An Optimization Method for Modeling the Uncertainty of Frontal Collision of Electric Buses

Abstract

The present invention relates to the technical field of new energy vehicle collisions, and discloses an optimization method for modeling the uncertainty of a frontal collision of an electric bus. First, a finite element model of the frontal collision of the electric bus is established to obtain design variables. Based on the polynomial chaos expansion method, the relationship between the design variables and the responses is established. Combining the Sobol’ index method for sensitivity analysis, the design variables with high sensitivity are obtained. An uncertainty optimization mathematical model for the frontal collision of the electric bus is established, and it is solved by combining a multi-objective optimization algorithm. This method optimizes the frontal collision cross-section with a 100% overlap rate considering uncertainty for the front end of a certain electric bus. Taking the cross-sectional dimensions of the tube beam components as the optimization variables and the collision intrusion speed as the optimization objective, the uncertainty analysis and optimization design research of the electric bus are carried out by combining the polynomial chaos expansion method and the maximum entropy method, reducing the influence of uncertainty factors on the optimization results, and ensuring the robustness and reliability of the design scheme.

Description

Technical Field

[0001] The present invention relates to the technical field of new energy vehicle collisions, and specifically to an optimization method for modeling the uncertainty of a frontal collision of an electric bus. Background Art

[0002] The collision safety and crashworthiness performance, as an important part of investigating the body safety of electric vehicles, have become an important research topic. Among them, the electric bus, as an important tool for public transportation and urban commuting, has the characteristics of high load requirements and a much larger mass and volume of the power system than household new energy vehicles. The stability and crashworthiness of its power system after being collided are gradually attracting more attention from society. In the traditional vehicle collision design and optimization process, some parameters are often assumed to be certain. However, in the actual manufacturing and experimental processes, due to the influence of uncertain factors such as manufacturing processes and material properties, some certain parameters often become uncertain. These uncertainties may cause changes or fluctuations in the final crashworthiness optimization results, or even lead to system performance failure. Due to the complex structure of vehicle body components and the large fitting difficulty, the crashworthiness optimization design of electric buses mostly focuses on deterministic optimization, and there is less uncertainty optimization.

[0003] At present, the uncertainty optimization design method for electric buses is not yet mature. First of all, designers lack in-depth analysis of the characteristics of uncertain factors, classify all uncertain factors as random uncertainties, and use the probability method for processing, without preliminary classification according to the characteristics of uncertain factors. Secondly, for the uncertainty quantification, analysis, and transmission under random uncertainties, most designers still use the traditional time-consuming Monte Carlo method, which is unrealistic for the uncertainty optimization design of electric buses that requires a large amount of simulation calculations and uncertainty analysis. In addition, almost all current research on the uncertainty optimization design of electric buses focuses on random uncertainty optimization, ignoring the modeling and subsequent optimization of epistemic uncertainties caused by designers' insufficient knowledge, lack of data, and incomplete information. It has its limitations. At the same time, the current uncertainty optimization of electric buses usually adopts a single-source uncertainty analysis and transmission method, without considering the combined quantification of coexisting random and epistemic uncertainties. Therefore, how to perform uncertainty analysis on two different forms of uncertainties within a unified framework after modeling random and epistemic uncertainties using different methods is a major difficulty.

[0004] In summary, the deterministic and traditional uncertainty optimization design methods cannot meet the current design requirements. Therefore, comprehensively considering uncertain factors and facing the uncertainty modeling of complex engineering optimization problems during vehicle crashworthiness research are the key issues that urgently need to be solved for the uncertainty optimization of electric buses. Summary of the Invention

[0005] (1) Technical problem to be solved

[0006] In view of the deficiencies of the prior art, the present invention provides an optimization method for modeling the uncertainty of a frontal collision of an electric bus, which has the advantages of robustness and reliability of the design scheme and solves the above technical problems.

[0007] (2) Technical solution

[0008] To achieve the above object, the present invention provides the following technical solution: an optimization method for modeling the uncertainty of a frontal collision of an electric bus, comprising the following steps:

[0009] S1. Establish a finite element model of the frontal collision of the electric bus to obtain design variables;

[0010] S2. Establish the relationship between the design variables and the response based on the polynomial chaos expansion method, specifically including the following steps:

[0011] S2.1 The uncertainty influence of the design variables is represented by a true random variable with a probability density function, and the specific expression is as follows:

[0012] S i = σξ i + μ

[0013] where S i represents the i-th uncertainty design variable, σ represents the variance of the design variable, ξ i represents the i-th design variable, and μ represents the mean of the design variable;

[0014] S2.2. The polynomial chaos expansion of any output response is as follows:

[0015]

[0016] where F(ξ) represents the output response function, represents the polynomial chaos expansion of the output response function, ξ represents the design variable, α d represents the polynomial chaos expansion term coefficient, Ψ d (ξ) represents the random part of the d-th order of the polynomial chaos expansion, represents the sum of M data, M represents the expansion order, and the specific expression is as follows:

[0017]

[0018] where n is the order of the random variable, p is the number of independent input variables, p! represents the factorial of p, n! represents the factorial of n, and (p + n)! represents the factorial of p + n;

[0019] S2.3. Solve the polynomial chaos expansion obtained in step S2.2 based on the point collocation regression method. The specific expression is as follows:

[0020]

[0021] where n is the order of the random variable; F(ξ i ) is the output response function; Ψ d (ξ i ) is the random part of the d-th order of the polynomial chaos expansion; α i represents the coefficient of the polynomial chaos expansion term, R represents the approximation of the expansion term coefficient vector, ξ i represents the i-th design variable. At the same time, set the matrix Γ, and Γ T Γ is the Fisher matrix, Γ T is the transpose of the matrix Γ;

[0022] Combining the two expressions of R and the matrix Γ can obtain the following formula:

[0023]

[0024] where α0, α1, α2, …, α M respectively represent M expansion term coefficient vectors, and finally obtain the following expression:

[0025] R = (Γ T Γ) -1 (Γ T F(ξ i )

[0026] where R is the approximation of the expansion term coefficient vector, F(ξ i ) is the output response function, M is the expansion order, and ξ i is the i-th design variable.

