value is a real number At that time, the corresponding proxy model response result; For the first The coefficients of the polynomial surrogate model for the deflection response at the tip of a stepped cantilever beam.

[0063] Due to the polynomial surrogate model of the deflection response at the tip of the stepped cantilever beam in a one-dimensional problem The highest polynomial order of the variance is Based on the algebraic precision characteristics of Gaussian integrals, their mean and variance can be accurately calculated; the polynomial surrogate model coefficients for the deflection response at the tip of the stepped cantilever beam are obtained by solving the following system of linear equations, specifically: ; in, For one-dimensional moment integration nodes The corresponding deflection response at the tip of the stepped cantilever beam; For the first Moment integral nodes Multiplication Second-rate.

[0064] The first term on the right-hand side of the above system of equations is a one-dimensional moment integral node matrix. The deflection response at the tip of the stepped cantilever beam at the left end is denoted as Then the polynomial surrogate model coefficients of the mechanical response of the mechanical structure for: ; in, These are the coefficients of the polynomial surrogate model for the mechanical response of the mechanical structure; The matrix is ​​the moment integral node matrix; Let be the response vector corresponding to the point of the moment integral structure.

[0065] Based on the above, the parameters for considering only one stepped cantilever beam are obtained. A polynomial surrogate model for the deflection response at the tip of a stepped cantilever beam under stochastic influences for: ; Step S52: Assemble the moment integral structural point set of the stepped cantilever beam parameter group. The moment integral structural point set of the stepped cantilever beam parameter group is obtained by assembling the one-dimensional moment integral structural point set of each stepped cantilever beam parameter in step S51 through tensor product operation.

[0066] For N segments of a stepped cantilever beam, consider the parameter vector of the stepped cantilever beam under the influence of uncertainties. The components are The elements are, in order, the width, height, and length of each rectangular cross-section. The total number of parameters for the stepped cantilever beam is... For the first Parameters of a stepped cantilever beam The one-dimensional deflection response at the tip of the stepped cantilever beam generated in step S512 The point set of the integral structure of the first moment, composed of The point set of the moment integral structure is: ; in, One-dimensional deflection response at the tip of a stepped cantilever beam Point set of integral structure of order moment; For the first The number of moment integral nodes corresponding to the parameters of a stepped cantilever beam. This represents the total number of moment integral structure points involved in the multidimensional real-world scenario. For example, the tensor product operator. .

[0067] If there is a linear correlation between the width, height, and length of the rectangular cross-section segments of each stepped cantilever beam, calculate its variance-covariance matrix. The lower triangular matrix is ​​obtained by performing the Cholliski decomposition. Specifically: in, The variance-covariance matrix of the parameters of the stepped cantilever beam; It is the lower triangular decomposition matrix of the variance-covariance matrix; For the first The parameters of the stepped cantilever beam and the first The covariance of the parameters of a stepped cantilever beam.

[0068] In the variance-covariance matrix, The moment integral structure points of a multidimensional correlated random variable are obtained according to the following formula: in, This is the lower triangular decomposition matrix of the variance-covariance matrix of the parameters of a stepped cantilever beam with a linear correlation. The variance and covariance matrix of the parameters of the stepped cantilever beam is as follows: And the total number of moment integral structure points is The point set of the moment integral structure.

[0069] Step S53: Construct a surrogate model for the deflection response at the tip of a stepped cantilever beam under the influence of multiple random variables. A surrogate model for the deflection response at the tip of a stepped cantilever beam under the influence of a single random variable is a surrogate model for a single stepped cantilever beam parameter. For example, a surrogate model considering only a single stepped cantilever beam parameter X, i.e., the model obtained in step S512... At this time, a complete The polynomial basis set of the surrogate model for the deflection response at the tip of a stepped cantilever beam using polynomials of order 1 is: ; in, For the polynomial basis set of the proxy model of the deflection response at the tip of the stepped cantilever beam; Let X be the value space of the parameter X of the stepped cantilever beam; It is the highest monomial order.

