CN111709082B - An efficient design optimization method for safety and reliability of vehicle side collision - Google Patents

Abstract

The invention relates to the technical field of automobile reliability design optimization, and discloses a high-efficiency automobile side collision safety reliability design optimization method. And then, carrying out deterministic optimization, solving the translation distance by using a function moving method based on response PDF, constructing an equivalent deterministic optimization model and solving to obtain a new optimal solution. And (4) repeatedly executing the reliability analysis and the deterministic optimization process by taking the new optimal solution as input until the convergence condition is met, and finally obtaining the optimal solution of the original optimization model. The invention decouples reliability analysis and optimization design by using the function moving method based on the response PDF, thereby improving efficiency; the reliability analysis uses the information of the first four moments of the response, can obtain higher precision, and has the double characteristics of high precision and high efficiency.

Description

An efficient design optimization method for safety and reliability of automobile side collision

technical field

The invention relates to the technical field of vehicle reliability design optimization, in particular to an efficient vehicle side collision safety reliability design optimization method.

Background technique

With the advent of high-performance computers, the automotive industry has used multidisciplinary optimization and crashworthiness simulation to address vehicle side crash safety issues, thereby reducing the cost and time of new vehicle development. However, simulation-based optimizations are typically deterministic optimal designs that typically employ the limits of design constraint boundaries. In practical engineering, there are a lot of uncertain factors such as material properties, geometric characteristics, and collision environment in the vehicle structure, which directly affect the reliability of vehicle side collision. Reliability-based design optimization (RBDO) can not only obtain the best design solution, but also fully consider the influence of these uncertain factors, which can effectively balance the best design goal and reliability. RBDO has become one of the most powerful methods for solving complex problems in engineering design, and many advanced theories and methods have been developed. Traditional RBDO solutions generally fall into three categories. (1) Double circulation method. This type of method reliability analysis is nested inside the design optimization and needs to be performed each time a new point in the design space is iterated through the optimization algorithm. The double-cycle method is very inefficient and cannot meet the requirements of engineering applications. (2) Single cycle method. These methods use approximation techniques or Karush-Kuhn-Tucker (KKT) optimality conditions instead of reliability analysis, thereby transforming nested double loops into single loop problems. It is very computationally efficient, but may not converge when the functional function is highly nonlinear. (3) Sequence optimization method. This type of approach decouples reliability analysis from design optimization, which are performed sequentially until convergence. The sequence optimization method is one of the most effective RBDO methods, which has the characteristics of high precision, high precision and strong stability.

The above-mentioned traditional RBDO solution method basically adopts the first-order reliability analysis method, which requires iterative calculation of MPP points, which may lead to reduced efficiency or insufficient accuracy when dealing with high-dimensional and highly nonlinear problems. Considering the high complexity, extreme working conditions (environment), and high degree of nonlinearity of the vehicle side collision problem, traditional methods cannot effectively solve this problem. The above methods are also unable to integrate many advanced reliability analysis methods, such as dimensionality reduction integration method (DRM), sparse grid integration (SGI), etc. Compared to MPP-based reliability analysis methods, these methods do not need to compute derivatives of functional functions, do not perform iterative solution processes, and are computationally more efficient in some cases. If these advanced methods are integrated into the sequence optimization method, it is hopeful to develop an efficient and high-accuracy RBDO solution method, which is of great significance to the research on the optimization design problem of vehicle side crash reliability.

For example, Chinese Patent Announcement No. CN 104036100 B, authorized announcement date 20170510, discloses an optimization method for automotive reliability design based on Bayesian deviation correction under uncertainty, which belongs to the technical field of automotive reliability design optimization. The method includes the following steps: step 1: define a reliability-based design optimization (RBDO) problem; step 2: construct a design of experiments (DOE) matrix for the Bayesian inference bias model and the initial response surface model; step 3: use the The described deviation model corrects the initial response surface model and quantifies the uncertainty from repeated experiments and CAE simulation; Step 4: Run the RBDO optimization program to find the optimal and most reliable solution; Step 5: Perform Monte Carlo simulation (MCS) Verify the reliability of the obtained solution. This method has many operations, time-consuming and low efficiency.

SUMMARY OF THE INVENTION

The technical problem to be solved by the present invention is that under the condition of completing the optimization target, for example, the optimization target is set to minimize the weight of the whole vehicle on the premise of ensuring the performance of side collision, further reduce the number of operations, and maintain higher precision under the condition of, Save optimization design time and improve optimization design efficiency.

In order to solve the above problems, a high-performance vehicle crash safety reliability design optimization method is adopted. The reliability analysis and deterministic optimization of this method are carried out in sequence. And the PDF of the response is obtained according to the principle of maximum entropy. Then, the deterministic optimization is carried out, and the translation distance is obtained by using the function function movement method based on the response PDF, and an equivalent deterministic optimization model is constructed and solved to obtain a new optimal solution. Taking the new optimal solution as input, the process of reliability analysis and deterministic optimization is repeated until the convergence conditions are met, and finally the optimal solution of the original optimization model is obtained.

