Evaluation method and system for quantifying reliability of mechanical truss structure based on nested integral

Abstract

The invention belongs to the technical field of mechanical truss structure reliability quantitative analysis, and relates to an evaluation method and system for quantifying the reliability of a mechanical truss structure based on nested integration, and the method comprises the steps: generating an initial moment integral of a reliability parameter of the mechanical truss structure; establishing a continuous nested integral model for quantifying the reliability of the mechanical truss structure, and outputting a nested integral rule; and calculating a numerical integration form of the reliability parameters of the mechanical truss structure according to a nested integration rule to obtain reliability evaluation of the mechanical truss structure. According to the method, continuous nested integral nodes and weights of any probability density function for mechanical truss structure reliability parameters are generated, unknown variable values such as nested integral nodes, nested integral weights and integral nodes are obtained by solving a minimization problem, and small singular values and numerical error influences are reduced by using regularization; and a nested integral rule with the highest polynomial algebraic precision under any probability density function can be obtained, and the stochastic analysis problem in the mechanical truss structure response can be effectively solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of reliability quantitative analysis of mechanical truss structures, and in particular relates to a reliability evaluation method and system for mechanical truss structures based on nested integral quantification. Background Art

[0002] The core idea of ​​the moment integral method is to calculate the corresponding Gauss quadrature rule based on a given probability density function. In practical applications, existing methods cannot quantify and analyze random variables, resulting in large errors in reliability assessment results and problems such as complex calculations and slow speed.

[0003] Since the moment integration nodes of different numbers of points are completely different, when adding integration nodes to improve the polynomial algebraic accuracy and verify the convergence of the results, the data calculated by the current integration node cannot be reused, resulting in a waste of resources. The nested integration rule allows you to start from the integration rule with a small number of points and gradually increase the number of points to improve the polynomial algebraic accuracy of the numerical integration rule. The integration nodes of each step are expanded on the basis of retaining the nodes of the previous step, so in actual calculations, all the numerical results obtained before can be reused.

[0004] The Gauss-Kronrod quadrature rule is the most commonly used nested integral rule and the one that requires the least amount of additional computation based on the Gaussian integral rule. By adding n+1 points to the n-point Gaussian integral rule, the Gauss-Kronrod quadrature rule can achieve a polynomial algebraic accuracy of up to 3n+2. However, the traditional Gauss-Kronrod quadrature rule is based on a uniformly distributed weight function and cannot be directly applied to weight functions corresponding to other distributions.

[0005] For general continuous probability distributions, the problem of solving the nodes and weights of continuous nested integration rules can be transformed into the problem of numerically solving nonlinear equations using iterative methods. However, negative weights may appear in the numerical results, and the existence of negative weights may cause the integration rules to fluctuate or converge in actual use.

[0006] Based on this, a systematic method is developed in this study to generate continuous nested integration nodes and weights for arbitrary probability density functions. Starting from the initial nodes and weights generated by the conventional regular integration method, a minimization problem is solved to obtain the values ​​of 3n+2 unknown variables, which include n+1 nested integration nodes, n+1 nested integration weights, and updated weights of n existing integration nodes. To ensure that the numerical solution falls within the specified domain, a strategy of adaptive algebraic accuracy is proposed, and regularization techniques are used to reduce the influence of small singular values ​​and numerical errors. For symmetric distributions, a semi-axis iteration method is proposed to speed up the solution process. In addition, a negative weight constraint optimization strategy is proposed to alleviate and quantify the fluctuation and convergence problems caused by negative weights. Using the developed method and program, the nested integration rule with the highest polynomial algebraic accuracy under arbitrary probability density functions can be dynamically obtained, which can effectively handle the random analysis problems in the response of mechanical truss structures. Summary of the invention

[0007] In view of the deficiencies in the prior art, the present invention provides a reliability evaluation method and system for a mechanical truss structure based on nested integral quantification, which can generate continuous nested integral nodes and weights for any probability density function of the reliability parameters of the mechanical truss structure, obtain the values ​​of unknown variables such as nested integral nodes, nested integral weights and integral nodes by solving the minimization problem, and use regularization techniques to reduce the influence of small singular values ​​and numerical errors; the present invention can obtain the nested integral rule with the highest polynomial algebraic precision under any probability density function, and effectively deal with the random analysis problems in the response of the mechanical truss structure.

[0008] To achieve the above object, the present invention discloses the following technical solution: a method for evaluating the reliability of a mechanical truss structure based on nested integral quantification, comprising:

[0009] S1: Generate initial moment integral of mechanical truss structure reliability parameters;

[0010] S11: Set the number of integration nodes of the mechanical truss structure reliability parameter integration rule to k, calculate the first 2k-order origin moments of the mechanical truss structure random variables, and construct the Hankel matrix H of the mechanical truss structure reliability parameters k ;

[0011] S12: Hankel matrix H for reliability parameters of mechanical truss structures k Perform Cholesky decomposition to obtain the upper triangular matrix R and construct the moment characteristic matrix J of the reliability parameters of the mechanical truss structure;

[0012] S13: Solve the eigenvalue λ of the moment characteristic matrix J of the reliability parameters of the mechanical truss structure j With the feature vector v j =(v j1,…,v jk ) T , j = 1,…,k; the integration node in the initial moment integration rule is x j =λ j , the corresponding integration node weight is

[0013] S2: Establish a continuous nested integral model to quantify the reliability of mechanical truss structures and output nested integral rules;

[0014] S21: Initialize the algebraic precision m of the continuous nested integral model, calculate the first m-order origin moments under the probability density function of the random variable of the reliability parameter of the mechanical truss structure, and the algebraic precision m is initially set to 3k+1;

[0015] S22: Initial integration node of nested mechanical truss structure reliability and determine the nested integration rule boundary; Initialize the integration node weight to ω 1 =…=ω 2k+1 =1 / (2k+1);

[0016] S23: Constructing an objective function to quantify the reliability of mechanical truss structures And initialize the termination condition

[0017] S24: Perform regularized Gauss-Newton iterative optimization on the objective function in step S23 Calculate the Newton iteration reduction η; for the objective function Find the partial derivatives to get the Jacobian matrix with N rows and M columns Perform singular value decomposition to obtain:

[0018]

[0019] Among them, u i represents the i-th left singular column vector; v i represents the i-th right singular column vector; σ i represents the i-th singular value; N represents the number of rows of the continuous nested integral model; M represents the number of columns of the continuous nested integral model; i represents the number of rows of the continuous nested integral model;

[0020] S25: Combine the initial integration nested nodes, integration node weights and termination conditions to set the judgment output conditions, and output the nested integration rule as {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1};

[0021] S3: According to the nested integral rule {x1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}, calculate the numerical integral form of the mechanical truss structure reliability parameters, and obtain the mechanical truss structure reliability evaluation results

[0022]

[0023] Among them, ψ(x i ) represents the numerical integration form of the integration node; ψ(z j ) represents the numerical integration form of the integral nested node; ω represents the integration node weight.

