CN112507458A - Chebyshev method-based automobile collision safety and reliability design method - Google Patents

Chebyshev method-based automobile collision safety and reliability design method Download PDF

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CN112507458A
CN112507458A CN202011455477.0A CN202011455477A CN112507458A CN 112507458 A CN112507458 A CN 112507458A CN 202011455477 A CN202011455477 A CN 202011455477A CN 112507458 A CN112507458 A CN 112507458A
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谷先广
陈瑞
张代胜
黄岳竹
高梦琳
张宇
李翼良
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Intelligent Manufacturing Institute of Hefei University Technology
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Abstract

The invention relates to the field of vehicle structure optimization design, in particular to a method for designing the safety and reliability of automobile collision based on a Chebyshev method. The optimal design method for the safety and reliability of the automobile collision comprises the following steps: s1: establishing an automobile collision finite element model; s2: defining an optimization problem comprising: selecting design variables, defining variable types, selecting an optimization target, a constraint target and the like; s3: obtaining design sample points and test sample points by adopting DOE; s4: building an approximate model, and predicting response by using the approximate model; and carrying out precision verification on the built approximate model; s5: acquiring upper and lower response boundaries by using a Chebyshev analysis method; s6: performing reliability optimization on the upper bound of the response by using an optimization algorithm; s7: and (4) performing reliability evaluation on the optimized design solution by using a Monte Carlo method. Compared with the traditional deterministic optimization design, the design method can realize light weight and improve the reliability of the design solution on the basis of meeting the collision performance.

Description

Chebyshev method-based automobile collision safety and reliability design method
Technical Field
The invention relates to the field of vehicle structure optimization design, in particular to a method for designing the safety and reliability of automobile collision based on a Chebyshev method.
Background
With the advent of high performance computers, the automotive industry has been able to solve vehicle structural design problems using multidisciplinary optimization and structural simulation approaches. In the design process of the automobile lightweight structure, a vehicle approximate model can be constructed through a computer, and simulation test and optimization are carried out based on the approximate model; meanwhile, the application OF methods such as DOE (DESIGN OF EXPERIMENT) and the like can provide convenience for data acquisition; these all greatly reduce the economic and time costs of vehicle development.
The same is true for the optimized design of the safety and reliability of the automobile collision, and the design process can be completed by a high-performance computer. On the one hand, however, in the conventional vehicle crash safety reliability optimization design, the default design variables are usually simple random variables, which may cause the obtained optimization design solution not to satisfy the constraints or to be inapplicable in actual engineering. On the other hand, in the traditional optimization design of the safety and reliability of the automobile collision, reliability optimization is usually performed only aiming at specific probability distribution of known variables, but the problem that uncertain interval variables exist in the design variables is ignored, and therefore the reliability of the obtained optimization design result is reduced. In addition, in the process of the optimized design of the safety and reliability of the collision of the automobile, due to great complexity of engineering problems, most designs cannot obtain a detailed probability distribution function of design variables; this also makes the finished reliability optimized design generally not of practical engineering significance.
Zhanhuai, luxiaojiang, zhou dao yong, etc. in the journal 2020, 42(2) of automotive engineering: in the article 222-; the authors of lv xiao jiang, zhou dao yong, sun light yong, etc. published "automotive engineering" in the journal 2018, 40 (7): an article in 790-794, namely 'multi-objective reliability optimization-based mule vehicle body crashworthiness and lightweight design', establishes a mule vehicle multi-objective reliability optimization process, and takes reliability into consideration in the optimization process. However, the design variables in the above documents are all known to have detailed probability distribution or are defined as random variables, which makes the optimization design under deterministic conditions, and the obtained optimization result is often close to the constraint boundary or even in an infeasible domain. The traditional optimization design can be simply understood as the extreme value which can be reached by the objective function under the constraint condition.
Therefore, how to solve the problems in the optimization design process of the safety and reliability of the automobile collision; the collision safety reliability optimization design solution obtained in the vehicle lightweight design process has higher reliability and practical significance; becomes a difficult problem to be overcome in the field of vehicle structure optimization design.