Specific steps are as follows:

Step 1: Define a mathematical optimization model according to the requirements of reliability design optimization of vehicle side collision safety. Defining a mathematical optimization model includes determining the deterministic design variables, random design variables and random parameters of the system. According to the probability of random variables and parameters Obtain the probability distribution of statistical characteristics; at the same time, determine the objective function, establish the system function function, and set the target reliability;

初始点

设置允许误差ε(一个较小的正数)，令a＝1来记录功能函数编号；Step 2: Set the number of iterations k=1, the moving distance of the function function PDF

initial point

Set the allowable error ε (a small positive number), let a=1 to record the function function number;

The third step: introducing a univariate dimensionality reduction method to decompose the system function function into a subsystem of a single random parameter, which is used to convert the high-dimensional integral of calculating the response origin moment into calculating a one-dimensional integral Q _ij ;

The fourth step: calculate the one-dimensional integral Q _ij by using the Gaussian series numerical integration method, and use the binomial theorem combination Q _ij to calculate the origin moment m _a,l .

使用最大熵方法求取

得到ρ_a(y)的解析式，在得到响应y的PDFρ(y)后，通过对ρ(y)进行积分来计算约束的可靠度R_a；Step 5: Suppose the probability density function of the response y to be estimated is

Use the maximum entropy method to find

The analytical formula of ρ _a (y) is obtained, and after obtaining the PDFρ(y) of the response y, the reliability of the constraint _Ra is calculated by integrating ρ(y);

The sixth step: set a=a+1, and repeat the third to fifth steps until the probability density function ρ _a and reliability R _a (a=1,2,...,n _g ) of all functional function responses are obtained;

Step 7: Calculate the moving distance using the functional function movement method based on the response PDF

和最小目标函数值

Step 8: Build a deterministic optimization model and solve it to obtain the optimal solution for the k-th iteration

and the minimum objective function value

是否成立，若成立执行第十步；若不成立，令k＝k+1，a＝1，重复第三～九歩。Step 9: Judgment

Whether it is established, if so, go to step ten; if not, set k=k+1, a=1, and repeat the third to ninth steps.

和最小目标函数值

结束。Step 10: Output the optimal solution

and the minimum objective function value

Finish.

As a further improvement of the present invention, in the first step, the mathematical model of reliability design optimization is:

是目标可靠度，d^L,d^U,

是确定性设计变量d和随机设计变量X各自均值的上下界，用Z＝(X,P)代表所有的随机变量/参数，则g_a(d,X,P)简写为g_a(d,Z)；由于进行不确定性分析时，d不变，因此可将g_a(d,Z)简写为g_a(Z)。Among them, C(d, μ _X ) is the objective function, P{g _a (d, X, P)≥0} is the reliability of the a-th functional function, and g _a (d, X, P) is the a-th function Function function, d is a deterministic design variable, X is a random design variable, P is a random parameter, μ _X , μ _P are the mean values of X and P, respectively,

is the target reliability, d ^L , d ^U ,

is the upper and lower bounds of the respective mean values of the deterministic design variable d and the random design variable X. Use Z=(X, P) to represent all random variables/parameters, then g _a (d, X, P) is abbreviated as g _a (d, Z); since d does not change during uncertainty analysis, ga ( _d , Z) can be abbreviated as _ga (Z).

As a further improvement of the present invention, in the first step, when the optimization goal is to minimize the vehicle weight under the premise of ensuring the performance of side impact, the side impact test standard of the European Enhanced Vehicle Safety Committee is adopted, then the reliability design is optimized. The mathematical model of is:

stP (abdominal load: F _Abdom ≤ 1.0kN) ≥ R ^t

P(Rib deformation (upper/middle/lower):Def _{rib_l/rib_m/rib_u} ≤32mm)≥R ^t

P (viscosity standard (upper/middle/lower): VC _{upper/middle/lower} ≤0.32m/s)≥R ^t

P(Comprehensive pubic force: Force _pubic ≤4.0kN)≥R ^t

P(B-pillar midpoint speed: Vel _B-pillar ≤9.9mm/ms)≥R ^t

P(B-pillar speed at front door: Vel _door ≤15.7mm/ms)≥R ^t

As a further improvement of the present invention, the optimal Latin hypercube sampling and quadratic reverse stepwise regression are used to establish Weight, F _Abdom , Def _{rib_l} , Def _{rib_m} , Def _{rib_u} , VC _upper , VC _middle , VC _lower , Force _pubic , Vel Global response surface model for _B-pillar and Vel _door .

In addition, in the above technical solution, the random design variable X includes X ₁ : the thickness of the inner wall of the B-pillar, X ₂ : the thickness of the reinforcement of the B-pillar, X ₃ : the thickness of the inner wall of the floor side, X ₄ : the thickness of the beam, X ₅ : thickness of door beam, X ₆ : thickness of door belt line reinforcement, X ₇ : thickness of roof rail, X ₈ : inner wall material of B-pillar and X ₉ : material of floor side inner wall; the random parameter P includes X ₁₀ : Move the barrier height and X ₁₁ : impact position.

