CN102521456A

CN102521456A - Optimal design method of arched girder based on neural network and genetic algorithm

Info

Publication number: CN102521456A
Application number: CN2011104195486A
Authority: CN
Inventors: 苑明海; 纪爱敏; 丁月; 郭平芳
Original assignee: Hohai University HHU
Current assignee: Hohai University HHU
Priority date: 2011-12-15
Filing date: 2011-12-15
Publication date: 2012-06-27

Abstract

The invention relates to an optimal design method of an arched girder based on neural network and genetic algorithm, belonging to the machine design and automatic technical field. The method comprises the steps of using the uniform test method to obtain a sample according to the design characteristic of the arched girder; training the BP neural network containing singer hidden layer; and using the BP neutral network to replace the finite element to judge constraint condition and determine fitness in the genetic algorithm. The global optimal solution is obtained by the genetic algorithm and the arched girder is structured, so that the weight of the arched girder is the minimum in the deformation condition. The method of the invention has fast operation speed, high precision, good effect in usage and good application prospect.

Description

Optimal Design Method of Arch Beam Based on Neural Network and Genetic Algorithm

技术领域 technical field

本发明涉及一种基于神经网络和遗传算法的拱梁优化设计方法，属于机械设计与自动化领域。The invention relates to an optimal design method of an arch beam based on a neural network and a genetic algorithm, and belongs to the field of mechanical design and automation.

背景技术 Background technique

拱结构由于外型美观、受力明确，在建筑、桥梁等工程实践中得到广泛应用。现有技术中，有的拱梁运用在天文观测中用来支撑天文观测设备，这种拱梁是由型材焊接而成，其结构和受力基本恒定不变。这种结构适合于参数化设计，即设计时只要根据拱梁的半径和变形要求改变其跨度和型材的截面尺寸。传统的设计方法是根据经验进行类比分析，在满足设计要求的情况下，很难达到最佳的结果。Arch structures are widely used in engineering practice such as buildings and bridges due to their beautiful appearance and clear force. In the prior art, some arched beams are used in astronomical observation to support astronomical observation equipment. Such arched beams are welded by profiles, and their structure and force are basically constant. This structure is suitable for parametric design, that is, it only needs to change the span and section size of the profile according to the radius and deformation requirements of the arch beam during design. The traditional design method is to conduct analogy analysis based on experience, and it is difficult to achieve the best result when the design requirements are met.

发明内容 Contents of the invention

本发明的目的是提供一种基于神经网络和遗传算法的拱梁优化设计方法，该方法用均匀设计的思想采集样本训练BP神经网络，并且用BP网络代替有限元分析计算遗传算法的适应度对拱梁结构进行优化，提高了优化速度而且精度高。The purpose of the present invention is to provide a kind of arch beam optimal design method based on neural network and genetic algorithm, this method uses the thought collection sample training BP neural network of uniform design, and replaces the fitness of finite element analysis calculation genetic algorithm with BP network to The arch beam structure is optimized, which improves the optimization speed and high precision.

为解决上述问题，本发明所采用的技术方案是提供一种基于神经网络和遗传算法的拱梁优化设计方法，其特征在于，包含以下步骤：In order to solve the above problems, the technical solution adopted in the present invention is to provide a kind of arch beam optimal design method based on neural network and genetic algorithm, it is characterized in that, comprises the following steps:

(1)建立拱梁优化模型(1) Establish the arch beam optimization model

跨度和所受载荷一定的拱梁优化准则为就是要求在满足刚度、强度的情况下使得重量最轻。拱梁结构中包括等边角钢和扁铁，其优化的数学模型表达为：The optimization criterion of the arch beam with a certain span and load is to make the weight the lightest while satisfying the rigidity and strength. The arch beam structure includes equilateral angle steel and flat iron, and its optimized mathematical model is expressed as:

$\{\begin{matrix} X x = = {[[{B B}_{11},, {B B}_{22},, {t t}_{11},, {t t}_{22}]]}^{T T} \\ min min W W ((X x)) \\ {σ σ}_{max max} \leq \leq [[σ σ]] \\ {d d}_{max max} \leq \leq [[d d]] \\ {X x}_{min min} \leq \leq X x \leq \leq {X x}_{max max} \end{matrix} - - - - - - ((11))$