Disclosure of Invention
In order to overcome the problems in the prior art, the invention provides a method for designing the safety and reliability of automobile collision based on a Chebyshev method.
An automobile collision reliability optimization design method comprises the following steps:
s1: establishing an automobile collision finite element model; the whole vehicle finite element model comprises a body in white, an opening part, a chassis, a power system, an electronic appliance and a sensor for testing; the mass of the dummy system and the safety belt system is added by means of a mass point;
s2: defining an optimization problem comprising: selecting design variables, defining the variable range of the variables, and selecting an optimization target and a constraint target;
s3: acquiring data of a design sample point and a test sample point through DOE design;
s4: establishing a functional relation between design variables and responses through data fitting of data acquired in DOE design; thereby establishing an approximate model and predicting response by using the approximate model; and carrying out precision verification on the built approximate model;
s5: and (2) acquiring an upper bound and a lower bound of the response predicted in the previous step by using a Chebyshev interval analysis method, wherein the searching process of the Chebyshev interval analysis method comprises the following steps:
s51: constructing one-dimensional n-times Chebyshev series of design variable x and using function Cn(x)Represents;
s52: transforming the Chebyshev series for the design variable x into a polynomial form with a recurrence relation;
s53: considering that the design variable x is in the interval [ -1,1 [)]Upper, n-th and m-th Chebyshev series Cn(x)And Cm(x)Has orthogonality; in a continuous closed interval, a continuous function can be represented by a polynomial satisfying precision; converting the function f (x) into an m-dimensional Chebyshev polynomial of order no more than n
Figure BDA0002828062250000021
An approximate form of (a);
s54: popularizing a Chebyshev polynomial expression of a continuous function f (x) into an interval algorithm, so that a variable x is replaced by an interval variable [ x ], and obtaining an interval expression f ([ x ]) of the continuous function;
s55: selecting proper Chebyshev polynomial order, and acquiring the corresponding upper and lower bounds of the approximate model by using a function f ([ x ]);
s6: performing reliability optimization on the upper bound of the response obtained in the last step by adopting a multi-objective optimization algorithm;
s7: and adopting a Monte Carlo method to carry out reliability evaluation on the solved optimization design solution.
Further, in step S2, the thickness of a key energy absorption component in the automobile mechanism is selected as a design variable, a defined variable is an interval variable, the energy absorption and the quality of an optimized component are selected as optimization targets, and the intrusion amount of the firewall and the maximum acceleration of the whole automobile are selected as constraint targets.
Further, in step S3, the DOE design method may select any one of full factor test design, orthogonal test design, latin hypercube test design, and optimized latin hypercube test design.
Further, the approximation model in step S4 may be any one of a kriging model, a response surface model, a radial basis model, and the like; after the model is built, calculating the decision coefficient R of the approximate model2And maximum relative error max (RE), verifying the precision of the approximate model, and determining coefficient R2And the maximum relative error max (re) is calculated as follows:
Figure BDA0002828062250000031
Figure BDA0002828062250000032
in the above formula, YIRepresenting finite element simulation values;
Figure BDA0002828062250000033
representing an approximate model response value;
Figure BDA0002828062250000034
represents the mean of the finite element simulation and Q represents the number of sample points.
Further, the requirements for accuracy in the approximation model are: determining the coefficient R2Not less than 0.9, and the maximum relative error max (RE) is not more than 5%.