As a further improvement of the present invention, the specific steps of the third step are: calculating the first-order origin moment m _a,l of the a-th functional function _ga (Z), where l=1,2,3,4 expression for:

where E{·} represents the mathematical expectation operator;

Additive decomposition of the functional function y = g _a (Z):

where μ _i represents the mean value of the random variable Z _i , and N represents the number of random variables;

The formula for the lth order origin moment of the response y after using the univariate dimensionality reduction method:

Expand Equation (4) using the binomial theorem to get:

Make the following definitions:

The formula (4) can be simplified to:

可通过下列递归公式求出：above

It can be found by the following recursive formula:

这是一个一维积分；为方便描述，用Q_ij表示该积分，则有：By formulas (3) to (8), high-dimensional integrals can be transformed into mathematical expectations

This is a one-dimensional integral; for the convenience of description, the integral is represented by Q _ij , then:

是Z_i的概率密度函数。in the formula

is the probability density function of Z _i .

As a further improvement of the present invention, the concrete steps of the 4th step are:

The Gaussian series numerical integration formula is:

In the formula, ω _i,m represents the weight, vi _,h represents the integration node, and m represents the number of integration nodes.

Calculate all Q _ij by the Gaussian series numerical integration, and combine the formulas (7) to (8), the origin moment m _a,l can be obtained. Each one-dimensional integration uses m integration nodes. For the N-dimensional functional function, all The number of nodes to be integrated is m×N+1, that is, m×N+1 functional functions need to be calculated.

As a further improvement of the present invention, the concrete steps of the 5th step are:

则其山农熵计算公式为：Suppose the probability density function of the response y to be estimated is

Then its Shannong entropy calculation formula is:

可描述为以下优化问题：Use the maximum entropy method (MEM) to find

It can be described as the following optimization problem:

Use the Lagrangian multiplier method to solve, and construct the Lagrangian function as follows:

When the partial derivative of the Lagrangian function with respect to the probability density function is equal to 0, Equation (9) obtains the extreme value, and the analytical formula of ρ _a (y) is obtained as follows:

After finding the Lagrangian multiplier λ _l (l=0,1,2,3,4) and obtaining the PDFρ(y) of the response y, the reliability R of the constraint can be calculated by integrating ρ(y) _a :

As a further improvement of the present invention, the concrete steps of the 7th step are:

Rewrite the reliability optimization model as:

代表第a个功能函数响应的PDF，对于优化过程中的每一组试探点(d,μ_X)，都可以用第三～五步求出响应的PDF，因此

是设计变量的函数，可以表示为

In the formula,

The PDF representing the response of the a-th functional function, for each set of test points (d, μ _X ) in the optimization process, the PDF of the response can be obtained in steps 3 to 5, so

is a function of the design variables and can be expressed as

The basic idea of the functional function movement method based on response PDF is: Usually, the reliability requirement in reliability design (≥0.90) is usually much higher than the reliability achieved by deterministic design (≈0.5), that is, the probability constraint is higher than the deterministic Constraints are tighter. Geometrically, when the target reliability R ^t >0.5, the feasible region of probabilistic design optimization is narrower than that of deterministic optimization. The key to establishing an equivalent deterministic optimization model lies in the conversion from probabilistic constraints to deterministic constraints, and the translation distance from the deterministic constraint boundary to the probability constraint boundary needs to be calculated. By translation, the actual constraint boundary is moved from the deterministic constraint boundary to the probabilistic constraint boundary, thereby improving the reliability of the constraint.

和

分别是第k次和第(k+1)次迭代得到的响应的PDF，

代表第k次的最优点，(d,μ_X)代表第(k+1)次迭代的试探点；Assumption

and

are the PDFs of the responses obtained at the kth and (k+1)th iterations, respectively,

represents the kth optimal point, (d, μ _X ) represents the test point of the (k+1)th iteration;

The relationship between the PDFs of the k-th and k+1-th iteration responses is:

代表设计变量从

变为(d,μ_X)时ρ_a(y|d,μ_X)的移动距离，这个距离等于功能函数值之差：In the formula,

represent design variables from

When it becomes (d, μ _X ), the moving distance of ρ _a (y|d, μ _X ) is equal to the difference of function function values:

In order to ensure that the probability constraint is satisfied, the constraint reliability of the current iteration step needs to be greater than or equal to the target reliability:

Substituting equation (17) into equation (19), we get:

表达式已经求出，所以式(20)不等号左边项的大小只与

的取值有关，可以定义为

的函数：because at the kth iteration

The expression has been calculated, so the size of the left-hand term of the inequality sign in formula (20) is only the same as

is related to the value of , which can be defined as

The function:

Equation (20) can be rewritten as:

By solving the above equation, the translation distance can be obtained

In the formula, arch represents the inverse function of the function h, and substituting the formula (18) into the formula (23) can get:

Considering that the unequal sign of the deterministic optimization constraint and the probability constraint is the same, the equation (24) can be taken as the equal sign, and the final translation distance formula is:

As a further improvement of the present invention, the concrete steps of the eighth step are:

The formula of the k-th iteration deterministic optimization model is:

和最小目标函数值

Solve the deterministic optimization model to get the optimal solution for the k-th iteration

and the minimum objective function value

The advantages and beneficial effects of the present invention lie in the following points:

1. The method of the present invention decomposes the design optimization of reliability into a sequential solution process performed in turn by reliability evaluation and deterministic optimization, avoiding a double-nested solution process; Passing the conflicting constraints through the function function based on the response PDF The move method moves to the reliable region, guaranteeing to find the optimal solution that satisfies the probability constraints. This method combines the advantages of high precision and high efficiency.