[0024] Preferably, in step S11, the Hankel matrix H of the reliability parameters of the mechanical truss structure is constructed. k , specifically:

[0025] When the probability density function of the random variable X of the mechanical truss structure reliability parameter is p(x), the k-order origin moment μ of the mechanical truss structure reliability parameter is k for:

[0026] μ k =E[X k ]=∫ x∈Ω x k p(x)dx (3)

[0027] Where, E,p(x),Ω represents the mathematical expectation, probability density function and setting domain of the random variable X of the reliability parameter of the mechanical truss structure; X,k,k represent the random variable, random variable value and origin moment order of the reliability parameter of the mechanical truss structure respectively;

[0028] When the probability density function of the random variable of the mechanical truss structure reliability parameter is unknown, the k-order origin moment μ of the mechanical truss structure reliability parameter is k Equal to the sample moment of the reliability parameter of the mechanical truss structure

[0029]

[0030] Among them, p n Represents the number of samples of reliability parameters of mechanical truss structure; x i represents the sample value of the reliability parameter of the i-th mechanical truss structure;

[0031] According to the k-order origin moment μ of the mechanical truss structure reliability parameter k , construct the Hankel matrix H of the reliability parameters of the mechanical truss structurek :

[0032]

[0033] Preferably, the Hankel matrix H of the reliability parameters of the mechanical truss structure in step S12 is k Perform Cholesky decomposition to obtain the upper triangular matrix R, and construct the moment characteristic matrix J of the reliability parameters of the mechanical truss structure, which is:

[0034] The Cholesky decomposition result is:

[0035]

[0036] Where R represents the upper triangular matrix of the reliability parameters of the mechanical truss structure; r k+1,k+1 The k+1th row and k+1th column element of the upper triangular matrix representing the reliability parameters of the mechanical truss structure;

[0037] According to the upper triangular matrix R of the mechanical truss structure reliability parameters, the moment characteristic matrix J of the mechanical truss structure reliability parameters is obtained:

[0038]

[0039] Among them, α j The first diagonal element of the Jacobian matrix representing the reliability parameters of the mechanical truss structure, and β j The second diagonal element of the Jacobian matrix representing the reliability parameters of the mechanical truss structure, and j represents the diagonal element number of the Jacobian matrix of the reliability parameters of the mechanical truss structure, and j=1,…,k.

[0040] Preferably, in step S22, the initial integration node of the mechanical truss structure reliability is nested, and the nested integration rule boundary is determined, specifically:

[0041] Get the initial integration node x obtained in step S1 1 ,…,x k , initialize the integral nested node z 1 ,…,z k+1 and the integration node weight ω 1 ,…,ω 2k+1 ; respectively find and 2k+1 Gaussian integration nodes, and set the Gaussian integration scaling factor s to:

[0042]

[0043] Where s represents the Gaussian integral scaling factor; represents the maximum value in the set of integration nodes corresponding to the first integration rule; represents the maximum value in the set of integration nodes corresponding to the second integration rule;

[0044] After multiplying the Gaussian integral node at point 2k+1 by the Gaussian integral scaling factor s, the points that are staggered with the initial integral nodes are selected as the initial integral nested nodes, otherwise the median of the two adjacent initial nodes is used as the initial integral nested node; the nested integral rule boundary is determined. If the set domain of the probability density function for evaluating the reliability parameters of the mechanical truss structure is bounded, then its set domain is the nested integral rule boundary; if the set domain of the probability density function for evaluating the reliability parameters of the mechanical truss structure is unbounded, then the maximum node and the minimum node in the Gaussian formula at point 2k+1 are taken as the nested integral rule boundary.

[0045] Preferably, in step S23, an objective function for quantifying the reliability of the mechanical truss structure is constructed: And initialize the termination condition Specifically:

[0046] The objective function of the mechanical truss structure reliability It is composed of the residual r of the origin moment of the objective function and the penalty term P for violating the constraint conditions;

[0047] The residual r of the origin moment of the objective function is:

[0048]

[0049] Where w represents the integration node weight; ω i represents the weight of the i-th integration node; μ represents the origin moment of the reliability parameter of the mechanical truss structure; z j represents the jth integral nested node; a and b represent the upper and lower bounds of the set domain respectively; P j represents the penalty term for the jth violation of the constraint; represents the mth power of the kth initial integration node; represents the mth power of the k+1th integral nested node;

[0050] The penalty term P for violating the constraint condition is determined according to the penalty function method, specifically:

[0051] P=[P 1 …P j …P 3k+2 ],c=max{10 3 ,1 / ||r|| 2} (11)

[0052] Among them, c represents the penalty factor for violating the constraint conditions; max represents the maximum value function;

[0053] The termination conditions Set as the objective function The 2-norm after partial normalization is:

[0054]

[0055] Among them, r j Represents the jth component of the residual of the origin moment of the objective function, μ j represents the j-th origin moment, P j represents the j-th column element of the penalty term, Represents the termination condition for the iterative optimization generated by the nested integration rule.

[0056] Preferably, in step S24, regularized Gauss-Newton iterative optimization is performed on the objective function in step S23. Calculate the Newton iteration decrement η, specifically:

[0057] S241: Construct a combined vector of the integral nested node and the integral node weight as the Newton iteration vector y=(z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1 ) T ; Combining the Gauss-Newton method and regularization to solve the Newton iteration step length Δy is:

[0058]

[0059] Among them, λ represents the regularization parameter; They represent the Jacobian matrix and objective function of the Newton iteration step y respectively; argmin represents the minimum independent variable point set; represents the square of the 2-norm of the corresponding vector;

[0060] S242: The L-curve method is used to set the regularization parameter λ. The optimal regularization parameter λ is the inflection point of the L-curve. The horizontal coordinate of the L-curve is The vertical axis is When the explicit expression of the L-curve is unknown, a cubic spline interpolation is constructed through the characteristic points of the L-curve to identify the regularization parameter λ;

[0061] S243: Calculate the update step length Δy of the regularized Gauss-Newton iteration according to the regularization parameter λ:

[0062]

[0063] Among them, σ i represents the i-th singular value;

[0064] S244: Update the value of the Newton iteration vector y to y-Δy, and then calculate the updated termination condition The Newton iteration decrement η is obtained as:

[0065]

[0066] Preferably, in step S25, the output condition is determined by combining the initial integral nested node, the integral node weight and the termination condition, and the output nested integral rule is {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}, specifically:

[0067] The first judgment output condition is: whether all integral nested nodes are within the set domain;

[0068] The second judgment output condition is: whether the weights of negative integral nodes are all greater than -10 -4 , whether the proportion of negative integral node weights is less than 50%;

[0069] The third judgment output condition is: and calculate the updated termination condition Is it less than the preset iteration precision∈?