Further, in steps S51 and S52, the functional expression of the one-dimensional n-degree Chebyshev series of the design variable x is:
Cn(x)＝cosnθ，
in the above formula, θ ═ arccos (x), θ ∈ [0, pi ];
wherein the polynomial form of the Chebyshev series for variable x is:
Figure BDA0002828062250000035
further, in step S53, when the argument x is within the interval [ -1,1 [ -1 [ ], 1 [ ]]Upper, n-th and m-th Chebyshev series Cn(x)And Cm(x)With orthogonality, there are:
Figure BDA0002828062250000036
in the above formula, the first and second carbon atoms are,
Figure BDA0002828062250000041
representing a weight function;
the function f (x) is formed by an m-dimensional Chebyshev polynomial of order no more than n
Figure BDA0002828062250000042
Approximately, the expression is:
Figure BDA0002828062250000043
in the above formula, k represents a subscript i1,i2,…imWhere zero is present, n represents the order of the polynomial, m represents the dimension of the polynomial,
Figure BDA0002828062250000044
represents a constant coefficient;
in the approximate expression, the expression is,
Figure BDA0002828062250000045
in the above formula, xjExpression polynomial
Figure BDA0002828062250000046
The zero point of (a) is,
considering that the tensor product of a one-dimensional Chebyshev polynomial can be used to represent a multidimensional series, there are:
Figure BDA0002828062250000047
then the polynomial
Figure BDA0002828062250000048
Zero point x ofjThe expression of (c) is transformed into:
xj＝cosθj，
in the above formula, the first and second carbon atoms are,
Figure BDA0002828062250000049
the magnitude of l is determined by the order n of the polynomial, defined as: and l is n + 1.
Further, in step S54, after the Chebyshev polynomial expression of the continuous function f (x) is generalized to the interval algorithm, the interval expression of the continuous function is:
Figure BDA00028280622500000410
in the above formula, k represents a subscript i1,i2,…imThe number of zeros appearing in the sequence; [ theta ] of]Is in the value range of [0, pi ]]And cosi1[θ]·cosi2[θ]…cosin[θ]＝[-1,1]This is always true.
Further, in step S55, the interval expression f ([ x ]) of the continuous function may be expressed as:
Figure BDA00028280622500000411
in the above formula, k represents a subscript i1,i2,…imThe number of zeros appearing therein, n representing the order of the polynomial;
in the process of acquiring the upper and lower bounds of the approximate model phase response, the order of the Chebyshev polynomial is selected to be 4.
Further, in step S6, the multi-objective optimization algorithm used may select a multi-island genetic algorithm or a non-dominated sorting genetic algorithm.
The design method for the safety and reliability of automobile collision based on the Chebyshev method has the following beneficial effects:
1. according to the method, a Chebyshev interval analysis method is introduced into an automobile collision safety reliability design method, and the upper and lower bounds of response are solved in an approximate model by using a Chebyshev polynomial approximation mode, so that the problem of unreliable design results caused by uncertain design variables in the traditional automobile structure collision resistance optimization design is solved, meanwhile, the workload of a calculation process is greatly simplified, the calculation cost is reduced, and the network responsiveness is improved; has great engineering significance.
2. In the invention, in the process of solving the problem of the optimal design of the automobile collision structure, only the upper and lower bounds of the design variable are needed to be known, and the accurate probability distribution is not needed to be determined, so that the method has more practical engineering significance compared with the traditional probability optimization method.
3. The Chebyshev uncertain analysis method is introduced into the front collision crashworthiness optimization design process, and the test design, the approximate model technology and the multi-objective deterministic optimization theory are systematically applied to the whole optimization design process, so that the interval reliability optimization design method suitable for the front collision of the automobile is established, and the lightweight of parts is realized while the crashworthiness of the whole automobile is improved.
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FIG. 1 is a flow chart of the method for optimally designing the collision reliability of the automobile in the embodiment 1;
FIG. 2 is a graph showing the results of the actual vehicle collision test in the present embodiment 1;
FIG. 3 is a diagram of finite element simulation results of a vehicle crash in this embodiment 1
FIG. 4 is a comparison graph of the acceleration change curves of the front crash test and the simulated whole vehicle in the embodiment 1;
FIG. 5 is a structural distribution diagram of sampling points of design variables in the vehicle structure in the present embodiment 1;
FIG. 6 is a comparison graph of the deterministic optimized pareto front and the interval optimized pareto front obtained by the multi-objective genetic algorithm in this example 1;
fig. 7 is a comparison graph of acceleration curves of the vehicle model of the initial design and the interval optimization design in the embodiment 1.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.