2. The method of the present invention uses the first four-order statistical moments of the functional function response in the reliability analysis. Compared with the traditional first-order reliability method that only uses the first-order and second-order moments, more information is used; although Both of them require less calculation amount, but the method of the present invention can relatively obtain more accurate calculation results.

3. The method of the present invention can integrate many advanced reliability analysis methods, such as bivariate (or multivariate) dimensionality reduction integration method, polynomial chaotic expansion method, sparse grid numerical integration method, etc., which can not only obtain higher accuracy, but also It can expand the scope of application and solve engineering problems with higher complexity and stronger nonlinearity.

Description of drawings

FIG. 1 is a flow chart of the method of the present invention.

Figure 2 is a finite element (FEM) model of an automobile side impact.

FIG. 3 is a schematic diagram of a functional function movement method based on response PDF.

Figure 4 is a schematic diagram of the PDF deformation approximation of the response for two consecutive iterations.

FIG. 5 shows the change of the objective function with respect to the number of times of functional function calculation.

Detailed ways

The present invention will be further described in detail below with reference to the accompanying drawings and specific examples, using a method compared with Monte Carlo simulation (MCS). Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

The first step: define a mathematical model according to the requirements of reliability design optimization of automotive lightweight design. This step needs to determine the deterministic design variables, random design variables and random parameters of the system, and obtain the probability distribution according to the probability and statistical characteristics of the random variables and parameters. At the same time, it is necessary to determine the objective function, establish the system function function, and set the target reliability.

For side impact protection, vehicle designs must meet government and automotive industry specifications for side impact protection. At present, there are three main side impact protection standards that are referenced or used in China: (1) the Chinese national standard "Occupant Protection for Vehicle Side Pillar Collision" (GB/T37337-2019); Safety Standard Side Impact Test Standard (FMVSS); (3) European Enhanced Vehicle Safety Committee (EEVC) Side Impact Test Standard. In this example, the European Enhanced Vehicle Safety Council (EEVC) side impact test standard is used. The main indicators of dummy response to side impact research, the EEVC standard specifies the response of dummy mainly including peak abdominal force (AbdomenLoad), rib deformation (Rib Deflection), viscosity index (VC: viscous criteria), pubicsymphysis force (Pubicsymphysis force) and Head Injury Criterion (HIC: Head injury criterion) in five parts. In addition, since the midpoint velocity of the B-pillar and the velocity of the B-pillar at the front door are also two indicators of great interest, these two indicators will also be used in this example. Table 1 is the details of this standard.

Table 1 Side impact test standards of the European Enhanced Vehicle Safety Council (EEVC)

The optimization goal of this example is to minimize the vehicle weight under the premise of ensuring the performance of side collision, and the target reliability is taken as R ^t =0.99. According to the safety standards shown in Table 1 and the standards of the midpoint speed of the B-pillar and the speed of the B-pillar at the front door, the formula for the RBDO problem in this example is:

The system model includes the vehicle finite element model, the side impact virtual finite element model and the side moving barrier model. Figure 2 is a finite element model of the vehicle, which contains 85,941 shell elements and 96,122 nodes. In the finite element simulation of the side collision accident, the initial velocity of the moving barrier is set to be 49.89 km/h. On the SGIOrigin2000 computer, the time for a finite element simulation is about 20 hours. In view of the high computational cost of the finite element method, in order to facilitate the calculation, this example adopts the optimal Latin Hypercube Sampling and quadratic backward-stepwise regression to establish the global response surface model. The formula of the response surface established in this example is:

Weight=1.98+4.90x ₁ +6.67x ₂ +6.98x ₃ +4.01x ₄ +1.78x ₅ +2.73x ₇

F _Abdom = 1.16-0.3717x ₂ x ₄ -0.00931x ₂ x ₁₀ -0.484x ₃ x ₉ +0.01343x ₆ x ₁₀

Def _{rib_l} = 46.36-9.9x ₂ -12.9x ₁ x ₈ +0.1107x ₃ x ₁₀

Def _{rib_m} = 33.86+2.95x ₃ +0.1792x ₁₀ +5.057x ₁ x ₂ -11.0x ₂ x ₈ -0.0215x ₅ x ₁₀ -9.98x ₇ x ₈ +22.0x ₈ x ₉

Def _{rib_u} = 28.98+3.818x ₃ -4.2x ₁ x ₂ +0.0207x ₅ x ₁₀ +6.63x ₆ x ₉ -7.7x ₇ x ₈ +0.32x ₉ x ₁₀

VC _upper = 0.261-0.0159x ₁ x ₂ -0.188x ₁ x ₈ -0.019x ₂ x ₇ +0.0144x ₃ x ₅ +0.0008757x ₅ x ₁₀ +0.08045x ₆ x ₉ +0.00139x ₈ x ₁₁ +0.00001575x ₁₀ x ₁₁

Vel _B-pillar = 10.85-0.674x ₁ x ₂ -1.95x ₂ x ₈ +0.02054x ₃ x ₁ 0-0.0198x ₄ x ₁₀ +0.028x ₆ x ₁₀

This example has 10 probabilistic constraints, including 11 random design variables and parameters such as geometric dimensions, material properties of key parts, height of moving barriers and impact positions. There are 0 deterministic design variables, 9 random design variables (X ₁ -X ₉ ) and 2 random parameters (X ₁₀ , X ₁₁ ). The names of random design variables and parameters, distribution types and parameters, and the range of variation of the mean are shown in Table 2.