[0027] The relationship between the design variables and the three intrusion points in the frontal collision, the vehicle mass, and the frontal collision energy absorption can be fitted through the expansion form of R with the data of the three intrusion points in the frontal collision, the vehicle mass, and the frontal collision energy absorption.

[0028] S2.4. Obtain the expansion coefficients by the Latin hypercube sampling method to get the mean and standard deviation of the crashworthiness performance in the output response;

[0029] S3. According to the relationship R established in step S2, perform sensitivity analysis by the Sobol’ index method to obtain the design variables with high sensitivity, express the design variables with high sensitivity as uncertain variables, and then obtain the mathematical model of the target, which specifically includes the following steps:

[0030] S3.1. The specific expression of the total variance of the Sobol’ index method is as follows:

[0031]

[0032] Among them, n represents a natural number, and A S represents the total variance of the output response, and A i represents the influence of the i-th design variable on the overall output variance of the objective function model R, and A 12…n represents the influence of the interaction among a total of n design variables on the objective, and A ij represents the interaction between the i-th and j-th design variables, and A i and A ij The expressions of are as follows:

[0033]

[0034] Among them, is the partial variance of the i-th single variable, F is the output response function, represents the variance, represents the mean of the conditional variance of the F function within the variable range of S i , represents that under the condition of S j , the mean of the conditional variance of the F function within the variable range of the variable S i , and A i represents the influence of the i-th input parameter on the overall output variance of the objective function model R, and A j represents the influence of the j-th input parameter on the overall output variance of the model R.

[0035] S3.2. The Sobol’ index based on variance decomposition is specifically expressed as follows:

[0036]

[0037] Among them, L i is the contribution degree of the i-th design variable to the objective function, and A i represents the influence of the i-th design variable on the overall output variance of the model R, and A(F) is the total variance of the objective function. By calculating the remaining terms in the same way of dividing the partial variance by the total variance, the high-order interaction Sobol’ index can be obtained as follows:

[0038]

[0039] Among them, L i represents the contribution degree of the single variable to the objective function, and L ij represents the interactive contribution degree of the i-th and j-th variables to the objective function, and L ijkIndicates the interactive contribution degree of the three design variables ijk to the objective function, L 12…n Indicates the interactive contribution degree of multiple variables to the objective function, L i To judge the sensitivity of the i-th design variable. L i The larger it is, the higher the sensitivity of the i-th design variable;

[0040] S3.3. The Sobol' index based on the polynomial chaos expansion method can be expressed as:

[0041]

[0042] Among them, A i Indicates the influence of the i-th input parameter on the overall output variance of the model R, α k Is the coefficient of the polynomial chaos expansion term, Ψ k Is the random part of the polynomial chaos expansion in step S3.2 L i Of, E(Ψ k 2 ) represents the mean value of Ψ k 2 Of, A PCE Is the polynomial chaos expansion of the total variance of the objective function, Indicates the summation for k belonging to different target values, Respectively represent different target values;

[0043] S4. Based on the expression R established in step S2, solve the failure probability of the constraint based on the maximum entropy method, so as to obtain the mathematical model of the constraint. The specific steps are as follows:

[0044] S4.1. Through statistical analysis, the expressions of the first l-order origin moments of the crashworthiness constraint g(S c ) are as follows:

[0045]

[0046] Among them, Z g,l Represents the first l-order origin moments of the constraint function g(S c ), E(g l ) represents the mean value of g l , ω c Represents the weight coefficient, S c Represents the uncertain design variable, l represents the order of the origin moment, Represents the summation of r internal data to obtain the first l-order origin moments of the constraint function g(S c ), and then combined with the maximum entropy method for the overall reliability analysis of the crashworthiness. The entropy can be defined as:

[0047]

[0048] Among them, H(g(S c )) represents the entropy of the constraint function g(S c ), C represents a constant not less than 0, g(S c ) represents the constraint function, S c represents the uncertainty design variable, ln* represents the logarithmic function with the natural logarithm e as the base, represents the integration of the uncertainty design variable S c from -∞ to +∞;

[0049] S4.2. According to the maximum entropy principle, with the maximum entropy as the objective function and the origin moment as the constraint condition, the optimization model is established as follows:

[0050]

[0051] Among them, s.t. represents the constraint, H(g(S c )) represents the entropy of the constraint function g(S c ), C represents a constant not less than 0, g(S c ) represents the constraint function, S c represents the uncertainty design variable, l is the order of the origin moment, S c l represents the l-th origin moment of the random variable S c , Z g,l represents finding the first l origin moments of the constraint function g(S c ), maxH(g(S c )) represents taking the maximum value of H(g(S c ), ln* represents the logarithmic function with the natural logarithm e as the base, represents the integration of the uncertainty design variable S c from -∞ to +∞;

[0052] S4.3. Based on the Lagrange method, the constrained optimization problem in step S4.2 is transformed into an unconstrained optimization problem and solved using the Newton method. The specific expression is as follows:

[0053]

[0054] Among them, L represents the Lagrange function, C represents a constant not less than 0, ln* represents the logarithmic function with the natural logarithm e as the base, l is the order of the origin moment, λ l represents the undetermined constant corresponding to the l-th origin moment of the uncertainty design variable S c , represents the integration of the uncertainty design variable S c from -∞ to +∞, m represents a positive integer, and at the extreme point, we can obtain:

[0055]

[0056] Among them, The function takes the partial derivative of the constraint function g(S c ), C represents a constant not less than 0, and λ l represents a constant to be determined, and S c l represents the l-th order origin moment of the uncertain design variable;

[0057] S4.4. Establish the relationship between the assumed variables and the constants to be determined. The specific expression is as follows:

[0058]