Example 1
In the method for designing the safety and reliability of the collision of the automobile based on the chebyshev method provided in the embodiment, the flow of the method is shown in fig. 1, and the method comprises the following processes:
firstly, building and verifying simulation model
In the embodiment, an Ls-dyna module in a Hypermesh software package is used for establishing an automobile collision finite element model; carrying out simulation calculation in a Ls-dyna solver; and importing the solved result into Hyperwords for post-processing result analysis, and checking the accuracy of the established finite element model. The whole vehicle finite element model comprises a body in white, an opening part, a chassis, a power system, electronic appliances, various sensors for testing and the like; in the automobile structure optimization, only energy absorption and vehicle acceleration in the collision process are considered, and injury indexes of passengers are not considered, so that a dummy system, a safety belt system and the like do not need to be modeled, and the mass of the dummy system and the quality of the safety belt system are added in a mass point mode.
According to the test requirements of NTHSA, the collision working condition set in the optimization problem design stage of the embodiment is 100% of completely overlapped rigid wall collision, and the collision speed is 56.3 km/h; fig. 2 shows the deformation of a car after a car crash test, and fig. 3 shows the deformation of the car after a crash in a finite element simulation result. The acceleration comparison curve of the front-impact test and the simulated whole vehicle is shown in figure 4. The results of analyzing FIGS. 2-4 show that: the simulation after the front collision of the model in the embodiment is basically consistent with the deformation mode of the test, and the acceleration change trend of the whole vehicle is basically the same. Therefore, the simulation model built in the embodiment has high reliability and good simulation effect, and can be used for subsequent optimization design.
Design variables and response definitions
To improve the crashworthiness of the automobile and achieve the goal of light weight, as shown in fig. 5, the thicknesses of the front anti-collision beam, the left and right crash boxes, the longitudinal beam and the inner and outer plates are selected as design variables and are respectively marked as x1-x8. The method is characterized in that the total energy absorption (E) and the total MASS (MASS) of the parts are selected as optimization targets, the maximum acceleration (A) of the whole vehicle and the collision intrusion amount (I) of a firewall are used as constraint conditions, and meanwhile, the aim of realizing light weight as far as possible is also the key target of the structural design of the vehicle on the premise of improving the collision resistance. The specific information of the optimization target and the constraint condition is shown in table 1:
table 1: optimizing objectives and constraints
Figure BDA0002828062250000061
Considering the constraints of factors such as materials and processing techniques in engineering, the present embodiment defines the design variables as interval parameters, and the intervals and initial values of the design variables are shown in table 2:
table 2: design variables and initial values
Figure BDA0002828062250000071
Third, construction of PSO-SVR approximate model
And optimizing the kernel function parameters of the SVR (support vector regression) approximate model by using a PSO (particle swarm optimization) algorithm, wherein the response MASS is the sum of the component quality, so that the high-precision model building can be completed by selecting a linear kernel function. The responses A, E, I were selected from K-type, and gaussian kernel functions, respectively, with specific optimization values and errors as shown in table 3:
table 3: kernel function and error statistics
Figure BDA0002828062250000072
Generally, in the accuracy index of the approximate model, the coefficient R is determined2The maximum relative error max (RE) is less than or equal to 5, and the precision requirement of the subsequent optimization can be met. According to the results in table 3, it can be found that the accuracy of the PSO-SVR approximation model established in this embodiment meets the requirement, and can be used for subsequent optimization design.
Fourth, Chebyshev interval analysis method
In the Matlab, the upper and lower bounds of response are searched by a Chebyshev substitution model containing high-order coefficients, and the method has the characteristics of low calculation cost and good response complexity; the process of obtaining the upper and lower bounds of the response predicted in the previous step by using the Chebyshev interval analysis method is as follows:
with Cn(x)Representing a one-dimensional n-times Chebyshev series of variable x, the expression of which is:
Cn(x)＝cosnθ；
in the above formula, θ ═ arccos (x), θ ∈ [0, pi ];
the Chebyshev series can be understood as a polynomial for the variable x, which has the following recursion:
Figure BDA0002828062250000081
the argument x of the function is in the interval [ -1,1 [ ]]Upper, n-th and m-th Chebyshev series Cn(x)And Cm(x)Has orthogonality:
Figure BDA0002828062250000082
in the above formula, the first and second carbon atoms are,
Figure BDA0002828062250000083
representing a weight function.