Table 2 Characteristics of random design variables and parameters

令a＝1来记录功能函数编号。The second step: set the number of iterations k=1, the initial point of the design variable (μ ₁ , μ ₂ ,...,μ ₉ )=(1,1,1,1,1,1,1,0.3,0.3), random Parameter mean (μ ₁₀ , μ ₁₁ )=(10,10) set the allowable error ε=0.01, the moving distance of the function function PDF

Let a=1 to record the function function number.

The third step: introducing a univariate dimensionality reduction method to decompose the system function function into a subsystem of a single random parameter, so that the high-dimensional integral of the calculated response origin moment is converted into a one-dimensional integral. The first l-order origin moment m _a,l (l=1,2,3,4) of the a-th functional function _ga (Z) is calculated as:

where E{·} represents the mathematical expectation operator.

Additive decomposition of the functional function y = g _a (Z)

where μ _j represents the mean value of random variable Z _j , and N represents the number of random variables.

The formula for the lth order origin moment of the response y after using the univariate dimensionality reduction method

Expand Equation (4) using the binomial theorem to get:

Make the following definitions:

The formula (4) can be simplified to:

可通过下列递归公式求出：above

It can be found by the following recursive formula:

这是一个一维积分。为方便描述，用Q_ij(i＝1,…,N,j＝1,…,l)表示该积分，则有：By formulas (3) to (8), high-dimensional integrals can be transformed into mathematical expectations

This is a one-dimensional integral. For the convenience of description, the integral is represented by Q _ij (i=1,...,N,j=1,...,l), then:

是Z_i的概率密度函数。in the formula

is the probability density function of Z _i .

The fourth step: calculate the one-dimensional integral Q _ij by using the Gaussian series numerical integration method, and use the binomial theorem combination Q _ij to calculate the origin moment m _a,l . The Gaussian series numerical integration formula is:

In the formula, ω _i,m represents the weight, vi _,h represents the integration node, and m represents the number of integration nodes.

Calculating all Q _ij by Gaussian series numerical integration, and combining equations (7) to (8), the origin moment ma _,l (l=1, 2, 3, 4) can be obtained. Each 1D integration uses 4 integration nodes. The model has a total of 10 functional functions, and the number of integration nodes is shown in Table 3:

Table 3 Calculation times of each functional function

则其山农熵计算公式为：Step 5: Suppose the probability density function of the response y to be estimated is

Then its Shannong entropy calculation formula is:

可描述为以下优化问题：Use the maximum entropy method (MEM) to find

It can be described as the following optimization problem:

Use the Lagrangian multiplier method to solve, and construct the Lagrangian function as follows:

When the partial derivative of the Lagrangian function with respect to the probability density function is equal to 0, Equation (9) obtains the extreme value, and the analytical formula of ρ _a (y) is obtained as follows:

After finding the Lagrangian multiplier λ _l (l=0,1,2,3,4) and obtaining the PDFρ(y) of the response y, the reliability of the constraint can be calculated by integrating ρ(y):

Step 6: Set a=a+1, and repeat steps 3 to 5 until the probability density function ρ _a and reliability R _a (a=1, 2, . . . , 10) of all functional function responses are obtained.

(a＝1,2,…,10)。Step 7: Calculate the moving distance using the function-based moving method based on the response PDF

(a=1,2,...,10).

Rewrite the reliability optimization model as:

代表第a个功能函数响应的PDF。对于优化过程中的每一组试探点(d,μ_X)，都可以用第三～五步求出响应的PDF，因此

是设计变量的函数，可以表示为

In the formula,

PDF representing the response of the a-th functional function. For each set of tentative points (d, μ _X ) in the optimization process, the third to fifth steps can be used to find the PDF of the response, so

is a function of the design variables and can be expressed as

The basic idea of the functional function movement method based on response PDF is: Usually, the reliability requirement in reliability design (≥0.90) is usually much higher than the reliability achieved by deterministic design (≈0.5), that is, the probability constraint is higher than the deterministic Constraints are tighter. Geometrically, when the target reliability R ^t >0.5, the feasible region of probabilistic design optimization is narrower than that of deterministic optimization. The key to establishing an equivalent deterministic optimization model lies in the conversion from probabilistic constraints to deterministic constraints, and the translation distance from the deterministic constraint boundary to the probability constraint boundary needs to be calculated. Through translation, the deterministic constraint boundary that does not satisfy the reliability condition is moved to the probability constraint boundary, thereby improving the reliability of the constraint.

和

组成；另一个是随机空间，由随机变量X₁和X₂组成。最内层虚线表示的曲面是确定性设计中的约束边界g(X₁,X₂)＝0，平移后，得到中间细实线代表的实际设计方案的约束边界。该曲面更加靠近最外层的概率约束边界，对应的可靠度也更高。经过数次平移后，实际设计方案边界与概率设计边界重合，得到满足约束的设计方案。Take the probability constraint P{g(X ₁ , X ₂ )≥0}=R ^t as an example, which has two random design variables. As shown in Figure 3, there are two coordinate systems in the figure: one is the design space, which is determined by the design variables

and

composition; the other is a random space, consisting _of random variables X1 and _X2 . The surface represented by the innermost dotted line is the constraint boundary g(X ₁ , X ₂ )=0 in the deterministic design. After translation, the constraint boundary of the actual design scheme represented by the middle thin solid line is obtained. The surface is closer to the outermost probability constraint boundary, and the corresponding reliability is higher. After several translations, the boundary of the actual design scheme coincides with the boundary of the probabilistic design scheme, and a design scheme that satisfies the constraints is obtained.