[0070] In this one-dimensional case, the number of MQDP-based structure points required to identify polynomial coefficients is: For cases involving multiple random variables, i.e., the 3N mechanical structural parameters involved in the original embodiment, this is expanded to a d=3N dimension using a singlet tensor product grid. If the number of moment integral structural points for each stepped cantilever beam parameter is set to n, then the polynomial basis set of its stepped cantilever beam tip deflection response surrogate model is: ; The polynomial basis set of the surrogate model for the deflection response at the tip of an N-segment stepped cantilever beam. The elements in the formula form the basis of the polynomial proxy model. This indicates that the value space for each of the d stepped cantilever beam parameters is... The number of terms in the polynomial basis set is The highest order is .

[0071] For example, if the number of moment integral structural points for each stepped cantilever beam parameter is set to 3, then the polynomial basis set of the surrogate model for the deflection response at the tip of the 17 stepped cantilever beam segments is: ; The polynomial basis set of the above 17-segment stepped cantilever beam tip deflection response surrogate model The elements in the formula form the basis of the multinomial proxy model, with the number of terms being . The highest order is 102.

[0072] Multiplying all elements of the polynomial basis set of the surrogate model for the deflection response at the tip of the stepped cantilever beam by their corresponding coefficients and summing them, we obtain the polynomial surrogate model for the deflection response at the tip of the stepped cantilever beam under the influence of multiple random variables as follows: ; in, The basis coefficients of each polynomial in the surrogate model for the deflection response at the tip of a stepped cantilever beam are given. It is the coefficient vector that needs to be identified; The polynomial basis set for the surrogate model of the deflection response at the tip of a stepped cantilever beam Number of elements in .

[0073] In this embodiment, the number of basis coefficients of each polynomial in the surrogate model for the deflection response at the tip of the stepped cantilever beam is consistent with the number of moment integral structural points (MQDPs). Therefore, by inputting the set of moment integral structural points obtained in step S52, the basis coefficients of each polynomial in the surrogate model for the deflection response at the tip of the stepped cantilever beam are obtained by solving the following system of linear equations: ; in, For moment integral structure points The corresponding mechanical structure mechanical response surrogate model response; For the surrogate model, all elements in the basis set of the polynomial are points of moment integral structure where the independent variable is a value of a moment integral. The response vector at that time; These are the basis coefficients of each polynomial in the surrogate model.

[0074] Solving the above equation, we obtain the basis coefficients of each polynomial in the surrogate model of the mechanical structure's mechanical response: ; in, For moment integral structure points The corresponding mechanical structure mechanical response surrogate model response; For the surrogate model, all elements in the basis set of the polynomial are points of moment integral structure where the independent variable is a value of a moment integral. The response vector at that time.

[0075] Since the first term on the right-hand side of the above linear equation system is a square matrix, an inverse matrix exists; thus, the matrix used for predicting the input is obtained. The polynomial surrogate model for the deflection response at the tip of the stepped cantilever beam is as follows: in, A polynomial surrogate model for the deflection response at the tip of a stepped cantilever beam; For the polynomial basis set of the proxy model of the deflection response at the tip of the stepped cantilever beam; The polynomial basis set of the surrogate model for the deflection response at the tip of a stepped cantilever beam contains all elements at the moment integral structure point. The response vector at that location; For moment integral structure points The corresponding mechanical structure mechanical response surrogate model response.

[0076] Theoretically, the steps for constructing a polynomial surrogate model of the deflection response at the tip of a stepped cantilever beam are as follows: Figure 4 As shown, although only three structural points are used for the parameters of each stepped cantilever beam, the dimension of this problem is d, resulting in a conventional method based on moment integrals of tensor products having a much larger number of structural points. Due to factors such as memory and computation time, it is not feasible to build its proxy model on a general computer. Therefore, it is necessary to build a sparse proxy model.

[0077] Step S54: Using the polynomial surrogate model of the deflection response at the tip of the stepped cantilever beam obtained in Step S53, a sparse surrogate model of the deflection response at the tip of the stepped cantilever beam is constructed. This yields a mechanical structure mechanical response surrogate model that meets the actual engineering requirements and is used for mechanical structure mechanical response analysis. The specific steps are as follows: Step S541: Sparsify the polynomial surrogate model of the deflection response at the tip of the stepped cantilever beam; the sparse moment integral method is implemented by introducing the sparse rule Smolyak into the standard moment integral method. The core idea of ​​the sparse rule Smolyak is to ignore terms of higher-order interactions in the linear functional approximating the true function. This is achieved by truncating the full tensor integral formula by pre-setting a level parameter. The level parameter and dimension together determine the maximum order in the truncated set. When constructing the surrogate model of the deflection response at the tip of the stepped cantilever beam, the corresponding sparse tensor product reflects a set of sparse monomials used to construct the sparse surrogate model of the deflection response at the tip of the stepped cantilever beam. By reducing the full tensor set to a sparse stepped cantilever beam parameter set, the number of terms is reduced from an exponential scale to a polynomial scale, and at most to a linear scale. The level parameter also affects the polynomial accuracy of the linear functional; therefore, increasing the level parameter improves the accuracy of the approximation.