Because in a continuous closed interval, the polynomial satisfying the precision can express a continuous function;
thus, the function f (x) may be formed of an m-dimensional Chebyshev polynomial of order no more than n
Figure BDA0002828062250000084
The approximate expression is:
Figure BDA0002828062250000085
in the above formula, k represents a subscript i1,i2,…imThe number of zeros appearing in the (c) bit,
Figure BDA0002828062250000086
represents a constant coefficient; n represents the order of the polynomial and m represents the dimension of the polynomial.
Wherein the content of the first and second substances,
Figure BDA0002828062250000087
in the above formula, xjExpression polynomial
Figure BDA0002828062250000088
Zero point of (d);
since the tensor product of the one-dimensional Chebyshev polynomial can be used to represent a multidimensional series, there are:
Figure BDA0002828062250000089
thus, xjThe expression of (c) can be expressed as:
xj＝cosθj，
in the above formula, the first and second carbon atoms are,
Figure BDA0002828062250000091
in general, the size of l is determined byThe order n of the polynomial is determined and is generally defined as l ═ n + 1.
The Chebyshev polynomial expression of the continuous function f (x) is popularized to an interval algorithm, namely the variable x is replaced by an interval variable [ x ], and then the interval expression of the continuous function is as follows:
Figure BDA0002828062250000092
in the above formula, [ theta ]]Is in the value range of [0, pi ]]And cosi1[θ]·cosi2[θ]…cosin[θ]＝[-1,1]This is always true.
Further, the above formula may be rewritten as:
Figure BDA0002828062250000093
in the optimization design of the crashworthiness of the automobile structure, a constraint function is implicitly expressed, an upper bound and a lower bound of response are solved by adopting an approximate model and a Chebyshev polynomial approximation mode, reliability optimization is carried out, the calculation cost can be reduced, and the calculation is simplified. Considering the calculated amount of the approximate model and the time cost required by solving, the order of the Chebyshev polynomial is selected to be 3-4, and the subsequent optimization requirements can be met. In this example, the order of the Chebyshev polynomial is chosen to be 4.
Fifthly, optimizing process and result
The optimization problem is globally optimized using a multi-objective genetic algorithm. The mathematical expression for deterministic optimization is:
Figure BDA0002828062250000094
wherein f is1(x, y) is the total energy absorption of the plate to be optimized; f. of2(x, y) is the total mass of the plate to be optimized; g1(x, y) is the maximum acceleration of the whole vehicle; g2(x, y) is the maximum intrusion of the firewall, xl,xuAre the lower and upper bounds of the design variable x.
The mathematical expression for interval reliability optimization of the problem is as follows:
Figure BDA0002828062250000101
deterministic optimized pareto fronts and interval optimized pareto fronts obtained by using a multi-objective genetic algorithm are shown in FIG. 6; analysis of the graph in fig. 6 shows that: the obtained interval optimization solution set in the embodiment is far away from the constraint boundary, and the reliability is obviously improved.
And evaluating the reliability of the deterministic optimal design solution and the interval reliability optimal design solution by using a Monte Carlo simulation method, and assuming that the fluctuation range of the design variable is 5 percent and obeying normal distribution. The result pair of deterministic optimization design and interval optimization design is shown in table 4:
table 4: deterministic and interval design result comparison
Figure BDA0002828062250000102
Analyzing the data in table 4, it can be found that: the reliability of the design result of the interval method is superior to that of the deterministic design result, wherein the reliability of the intrusion amount of the firewall is improved from 59.85% to 99%, and meanwhile, the interval optimization method can obtain a design solution with higher reliability without the probability distribution of design variables. Although the interval optimization result is reduced in design target quality MASS and energy absorption E compared with the deterministic optimization result, the interval optimization result is superior to the initial design and the reliability of the constraint item is improved. Therefore, the interval optimization design result is selected as the final design scheme in the example.