和

分别是第k次和第(k+1)次迭代得到的响应的PDF。

代表第k次的最优点，(d,μ_X)代表第(k+1)次迭代的试探点。一般而言，如附图4所示，

和

同时存在伸缩和平移两种变形。对于伸缩变形，PDF曲线表现为沿垂直轴伸展或收缩；而平移变形是PDF曲线显示沿水平轴平行移动。Assumption

and

are the PDFs of the responses obtained at the kth and (k+1)th iterations, respectively.

represents the k-th optimal point, and (d, μ _X ) represents the trial point of the (k+1)-th iteration. In general, as shown in Figure 4,

and

There are two kinds of deformations, telescopic and translational at the same time. For telescopic deformation, the PDF curve appears to expand or contract along the vertical axis; while for translation deformation, the PDF curve appears to move parallel to the horizontal axis.

和

仅存在平移变形而没有伸缩变形，如附图4所示。由于在两次连续迭代中获得的候选设计点通常位于较小的邻域内，尤其是当优化设计接近收敛时，在这种情况下，响应的PDF的形状非常接近，两者的差异可视为仅有平移变形，因此进行这样的近似是合理的。基于这个近似，第k次和第k+1次迭代响应的PDF的关系为：Introduce an approximation:

and

There is only translational deformation and no telescopic deformation, as shown in FIG. 4 . Since the candidate design points obtained in two consecutive iterations are usually located in a small neighborhood, especially when the optimized design is close to convergence, in this case the shape of the PDF of the response is very close, the difference between the two can be seen as There are only translational deformations, so it is reasonable to make such an approximation. Based on this approximation, the relationship between the PDFs of the k-th and k+1-th iteration responses is:

代表设计变量从

变为(d,μ_X)时ρ_a(y|d,μ_X)的移动距离，这个距离等于功能函数值之差:In the formula,

represent design variables from

When it becomes (d, μ _X ), the moving distance of ρ _a (y|d, μ _X ) is equal to the difference of function function values:

In order to ensure that the probability constraint is satisfied, the constraint reliability of the current iteration step needs to be greater than or equal to the target reliability:

Substituting equation (17) into equation (19), we get:

表达式已经求出，所以式(20)不等号左边项的大小只与

的取值有关，可以定义为

的函数because at the kth iteration

The expression has been calculated, so the size of the left-hand term of the inequality sign in formula (20) is only the same as

is related to the value of , which can be defined as

The function

Equation (20) can be rewritten as:

By solving the above equation, the translation distance can be obtained

where arch represents the inverse function of the function h. Substitute equation (18) into equation (23) to get:

Considering that the deterministic optimization constraints and the probability constraints have the same inequality sign, Equation (24) can take the equal sign. The final formula for the translation distance is:

第k次迭代确定性优化模型的公式为：Step 8: Build a deterministic optimization model and solve it to obtain the optimal solution for the k-th iteration

The formula of the k-th iteration deterministic optimization model is:

和最小目标函数值

Solve the deterministic optimization model to get the optimal solution for the k-th iteration

and the minimum objective function value

是否成立，若成立执行第十一步；若不成立，令k＝k+1，a＝1，重复第三～九歩。Step 9: Judgment

Whether it is established, if so, go to the eleventh step; if not, set k=k+1, a=1, and repeat the third to ninth steps.

和最小目标函数值

结束。Step 10: Output the optimal solution

and the minimum objective function value

Finish.

Using the above steps, convergence is achieved after three iterations, and the optimal solution, the minimum objective function value, and the number of functional function calculations obtained in each iteration are shown in Table 4. In order to compare the efficiency, this example is solved by using the double-circulation method, wherein the reliability analysis of the double-circulation method is the same as that of the method of the present invention, and Table 5 is the comparison of the results of the two methods. From Table 4, it can be seen that the proposed method only needs 2118 performance function evaluations to converge, which requires an average of 706 evaluations per cycle. Compared with the double-cycle method, the calculation amount of the method of the present invention is less than 1/15 of it, showing very high efficiency. At the same time, the minimum objective function value obtained by this method is very close to the double-loop method (the error is 0.9%), which shows that the accuracy of this method is similar to that of the double-loop method. Table 6 shows the reliability of each functional function during convergence. It can be seen that except the reliability of the 8th constraint is slightly less than the target reliability R ^t = 0.99 (the difference is only 0.04%), the reliability of other functional functions is very close to or exceeds The target reliability again shows that the method of the present invention has a very high precision. FIG. 5 describes the convergence process of the objective function value with respect to the scalar function of the number of computations in the optimization process. In the figure, each iteration cycle is clearly distinguished from each other. In each cycle, a reliability evaluation is performed before deterministic optimization is performed. To sum up, the method of the present invention has the dual characteristics of high precision and high efficiency, and is worthy of promotion.