其中，X为设计变量；B₁、t₁为拱梁中等边角钢的边长和厚度；B₂、t₂为拱梁中扁铁的宽度和厚度；d_max、[d]为拱梁变形时竖直方向的最大位移和允许位移；σ_max、[σ]为拱梁最大应力和许用应力；X_min、X_max为设计变量的上、下限；W(X)为BP网络计算的重量值。Among them, X is the design variable; B ₁ and t ₁ are the side length and thickness of the medium angle steel of the arch beam; B ₂ and t ₂ are the width and thickness of the flat iron in the arch beam; d _max and [d] are the vertical The maximum displacement and allowable displacement in the vertical direction; σ _max and [σ] are the maximum stress and allowable stress of the arch beam; X _min and X _max are the upper and lower limits of the design variables; W(X) is the weight value calculated by the BP network.

由上面的数学模型可以看出，这种拱梁的优化问题是单目标多约束的优化。用遗传算法对约束问题进行优化时需要设计惩罚函数，即约束条件函数：It can be seen from the above mathematical model that the optimization problem of this kind of arch beam is a single-objective multi-constraint optimization. When using genetic algorithm to optimize the constraint problem, it is necessary to design a penalty function, that is, a constraint function:

惩罚函数即约束条件函数设计为： $P (X) = W (X) + r Σ_{i = 1}^{2} (\max {[0, g_{i} (x)]}^{2})$ The penalty function, that is, the constraint function is designed as: $P (x) = W (x) + r Σ_{i = 1}^{2} (\max {[0, g_{i} (x)]}^{2})$

其中， $g_{1} (x) = \frac{σ_{\max}}{[σ]} - 1,$ $g_{2} (x) = \frac{d_{\max}}{[d]} - 1,$ r为正系数。in, $g_{1} (x) = \frac{σ_{\max}}{[σ]} - 1,$ $g_{2} (x) = \frac{d_{\max}}{[d]} - 1,$ r is a positive coefficient.

适应度函数为：f(X)＝C₀-P(X)，C₀为保证f(X)为正的常数。f(X)是关于W(X)、σ_max、d_max的函数，即f(X)＝f(W(X)，σ_max，d_max)，只要计算出W(X)、σ_max、d_max就可以得到此结构的适应度，进而可以进行结构的优化。The fitness function is: f(X)=C ₀ -P(X), and C ₀ is a constant to ensure that f(X) is positive. f(X) is a function about W(X), σ _max , d _max , that is, f(X)=f(W(X), σ _max , d _max ), as long as W(X), σ max , d _max are calculated d _max can get the fitness of this structure, and then can optimize the structure.

(2)拱梁优化的过程(2) The process of arch beam optimization

①样本点的获取① Acquisition of sample points

训练BP神经网络必需有一定数量的样本点，样本点的选取直接影响BP网络的质量。通过试验的方法来获得样本点显然不可取，因为构成梁的型材截面不断发生变化，实际中不可能生产出很多的成品来做试验。因此我们用有限元软件对不同尺寸的拱梁进行分析，获得样本点。但由于型材的种类繁多，组合每一种情况进行分析，工作量大，如果进行全面计算就失去了优化设计的意义。本发明采取均匀设计的思想确定样本点，进行有限元分析。这样就大大减少了样本点的数量也能满足训练BP网络的要求。A certain number of sample points are necessary for training BP neural network, and the selection of sample points directly affects the quality of BP network. It is obviously not advisable to obtain sample points by means of experiments, because the profiles of the beams are constantly changing, and it is impossible to produce many finished products for experiments in practice. Therefore, we use finite element software to analyze arch beams of different sizes to obtain sample points. However, due to the wide variety of profiles, combining each case for analysis requires a large workload. If a comprehensive calculation is performed, the meaning of optimal design will be lost. The present invention adopts the idea of uniform design to determine sample points and perform finite element analysis. This greatly reduces the number of sample points and can also meet the requirements of training BP network.