Sixthly, verifying the optimization result
The interval optimization results are subjected to simulation verification, and the comparison between the obtained initial design and the interval optimization design results is shown in table 5:
table 5: comparing initial design and interval optimization design results
Figure BDA0002828062250000103
Figure BDA0002828062250000111
Wherein the values of the variables in the initial design and final interval-optimized design are shown in table 6:
table 6: initial design versus interval design variables
Figure BDA0002828062250000112
Analysis of the data in table 5 can lead to the conclusion that: the maximum relative error between the predicted value of the approximate model and the calculated value of the finite element simulation in the embodiment is only 3.45%, so that the PSO-SVR approximate model built by the method has high accuracy. Compared with the initial design, after interval optimization, the mass is reduced by 3.69%, the peak acceleration is reduced by 5.03%, the energy absorption is increased by 3.51%, and the firewall intrusion is reduced by 3.87%. Meanwhile, the reliability is considered, the quality of the structure is reduced, and the crashworthiness of the automobile structure under the front collision working condition is improved.
Fig. 7 shows a comparison graph of the acceleration variation curve of the entire vehicle between the initial design and the interval optimization design in this example, and analysis of the curve in the comparison graph shows that the acceleration peak value in the interval optimization design is significantly lower than the initial design acceleration in this example, and the expected optimization requirement is met.
The present invention is not limited to the above preferred embodiments, and any modifications, equivalent substitutions and improvements made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims (10)

1. A method for designing the safety and reliability of automobile collision based on a Chebyshev method is characterized by comprising the following steps:
s1: establishing an automobile collision finite element model; the whole vehicle finite element model comprises a body in white, an opening piece, a chassis, a power system, an electronic appliance and a sensor for testing; the mass of the dummy system and the safety belt system is added by means of a mass point;
s2: defining an optimization problem comprising: selecting design variables, defining the variable range of the variables, and selecting an optimization target and a constraint target;
s3: acquiring data of a design sample point and a test sample point through DOE design;
s4: establishing a functional relation between design variables and responses through data fitting of data acquired in DOE design; thereby building an approximate model and predicting response by using the approximate model; and carrying out precision verification on the built approximate model;
s5: and (2) acquiring an upper bound and a lower bound of the response predicted in the previous step by using a Chebyshev interval analysis method, wherein the searching process of the Chebyshev interval analysis method comprises the following steps:
s51: constructing one-dimensional n-times Chebyshev series of design variable x and using function Cn(x)Represents;
s52: transforming the Chebyshev series for the design variable x into a polynomial form with a recurrence relation;
s53: considering that the design variable x is in the interval [ -1,1 [)]Upper, n-th and m-th Chebyshev series Cn(x)And Cm(x)Has orthogonality; in a continuous closed interval, a continuous function can be represented by a polynomial satisfying precision; converting the function f (x) into an m-dimensional Chebyshev polynomial of order no more than n
Figure FDA0002828062240000011
An approximate form of (a);
s54: popularizing a Chebyshev polynomial expression of a continuous function f (x) into an interval algorithm, so that a variable x is replaced by an interval variable [ x ], and obtaining an interval expression f ([ x ]) of the continuous function;
s55: selecting proper Chebyshev polynomial order, and acquiring the corresponding upper and lower bounds of the approximate model by using a function f ([ x ]);
s6: performing reliability optimization on the upper bound of the response obtained in the last step by adopting a multi-objective optimization algorithm;
s7: and adopting a Monte Carlo method to carry out reliability evaluation on the solved optimization design solution.
2. The chebyshev method-based automobile collision safety reliability design method as claimed in claim 1, characterized in that: in the step S2, the thickness of a key energy absorbing component in the automobile mechanism is selected as a design variable, a defined variable is an interval variable, the energy absorption and the quality of an optimized component are selected as optimization targets, and the intrusion amount of a firewall and the maximum acceleration of the whole automobile are selected as constraint targets.
3. The chebyshev method-based automobile collision safety reliability design method as claimed in claim 1, characterized in that: in step S3, the DOE design method selects any one of full factor test design, orthogonal test design, latin hypercube test design, and optimized latin hypercube test design.