[0070] When the above three judgment output conditions are met, the nested integral rule {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}; otherwise, execute step S24 to continue iterating and record the number of iterations; when m=3k+1, if the termination condition Stable at a value greater than the preset iteration precision ∈ but the Newton iteration decrement η is less than ∈ 2 , or after 2×10 5 Termination condition after iterative optimization If it is still greater than ∈, the algebraic precision is m-1 and then transferred to S21 to continue the calculation; if it is 2×10 5 Termination condition within iterations The current Newton iteration vector y is recorded, and then the algebraic precision is increased to m+1 before going to step S21 to continue the calculation; when m≠3k+1, if the termination condition If m<3k+1, the iteration is terminated and the nested integral rule is output as {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1,…,ω 2k+1};

[0071] If the termination condition If m>3k+1, the algebraic precision is m+1 and the process goes to step S21 to continue the calculation. 5 Termination condition after iterations If m>3k+1, the iteration is terminated and the corresponding output nested integral rule after outputting m-1 is {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1};

[0072] If in 2×10 5 Termination condition after iterations If m<3k+1, the algebraic precision is m-1 and the calculation is continued in S21.

[0073] In a second aspect of the present invention, a mechanical truss structure reliability assessment system is proposed based on the aforementioned nested integral quantitative mechanical truss structure reliability assessment method. Aiming at the nonlinear relationship of the mechanical truss structure reliability response, a nested integral rule of an arbitrary probability density function is generated, and the high-order reliability response of the mechanical truss structure is accurately estimated using the minimum calculation increment. The system includes: a moment estimation unit, a nested point configuration unit, a nested optimization unit, a precision adjustment unit, a symmetry dimension reduction unit, a solution space expansion unit, and a response analysis unit.

[0074] The moment estimation unit is used to generate an initial mechanical truss structure reliability response estimation based on a moment integration method, obtain an initial integration rule based on the moment integration method, substitute the points in the integration rule as an input parameter set of the mechanical truss structure response model into the calculation, and weight each result according to the weight in the integration rule to obtain an initial model estimation of the mechanical truss structure response;

[0075] The nested point configuration unit is used to initialize the integration nodes in the nested integration rule, and guide the selection and distribution of the initial values ​​of the nested integration nodes based on the arrangement properties of the integration nodes of the nested rule and the Gaussian rule, so as to improve the overall stability and performance of the algorithm;

[0076] The nested optimization unit is used to generate a nested integration rule, construct a target optimization function according to the initial nested integration node and the origin moment result of the sample data, convert the constrained problem under the corresponding constraint conditions into an unconstrained problem based on the penalty function method, and perform iterative calculation based on the regularized Gauss-Newton method until the corresponding iteration termination condition is met;

[0077] The precision adjustment unit is used to obtain the optimal nested integration rule, and dynamically adjust the algebraic precision under the current probability density function according to the result of the nested optimization unit;

[0078] The symmetric dimension reduction unit is used to accelerate the convergence rate of the symmetric probability density function. Based on the fact that the odd-order origin moment is always 0 when the probability density function is symmetric, the iteration is restricted to the semi-axis to enhance the numerical stability and accelerate the convergence rate.

[0079] The solution space expansion unit is used to expand the solution space of the conventional integration rule. The weights in the conventional integration rule are valid only when they are all positive numbers. The unit transforms the validity of the integration rule into a boundedness problem of the corresponding operator norm, thereby expanding the weight space that can only take positive values ​​to a weight space that can partially take negative values, further ensuring the optimality of the obtained nested integration rule at the algebraic precision level;

[0080] The response analysis unit is used to generate a model estimation result based on a nested integration rule, substitute the integration nodes of the nested integration rule as an input parameter set of the mechanical truss structure response model into the calculation, and weight each result according to the weight of the nested integration rule to obtain a nested model estimation of the mechanical truss structure response, calculate the relative change rate with the initial model estimation, and perform continuous nesting or termination based on the user's accuracy requirements for the model response estimation.

[0081] Compared with the prior art, the present invention has the following beneficial effects:

[0082] (1) Based on the Gauss-Kronrod integration rule, the present invention proposes an adaptive algebraic precision strategy, which extends its application range from uniform distribution to any probability density function. In addition, a negative weight constraint optimization strategy is proposed for the value of weights. On the basis of ensuring the stability and convergence of the obtained integration rule, the restrictions on weights in the integration rule are relaxed, so that the obtained nested rule has the highest algebraic precision.

[0083] (2) In order to solve the problem of numerical errors in the solution process, the present invention introduces regularization techniques to reduce the influence of small singular values ​​and numerical errors. At the same time, for symmetric probability density functions, a semi-axis iteration method is proposed to reduce the dimension of the target optimization function, thereby further alleviating the problem of numerical instability and accelerating the solution rate of the nested integration rule of the symmetric probability density function.

[0084] (3) The present invention utilizes the developed method and program, starting from the initial nodes and weights generated by the conventional regular integration method, to dynamically obtain the nested integration rule with the highest polynomial algebraic precision under any probability density function; this rule can effectively handle the random analysis problems in the response of mechanical truss structures, thereby improving the accuracy and reliability of the analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] Figure 1 It is a flow chart of the reliability evaluation method of a mechanical truss structure based on nested integral quantification of the present invention;

[0086] Figure 2 It is a schematic diagram of the semi-axis iteration method of the present invention;

[0087] Figure 3 A schematic diagram of a mechanical truss structure in a specific embodiment of the present invention;

[0088] Figure 4 It is a diagram of finite element simulation results of a mechanical truss structure in a specific embodiment of the present invention;

[0089] Figure 5 It is a schematic diagram of the one-point continuous nesting rule of the standard normal distribution of the present invention;

[0090] Figure 6 A schematic diagram of a cantilever beam structure in a specific embodiment of the present invention;

[0091] Figure 7 is the cantilever beam deflection M in a specific embodiment of the present invention 1 Result plots of the single model evaluation corresponding to the deflection;

[0092] Figure 8 is the cantilever beam deflection M in a specific embodiment of the present invention 2 It is in M 1 The result graph obtained by nesting it on the basis of

[0093] Fig. 9 is the cantilever beam deflection M in a specific embodiment of the present invention 3 It is in M 2 The result graph obtained by nesting it on the basis of

[0094] Fig.10 It is a schematic diagram of a main beam of a gantry crane in a specific embodiment of the present invention;

[0095] Fig.11 This is a diagram showing the finite element simulation results of a gantry crane main beam in a specific embodiment of the present invention. DETAILED DESCRIPTION

[0096] The exemplary embodiments, features and aspects of the present invention will be described in detail below with reference to the accompanying drawings. The same reference numerals in the accompanying drawings represent elements with the same or similar functions. Although various aspects of the embodiments are shown in the accompanying drawings, the drawings are not necessarily drawn to scale unless otherwise specified.