[0059] Substitute the relationship formula between the established assumed variables and the constants to be determined into the in step S4.3. By moving the terms of , the following formula can be further obtained:

[0060]

[0061] Among them, a0, a1, a2, …, a m respectively represent m assumed variables, λ0, λ1, λ2, …, λ m respectively represent m constants to be determined, c represents a constant not less than 0, a l represents the assumed variable corresponding to the l-th order origin moment of the random variable z, z l represents the l-th order origin moment of the random variable z, and exp(*) represents the exponential function with the natural logarithm e as the base;

[0062] S4.5. Substitute the formula obtained from g(S c ) in step S4.4 into Z g,l in step S4.1, and a nonlinear equation system about a l can be obtained. Using the 4th order origin moment to solve, the specific nonlinear equation system is as follows:

[0063]

[0064] Among them, S c represents the uncertain design variable, a l represents the assumed variable, and l represents the order of the origin moment;

[0065] S4.6. Transform the nonlinear equation in step S4.5 into an unconstrained optimization problem for solution. The specific expression is as follows:

[0066]

[0067] Among them, min represents the minimum value function, represents the sum of the internal 4 data, represents the maximum value of the first l-order origin moment of the constraint function g(S c ), and Z g,l represents finding the value of the first l-order origin moment of the constraint function g(S c ), and then can solve for a0, a1, a2, …, a in step S4.4 m and the Lagrange function in step S4.3;

[0068] S4.7. Standardize the output crashworthiness G, and the overall failure probability can be obtained from the probability density integral:

[0069]

[0070] Among them, P f represents the overall failure probability of the crashworthiness, and Pr(G≤0) represents the probability of G≤0, represents the probability of, Q represents the standardized result of the crashworthiness G, μ G represents the mean value of the crashworthiness, and σ G represents the standard deviation of the crashworthiness, and exp(*) represents the exponential function with the natural logarithm e as the base;

[0071] S5. Based on the uncertainty optimization mathematical model established in steps S3 and S4, and combined with the multi-objective optimization algorithm for solution, the specific steps are as follows:

[0072] S5.1. Establish the uncertainty optimization mathematical model of the electric bus;

[0073] S5.2. Solve the uncertainty optimization mathematical model established in step S5.1 based on the multi-objective optimization algorithm to obtain the result of the improvement of the crashworthiness and its robustness result.

[0074] As a preferred technical solution of the present invention, the specific expression of Γ in step S2.3 is as follows:

[0075]

[0076] Among them, Ψ0(ξ 0 ), Ψ1(ξ 0 ), Ψ2(ξ 0 ), …, Ψ P-1 (ξ nrespectively represent the random part of the first - order polynomial chaos expansion of the first design variable, the random part of the second - order polynomial chaos expansion of the first design variable, the random part of the third - order polynomial chaos expansion of the first design variable, …, the random part of the (P - 1) - th order polynomial chaos expansion of the n - th design variable.

[0077] As a preferred technical solution of the present invention, the expressions for the mean and standard deviation of the crashworthiness performance in the output response in step S2.4 are as follows:

[0078]

[0079] Among them, E(F) is the mean of the output response vector, σ(F) represents the variance of the output response vector, F(ξ) is the output response function, ∫ ξ F(ξ)dξ represents the integral of the output response function with respect to the design variable ξ, ξ is the design variable, M is the expansion order, α d is the coefficient of the d - th polynomial chaos expansion term, Ψ d (ξ) is the random part of the i - th order of the polynomial chaos expansion, Ψ j (ξ) is expressed as the random part of the j - th order of the polynomial chaos expansion, Ψ j (ξ) and Ψ d (ξ) have exactly the same meaning and are listed in a separated form here for the convenience of derivation.

[0080] As a preferred technical solution of the present invention, the expression for normalizing the output crashworthiness performance G in step S4.7 is as follows:

[0081]

[0082] Among them, G represents the crashworthiness performance, μ G represents the mean of the crashworthiness performance, σ G represents the standard deviation of the crashworthiness performance, and Q represents the normalization result of the crashworthiness performance G.

[0083] As a preferred technical solution of the present invention, the specific expression of the model established in step S5.1 is as follows:

[0084] min μ(I1), σ(I1)

[0085]

[0086] Among them, μ(I1) represents the mean of I1, σ(I1) represents the standard deviation of I1, I1, I2, and I3 respectively represent the intrusion amounts of three frontal - collision intrusion points, O represents the vehicle body mass, P f2 、P f3 、P f4 、P f5respectively represent the failure probabilities of I2, I3, O, and B. Pr(G2≤0), Pr(G3≤0), Pr(G4≤0), and Pr(G5≤0) respectively represent the probabilities of G2≤0, G3≤0, G4≤0, and G5≤0. G2 and G3 respectively represent the crashworthiness performance of the intrusion points of I2 and I3. G4 and G5 respectively represent the performance of the vehicle mass and the frontal collision energy absorption. B represents the frontal collision energy absorption, and respectively represent the maximum intrusion amount of the frontal collision intrusion point, O * represents the maximum value of the body mass, B * represents the minimum value of the frontal collision energy absorption of the bus, and respectively represent the maximum failure probability of the frontal collision intrusion point, represents the failure probability of the vehicle mass, represents the failure probability of the frontal collision energy absorption of the bus, respectively represent the failure probability of the frontal collision intrusion point 2, the failure probability of the frontal collision intrusion point 3, the failure probability of the vehicle mass, and the failure probability of the frontal collision energy absorption.

[0087] Compared with the prior art, the present invention provides an optimization method for modeling the uncertainty of the frontal collision of an electric bus, having the following beneficial effects:

[0088] 1. In view of the problems that the structure of the body components is complex, the fitting is difficult, the optimization design of the crashworthiness performance of electric buses mostly focuses on deterministic optimization, and there is less uncertainty optimization, the present invention proposes an uncertainty modeling and optimization method. Without affecting the structure of the vehicle body anti-collision system, a 100% overlap rate frontal collision cross-section optimization considering uncertainty is carried out on the front end of a certain electric bus. Taking the cross-sectional dimensions of the tube beam components as the optimization variables and the collision intrusion speed as the optimization target, the present invention combines the Polynomial Chaos Expansion (PCE) method and the Maximum Entropy Method (MEM) to conduct uncertainty analysis and optimization design research on the electric bus, reducing the influence of uncertainty factors on the optimization results and ensuring the robustness and reliability of the design scheme.