均匀设计(Uniform Design)是一种试验设计方法(Experimental Design Method)，称为均匀设计(Uniform Design)或均匀设计试验法(Uniform Design Experimentation)，或空间填充设计。它是只考虑试验点在试验范围内均匀散布的一种试验设计方法。它由方开泰教授和数学家王元在1978年共同提出，是数论方法中的“伪蒙特卡罗方法”的一个应用。Uniform Design (Uniform Design) is an experimental design method (Experimental Design Method), known as Uniform Design (Uniform Design) or Uniform Design Experimentation (Uniform Design Experimentation), or space-filling design. It is an experimental design method that only considers the uniform distribution of experimental points in the experimental range. It was jointly proposed by Professor Fang Kaitai and mathematician Wang Yuan in 1978. It is an application of the "pseudo-Monte Carlo method" in the number theory method.

②BP神经网络的设计②Design of BP neural network

根据拱梁优化模型可以确定输入结点数为四个即B₁、t₁、B₂、t₂四个因素，输出结点为三个即W、σ、d。对上述的优化问题，采用单隐层16个结点。利用样本对BP网络进行训练。According to the arch beam optimization model, it can be determined that the number of input nodes is four, that is, B ₁ , t ₁ , B ₂ , and t ₂ , and the output nodes are three, namely, W, σ, and d. For the above optimization problem, a single hidden layer with 16 nodes is used. Use samples to train the BP network.

③用遗传算法优化的流程③Optimized process with genetic algorithm

在训练好网络后进行遗传算法的优化，遗传算法优化的步骤为：a、产生初始种群；b、用BP神经网络计算适应度和约束条件值，同时满足优化准则和约束条件的就输出结果，否则转向c；c、选择适应度高的个体，执行遗传操作生成新的个体转向b。After the network is trained, optimize the genetic algorithm. The steps of genetic algorithm optimization are: a. Generate the initial population; b. Use the BP neural network to calculate the fitness and constraint values, and output the results if the optimization criteria and constraints are met at the same time. Otherwise, turn to c; c, select individuals with high fitness, and perform genetic operations to generate new individuals and turn to b.

本发明所具有的有益效果：The beneficial effects that the present invention has:

本发明针对天文观测中用来支撑天文观测设备的拱梁设计特点，用均匀试验的方法获取样本，对含单隐层的BP神经网络进行训练，用BP神经网络代替有限元来判断约束条件和确定遗传算法中的适应度。用遗传算法获得拱梁重量的全局最优解，对拱梁进行结构优化，使拱梁在满足变形的条件下重量最小。此方法运行速度快，精度高，在实际使用中取得良好的效果。Aiming at the design characteristics of the arch beam used to support astronomical observation equipment in astronomical observation, the present invention uses the method of uniform test to obtain samples, trains the BP neural network containing a single hidden layer, and uses the BP neural network instead of finite elements to judge the constraint conditions and Determining fitness in a genetic algorithm. The global optimal solution of the weight of the arch beam is obtained by genetic algorithm, and the structure of the arch beam is optimized, so that the weight of the arch beam can be minimized under the condition of satisfying the deformation. This method runs fast and has high precision, and has achieved good results in practical use.

附图说明 Description of drawings

图1拱梁的受力模型；The force model of the arch beam in Fig. 1;

图2遗传算法优化的程序流程图。Figure 2 The program flow chart of genetic algorithm optimization.

具体实施方式 Detailed ways

实施例1Example 1

下面结合附图和实例对本发明做进一步说明如下：Below in conjunction with accompanying drawing and example the present invention is described further as follows:

(1)拱梁优化模型的建立(1) Establishment of arch beam optimization model

拱梁受力模型如图1所示，这种拱梁的受力模型较特殊，梁和水平面成50°，梁上均匀分布若干个广角望远镜和辅助观测设备，梁受竖直向下的重力。梁受到竖直向下的分布力简化受力模型如图1。The force model of the arched beam is shown in Figure 1. The force model of the arched beam is quite special. The angle between the beam and the horizontal plane is 50°. Several wide-angle telescopes and auxiliary observation equipment are evenly distributed on the beam, and the beam is subjected to vertical gravity. The simplified force model of the beam subjected to the vertically downward distributed force is shown in Figure 1.