[0097] The embodiment of the present invention takes a mechanical truss structure as an analysis object and provides an evaluation method for mechanical structure reliability based on nested integral quantification, such as Figure 1 As shown, the initial moment integral of the reliability parameters of the mechanical truss structure is generated; a continuous nested integral model for quantifying the reliability of the mechanical truss structure is established, and the nested integral rule is output; the numerical integral form of the reliability parameters of the mechanical truss structure is calculated according to the nested integral rule, and the reliability evaluation of the mechanical truss structure is obtained; it includes:

[0098] Step S1: Generate initial moment integral of mechanical truss structure reliability parameters.

[0099] Step S11: Set the number of integration nodes of the mechanical truss structure reliability parameter integration rule to k, calculate the first 2k-order origin moments of the mechanical truss structure random variables, and construct the Hankel matrix H of the mechanical truss structure reliability parameters k , specifically:

[0100] When the probability density function of the random variable X of the mechanical truss structure reliability parameter is p(x), the k-order origin moment μ of the mechanical truss structure reliability parameter is k for:

[0101] μ k =E[X k ]=∫ x∈Ω x k p(x)dx (3)

[0102] Among them, E, p(x), Ω represent the mathematical expectation, probability density function and setting domain of the random variable X of the reliability parameter of the mechanical truss structure respectively; X, x, k represent the random variable, random variable value and origin moment order of the reliability parameter of the mechanical truss structure respectively.

[0103] When the probability density function of the random variable of the mechanical truss structure reliability parameter is unknown, the k-order origin moment μ of the mechanical truss structure reliability parameter is k Equal to the sample moment of the reliability parameter of the mechanical truss structure

[0104]

[0105] Among them, p n Represents the number of samples of reliability parameters of mechanical truss structure; x i Represents the sample value of the reliability parameter of the i-th mechanical truss structure.

[0106] According to the k-order origin moment μ of the mechanical truss structure reliability parameter k , construct the Hankel matrix H of the reliability parameters of the mechanical truss structure k :

[0107]

[0108] Step S12: Hankel matrix H of reliability parameters of mechanical truss structure k Perform Cholesky decomposition to obtain the upper triangular matrix R, and construct the moment characteristic matrix J of the reliability parameters of the mechanical truss structure, which is:

[0109] The Cholesky decomposition results in:

[0110]

[0111] Where R represents the upper triangular matrix of the reliability parameters of the mechanical truss structure; r k+1,k+1 The k+1th row and k+1th column element of the upper triangular matrix representing the reliability parameters of the mechanical truss structure.

[0112] According to the upper triangular matrix R of the mechanical truss structure reliability parameters, the moment characteristic matrix J of the mechanical truss structure reliability parameters is obtained:

[0113]

[0114] Among them, α j The first diagonal element of the Jacobian matrix representing the reliability parameters of the mechanical truss structure, and β j The second diagonal element of the Jacobian matrix representing the reliability parameters of the mechanical truss structure, and j represents the diagonal element number of the Jacobian matrix of the reliability parameters of the mechanical truss structure, and j=1,…,k.

[0115] Step S13: Solve the eigenvalue λ of the moment characteristic matrix J of the mechanical truss structure reliability parameter j With the feature vector v j =(v j1 ,…,v jk ) T , j = 1,…,k; the integration node in the initial moment integration rule is x j =λ j , the corresponding integration node weight is

[0116] Step S2: Establish a continuous nested integration model for quantifying the reliability of mechanical truss structures and output nested integration rules.

[0117] Step S21: Initialize the algebraic precision m of the continuous nested integral model, calculate the first m-order origin moments under the probability density function of the random variable of the mechanical truss structure reliability parameter, and the algebraic precision m is initially set to 3k+1.

[0118] Step S22: nesting the initial integration nodes of the mechanical truss structure reliability and determining the nested integration rule boundaries, specifically:

[0119] Get the initial integration node x obtained in step S1 1 ,…,x k , initialize the integral nested node z 1 ,…,z k+1 and the integration node weight ω 1 ,…,ω 2k+1 ; respectively find and 2k+1 Gaussian integration nodes, and set the Gaussian integration scaling factor s to:

[0120]

[0121] Where s represents the Gaussian integral scaling factor; represents the maximum value in the set of integration nodes corresponding to the first integration rule; Represents the maximum value in the set of integration nodes corresponding to the second integration rule.

[0122] After multiplying the Gaussian integral node at point 2k+1 by the Gaussian integral scaling factor s, the points that are staggered with the initial integral nodes are selected as the initial integral nested nodes, otherwise the median of the two adjacent initial nodes is used as the initial integral nested node; the nested integral rule boundary is determined. If the set domain of the probability density function for evaluating the reliability parameters of the mechanical truss structure is bounded, then its set domain is the nested integral rule boundary; if the set domain of the probability density function for evaluating the reliability parameters of the mechanical truss structure is unbounded, then the maximum node and the minimum node in the Gaussian formula at point 2k+1 are taken as the nested integral rule boundary.

[0123] Initialize the integration node weight to ω 1 =…=ω 2k+1 =1 / (2k+1).

[0124] Step S23: Constructing an objective function to quantify the reliability of the mechanical truss structure And initialize the termination condition Specifically:

[0125] Objective function of mechanical truss structure reliability It is composed of the residual r of the origin moment of the objective function and the penalty term P for violating the constraint conditions. The residual r of the origin moment of the objective function is:

[0126]

[0127] Where w represents the integration node weight; ω i represents the weight of the i-th integration node; μ represents the origin moment of the reliability parameter of the mechanical truss structure; z jrepresents the jth integral nested node; a and b represent the upper and lower bounds of the set domain respectively; P j represents the penalty term for the jth violation of the constraint; represents the mth power of the kth initial integration node; Represents the mth power of the k+1th integral nested node.

[0128] The penalty term P for violating the constraint condition is determined according to the penalty function method, which is:

[0129] P=[P 1 …P j …P 3k+2 ],c=max{10 3 ,1 / ||r|| 2} (11)

[0130] Among them, c represents the penalty factor for violating the constraint conditions; max represents the maximum value function.

[0131] Termination Condition Set as the objective function The 2-norm after partial normalization is:

[0132]

[0133] Among them, r j Represents the jth component of the residual of the origin moment of the objective function, μ j represents the j-th origin moment, P j represents the j-th column element of the penalty term, Represents the termination condition for the iterative optimization generated by the nested integration rule.

[0134] Step S24: Perform regularized Gauss-Newton iterative optimization of the objective function in step S23 Calculate the Newton iteration decrement η, specifically:

[0135] Step S241: construct a combined vector of the integral nested node and the integral node weight as the Newton iteration vector y=(z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1 ) T ; Combining the Gauss-Newton method and regularization to solve the Newton iteration step length Δy is:

[0136]

[0137] Among them, λ represents the regularization parameter; They represent the Jacobian matrix and objective function of the Newton iteration step Δy respectively; argmin represents the minimum independent variable point set; represents the square of the 2-norm of the corresponding vector.