[0089] 2. The present invention expands the optimization objectives and constraints by using the Polynomial Chaos Expansion method, and solves for the failure probability of the crashworthiness performance through the Maximum Entropy Method, establishing a proxy model for the frontal collision of a certain electric bus. Compared with the traditional MC method, the uncertainty design method of the present invention can effectively reduce the optimization calculation cost while ensuring basically the same accuracy;

[0090] 3. In the Sobol’ global sensitivity analysis method based on variance decomposition in the present invention, the sensitivity degrees of each design variable to the optimization response and constraints can be expressed under the condition of ensuring accuracy, so that after screening out the uncertain design variables with higher contribution degrees, the model can be dimensionally reduced to improve the optimization efficiency. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 It is a schematic flow chart of the present invention;

[0092] Figure 2 It is a schematic diagram of the finite element framework of the present invention;

[0093] Figure 3 It is a schematic diagram of the front collision energy curve of the finite element model of the present invention;

[0094] Figure 4 It is a schematic diagram of the robust optimization iteration of the present invention using the PCE method and the MC method;

[0095] Figure 5 It is a schematic diagram of the reliability-based robust optimization iteration of the present invention using the PCE method and the MC method. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0096] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0097] Please refer to Figures 1-5 , an uncertainty modeling and optimization method for the front collision of an electric bus, comprising the following steps:

[0098] S1. Establish a finite element model of the front collision of the electric bus to obtain design variables;

[0099] S2. Establish the relationship between the design variables and the response based on the polynomial chaos expansion method (PCE expansion), specifically including the following steps:

[0100] S2.1 The uncertainty influence of the design variables is represented by a true random variable with a probability density function, and the specific expression is as follows:

[0101] S i = σξ i + μ

[0102] Wherein, S i represents the i-th uncertain design variable, σ represents the variance of the design variable, and ξ iDenote the \(i\)-th design variable, and \(\mu\) represents the mean value of the design variables;

[0103] S2.2. For the polynomial chaos expansion of any output response, it is as follows:

[0104]

[0105] Among them, \(F(\xi)\) represents the output response function, represents the polynomial chaos expansion of the output response function, \(\xi\) represents the design variable, \(\alpha\) d represents the coefficient of the polynomial chaos expansion term, \(\Psi\) d \((\xi)\) represents the stochastic part of the \(d\)-th order of the polynomial chaos expansion, represents the sum over \(M\) data, \(M\) represents the expansion order, and the specific expression is as follows:

[0106]

[0107] Among them, \(n\) is the order of the random variable, \(p\) is the number of independent input variables, \(p!\) represents the factorial of \(p\), \(n!\) represents the factorial of \(n\), and \((p + n)!\) represents the factorial of \(p + n\);

[0108] S2.3. Based on the point collocation regression method to solve the polynomial chaos expansion obtained in step S2.2, the point collocation regression method first selects \(N\) samples in the random space through deterministic sampling. For the spectral pattern of the random variable, a linear system consisting of \(N\) equations is established (\(N>M\), where \(M\) is the expansion order), and \(\xi\) N-1 represents the value of the standard random variable vector \(\xi\) at the \(N\)-th sampling point, \(\mathbf{a}=(a_0,a_1,a_2,\cdots,a_M)\) is the vector of expansion term coefficients, and the approximate value \(R\) of the vector of expansion term coefficients can be solved by the least squares method. The specific expression is as follows:

[0109]

[0110] Among them, \(n\) is the order of the random variable; \(F(\xi\) i ) is the output response function; \(\Psi\) d \((\xi\) i ) is the stochastic part of the \(i\)-th order of the polynomial chaos expansion; \(\alpha\) d represents the coefficient of the polynomial chaos expansion term, \(R\) represents the approximate value of the vector of expansion term coefficients, \(\xi\) i represents the \(i\)-th design variable, and at the same time, matrix \(n\) is set, and \(\Gamma\) T \(\Gamma\) is the Fisher matrix, \(\Gamma\) T is the transpose of matrix \(\Gamma\), and its specific expression is as follows:

[0111]

[0112] Among them, \(\Psi_0(\xi\)0 )、Ψ1(ξ 0 )、Σ2(ξ 0 )、…、Ψ P-1 (ξ n ) respectively represent the stochastic part of the first - order polynomial chaos expansion of the first design variable, the stochastic part of the second - order polynomial chaos expansion of the first design variable, the stochastic part of the third - order polynomial chaos expansion of the first design variable, …, the stochastic part of the (P - 1) - th order polynomial chaos expansion of the n - th design variable.

[0113] Combining the two expressions of R and matrix Γ can obtain the following formula:

[0114]

[0115] Among them, α0, α1, α2, …, α M respectively represent M coefficient vectors of expansion terms, and finally obtain the following expression:

[0116] R = (Γ T Γ) -1 Γ T F(ξ i )

[0117] Among them, R is the approximation of the coefficient vector of the expansion term, F(ξ i ) is the output response function, M is the expansion order, ξ i is the i - th design variable;

[0118] S2.4. Obtain the expansion coefficients by the Latin hypercube sampling method to get the mean and standard deviation of the crashworthiness performance in the output response. The specific expressions are as follows:

[0119]

[0120] Among them, E(F) is the mean of the output response vector, σ(F) represents the variance of the output response vector, F(ξ) is the output response function, ∫ ξ F(ξ)dξ represents the integral of the output response function with respect to the design variable ξ, ξ is the design variable, M is the expansion order, α d is the coefficient of the d - th polynomial chaos expansion term, Ψ d (ξ) is the stochastic part of the d - th order of the polynomial chaos expansion, Ψ j (ξ) is expressed as the stochastic part of the j - th order of the polynomial chaos expansion, Ψ j (ξ) and Ψ d (ξ) have exactly the same meaning and are listed in a separated form here for the convenience of derivation.