[0078] In the embodiments of the present invention, for In reality, let the number of moment integral structure points in each dimension be denoted as . Let the sum of the multidimensional index set be . Set the upper bound of the cutoff as ,in, For horizontal parameters, i.e., retain The terms are used as the parameter set for a sparse stepped cantilever beam. In this embodiment, we take... Proxy model polynomial basis set The polynomial in the expression can be obtained through the following expression: ; in, To obtain the operations of polynomial terms, when At that time, obtain the polynomial basis set of the proxy model. The polynomial operations are represented as: ; in, is the vector of the number of moment integral structure points in each dimension of space; Parameters of stepped cantilever beam Random input variables; This is the expression for the polynomial terms under the point number vector and variable vector of the corresponding moment integral structure.

[0079] In this embodiment, all polynomial terms in the polynomial base set of the sparse surrogate model can be obtained by iterating through all possible multi-index terms. To enumerate, so that .

[0080] Step S542: Obtain the total number of moment integral structure points based on the number of structure points for each moment integral. The least squares method is used to identify the fitting coefficients, and the coefficients of each polynomial basis of the surrogate model are determined as follows: Residual vector of polynomial surrogate model at time The optimal coefficients corresponding to each polynomial basis of the proxy model are obtained. The resulting sparse version of the step cantilever beam tip deflection response surrogate model based on moment integral structural point polynomials. .

[0081] In step S53, the number of moment integral structural points and the number of fitting coefficients for the stepped cantilever beam parameters are equal. However, in the sparse model, the number of sparse moment integral structural points for the mechanical structure parameters is all possible. The total number of moment integral structure points involved in the multidimensional actual situation of the number of each moment integral structure point in the set is specifically: in, For the first The number of point structures in the moment integral of a dimensional space. For horizontal parameters; The vector of the number of points in the moment integral structure. The set domain; This is an operation to find the maximum value.

[0082] In the embodiment, for the deflection response at the tip of the 17-segment stepped cantilever beam, the total number of moment integral structural points involved in the multidimensional actual situation of each moment integral structural point is as follows: The number of sparse version MQDPs based on moment integral structure points is always greater than that of the sparse version. The number of fitting coefficients. The least squares method is used to identify the fitting coefficients. Based on the model response and polynomial basis set response obtained from the sparse moment integral structural points of the stepped cantilever beam parameters, the formula to be solved is as follows: ; in, These are the optimal coefficients corresponding to the polynomial basis of the surrogate model; The operation is to find the parameters that make the function reach its minimum value; These are the possible values ​​of the basis coefficients of each polynomial in the surrogate model; The coefficients of the polynomial basis of the surrogate model are The model residual vector at that time; The coefficients of the polynomial basis of the surrogate model are The transpose of the model residual vector at that time.

[0083] The coefficients of each polynomial basis in the surrogate model are Model residual vector at time for: ; The coefficients of each polynomial basis in the proxy model in the embodiment are Model residual vector at time for: ; Optimal coefficients corresponding to each polynomial basis of the proxy model The transpose of can be expressed as: ; The optimal coefficients corresponding to each polynomial basis of the proxy model in the embodiment The transpose of can be expressed as: ; The resulting sparse version of the step cantilever beam tip deflection response surrogate model based on moment integral structural point MQDP polynomial is as follows: ; in, A proxy model for the deflection response at the tip of a stepped cantilever beam; These are the optimal coefficients corresponding to the polynomial basis of the surrogate model.