[0138] Step S242: Use the L-curve method to set the regularization parameter λ. The optimal regularization parameter λ is the inflection point of the L-curve. The horizontal coordinate of the L-curve is The vertical axis is When the explicit expression of the L-curve is unknown, a cubic spline interpolation is constructed through the characteristic points of the L-curve to identify the regularization parameter λ.

[0139] Step S243: Calculate the update step length Δy of the regularized Gauss-Newton iteration according to the regularization parameter:

[0140]

[0141] Among them, σ i represents the i-th singular value.

[0142] Step S244: Update the value of the Newton iteration vector y to y-Δy, and then calculate the updated termination condition The Newton iteration decrement η is obtained as:

[0143]

[0144] For the objective function Find the partial derivatives to get the Jacobian matrix with N rows and M columns Perform singular value decomposition to obtain:

[0145]

[0146] Among them, u i represents the i-th left singular column vector; v i represents the i-th right singular column vector; σ i represents the i-th singular value; N represents the number of rows of the continuous nested integral model; M represents the number of columns of the continuous nested integral model; i represents the row number of the continuous nested integral model.

[0147] In particular, when the probability density function is symmetric, a semi-axis iteration method is proposed. This method limits the iteration process to the positive semi-axis, thereby effectively reducing the dimension of the target vector and enhancing the numerical stability of the solution process. When the weight function is symmetric about the origin, ensuring the symmetry of the integration nodes can make the value of the odd-order integrand 0, so the odd-order rows and columns of the negative nodes in the node matrix X in the initial nonlinear system, as well as the weights corresponding to the negative nodes in the weight vector w and the odd-order elements in the moment vector μ can be deleted. Due to the symmetry of the distribution, whether it is the initial integral rule or the nested integral rule to be solved, there must be a node with a value of 0. The column of the node matrix X and the elements in the moment vector μ corresponding to the point 0 will be multiplied by 1 / 2. Figure 2This is a schematic diagram after the semi-axis iteration method is used in this specific embodiment, where the circle represents the integral node of the initial rule, and the cross represents the new node of the nested rule to be solved. The residual vector r at this time is:

[0148]

[0149] At this point, it is recommended to use a more sophisticated penalty term to ensure the interleaving nature of the new nodes:

[0150]

[0151] After completing all iterations, all nodes and weights of the nested integration rule can be obtained by selecting symmetrically on the negative semi-axis.

[0152] Step S25: Combine the initial integration nested nodes, integration node weights and termination conditions to set the judgment output conditions, and output the nested integration rule as {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}, specifically:

[0153] The first output judgment condition is: whether all integral nested nodes are within the set domain.

[0154] The second judgment output condition is: whether the weights of negative integral nodes are all greater than -10 -4 , whether the proportion of negative integral node weights is less than 50%.

[0155] The third judgment output condition is: and calculate the updated termination condition Is it less than the preset iteration precision ∈?

[0156] When the above three judgment output conditions are met, the nested integral rule {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}; otherwise, execute step S24 to continue iterating and record the number of iterations; when m=3k+1, if the termination condition Stable at a value greater than the preset iteration precision ∈ but the Newton iteration decrement η is less than ∈ 2 , or after 2×10 5 Termination condition after iterative optimization If it is still greater than ∈, the algebraic precision is m-1 and then transferred to S21 to continue the calculation; if it is 2×10 5 Termination condition within iterations The current Newton iteration vector y is recorded, and then the algebraic precision is increased to m+1 before going to step S21 to continue the calculation; when m≠3k+1, if the termination condition If m<3k+1, the iteration is terminated and the nested integral rule is output as {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}.

[0157] If the termination condition If m>3k+1, the algebraic precision is m+1 and the process goes to step S21 to continue the calculation. 5 Termination condition after iterations If m>3k+1, the iteration is terminated and the corresponding output nested integral rule after outputting m-1 is {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}.

[0158] If in 2×10 5 Termination condition after iterations If m<3k+1, the algebraic precision is m-1 and the calculation is continued in S21.

[0159] Step S3: According to the nested integral rule {x 1 ,…,x k ,z 1 ,…,z k+1 ,ω 1 ,…,ω 2k+1}, calculate the numerical integral form of the mechanical truss structure reliability parameters, and obtain the mechanical truss structure reliability evaluation results

[0160]

[0161] Among them, ψ(x i ) represents the numerical integration form of the integration node; ψ(z j ) represents the numerical integration form of the integral nested node; ω represents the integration node weight.

[0162] In a specific embodiment, for a mechanical truss structure problem, such as Figure 3 The truss structure with 8 random variables is shown in Figure 1. The structure consists of 7 horizontal bars and 8 diagonal bars, where the cross-sectional area and Young's modulus of the horizontal bars are A and B respectively. 1 ,E1 There are four vertical forces on the upper horizontal rod nodes: F 1 ,F 2 ,F 3 ,F 4 , the cross-sectional area and Young's modulus of the inclined rod are A 2 ,E 2 Each random variable is independent of each other, and the specific parameters of the random variables are shown in Table 1.

[0163]

[0164]

[0165] Table 1

[0166] Take the mean value of material parameters for finite element simulation, the results are as follows Figure 4 As shown in the figure, the maximum displacement occurs at the midpoint of the truss bottom structure, that is, Figure 2 Therefore, the displacement at point D is taken as the maximum deformation Δ of the structure, and the threshold is set to 0.018m. If the result exceeds this value, it is considered to be a failure. Figure 5 The continuous nested integration rule in the problem is used to obtain the model input parameter set in this problem. The mean and variance results of the maximum deformation of the structure calculated by the finite element method are shown in Table 2. 1 represents the result of a single model evaluation using the 1-point moment integration method, M 2 Indicates that in M 1 The result obtained by nesting it on the basis of M 3 Indicates that in M 2 The result obtained by nesting it on the basis of M 4 Indicates that in M 3 The results are obtained by nesting them on the basis of . At this time, the relative change rates of the mean and variance of the continuous nesting results are less than 0.1%, and the results are considered to have converged. 4 This is the final result, M 4 The corresponding failure probability is 0.8446%. The sample size is 10 6 The Monte Carlo method is used as a comparison, and the corresponding failure probability is calculated to be 0.8400%. It can be seen that the accuracy of the proposed method reaches the ten thousand level, and its calculation amount is only 0.3375% of the Monte Carlo method. The speed is greatly improved while ensuring the accuracy.

[0167]

[0168] Table 2

[0169] In another specific embodiment, the mechanical cantilever beam structure is analyzed, such as Figure 6The width w and height h of the cantilever beam are both 4, the length L is 150, and the elastic modulus E is 2.9×10 6 , the horizontal load X and the lateral load Y are independent random variables, subject to N(500,100) and N(1000,400) respectively. For reliability analysis, the failure criterion of the beam is defined as the maximum deflection Exceeding the specified value The calculation formula for the deflection of the beam is:

[0170]

[0171] in, represents the maximum deflection of the failure standard of the beam; L represents the length of the beam; E represents the elastic modulus of the beam; w represents the width of the beam; h represents the height of the beam; X represents the horizontal load; and Y represents the lateral load.