[0121] S3. Based on the relationship R established in step S2, perform sensitivity analysis using the Sobol’ index method to obtain design variables with high sensitivity. Represent the design variables with high sensitivity as uncertain variables, and then obtain the mathematical model of the objective, which specifically includes the following steps:

[0122] S3.1. The specific expression of the total variance of the Sobol’ index method is as follows:

[0123]

[0124] Among them, n represents a natural number, A S represents the total variance of the output response, A i represents the influence of the i-th design variable on the overall output variance of the objective function model R, A 12…n represents the influence of the interaction among n design variables on the objective, A ij represents the interaction between the i-th and j-th design variables, and A i and A ij The expressions of are as follows:

[0125]

[0126] Among them, is the partial variance of the i-th single variable, F is the output response function, represents the variance, represents the mean of the conditional variance of the F function within the variable range of S i , represents that under the condition of S j , within the variable range of the variable S i , the mean of the conditional variance of the F function, A i represents the influence of the i-th input parameter on the overall output variance of the objective function model R, A j represents the influence of the j-th input parameter on the overall output variance of the model R.

[0127] S3.2. The Sobol’ index based on variance decomposition is specifically expressed as follows:

[0128]

[0129] Among them, L i is the contribution degree of the i-th design variable to the objective function, A i represents the influence of the i-th design variable on the overall output variance of the model R, A(F) is the total variance of the objective function. The remaining terms are obtained by dividing the same partial variance by the total variance, and then the high-order interaction Sobol’ index can be obtained as follows:

[0130]

[0131] Among them, L i represents the contribution degree of a single variable to the objective function, and L ij represents the interactive contribution degree of the i-th and j-th variables to the objective function, and L ijk represents the interactive contribution degree of three variables to the objective function, and L 12…n represents the interactive contribution degree of multiple variables to the objective function;

[0132] S3.3. The Sobol’ index based on the polynomial chaos expansion method can be expressed as:

[0133]

[0134] Among them, A i represents the influence of the i-th input parameter on the overall output variance of the model R, α k is the coefficient of the polynomial chaos expansion term, Ψ k is the random part of the polynomial chaos expansion of L i in step S3.2, E(Ψ k 2 ) represents the mean value of Ψ k 2 , and A PCE is the polynomial chaos expansion of the total variance of the objective function, represents the summation for k belonging to different target values, represent different target values respectively;

[0135] S4. Based on the expression R established in step S2, solve the failure probability of the constraint based on the maximum entropy method to obtain the mathematical model of the constraint. The specific steps are as follows:

[0136] S4.1. Through statistical analysis, the expressions of the first l-order origin moments of the crashworthiness performance constraint g(S c ) are as follows:

[0137]

[0138] Among them, Z g,l represents obtaining the first l-order origin moments of the constraint function g(S c ), E(g l ) represents taking the mean value of g l , ω c represents the weight coefficient, S c represents the uncertain design variable, l represents the order of the origin moment, represents summing the internal r data. After obtaining the first l-order origin moments of the constraint function g(S c ), and combining the maximum entropy method for the overall reliability analysis of the crashworthiness performance, the entropy can be defined as:

[0139]

[0140] Among them, H(g(S c )) represents the entropy of the constraint function g(S c ), C represents a constant not less than 0, g(S c ) represents the constraint function, Sc represents the uncertainty design variable, and ln* represents the logarithmic function with the natural logarithm e as the base. represents the integral of the uncertainty design variable S c from -∞ to +∞;

[0141] S4.2. According to the principle of maximum entropy, with the entropy maximization as the objective function and the origin moment as the constraint condition, an optimization model is established as follows:

[0142]

[0143] Among them, s.t. represents the constraint, H(g(S c )) represents the entropy of the constraint function g(S c ), C represents a constant not less than 0, g(S c ) represents the constraint function, S c represents the uncertainty design variable, l is the order of the origin moment, S c l represents the l-th order origin moment of the random variable S c , Z g,l represents finding the first l-th order origin moments of the constraint function g(S c ), maxH(g(S c )) represents taking the maximum value of H(g(S c ), and ln* represents the logarithmic function with the natural logarithm e as the base. represents the integral of the uncertainty design variable S c from -∞ to +∞;

[0144] S4.3. Based on the Lagrange method, the constrained optimization problem in step S4.2 is transformed into an unconstrained optimization problem and solved using the Newton method, and the specific expression is as follows:

[0145]

[0146] Among them, L represents the Lagrange function, C represents a constant not less than 0, ln* represents the logarithmic function with the natural logarithm e as the base, l is the order of the origin moment, and λ l represents the undetermined constant corresponding to the l-th order origin moment of the uncertainty design variable S c , represents the integral of the uncertainty design variable S from -∞ to +∞.c Integrate, where \(m\) represents a positive integer. At the extreme point, we have:

[0147]

[0148] where The function takes the partial derivative with respect to the constraint function \(g(S c ), \(C\) represents a constant not less than 0, \(\lambda l represents a constant to be determined, and \(S c l represents the \(l\)-th origin moment of the uncertainty design variable;

[0149] S4.4. Establish the relationship between the assumed variables and the constants to be determined. The specific expression is as follows:

[0150]

[0151] Substitute the established relationship between the assumed variables and the constants to be determined into the in step S4.3. By rearranging the terms of , we can further obtain the following equation:

[0152]

[0153] where \(a_0, a_1, a_2, \cdots, a m respectively represent \(m\) assumed variables, \(\lambda_0, \lambda_1, \lambda_2, \cdots, \lambda m respectively represent \(m\) constants to be determined, \(c\) represents a constant not less than 0, \(a l represents the assumed variable corresponding to the \(l\)-th origin moment of the random variable \(z\), \(z l represents the \(l\)-th origin moment of the random variable \(z\), and \(\exp(*)\) represents the exponential function with the natural logarithm \(e\) as the base;