[0084] Step S55: Directly use the surrogate model of the deflection response at the tip of the stepped cantilever beam to obtain the indices of the deflection response at the tip of the stepped cantilever beam. One of the significant advantages of constructing a surrogate model for the moment integral structure point (MQDP) is that after evaluating the model response at the moment integral structure point, various important indices can be directly calculated using the weight combinations corresponding to these moment integral structure points, such as the mean and variance for uncertainty quantification, and the Sobol sensitivity index for sensitivity analysis. "Direct" means that no additional calculations are required by calling the original model or the surrogate model. This advantage stems from the fact that the moment integral structure points are essentially integral nodes, and the model response obtained during the surrogate model construction process can be directly combined with the integral weights to calculate the required statistics—because statistical moments and the Sobol index are essentially integral operations, which is the core computational objective of the moment integral method.

[0085] like Figure 9 The diagram shown is of a 17-segment stepped cantilever beam. A surrogate model based on the sparse moment integral structure point MQDP was established, with a total of 5356 moment integral structure points and 1378 singlets. To verify the performance of the surrogate model, randomly generated... Calculate the true value of a random sample. For example... Figure 10 The figure shown is a comparison between the surrogate model's predictions and the actual results in an embodiment of the present invention. The standard deviation of the model's prediction residuals is 0.0034, indicating that the constructed surrogate model can provide accurate prediction results. The figure displays a scatter plot with the actual response values ​​on the x-axis and the surrogate model's predicted values ​​on the y-axis. If the surrogate model fits well, the scatter plots should follow... linear distribution Figure 10 The results are consistent with this trend.

[0086] like Figure 11 The figure shows the histogram of the model prediction residuals in an embodiment of the present invention, which shows the empirical probability density function of the model prediction residuals. The histogram of the prediction residuals and its empirical probability density distribution are further given. The standard deviation of the residuals is 0.0034, which indicates that the constructed surrogate model has high prediction accuracy and can determine the mechanical response of the 17-segment stepped cantilever beam in engineering applications.

[0087] The second embodiment of this invention uses the sparse MQDP polynomial-based mechanical structure response surrogate model obtained in step S3 for the practical analysis of an 8-dimensional robotic arm. For the end-effector position problem, the external loads in all mechanical structure parameters relate to the robotic arm's rotation angle, and the geometric dimensions relate to the length of the robotic arm segments. The robotic arm function is a widely used practical application in neural network research. This function establishes a four-segment planar robotic arm model with the shoulder arm fixed at the origin of the plane; the length of each segment of the planar robotic arm model is... , , ,and The first line segment rotated by an absolute angle from the horizontal direction. And line segment Relative to line segment Rotated The end effector position of the robotic arm can be represented as: ; in, For the first The length of a single planar robotic arm model segment; For the first The rotation angle of a planar robotic arm model segment The total number of random variables is 8. Each dimension uses a 4-point MQDP-based moment integral structure point construction for the proxy model, resulting in a total of 65,535 moment integral structure points. Figure 12 The image shown is a comparison chart of the surrogate model prediction and the true value in an embodiment of the present invention, where the scatter plot represents randomly selected values. There are 10 samples to be predicted. The horizontal axis represents the model response of the samples to be predicted, and the vertical axis represents the response prediction result of the surrogate model constructed using the moment integral structure points. The solid line represents the baseline. The closer the distribution of the scattered points is to the baseline, the more accurate the prediction result is. Figure 13 The figure shown is the empirical probability density function graph of the model prediction residuals in an embodiment of the present invention. Ideally, the residuals of the model should be affected only by random Gaussian noise, exhibiting a normal distribution with a mean of 0. Figure 13 Consistency indicates the effectiveness of the model.

[0088] The third embodiment of the present invention applies the sparse moment integral-based structural point MQDP polynomial surrogate model of mechanical structural response obtained in step S3 to the actual situation of 15-dimensional torsional vibration, such as... Figure 14 The diagram shown is a schematic of a disc rotor in an embodiment of the present invention. For the torsional vibration problem of the disc rotor, the geometric dimensions among all mechanical structural parameters involve the diameter and length of the shaft, and the material properties involve Young's modulus and material density. The torsional vibration frequency of the disc rotor is determined by a 15-dimensional function, specifically: ; in, The coefficients of the first characteristic equation, The coefficients of the second characteristic equation, The coefficients of the third characteristic equation, For the torsional stiffness of the shaft segment, For the mass of the disk, Let the moment of inertia of the disk be... It is the acceleration due to gravity. It is a constant term.

[0089] Table 1 provides the nominal values ​​of the disk and shaft parameters. The parameters are uniform random variables centered at their nominal values, with a maximum deviation of 5% around these nominal values. The assumed deviation range reflects potential manufacturing uncertainties. The random variable in this actual situation is... .