[0172] Starting from the one-point moment integral rule of the standard normal distribution N(0,1), we start nesting and nest three times in a row to get the corresponding nesting rule as follows: Figure 4 As shown. For the distribution N(μ,σ 2 ) can use the following variable substitution formula:

[0173] t=μ+τσ (20)

[0174] Among them, τ represents the integration node of N(0,1); t represents the corresponding N(μ,σ 2 )’s integration node.

[0175] According to the variable substitution formula, the input parameter set corresponding to X and Y under the corresponding nested integration rule is obtained, and the estimated results of the mean and variance of the deflection of the beam are calculated, as shown in Table 3.

[0176]

[0177] Table 3

[0178] in, Figure 7 is the cantilever beam deflection M in this specific embodiment 1 Result plot of a single model evaluation of the initial integration rule corresponding to the deflection, M 1 represents the results of a single model evaluation using the 1-point moment integration method; Figure 8 is the cantilever beam deflection M in this specific embodiment 2 The corresponding 1 The result graph obtained by nesting it once on the basis of M 2 Indicates that in M 1 The result obtained by nesting it on the basis of Fig. 9 is the cantilever beam deflection M in this specific embodiment 3 The corresponding 2On the basis of M 1 The result graph obtained by nesting twice, M 3 Indicates that in M 2 The results are obtained by nesting them on the basis of . At this time, the relative change rates of the mean and variance of the continuous nesting results are less than 0.1%, and the results are considered to have converged. 3 This is the final result. 3 The reliability of the beam obtained by the results is 97.6848%. For comparison, the sample size is 10 6 The result obtained by the Monte Carlo method is 97.6830%. It can be seen that the accuracy of the proposed method reaches the ten thousand level, and its calculation amount is only It is also proved that the proposed method can greatly improve the calculation speed while ensuring accuracy.

[0179] In the third specific embodiment, the reliability of the main beam of the gantry crane is analyzed. The schematic diagram of the main beam of the gantry crane is as follows: Fig.10 As shown, the elastic modulus of the main beam structure is 7×10 10 Pa, Poisson's ratio is 0.33, and density is 2700 kg / m 3 , the load of a certain working condition obeys the normal distribution, and the mean is 2×10 5 N, standard deviation is 4×10 4 N, use ansys to simplify the modeling, take the load mean value for finite element simulation, and get the simulation results as follows Fig.11 As shown in the figure, the maximum displacement occurs in the middle of the top of the main beam, so the displacement of the middle of the top of the main beam is defined as the maximum deformation Δ of the main beam, and the threshold is 2.8mm. Figure 5 The continuous nested integration rule in the problem is used to obtain the model input parameter set in this problem. The mean and variance results of the maximum deformation of the structure calculated by the finite element method are shown in Table 4. 1 represents the result of a single model evaluation using the 1-point moment integration method, M 2 Indicates that in M 1 The result obtained by nesting it on the basis of M 3 Indicates that in M 2 The results are obtained by nesting them on the basis of . At this time, the relative change rates of the mean and variance of the continuous nesting results are less than 0.1%, and the results are considered to have converged. 3 This is the final result. At this time, the reliability of the main beam of the crane under this working condition is 99.7968%. For comparison, the sample size is 10 6 The result obtained by the Monte Carlo method is 99.7942%. It can be seen that the accuracy of the proposed method reaches the ten thousand level, and its calculation amount is only It is also proved that the proposed method can greatly improve the calculation speed while ensuring accuracy.

[0180]

[0181] Table 4

[0182] The second aspect of the present invention proposes a mechanical structure reliability assessment system based on a nested integral quantitative mechanical structure reliability assessment method. Aiming at the nonlinear relationship of the mechanical structure reliability response, a nested integral rule of an arbitrary probability density function is generated, and the high-order reliability response of the mechanical structure is accurately estimated using the minimum calculation increment. The system includes: a moment estimation unit, a nested point configuration unit, a nested optimization unit, a precision adjustment unit, a symmetry dimension reduction unit, a solution space expansion unit and a response analysis unit.

[0183] The moment estimation unit is used to generate an initial reliability response estimate based on the moment integration method, obtain an initial integration rule based on the moment integration method, substitute the points in the integration rule as the input parameter set of the mechanical structure response model into the calculation, and weight each result according to the weight in the integration rule to obtain an initial model estimate of the mechanical structure response.

[0184] The nested point configuration unit is used to initialize the integration nodes in the nested integration rule. Based on the arrangement properties of the integration nodes of the nested rule and the Gaussian rule, it guides the selection and distribution of the initial values ​​of the nested integration nodes to improve the overall stability and performance of the algorithm.

[0185] The nested optimization unit is used to generate nested integration rules, construct the target optimization function according to the initial nested integration nodes and the origin moment results of the sample data, convert the constrained problem under the corresponding constraint conditions into an unconstrained problem based on the penalty function method, and perform iterative calculations based on the regularized Gauss-Newton method until the corresponding iteration termination conditions are met.

[0186] The precision adjustment unit is used to obtain the optimal nested integration rule and dynamically adjust the algebraic precision under the current probability density function according to the result of the nested optimization unit.

[0187] The symmetric dimension reduction unit is used to speed up the convergence rate of the symmetric probability density function. Based on the fact that the odd-order origin moment is always 0 when the probability density function is symmetric, the iteration is restricted to the semi-axis, which enhances the numerical stability and speeds up the convergence rate.

[0188] The solution space expansion unit is used to expand the solution space of conventional integration rules. The weights in conventional integration rules are valid only when they are all positive numbers. This unit transforms the validity of the integration rule into a boundedness problem of the corresponding operator norm, thereby expanding the weight space that can only take positive values ​​to one that can partially take negative values, further ensuring the optimality of the resulting nested integration rule at the algebraic precision level.

[0189] The response analysis unit is used to generate model estimation results based on the nested integration rule, substitute the integration nodes of the nested integration rule as the input parameter set of the mechanical structure response model into the calculation, and weight each result according to the weight of the nested integration rule to obtain the nested model estimation of the mechanical structure response, calculate the relative change rate with the initial model estimation, and perform continuous nesting or termination based on the user's requirements for the accuracy of the model response estimation.

[0190] The beneficial effects of the present invention are as follows: the present invention provides an evaluation method for the reliability of a mechanical structure based on nested integration quantification. According to the initial nodes and weights generated by the conventional regular integration method, the value of the unknown variable is obtained by solving the minimization problem. In order to ensure that the numerical solution can fall within the definition domain, an adaptive algebraic precision strategy is proposed, and regularization techniques are used to reduce the influence of small singular values ​​and numerical errors. In addition, a negative weight constraint optimization strategy is proposed to alleviate and quantify the fluctuation and convergence problems caused by negative weights. At the same time, the present invention utilizes the developed method and program to dynamically obtain the nested integration rule with the highest polynomial algebraic precision under any probability density function, and the rule can effectively handle the random analysis problems in the response of the mechanical structure; the embodiments of the present invention perform computational analysis on trusses and cantilever beams to prove that the method can meet the actual engineering needs and has a good use effect.