[0154] S4.5. Substitute the equation obtained from \(g(S c ) in step S4.4 into \(Z g,l in step S4.1 to obtain a non-linear equation system about \(a l . Solve it using the 4th-order origin moment to obtain the specific non-linear equation system as follows:

[0155]

[0156] where \(S c represents the uncertainty design variable, \(a l represents the assumed variable, and \(l\) represents the order of the origin moment;

[0157] S4.6. Transform the non-linear equation in step S4.5 into an unconstrained optimization problem for solution. The specific expression is as follows:

[0158]

[0159] Among them, min represents the minimum value function, represents the sum of the internal four data, represents the constraint function g(S c ) the first l the maximum value of the l-th order origin moment, Z g,l represents finding the first l-th order origin moment values of the constraint function g(S c ), and then can solve for a0, a1, a2,..., a m in step S4.4 and the Lagrange function in step S4.3;

[0160] S4.7. Standardize the output crashworthiness G, and the overall failure probability can be obtained from the probability density integral:

[0161]

[0162] Among them, P f represents the overall failure probability of the crashworthiness, Pr(G≤0) represents the probability of G≤0, represents the probability of, Q represents the standardized result of the crashworthiness G, μ G represents the mean value of the crashworthiness, σ G represents the standard deviation of the crashworthiness, exp(*) represents the exponential function with the natural logarithm e as the base;

[0163] S5. Based on the uncertainty optimization mathematical model of the electric bus established in steps S3 and S4, and combined with the multi-objective optimization algorithm for solution, the specific steps are as follows:

[0164] S5.1. Establish the uncertainty optimization mathematical model of the electric bus, and the specific expression is as follows:

[0165] min μ(I1), σ(I1)

[0166]

[0167] Among them, μ(I1) represents the mean value of I1, σ(I1) represents the standard deviation of I1, I1, I2, and I3 respectively represent the intrusion amounts of three frontal collision intrusion points, O represents the vehicle body mass, P f2 , P f3 , P f4 , P f5They respectively represent the failure probabilities of I2, I3, O, and B. Pr(G2≤0), Pr(G3≤0), Pr(G4≤0), and Pr(G5≤0) respectively represent the probabilities of G2≤0, G3≤0, G4≤0, and G5≤0. G2 and G3 respectively represent the crashworthiness of the intrusion points of I2 and I3. G4 and G5 respectively represent the performance of the vehicle mass and the frontal collision energy absorption. B represents the frontal collision energy absorption. and respectively represent the maximum intrusion of the frontal collision intrusion point, O * represents the maximum value of the body mass, B * represents the minimum value of the passenger car frontal collision energy absorption, and respectively represent the maximum failure probability of the frontal collision intrusion point, represents the failure probability of the vehicle mass, represents the failure probability of the passenger car frontal collision energy absorption, respectively represent the failure probability of the frontal collision intrusion point 2, the failure probability of the frontal collision intrusion point 3, the failure probability of the vehicle mass, and the failure probability of the frontal collision energy absorption. ;

[0168] S5.2. Solve the uncertainty optimization mathematical model established in step S5.1 based on the multi-objective optimization algorithm to obtain the result of the improved crashworthiness and its robustness result.

[0169] The order l of the origin moment is set to 4. Based on the Sobol’ index method, the number of deterministic design variables is 5, and the number of uncertain design variables is 3. The sampling points of the Monte Carlo method are set to 120, and the sampling points of the PCE method are 16. As described above, the parameters adopted in the embodiments of the present invention are shown in Table I and Table II. In the tables, S1 to S7 are design variables.

[0170] Table I

[0171]

[0172] Table II

[0173]

[0174] The accuracy results of the predicted mean and standard deviation by the MC method and the PCE method are compared as shown in Table III

[0175] Table III

[0176]

[0177] As can be seen from Table II, the mean and standard deviation values of the PCE method are close to those obtained by the MC method, and the errors are all controlled within 3%, indicating that the accuracy of the PCE method is close to that of the MC method, which proves that the PCE method can be used for subsequent optimization. The uncertainty optimization mathematical model of the electric bus frontal collision in step S5 is established according to the parameters in Table I, and the traditional uncertainty optimization results and the uncertainty optimization results proposed in the present invention can be obtained by solving based on the NSGA-II optimization algorithm. The NSGA-II optimization algorithm is set to iterate 2000 generations to obtain the optimization results, and the optimization convergence times of the PCE method and the MC method are counted. The specific results are shown in Table IV.

[0178] Table IV

[0179]

[0180] As can be seen from Table IV:

[0181] (1) The optimization scheme of the present invention reduces I1 by 5.5%. Compared with the optimization method that does not consider the constraint failure probability, the failure probabilities of each crashworthiness performance are relatively small and lower than 0.5%, indicating that the present invention can better ensure the reliability and robustness of the system while improving the frontal collision of the electric bus; when the electric bus encounters a frontal collision, it can effectively ensure that the collision system does not fail and reduce the fluctuation or even deterioration of the crashworthiness performance.

[0182] The optimization convergence time of the PCE method used in the present invention is less than that of the MC method, and the failure probabilities are basically the same, indicating that the calculation efficiency of the present invention is relatively high.