4. The chebyshev method-based automobile collision safety reliability design method as claimed in claim 1, characterized in that: the approximate model in the step S4 may be any one of a kriging model, a response surface model, and a radial basis model; after the model is built, calculating the decision coefficient R of the approximate model2And a maximum relative error max (RE), verifying the accuracy of the approximation model, and determining the coefficient R2And the maximum relative error max (re) is calculated as follows:
Figure FDA0002828062240000021
Figure FDA0002828062240000022
in the above formula, YIRepresenting finite element simulation values;
Figure FDA0002828062240000023
representing an approximate model response value;
Figure FDA0002828062240000024
represents the mean of the finite element simulation and Q represents the number of sample points.
5. The chebyshev method-based automobile collision safety reliability design method according to claim 4, characterized in that: the accuracy requirements of the approximation model are: determining the coefficient R2Not less than 0.9, and the maximum relative error max (RE) is not more than 5%.
6. The chebyshev method-based automobile collision safety reliability design method as claimed in claim 1, characterized in that: in the steps S51 and S52, the functional expression of the one-dimensional n-times Chebyshev series of the design variable x is:
Cn(x)＝cosnθ，
in the above formula, θ ═ arccos (x), θ ∈ [0, pi ];
wherein the polynomial form of the Chebyshev series for variable x is:
Figure FDA0002828062240000025
7. the chebyshev method-based automobile collision safety reliability design method according to claim 6, characterized in that: in the step S53, when the argument x is in the range [ -1,1 [ ]]Upper, n-th and m-th Chebyshev series Cn(x)And Cm(x)With orthogonality, there are:
Figure FDA0002828062240000026
in the above formula, the first and second carbon atoms are,
Figure FDA0002828062240000031
representing a weight function;
the function f (x) is formed by an m-dimensional Chebyshev polynomial of order no more than n
Figure FDA0002828062240000032
Approximately, the expression is:
Figure FDA0002828062240000033
in the above formula, k represents a subscript i1，i2，…imWhere zero is present, n represents the order of the polynomial, m represents the dimension of the polynomial,
Figure FDA0002828062240000034
represents a constant coefficient;
in the approximate expression, the expression is,
Figure FDA0002828062240000035
in the above formula, xjExpression polynomial
Figure FDA0002828062240000036
The zero point of (a) is,
considering that the tensor product of a one-dimensional Chebyshev polynomial can be used to represent a multidimensional series, there are:
Figure FDA0002828062240000037
then the polynomial
Figure FDA0002828062240000038
Zero point x ofjThe expression of (c) is transformed into:
xj＝cosθj，
in the above formula, the first and second carbon atoms are,
Figure FDA0002828062240000039
the magnitude of l is determined by the order n of the polynomial, defined as: and l is n + 1.
8. The chebyshev method-based automobile collision safety reliability design method according to claim 7, characterized in that: in step S54, after the Chebyshev polynomial expression of the continuous function f (x) is generalized to the interval algorithm, the interval expression of the continuous function is:
Figure FDA00028280622400000310
in the above formula, k represents a subscript i1，i2，…imThe number of zeros appearing in the sequence; [ theta ] of]Is in the value range of [0, pi ]]And cos i1[θ]·cos i2[θ]…cos in[θ]＝[-1，1]This is always true.
9. The chebyshev method-based automobile collision safety reliability design method according to claim 8, characterized in that: in step S55, the interval expression f ([ x ]) of the continuous function may be expressed as:
Figure FDA0002828062240000041
in the above formula, k represents a subscript i1，i2，…imThe number of zeros appearing therein, n representing the order of the polynomial;
in the process of acquiring the upper and lower bounds of the approximate model phase response, the order of the Chebyshev polynomial is selected to be 4.
10. The chebyshev method-based automobile collision safety reliability design method as claimed in claim 1, characterized in that: in step S6, the multi-objective optimization algorithm used includes a multi-island genetic algorithm or a non-dominated sorting genetic algorithm.
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