[0191] The embodiments described above are only descriptions of the preferred implementation modes of the present invention, and are not intended to limit the scope of the present invention. Without departing from the design spirit of the present invention, various modifications and improvements made to the technical solutions of the present invention by ordinary technicians in this field should all fall within the protection scope determined by the claims of the present invention.

Claims

1. A reliability assessment method for mechanical truss structures based on nested integral quantification, characterized in that: It includes: S1: Generate initial moment integral of mechanical truss structure reliability parameters; S11: Set the number of integration nodes of the mechanical truss structure reliability parameter integration rule to k, calculate the first 2k-order origin moments of the mechanical truss structure random variables, and construct the Hankel matrix H of the mechanical truss structure reliability parameters k ; S12: Hankel matrix H for reliability parameters of mechanical truss structures k Perform Cholesky decomposition to obtain the upper triangular matrix R and construct the moment characteristic matrix J of the reliability parameters of the mechanical truss structure; S13: Solve the eigenvalue λ of the moment characteristic matrix J of the reliability parameters of the mechanical truss structure j With the feature vector v j =v j1 ,…,v jk T , j = 1,…,k; the integration node in the initial moment integration rule is x j =λ j , the corresponding integration node weight is S2: Establish a continuous nested integral model to quantify the reliability of mechanical truss structures and output nested integral rules; S21: Initialize the algebraic precision m of the continuous nested integral model, calculate the first m-order origin moments under the probability density function of the random variable of the reliability parameter of the mechanical truss structure, and the algebraic precision m is initially set to 3k+1; S22: Initial integration nodes of nested mechanical truss structure reliability and determine the nested integration rule boundary; Initialize the integration node weights to ω1=…=ω 2k+1 =1 / 2k+1; S23: Constructing an objective function to quantify the reliability of mechanical truss structures And initialize the termination condition S24: Perform regularized Gauss-Newton iterative optimization on the objective function in step S23 Calculate the Newton iteration reduction η; for the objective function Find the partial derivatives to get the Jacobian matrix with N rows and M columns Perform singular value decomposition to obtain: Among them, u i represents the i-th left singular column vector; v i represents the i-th right singular column vector; σ i represents the i-th singular value; N represents the number of rows of the continuous nested integral model; M represents the number of columns of the continuous nested integral model; i represents the number of rows of the continuous nested integral model; S25: Combine the initial integration nested nodes, integration node weights and termination conditions to set the judgment output conditions, and output the nested integration rule as {x1,…,x k ,z1,…,z k+1 ,ω1,…,ω 2k+1 }; S3: According to the nested integration rule {x1,…,x k ,z1,…,z k+1 ,ω1,…,ω 2k+1 }, calculate the numerical integral form of the mechanical truss structure reliability parameters, and obtain the mechanical truss structure reliability evaluation results Among them, ψx i Indicates the numerical integration form of the integration node; ψz j Represents the numerical integration form of the integral nested node; ω represents the integration node weight.

2. The reliability assessment method of a mechanical truss structure based on nested integral quantification according to claim 1 is characterized in that: In step S11, the Hankel matrix H of the reliability parameters of the mechanical truss structure is constructed. k , specifically: When the probability density function of the random variable X of the mechanical truss structure reliability parameter is p(x), the k-order origin moment μ of the mechanical truss structure reliability parameter is k for: μ k =EX k =∫ x∈Ω x k pxdx(3) Where, E, p(x), Ω represent the mathematical expectation, probability density function and setting domain of the random variable X of the reliability parameter of the mechanical truss structure respectively; X, x, k represent the random variable, random variable value and origin moment order of the reliability parameter of the mechanical truss structure respectively; When the probability density function of the random variable of the mechanical truss structure reliability parameter is unknown, the k-order origin moment μ of the mechanical truss structure reliability parameter is k Equal to the sample moment of the reliability parameter of the mechanical truss structure Among them, p n Represents the number of samples of reliability parameters of mechanical truss structure; x i represents the sample value of the reliability parameter of the i-th mechanical truss structure; According to the k-order origin moment μ of the mechanical truss structure reliability parameter k , construct the Hankel matrix H of the reliability parameters of the mechanical truss structure k :

3. The reliability assessment method of mechanical truss structure based on nested integral quantification according to claim 1, characterized in that: In step S12, the Hankel matrix H of the reliability parameters of the mechanical truss structure is k Perform Cholesky decomposition to obtain the upper triangular matrix R, and construct the moment characteristic matrix J of the reliability parameters of the mechanical truss structure, which is: The Cholesky decomposition result is: Where R represents the upper triangular matrix of the reliability parameters of the mechanical truss structure; r k+1,k+1 The k+1th row and k+1th column element of the upper triangular matrix representing the mechanical structure reliability parameters; According to the upper triangular matrix R of the mechanical truss structure reliability parameters, the moment characteristic matrix J of the mechanical truss structure reliability parameters is obtained: Among them, α j The first diagonal element of the Jacobian matrix representing the reliability parameters of the mechanical truss structure, and β j The second diagonal element of the Jacobian matrix representing the reliability parameters of the mechanical truss structure, and j represents the diagonal element number of the Jacobian matrix of the reliability parameters of the mechanical truss structure, and j=1,…,k.

4. The method for evaluating the reliability of a mechanical truss structure based on nested integral quantification according to claim 1, characterized in that: In step S22, the initial integration node of the mechanical truss structure reliability is nested, and the nested integration rule boundary is determined, specifically: Get the initial integration nodes x1,…,x obtained in step S1 k , initialize the integration nested nodes z1,…,z k+1 and the integration node weights ω1,…,ω 2k+1 ; respectively find and 2k+1 Gaussian integration nodes, and set the Gaussian integration scaling factor s to: Where s represents the Gaussian integral scaling factor; represents the maximum value in the set of integration nodes corresponding to the first integration rule; represents the maximum value in the set of integration nodes corresponding to the second integration rule; After multiplying the Gaussian integral node at point 2k+1 by the Gaussian integral scaling factor s, the points that are staggered with the initial integral nodes are selected as the initial integral nested nodes, otherwise the median of the two adjacent initial nodes is used as the initial integral nested node; the nested integral rule boundary is determined. If the set domain of the probability density function for evaluating the reliability parameters of the mechanical truss structure is bounded, then its set domain is the nested integral rule boundary; if the set domain of the probability density function for evaluating the reliability parameters of the mechanical truss structure is unbounded, then the maximum node and the minimum node in the Gaussian formula at point 2k+1 are taken as the nested integral rule boundary.