Table 4 Reliability design optimization iterative process

Table 5 Comparison of the two methods

Table 6 Probabilistic Constraint Reliability

function function g₁ g₂ g₃ g₄ g₅ g₆ g₇ g₈ g₉ g₁₀ reliability 1.0 0.9901 1.0 1.0 1.0 1.0 1.0 0.9896 0.9997 0.9903

The present invention firstly defines a mathematical optimization model for reliability design optimization of vehicle side collision safety, obtains the probability distribution characteristics of random design variables and parameters by means of probability statistics and the like, and establishes the response surface model of the function function of the vehicle side collision system ; Then, develop a functional function movement method based on response PDF to decouple the traditional nested reliability optimization design into reliability analysis and deterministic optimization sequence execution, that is, use a univariate dimensionality reduction method to solve the response PDF for reliability analysis, And calculate the movement amount of the function function to construct equivalent deterministic optimization to obtain better design variables. The above process is repeated until the convergence conditions are met to obtain the optimal solution of the problem. Finally, a specific example of vehicle side impact safety analysis is used to verify the above Feasibility and efficiency of the method.

The present invention proposes an efficient reliability design optimization method for vehicle side collision safety by combining the function function movement method based on response PDF, the univariate dimension reduction method (UDRM) and the maximum entropy method (MEM). The difference between the method of the present invention and the traditional method is that: the proposed method of moving function functions based on the response PDF is used to decouple the reliability analysis and the optimal design, and the efficiency is improved; the reliability analysis uses the first fourth order moment of the response. information, can obtain higher precision, and has the dual characteristics of high precision and high efficiency.

The above content is a further detailed description of the present invention in conjunction with specific preferred embodiments, and it cannot be considered that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art to which the present invention belongs, without departing from the concept of the present invention, several equivalent substitutions or obvious modifications can be made, and the performance or use is the same, which should be regarded as belonging to the protection scope of the present invention. Inside.

Claims (10)

1. A high-efficiency design optimization method for safety and reliability of side collision of an automobile is characterized by comprising the following steps:

the first step is as follows: according to the reliability design optimization requirement of the side collision safety of the automobile, defining a mathematical optimization model, wherein the mathematical optimization model comprises deterministic design variables, random design variables and random parameters of a determination system, and obtaining probability distribution according to the probability statistical characteristics of the random variables and the random parameters; meanwhile, determining a target function, establishing a system function, and setting target reliability;

the second step is that: setting the iteration number k as 1, and the moving distance of the function PDF

a＝1,2,…,n_gInitial point

Setting an allowable error epsilon, and recording the serial number of the function by making a equal to 1;

the third step: introducing a univariate dimension reduction method, decomposing a system function into a subsystem of single random parameters, and converting the high-dimensional integral for calculating the response origin moment into a one-dimensional integral Q _ij；

The fourth step: one-dimensional integral Q is calculated by utilizing a Gaussian series numerical integration method_ijUsing binomial theorem to combine Q_ijCalculating the origin moment m_a,l；

The fifth step: assume that the probability density function of the response y to be estimated is

Solving using the maximum entropy method

To obtain rho_aThe analytical expression (y) calculates the reliability R of the constraint by integrating ρ (y) after obtaining the PDF ρ (y) of the response y_a；

And a sixth step: the third to the fifth steps are repeated until the probability density function rho of all the function responses is obtained_aAnd degree of reliability R_a，a＝1,2,…,n_g；

The seventh step: calculating a moving distance by using a function moving method based on a response PDF

Eighth step: constructing a deterministic optimization model and solving to obtain the optimal solution of the kth iteration

And minimum objective function value

The ninth step: judgment of

If yes, executing the tenth step; if the result is not true, making k equal to k +1 and a equal to 1, and repeating the third step to the ninth step;

the tenth step: outputting an optimal solution

And minimum objective function value

And (6) ending.

2. The efficient design optimization method for the safety and reliability of the side collision of the automobile according to claim 1, wherein in the first step, the mathematical model for the reliability design optimization is as follows:

wherein,

is an objective function, P { g }_a(d, X, P) ≧ 0} is the reliability of the a-th function, g _a(d, X, P) is the a-th function, d is a deterministic design variable, X is a random design variable, P is a random parameter,. mu._X，μ_PAre the average values of X, P,

target reliability, d^L，d^U，

The upper and lower bounds of the mean values of the deterministic design variable d and the random design variable X, where Z ═ X, P represents all random variables/parameters, then g_a(d, X, P) is abbreviated as g_a(d, Z); since d is not changed when uncertainty analysis is performed, g can be set_a(d, Z) is abbreviated to g_a(Z)。

3. The method as claimed in claim 2, wherein in the first step, when the optimization goal is to minimize the weight of the whole vehicle under the premise of ensuring the performance of the side collision, the side collision test standard of the european commission on enhanced vehicle safety is adopted, and then the mathematical model for the reliability design optimization is:

s.t.P (abdominal load: F)_Abdom≤1.0kN)≥R^t

P (Rib deformation)_{Up/middle/down}：Def_{rib_1/rib_m/rib_u}≤32mm)≥R^t

P (viscosity standard)_{Up/middle/down})：VC_{upper/middle/lower}≤0.32m/s)≥R^t

P (pubis comprehensive Force)_pubic≤4.0kN)≥R^t

P (B column midpoint velocity: Vel)_B-pillar≤9.9mm/ms)≥R^t

P (B-pillar speed at front door: Vel)_door≤15.7mm/ms)≥R^t

4. The efficient design optimization method for the safety and reliability of the automobile side collision as claimed in claim 3, wherein the optimal Latin hypercube sampling and the quadratic reverse stepwise regression construction including Weight and F are adopted _Abdom、Def_{rib_1}、Def_{rib_m}、Def_{rib_u}、VC_upper、VC_middle、VC_lower、Force_pubic、Vel_B-pillarAnd Vel_doorThe global response surface model of (2).