跨度和所受载荷(天文观测辅助设备重量)一定的拱梁优化就是要求在满足刚度、强度的情况下使得重量最轻。其优化的数学模型可表达为：The optimization of the span and the load (the weight of the astronomical observation auxiliary equipment) of a certain arch beam is to make the weight the lightest under the condition of satisfying the rigidity and strength. Its optimized mathematical model can be expressed as:

其中，B₁、t₁为等边角钢1的边长和厚度；B₂、t₂为扁铁2的宽度和厚度；d_max、[d]为拱梁变形时竖直方向的最大位移和允许位移；σ_max、[σ]为拱梁最大应力和许用应力；X_min、X_max为设计变量的上、下限。由上面的数学模型可以看出，这种拱梁的优化问题是单目标多约束的优化。用遗传算法对约束问题进行优化时需要设计惩罚函数。Among them, B ₁ and t ₁ are the side length and thickness of the equilateral angle steel 1; B ₂ and t ₂ are the width and thickness of the flat iron 2; d _max and [d] are the maximum vertical displacement and Allowable displacement; σ _max and [σ] are the maximum stress and allowable stress of the arch beam; X _min and X _max are the upper and lower limits of the design variables. It can be seen from the above mathematical model that the optimization problem of this kind of arch beam is a single-objective multi-constraint optimization. The penalty function needs to be designed when genetic algorithm is used to optimize the constrained problem.

惩罚函数设计为： $P (X) = W (X) + r Σ_{i = 1}^{2} (\max {[0, g_{i} (x)]}^{2})$ The penalty function is designed as: $P (x) = W (x) + r Σ_{i = 1}^{2} (\max {[0, g_{i} (x)]}^{2})$

其中，W(X)为BP网络计算的重量值， $g_{1} (x) = \frac{σ_{\max}}{[σ]} - 1,$ $g_{2} (x) = \frac{d_{\max}}{[d]} - 1,$ r为正系数；Among them, W(X) is the weight value calculated by the BP network, $g_{1} (x) = \frac{σ_{\max}}{[σ]} - 1,$ $g_{2} (x) = \frac{d_{\max}}{[d]} - 1,$ r is a positive coefficient;

适应度函数为：f(X)＝C₀-P(X)，C₀为保证f(X)为正的常数。f(X)关于W(X)，σ_max，d_max的函数，即f(X)＝f(W(X)，σ_max，d_max)，只要计算出W(X)、σ_max、d_max就可以得到此结构的适应度，进而可以进行结构的优化。The fitness function is: f(X)=C ₀ -P(X), and C ₀ is a constant to ensure that f(X) is positive. The function of f(X) about W(X), σ _max , d _max , that is, f(X)=f(W(X), σ _max , d _max ), as long as W(X), σ _{max , d max} are calculated _max can get the fitness of this structure, and then optimize the structure.