[0183] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principle and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. An optimization method for modeling the uncertainty of frontal collision of electric buses, characterized in that: It includes the following steps: S1. Establish a finite element model of the frontal collision of an electric bus to obtain design variables; S2. Based on the polynomial chaos expansion method, establish the relationship between design variables and responses, which specifically includes the following steps: S2.1 The uncertainty influence of design variables is represented by real random variables with probability density functions, and the specific expression is as follows: S i = σξ i + μ Among them, S i represents the i-th uncertainty design variable, σ represents the variance of the design variable, and ξ i represents the i-th design variable, and μ represents the mean value of the design variable; S2.2 The polynomial chaos expansion formula for any output response is as follows: Among them, F(ξ) represents the output response function, represents the polynomial chaos expansion of the output response function, ξ represents the design variable, α d represents the coefficient of the polynomial chaos expansion term, Ψ d (ξ) represents the stochastic part of the d-th order of the polynomial chaos expansion, represents the summation of M data, M represents the expansion order, and the specific expression is as follows: Where, n is the order of the random variable, p is the number of independent input variables, p! represents the factorial of p, n! represents the factorial of n, and (p + n)! represents the factorial of p + n; S2.3 Based on the point collocation regression method, solve the polynomial chaos expansion formula obtained in step S2.2, and the specific expression is as follows: where n is the order of the random variable; F(ξ i ) is the output response function; Ψ d (ξ i ) is the random part of the d-th order of the polynomial chaos expansion; α d represents the coefficient of the polynomial chaos expansion term, R represents the approximation of the expansion term coefficient vector, ξ i represents the i-th design variable, and at the same time, the matrix Γ is set, and Γ T Γ is the Fisher matrix, Γ T is the transpose of the matrix Γ; Combining the two expressions of R and matrix Γ can obtain the following formula: Among them, α0, α1, α2, …, α M respectively represent the coefficient vectors of M expansion terms, and finally the following expression is obtained: R = (Γ T Γ) -1 Γ T F(ξ i ) where R is the approximation of the expansion term coefficient vector, F(ξ i ) is the output response function, M is the expansion order, and ξ i is the i-th design variable; The relationship between the design variables and the three intrusion points, the vehicle mass, and the frontal collision energy absorption during frontal collision can be fitted through the expansion form of R with the data of the three intrusion points, the vehicle mass, and the frontal collision energy absorption during frontal collision; S2.4 Use the Latin hypercube sampling method to obtain the expansion coefficients, and obtain the mean and standard deviation of the crashworthiness performance in the output response; S3. According to the relationship R established in step S2, combined with the Sobol’ index method, conduct sensitivity analysis to obtain high-sensitivity design variables, and represent the design variables with high sensitivity as uncertain variables to obtain the mathematical model of the target, which specifically includes the following steps: S3.1 The specific expression of the total variance of the Sobol’ index method is as follows: where n represents a natural number, A S represents the total variance of the output response, A i represents the influence of the i-th input parameter on the overall output variance of the objective function model R, A 12…n represents the influence of the interaction between n design variables on the objective, A ij represents the interaction between the i-th and j-th design variables, and A i and A ij The expressions of are as follows: Among them, is the partial variance of the i-th single variable, F is the output response function, represents variance, represents the mean value of the conditional variance of the F function within the variable range of variable S i , represents, under the condition of S j , the mean value of the conditional variance of the F function within the variable range of variable S i . A i represents the influence of the i-th input parameter on the overall output variance of the objective function model R, and A j represents the influence of the j-th input parameter on the overall output variance of model R; S3.2 The Sobol’ index based on variance decomposition is specifically expressed as follows: Among them, L i is the contribution degree of the i-th design variable to the objective function, and A i represents the influence of the i-th design variable on the overall output variance of the model R. A(F) is the total variance of the objective function. By obtaining the remaining terms in the same way of dividing the partial variance by the total variance, the high-order interaction Sobol’ index can be obtained as follows: Among them, L i represents the contribution degree of a single variable to the objective function, and L ij represents the interactive contribution degree of the i-th and j-th variables to the objective function, and L ijk represents the interactive contribution degree of three variables to the objective function, and L 12…n represents the interactive contribution degree of multiple variables to the objective function; The Sobol’ index based on the polynomial chaos expansion method can be expressed as: Among them, A i represents the influence of the i-th input parameter on the overall output variance of the model R, and α k is the coefficient of the polynomial chaos expansion term, and Ψ k is the random part of the polynomial chaos expansion of L in step S3.

2. E(Ψ i k 2 ) represents the mean value of Ψ k 2 PCE , and A PCE is the polynomial chaos expansion of the total variance of the objective function. represents the summation for k belonging to different target values, respectively represent different target values; S4. Based on the expression R established in step S2, solve the failure probability of the constraint based on the maximum entropy method to obtain the mathematical model of the constraint. The specific steps are as follows: S4.

1. Obtain the expression of the first l-order origin moments of the crashworthiness constraint g(S c ) as follows: Among them, Z g,l represents the first l-order origin moment of the constraint function g(S c ), E(g(S c )) l represents the mean value of g(S c ), l ω c represents the weight coefficient, S c represents the uncertainty design variable, l represents the order of the origin moment, represents the summation of r internal data. After obtaining the statistical moment and combining the maximum entropy method, the overall reliability analysis of the crashworthiness performance is carried out. The entropy can be defined as: Among them, H(g(S c )) represents the entropy of the constraint function g(S c ), C represents a constant not less than 0, G(S c ) represents the constraint function, S c represents the uncertainty design variable, ln* represents the logarithmic function with the natural logarithm e as the base, represents the integral of the uncertainty design variable S c from -∞ to +∞; S4.2 According to the maximum entropy principle, with the maximum entropy as the objective function and the origin moment as the constraint condition, establish an optimization model, which is specifically expressed as follows: Among them, s.t. represents a constraint, and H(g(S c )) represents the entropy of the constraint function g(S c ), C represents a constant not less than 0, g(S c ) represents a constraint function, S c represents an uncertainty design variable, l is the order of the origin moment, S c l represents the l-th order origin moment of the random variable S c , Z g,l represents finding the first l-th order origin moments of the constraint function g(S c ), maxH(g(S c )) represents taking the maximum value of H(g(S c ), ln* represents the logarithmic function with the natural logarithm e as the base, represents integrating the uncertainty design variable S c from -∞ to +∞; S4.3 Based on the Lagrange method, transform the constrained optimization problem in step S4.2 into an unconstrained optimization problem, and use the Newton method for solution. The specific expression is as follows: where \(L\) represents the Lagrange function, \(C\) represents a constant not less than \(0\), \(\ln^*\) represents the logarithmic function with the natural logarithm \(e\) as the base, \(l\) is the order of the origin moment, and \(\lambda\) l represents the undetermined constant corresponding to the \(l\)-th order origin moment of the uncertainty design variable \(S\) c , represents the integral of the uncertainty design variable \(S\) from \(-\infty\) to \(+\infty\), \(m\) represents a positive integer, and at the extreme point, we have: c ​ Among them, The function takes the partial derivative of the constraint function g(S c ), C represents a constant not less than 0, and λ l represents a constant to be determined, and S c l represents the l-th order origin moment of the uncertain design variable; S4.4 Establish the relationship between the hypothetical variable and the undetermined constant, and the specific expression is as follows: Substitute the relationship between the established hypothetical variables and undetermined constants into the formula in step S4.3 In, by Moving the terms, the following formula can be further obtained: where, a0, a1, a2, …, a m respectively represent m hypothetical variables, λ0, λ1, λ2, …, λ m respectively represent m undetermined constants, c represents a constant not less than 0, a l represents the hypothetical variable corresponding to the l-th origin moment of the random variable z, z l represents the l-th origin moment of the random variable z, and exp(*) represents the exponential function with the natural logarithm e as the base; S4.