[0090] Table 1. Nominal values ​​of disk and shaft parameters use The sparse surrogate model is built based on the MQDP (Moment Integral Point Stratification) structure, with a total of 31 sparse moment integral structure points. The constructed surrogate model is a 16-term linear model. For example... Figure 15 The image shown is a comparison chart of the surrogate model prediction and the true value in an embodiment of the present invention, where the scatter plot represents randomly selected values. There are 10 samples to be predicted. The horizontal axis represents the model response result of the sample to be predicted, and the vertical axis represents the response prediction result of the surrogate model constructed using the moment integral structure points. The solid line represents the baseline. The closer the distribution of the scattered points is to the baseline, the more accurate the prediction result is. Figure 16 The figure shown is the empirical probability density function graph of the model prediction residuals in an embodiment of the present invention. Ideally, the residuals of the model should be affected only by random Gaussian noise, exhibiting a normal distribution with a mean of 0. Figure 16 Consistency indicates the effectiveness of the model; in this case, all prediction errors of the surrogate model are less than 1% of the nominal frequency.

[0091] The fourth embodiment of this invention uses the sparse mechanical structural response surrogate model based on the moment integral structure point MQDP polynomial obtained in step S3 for wing weight analysis. For the wing weight problem, the external loads among all mechanical structural parameters involve cruise dynamic pressure and ultimate load sparseness, the geometric dimensions involve wing area, aspect ratio, quarter-chord sweep angle, aspect ratio, and airfoil thickness chord ratio, and the mass distribution involves flight design gross weight, wing internal fuel weight, and paint weight. The following analytical expression is used as a conceptual level estimate of the aircraft wing weight: ; in, Wing area; This refers to the weight of the fuel inside the wing; This is the total weight of the paint; Aspect ratio; For cruise dynamic pressure; For aspect ratio; The airfoil thickness chord ratio; It is a quarter-chord sweep angle; This is the ultimate load factor; Total weight for flight design.

[0092] The range of values ​​for the 10 random variables involved in the conceptual level estimation of aircraft wing weight is shown in Table 2.

[0093] Table 2 Random Variables and Their Ranges The embodiments of the present invention adopt The sparse surrogate model was constructed using the Moment Integral (MII) structure point MQDP, generating a total of 1771 MII structure points. The model contains 117 effective monomials. Figure 17 This is a comparison chart of the model prediction and the true values ​​of 10^4 random samples in an embodiment of the present invention. Figure 18 The figure shown is an empirical probability density function graph of the predicted residuals of the model in an embodiment of the present invention. The mean weight is 626.3 lb, while the standard deviation of the predicted residuals is 6.65 lb, which is about two orders of magnitude smaller than the mean, indicating that the model can produce accurate predictions.

[0094] The second aspect of this invention proposes an automatic generation system for mechanical structure mechanical response surrogate models based on a method for generating mechanical structure mechanical response surrogate models under multi-source random variables. The system includes: a mechanical structure parameter point set generation module, a mechanical structure parameter moment integral point set assembly module, a mechanical structure mechanical response surrogate model construction module, and a mechanical structure mechanical response analysis module.

[0095] The mechanical structure parameter point set generation module generates the corresponding moment integral structural points for each mechanical structure parameter, constructing a one-dimensional surrogate model. It then iterates through all external loads, geometric dimensions, material properties, and mass distribution parameters involved in the actual mechanical structure response problem to obtain the moment integral structural points for each mechanical structure parameter. Based on the mechanical structure mechanical response surrogate model responses corresponding to the moment integral structural points of the mechanical structure parameters, a polynomial surrogate model of the mechanical structure mechanical response is constructed. The coefficients of the polynomial surrogate model of the mechanical structure mechanical response can be obtained by solving the following system of linear equations, thus determining the polynomial surrogate model of the mechanical structure mechanical response.