5. The reliability assessment method of mechanical truss structure based on nested integral quantification according to claim 1, characterized in that: In step S23, an objective function for quantifying the reliability of the mechanical truss structure is constructed. And initialize the termination condition Specifically: The objective function of the mechanical truss structure reliability It is composed of the residual r of the origin moment of the objective function and the penalty term P for violating the constraint conditions; The residual r of the origin moment of the objective function is: Where w represents the integration node weight; ω i represents the weight of the i-th integration node; μ represents the origin moment of the reliability parameter of the mechanical truss structure; z j represents the jth integral nested node; a and b represent the upper and lower bounds of the set domain respectively; P j represents the penalty term for the jth violation of the constraint; represents the mth power of the kth initial integration node; represents the mth power of the k+1th integral nested node; The penalty term P for violating the constraint condition is determined according to the penalty function method, specifically: P=[P1…P j …P 3k+2 ],c=max{10 3 ,1 / r2}(11) Among them, c represents the penalty factor for violating the constraint conditions; max represents the maximum value function; The termination conditions Set as the objective function The 2-norm after partial normalization is: Among them, r j Represents the jth component of the residual of the origin moment of the objective function, μ j represents the j-th origin moment, P j represents the j-th column element of the penalty term, Represents the termination condition for the iterative optimization generated by the nested integration rule.

6. The reliability assessment method of mechanical truss structure based on nested integral quantification according to claim 1, characterized in that: In step S24, regularized Gauss-Newton iteration is performed to optimize the objective function in step S23. Calculate the Newton iteration decrement η, specifically: S241: Construct the combined vector of the integral nested node and the integral node weight as the Newton iteration vector y=z1,…,z k+1 ,ω1,…,ω 2k+1 T ; Combining the Gauss-Newton method and regularization to solve the Newton iteration step length Δy is: Among them, λ represents the regularization parameter; They represent the Jacobian matrix and objective function of the Newton iteration step y respectively; argmin represents the minimum independent variable point set; represents the square of the 2-norm of the corresponding vector; S242: The L-curve method is used to set the regularization parameter λ. The optimal regularization parameter λ is the inflection point of the L-curve. The horizontal coordinate of the L-curve is The vertical axis is When the explicit expression of the L-curve is unknown, a cubic spline interpolation is constructed through the characteristic points of the L-curve to identify the regularization parameter λ; S243: Calculate the update step length Δy of the regularized Gauss-Newton iteration according to the regularization parameter λ: Among them, σ i represents the i-th singular value; S244: Update the value of the Newton iteration vector y to y-Δy, and then calculate the updated termination condition The Newton iteration decrement η is obtained as:

7. The reliability assessment method of mechanical truss structure based on nested integral quantification according to claim 1, characterized in that: In step S25, the output condition is determined by combining the initial integration nested node, the integration node weight and the termination condition, and the output nested integration rule is {x1,…,x k ,z1,…,z k+1 ,ω1,…,ω 2k+1 }, specifically: The first judgment output condition is: whether all integral nested nodes are within the set domain; The second judgment output condition is: whether the weights of negative integral nodes are all greater than -10 -4 , whether the proportion of negative integral node weights is less than 50%; The third judgment output condition is: and calculate the updated termination condition Is it less than the preset iteration precision∈? When the above three judgment output conditions are met, the nested integral rule {x1,…,x k ,z1,…,z k+1 ,ω1,…,ω 2k+1 }; Otherwise, execute step S24 to continue iterating and record the number of iterations; when m=3k+1, if the termination condition Stable at a value greater than the preset iteration precision ∈ but the Newton iteration decrement η is less than ∈ 2 , or after 2×10 5 Termination condition after iterative optimization If it is still greater than ∈, the algebraic precision is m-1 and then transferred to S21 to continue the calculation; if it is 2×10 5 Termination condition within iterations The current Newton iteration vector y is recorded, and then the algebraic precision is increased to m+1 before going to step S21 to continue the calculation; when m≠3k+1, if the termination condition If m<3k+1, the iteration is terminated and the nested integration rule {x1,…,x k ,z1,…,z k+1 ,ω1,…,ω 2k+1 }; If the termination condition If m>3k+1, the algebraic precision is m+1 and the process goes to step S21 to continue the calculation. 5 Termination condition after iterations If m>3k+1, the iteration is terminated and the corresponding output nested integral rule after output m-1 is {x1,…,x k ,z1,…,z k+1 ,ω1,…,ω 2k+1 }; If in 2×10 5 Termination condition after iterations If m<3k+1, the algebraic precision is m-1 and the calculation is continued in S21.

8. A mechanical truss structure reliability assessment system according to any one of claims 1 to 7, characterized in that: Aiming at the nonlinear relationship of reliability response of mechanical truss structure, a nested integration rule of arbitrary probability density function is generated, and the high-order reliability response of mechanical truss structure is accurately estimated by using the minimum calculation increment, which includes: moment estimation unit, nested point configuration unit, nested optimization unit, precision adjustment unit, symmetry dimension reduction unit, solution space expansion unit and response analysis unit; The moment estimation unit is used to generate an initial mechanical truss structure reliability response estimation based on a moment integration method, obtain an initial integration rule based on the moment integration method, substitute the points in the integration rule as an input parameter set of the mechanical truss structure response model into the calculation, and weight each result according to the weight in the integration rule to obtain an initial model estimation of the mechanical truss structure response; The nested point configuration unit is used to initialize the integration nodes in the nested integration rule, and guide the selection and distribution of the initial values ​​of the nested integration nodes based on the arrangement properties of the integration nodes of the nested rule and the Gaussian rule, so as to improve the overall stability and performance of the algorithm; The nested optimization unit is used to generate a nested integration rule, construct a target optimization function according to the initial nested integration node and the origin moment result of the sample data, convert the constrained problem under the corresponding constraint conditions into an unconstrained problem based on the penalty function method, and perform iterative calculation based on the regularized Gauss-Newton method until the corresponding iteration termination condition is met; The precision adjustment unit is used to obtain the optimal nested integration rule, and dynamically adjust the algebraic precision under the current probability density function according to the result of the nested optimization unit; The symmetric dimension reduction unit is used to accelerate the convergence rate of the symmetric probability density function. Based on the fact that the odd-order origin moment is always 0 when the probability density function is symmetric, the iteration is restricted to the semi-axis to enhance the numerical stability and accelerate the convergence rate. The solution space expansion unit is used to expand the solution space of the conventional integration rule. The weights in the conventional integration rule are valid only when they are all positive numbers. The unit transforms the validity of the integration rule into a boundedness problem of the corresponding operator norm, thereby expanding the weight space that can only take positive values ​​to a weight space that can partially take negative values, further ensuring the optimality of the obtained nested integration rule at the algebraic precision level; The response analysis unit is used to generate a model estimation result based on a nested integration rule, substitute the integration nodes of the nested integration rule as an input parameter set of the mechanical truss structure response model into the calculation, and weight each result according to the weight of the nested integration rule to obtain a nested model estimation of the mechanical truss structure response, calculate the relative change rate with the initial model estimation, and perform continuous nesting or termination based on the user's accuracy requirements for the model response estimation.

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