5. Substitute the expression obtained in step S4.4 for g(S c ) into Z in step S4.1 g,l to obtain a system of non-linear equations about a l . Solve using the fourth-order raw moment to obtain the following specific system of non-linear equations: Among them, S c represents the uncertain design variable, a l represents the assumed variable, and l represents the order of the origin moment; S4.6 Transform the nonlinear equation in step S4.5 into an unconstrained optimization problem for solution, and the specific expression is as follows: where min represents the minimum value function, represents the summation of 4 internal data, represents the maximum value of the first l - order origin moments of the constraint function G(S c ), and Z g,l represents the value of the first l - order origin moments of the constraint function g(S c ), and thus can solve for a0, a1, a2, …, a m in step S4.4 and the Lagrange function in step S4.3; S4.7 Standardize the output crashworthiness performance G, and the overall failure probability can be obtained by probability density integration: Among them, P f represents the overall failure probability of crashworthiness performance, Pr(G≤0) represents the probability of G≤0, represents the probability of, Q represents the standardized result of the crashworthiness performance G, μ G represents the mean value of the crashworthiness performance, σ G represents the standard deviation of the crashworthiness performance, exp(*) represents the exponential function with the natural logarithm e as the base; S5. Based on the uncertainty optimization mathematical model of the frontal collision of the electric bus established in steps S3 and S4, and combined with the multi-objective optimization algorithm for solution. The specific steps are as follows: S5.1 Establish the uncertainty optimization mathematical model of the electric bus; S5.2 Based on the multi-objective optimization algorithm, solve the uncertainty optimization mathematical model established in step S5.1 to obtain the result of the improvement of the crashworthiness performance and its robustness result; The specific expression of Γ in step S2.3 is as follows: Among them, Ψ0(ξ 0 ), Ψ1(ξ 0 ), Ψ2(ξ 0 ), …, Ψ P-1 (ξ n ) respectively represent the random part of the first-order polynomial chaos expansion of the first design variable, the random part of the second-order polynomial chaos expansion of the first design variable, the random part of the third-order polynomial chaos expansion of the first design variable, …, the random part of the (P-1)-th order polynomial chaos expansion of the n-th design variable; The expressions for the mean and standard deviation of the crashworthiness performance in the output response in step S2.4 are as follows: where, E(F) is the mean of the output response vector, σ(F) represents the variance of the output response vector, F(ξ) is the output response function, and ∫ ξ F(ξ)dξ represents the integral of the output response function with respect to the design variable ξ, ξ is the design variable, M is the expansion order, α d is the coefficient of the d-th polynomial chaos expansion term, Ψ d (ξ) is the random part of the d-th order of the polynomial chaos expansion, and Ψ j (ξ) is expressed as the random part of the j-th order of the polynomial chaos expansion. The meanings of Ψ j (ξ) and Ψ d (ξ) are exactly the same. For the convenience of derivation, they are listed in a separated form here; The expression for normalizing the output crashworthiness performance G in step S4.7 is as follows: Among them, G represents the crashworthiness performance, μ G represents the mean value of the crashworthiness performance, σ G represents the standard deviation of the crashworthiness performance, and Q represents the standardized result of the crashworthiness performance G; The specific expression of the model established in step S5.1 is as follows: minμ(I1),σ(I1) Among them, μ(I1) represents the mean value of I1, σ(I1) represents the standard deviation of I1, I1, I2, and I3 respectively represent the intrusion amounts at three frontal collision intrusion points, and O represents the vehicle body mass. respectively represent the failure probabilities of I2, I3, O, and B, Pr(G2≤0), Pr(G3≤0), Pr(G4≤0), Pr(G5≤0) respectively represent the probabilities of G2≤0, G3≤0, G4≤0, G5≤0, G2 and G3 respectively represent the crashworthiness of the intrusion points of I2 and I3, G4 and G5 respectively represent the performance of the vehicle mass and frontal collision energy absorption, and B represents the frontal collision energy absorption. and respectively represent the maximum intrusion amounts at the frontal collision intrusion points, O * represents the maximum value of the vehicle body mass, B * represents the minimum value of the frontal collision energy absorption of the passenger car. and respectively represent the maximum failure probabilities at the frontal collision intrusion points, represents the failure probability of the vehicle mass, represents the failure probability of the frontal collision energy absorption of the passenger car. respectively represent the failure probability of the frontal collision intrusion point 2, the failure probability of the frontal collision intrusion point 3, the failure probability of the vehicle mass, and the failure probability of the frontal collision energy absorption.

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