[0096] The mechanical structure parameter moment integral structure point set assembly module assembles the one-dimensional moment integral structure point set of each mechanical structure parameter through tensor product operation to obtain the moment integral structure point set of the mechanical structure parameter. First, it generates moment integral structure points for the external load, geometric dimensions, material properties, and mass distribution of the mechanical structure. The probability density function and number of nodes for the external load, geometric dimensions, material properties, and mass distribution are set according to the actual situation. Then, the moment integral structure point set is assembled based on the tensor product. A random vector of mechanical response of the mechanical structure is set, in which the elements are, in order, the mechanical structure parameters involved in the actual situation, namely the lateral force applied at the tip, the width, height, and length of each rectangular section segment, and Young's modulus. The generated moment integral structure point set of mechanical response of the mechanical structure is composed of the one-dimensional moment integral structure point set.

[0097] The mechanical structure mechanical response surrogate model construction module constructs a surrogate model of the mechanical structure mechanical response under the influence of multiple random variables based on the surrogate model of a single mechanical structure parameter; for cases with the influence of multiple random variables, it expands the model using a one-term tensor product grid. In the case of dimensionality, the elements in the polynomial basis set of the surrogate model for mechanical structural mechanical response constitute the basis of the polynomial surrogate model. Multiplying all elements in the polynomial basis set of the surrogate model for mechanical structural mechanical response by their corresponding coefficients and summing them, we obtain the polynomial surrogate model for mechanical structural mechanical response under the influence of multiple random variables. Solving for the polynomial basis coefficients of the surrogate model for mechanical structural mechanical response yields the polynomial surrogate model for mechanical structural mechanical response. The accuracy of the surrogate model depends on the Weierstrass approximation theorem, which states that any continuous real-valued function set on a closed interval can be approximated by a polynomial function. That is, any continuously changing mechanical structural mechanical response problem can be constructed into a surrogate model with a certain accuracy using this method. Since the number of moment integral structure points (MQDPs) is equal to the number of moment integral nodes, and the number of moment integral nodes determines the polynomial accuracy of the surrogate, the accuracy of the surrogate increases monotonically with the increase of the number of moment integral nodes.

[0098] The mechanical structure mechanical response analysis module constructs a sparse mechanical structure mechanical response surrogate model based on Moment Integral (MII) structural point MQDPs to obtain a mechanical structure mechanical response surrogate model that meets the needs of actual engineering and is used for mechanical structure mechanical response analysis. As the number of mechanical structure parameters increases, the dimensionality of the mechanical structure mechanical response problem continuously increases, and the construction of multidimensional surrogate models based on tensor product operations is limited by the curse of dimensionality. To alleviate this, a sparse moment integral method is used to generate a sparse mechanical structure parameter set based on MII structural point MQDPs to construct a sparse mechanical structure mechanical response surrogate model based on MII structural point MQDP polynomials. The sparse rule Smolyak is introduced into the standard moment integral method for implementation. When constructing the mechanical structure mechanical response surrogate model, sparse tensor product is used to construct the sparse monomials of the sparse mechanical structure mechanical response surrogate model. By reducing the full tensor set to a sparse mechanical structure parameter set, the number of terms is reduced from an exponential scale to a polynomial scale, and at most to a linear scale. Level parameters also affect the polynomial accuracy of linear functionals; therefore, increasing level parameters improves the approximation accuracy.

[0099] The beneficial effects of this invention are as follows: This invention proposes a method and system for generating surrogate models of mechanical response under multi-source random variables. By establishing a unified moment integral structural point generation method, it can deterministically generate structural points for constructing surrogate models of mechanical response based on a given input distribution. It provides a stable and reliable structural point generation scheme for complex mechanical structure response analysis. Compared with traditional random sampling methods, this method completely avoids result fluctuations caused by random sampling, making it particularly suitable for practical engineering scenarios with high requirements for robustness of analysis results, such as vibration analysis of aero-engine blades and life prediction of heavy equipment support structures, ensuring repeatable and reliable input for each modeling. The framework of the Moment Integral Structural Point (MQDP) is established, directly utilizing the algebraic accuracy of Gaussian integrals, enabling the constructed surrogate model to accurately reproduce the key statistical moments of the response; achieving high-precision reproduction of response statistical characteristics and supporting engineering decision-making; this characteristic allows the model to accurately predict the fluctuation range and extreme value tendency of the response in structural safety assessments of aviation, vehicles, and ships, providing a direct basis for reliability design and risk assessment, and achieving optimal polynomial approximation under the same sample size conditions. By employing tensor product and covariance transformation, the method is extended to multivariate and correlated input scenarios, and sparse rules are introduced to address high-dimensional problems. It is suitable for efficient modeling of multi-parameter, high-dimensional practical engineering problems. While maintaining accuracy, it significantly reduces the required sample size, effectively relieving computational burden on complex mechanical systems with multiple design variables and uncertain parameters. This invention achieves an automated proxy model construction process from polynomial basis function generation and coefficient regression to model simplification, providing an integrated modeling and analysis workflow and improving engineering application efficiency. Furthermore, thanks to the numerical integration characteristics of moment integral structural points, the same set of model results can be directly used for uncertainty quantification and global sensitivity analysis without additional simulation calculations. This integrated advantage is particularly prominent in engineering optimization and diagnostic scenarios, significantly accelerating practical applications such as uncertainty analysis of aerospace structural loads and sensitivity ranking of dynamic characteristic parameters of machine tool spindles.