5. The efficient design optimization method for automobile side collision safety reliability as claimed in claim 2, wherein the random design variables X comprise X₁: thickness of inner wall of B column, X₂: thickness, X, of B-pillar reinforcement₃: thickness, X, of the inner wall of the floor side₄: thickness, X, of the Beam₅: thickness, X of door beam₆: door belt line reinforcement thickness, X₇: thickness, X, of roof rails₈: b column inner wall material and X₉: floor side inner wall material; the random parameter P comprises X₁₀: moving barrier height and X₁₁The impact position.

6. The efficient design optimization method for the safety and reliability of the side collision of the automobile as claimed in claim 2, wherein the third step comprises the following specific steps:

computing the a-th function g_aFirst l order origin moment m of (Z)_a,l1,2,3,4 is expressed as:

wherein E {. represents a mathematical expectation operator;

for function y ═ g_a(Z) carrying out additive decomposition:

in the formula of_iRepresenting a random variable Z_iN represents the number of random variables;

the ith order origin moment formula of the response y after using the univariate dimension reduction method:

and (3) expanding the formula (4) by using a binomial theorem to obtain:

the following definitions are made:

equation (4) can be simplified to:

of the above formula

This can be found by the following recursive formula:

through the formulas (3) to (8), the high-dimensional integral can be converted into the solution mathematical expectation

This is a one-dimensional integral; by Q_ijRepresenting this integral, then:

in the formula

Is Z_iIs determined.

7. The design optimization method for high-efficiency automobile side collision safety reliability as claimed in claim 6, wherein the fourth step comprises the following specific steps:

the gaussian series numerical integration formula is:

in the formula of omega_i,mRepresents a weight, v_i,hRepresents the number of integral nodes, and m represents the number of integral nodes;

calculation of all Q by Gaussian series numerical integration_ijCombining equations (7) to (8) to determine the origin moment m_a,lFor the N-dimensional function, the number of required integration nodes is m × N +1, that is, m × N +1 times of function functions need to be calculated.

8. The efficient design optimization method for the safety and reliability of the side collision of the automobile according to claim 7, wherein the fifth step comprises the following specific steps:

assume that the probability density function of the response y to be estimated is

The Shannon entropy calculation formula is as follows:

solving using the maximum entropy method

The following optimization problem can be described:

and (3) solving by using a Lagrange multiplier method, and constructing a Lagrange function as follows:

When the partial derivative of the Lagrangian function with respect to the probability density function is equal to 0, equation (9) takes an extremum value, resulting in ρ_aThe analytical formula of (y) is as follows:

solving lagrange multiplier lambda_lAfter the PDF rho (y) of the response y is obtained, the reliability R of the constraint can be calculated by integrating the rho (y)_a：

9. The efficient design optimization method for the safety and reliability of the side collision of the automobile according to claim 8, wherein the seventh step comprises the following specific steps:

the reliability optimization model is rewritten as:

in the formula,

PDF representing the a-th function response, for each set of heuristic points (d, μ) in the optimization process_X) The PDF of the response can be found in the third to fifth steps, therefore

Is a function of a design variable and can be expressed as

Suppose that

And

the PDF of the response from the k and (k +1) th iterations respectively,

represents the optimal point of the k-th order, (d, μ_X) Probe points representing the (k +1) th iteration;

the relation of the PDFs of the kth and (k +1) th iterative responses is as follows:

in the formula,

represents a design variable from

Becomes (d, mu)_X) Time rho_a(y|d,μ_X) Is equal to the difference between the function values:

in order to ensure that the probability constraint is satisfied, the constraint reliability of the current iteration step needs to be greater than or equal to the target reliability:

By substituting formula (17) for formula (19), we obtain:

because at the kth iteration

The expression has been solved so that the left-hand terms of the equal sign of equation (20) are only equal in magnitude to

Is related to and can be defined as

Function of (c):

equation (20) can be rewritten as:

by solving the above equation, the translation distance can be obtained

In the formula, the arch represents an inverse function of the function h, and formula (18) is substituted for formula (23) to obtain:

considering that the deterministic optimization constraint is the same as the unequal sign of the probabilistic constraint, equation (24) can take an equal sign, and the equation for finally obtaining the translation distance is as follows:

10. the efficient design optimization method for the safety and reliability of the side collision of the automobile according to claim 9, wherein the eighth step comprises the following specific steps:

the formula of the kth iteration deterministic optimization model is as follows:

solving the deterministic optimization model to obtain the optimal solution of the kth iteration

And minimum objective function value

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