[0100] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made by those skilled in the art to the technical solutions of the present invention without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for generating a surrogate model of the mechanical response of a mechanical structure under multi-source random variables, characterized in that: It includes the following steps: S1: A set of structural points for generating mechanical structural parameters based on moment integrals; By generating moment integral structural points corresponding to each mechanical structure parameter, a polynomial surrogate model of the mechanical structure's mechanical response is obtained. ; S2: Obtain the one-dimensional moment integral structure point set for each mechanical structure parameter in step S1, and assemble it through tensor product operation to obtain the moment integral structure point set of the mechanical structure parameter as follows: in, The variance-covariance matrix of the mechanical structure parameters is as follows And the total number of moment integral structure points is The point set of the moment integral structure; The total number of moment integral structure points involved in the multidimensional actual situation; Let be the variance and covariance matrix of the mechanical structure parameters; It is the lower triangular decomposition matrix of the variance-covariance matrix; This is the matrix transpose symbol; These are points on the moment integral structure; This represents the number of moment integral nodes corresponding to the mechanical structure parameters; Mechanical structure parameters Random input variables; This is an assignment operator; This represents the total number of mechanical structure parameters. This is the tensor product operator; S3: Construct a surrogate model of the mechanical response of a mechanical structure under the influence of multiple source random variables; input the moment integral structure point set obtained in step S2 to obtain the moment integral structure points. Corresponding mechanical structure mechanical response surrogate model response Solve the system of linear equations to obtain the results used for predicting the input. The polynomial surrogate model for the mechanical response of the structure is as follows: ； in, A polynomial surrogate model for the mechanical response of a mechanical structure; For the polynomial basis set of the surrogate model of mechanical response of mechanical structure; These are the basis coefficients of the surrogate model polynomial; S4: Use the polynomial surrogate model of the mechanical response of the mechanical structure obtained in step S3 to construct a sparse surrogate model of the mechanical response of the mechanical structure, which is used for mechanical response analysis of specific mechanical structures in actual engineering application scenarios.

2. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 1, characterized in that: Step S1 is as follows: S11: Generate the moment integral structure points corresponding to each mechanical structure parameter; let a certain mechanical structure parameter... The probability density function is Calculate its predecessor First-order origin moment Constructing mechanical structure parameters Hankel matrix Perform the Chollisky decomposition to make Obtain mechanical structure parameters Decomposition matrix Obtain mechanical structure parameters Moment integral structure points and weight This yields the moment integral structure points for each mechanical structure parameter; S12: One-dimensional generation of each mechanical structure parameter The point set of the integral structure of the first moment is used to construct a one-dimensional surrogate model; Based on the mechanical structure parameters obtained in step S11 Moment integral structure points Then the mechanical structure parameters Step moment integral structure point set ; Polynomial surrogate model for obtaining mechanical response of mechanical structure .

3. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 2, characterized in that: The polynomial surrogate model for the mechanical response of the mechanical structure in step S12 is as follows: ； in, A polynomial surrogate model for the mechanical response of a mechanical structure; These are the coefficients of the polynomial surrogate model for the mechanical response of the mechanical structure; These are the nodes for moment integrals.

4. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 1, characterized in that: The variance-covariance matrix between mechanical structure parameters in step S2 for: in, The variance-covariance matrix is ​​the first... Line 1 Column elements.

5. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 1, characterized in that: The surrogate model polynomial basis coefficients in step S3 for: ； in, For moment integral structure points The corresponding mechanical structure mechanical response surrogate model response; For all elements in the basis set of the surrogate model polynomial at the moment integral structure point The response vector at that location; These are the basis coefficients of each polynomial in the surrogate model.

6. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 1, characterized in that: Step S4 is as follows: S41: Sparsify the polynomial surrogate model of the mechanical response of the mechanical structure by introducing the sparse rule Smolyak into the standard moment integral method; by reducing the full tensor set to a sparse mechanical structure parameter set. S42: Obtain the total number of moment integral structure points involved in the multidimensional actual situation for each moment integral structure point. The number of sparse version MQDPs based on moment integral structure points is always greater than that of the sparse version. The number of fitting coefficients; using the least squares method to identify the fitting coefficients, and determining the coefficients of each polynomial basis of the surrogate model. Model residual vector at time The optimal coefficients corresponding to each polynomial basis of the proxy model are obtained. ; The resulting sparse version of the mechanical structure mechanical response surrogate model based on moment integral structure point MQDP polynomials .

7. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 6, characterized in that: The polynomial basis set for the mechanical structural response problem corresponding to the sparse mechanical structural parameter set in step S41 is: ； ； ； in, To obtain the operations of polynomial terms; is the vector of the number of moment integral structure points in each dimension of space; Mechanical structure parameters The One random input variable; For the polynomial terms under the corresponding moment integral structure point number vector and variable vector; For the first The number of moment integral structure points in a dimensional space.

8. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 6, characterized in that: The sparse version of the mechanical response surrogate model based on moment integral structural point MQDP polynomials in step S42 for: ； ； in, A proxy model for the mechanical response of a mechanical structure; These are the optimal coefficients corresponding to the polynomial basis of the surrogate model.

9. The method for generating a surrogate model of mechanical structure mechanical response under multi-source random variables according to claim 1, characterized in that, It includes: S51: Generate the structural point set of stepped cantilever beam parameters based on moment integrals; generate the moment integral structural points corresponding to each stepped cantilever beam parameter to obtain a polynomial surrogate model of the deflection response at the tip of the stepped cantilever beam. ; S52: Obtain the one-dimensional moment integral structural point set of each stepped cantilever beam parameter in step S51, and assemble it through tensor product operation to obtain the moment integral structural point set of the stepped cantilever beam parameter. S53: Construct a proxy model for the deflection response at the tip of a stepped cantilever beam under the influence of multiple random variables; input the moment integral structure point set obtained in step S52 to obtain the moment integral structure points. Corresponding mechanical structure mechanical response surrogate model response ; Solving the system of linear equations yields the results used for predicting the input. Polynomial surrogate model of deflection response at the tip of a stepped cantilever beam ; S54: Use the polynomial surrogate model of the deflection response at the tip of the stepped cantilever beam obtained in step S53 to construct a sparse surrogate model of the deflection response at the tip of the stepped cantilever beam, which is used for mechanical response analysis of the stepped cantilever beam in actual engineering application scenarios. S55: Use the surrogate model of the deflection response at the tip of the stepped cantilever beam to obtain the index of the deflection response at the tip of the stepped cantilever beam.

10. An automatic generation system for a surrogate model of mechanical structure mechanical response under multi-source random variables as described in any one of claims 1 to 9, characterized in that, include: The module includes: mechanical structure parameter point set generation module, mechanical structure parameter moment integral point set assembly module, mechanical structure mechanical response proxy model construction module, and mechanical structure mechanical response analysis module. The mechanical structure parameter point set generation module generates the corresponding moment integral structural points for each mechanical structure parameter and constructs a one-dimensional surrogate model; based on the mechanical mechanical response surrogate model response corresponding to the moment integral structural points of the mechanical structure parameter, a polynomial surrogate model of the mechanical mechanical response is constructed. The mechanical structure parameter moment integral structure point set assembly module assembles the one-dimensional moment integral structure point set of each mechanical structure parameter through tensor product operation to obtain the moment integral structure point set of the mechanical structure parameter. The mechanical structure mechanical response surrogate model construction module constructs a mechanical structure mechanical response surrogate model under the influence of multiple source random variables based on the surrogate model of a single mechanical structure parameter; The mechanical structure mechanical response analysis module constructs a sparse mechanical structure mechanical response proxy model based on moment integral structural point MQDP, thereby obtaining a mechanical structure mechanical response proxy model that meets the actual engineering requirements and is used for mechanical structure mechanical response analysis.

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