(2)拱梁优化的过程(2) The process of arch beam optimization

①样本点的获取① Acquisition of sample points

训练BP神经网络必需有一定数量的样本点，样本点的选取直接影响BP网络的质量。通过试验的方法来获得样本点显然不可取，因为构成梁的型材截面不断发生变化，实际中不可能生产出很多的成品来做试验。因此我们用有限元软件对不同尺寸的拱梁进行分析，获得样本点。但由于型材的种类繁多，组合每一种情况进行分析，工作量大，如果进行全面计算就失去了此工作的意义。本文采取均匀设计的思想确定样本点，进行有限元分析。这样就大大减少了样本点的数量也能满足训练BP网络的要求。由上述的优化模型，有B₁、t₁、B₂、t₂四个因素影响，其中按国标等边角钢不同边长时有不同的厚度范围，而扁铁不同宽度厚度范围变化不大。因此在设计均匀表格时，根据国标取B₁(36～90mm)、B₂(35～80mm)、t₂(3mm～12mm)为十个水平，而t₁(4mm～8mm)取五个水平。上述尺寸按国标取值，采用混合水平均匀设计表U₁₀(10³×5)。则选取十个训练样本(表1中1～10)和(表1中11～12)两个测试样本点，如表1：A certain number of sample points are necessary for training BP neural network, and the selection of sample points directly affects the quality of BP network. It is obviously not advisable to obtain sample points by means of experiments, because the profiles of the beams are constantly changing, and it is impossible to produce many finished products for experiments in practice. Therefore, we use finite element software to analyze arch beams of different sizes to obtain sample points. However, due to the wide variety of profiles, combining each situation for analysis requires a large workload, and the meaning of this work will be lost if a comprehensive calculation is performed. In this paper, the idea of uniform design is adopted to determine the sample points and carry out finite element analysis. This greatly reduces the number of sample points and can also meet the requirements of training BP network. According to the above optimization model, there are four factors B ₁ , t ₁ , B ₂ , and t ₂ . According to the national standard, equilateral angle steels have different thickness ranges with different side lengths, while flat irons with different widths have little change in thickness ranges. Therefore, when designing a uniform table, take B ₁ (36-90mm), B ₂ (35-80mm), and t ₂ (3mm-12mm) as ten levels according to the national standard, and take five levels for t ₁ (4mm-8mm) . The above dimensions are taken according to the national standard, and the mixed level uniform design table U ₁₀ (10 ³ ×5) is adopted. Then select ten training samples (1-10 in Table 1) and two test sample points (11-12 in Table 1), as shown in Table 1:

表1用混合水平均匀设计的样本点Table 1 Sample points with mixed level uniform design

②BP网络的设计②Design of BP network

根据拱梁优化模型可以确定输入结点数为四个即B₁、t₁、B₂、t₂四个因素，输出结点为三个即W、σ、d。对上述的优化问题，采用单隐层16个结点。利用表1中的样本对BP网络进行训练。在总误差小于0.0001，或学习次数大于10000次后训练完毕，保存BP网络的权值和阈值。表2为对表1中的样本11～12测试的结果，括号内为误差率。According to the arch beam optimization model, it can be determined that the number of input nodes is four, that is, B ₁ , t ₁ , B ₂ , and t ₂ , and the output nodes are three, namely, W, σ, and d. For the above optimization problem, a single hidden layer with 16 nodes is used. Use the samples in Table 1 to train the BP network. After the total error is less than 0.0001, or the number of learning times is greater than 10,000, the training is completed, and the weight and threshold of the BP network are saved. Table 2 shows the test results of samples 11-12 in Table 1, and the error rate is in brackets.

表2网络测试结果Table 2 Network test results

由表2的测试结果看出，误差率较小，网络的训练结果比较理想，可以替代有限元计算遗传算法的适应度。From the test results in Table 2, it can be seen that the error rate is small, and the training results of the network are ideal, which can replace the fitness of the finite element calculation genetic algorithm.

③用遗传算法优化的流程③Optimized process with genetic algorithm

在训练好网络后进行遗传算法的优化，适应度fitness选样本中的最大重量176.89减去网络计算出的重量fitness＝176.89-W(X)，遗传算法的其他初始参数见表3。After the network is trained, the genetic algorithm is optimized. The maximum weight in the fitness selection sample is 176.89 minus the weight calculated by the network fitness=176.89-W(X). Other initial parameters of the genetic algorithm are shown in Table 3.

表3遗传算法的参数Table 3 Parameters of Genetic Algorithm

遗传算法优化的步骤为：(1)、产生初始种群；(2)、用BP神经网络计算适应度和约束条件值，同时满足优化准则和约束条件的就输出结果，否则转向(3)；(3)、选择适应度高的个体，执行遗传操作生成新的个体转向(2)。具体的程序流程见图2。The steps of genetic algorithm optimization are: (1), generate the initial population; (2), use BP neural network to calculate the fitness and constraint value, and output the result if it meets the optimization criteria and constraint conditions at the same time, otherwise turn to (3); ( 3) Select individuals with high fitness, and perform genetic operations to generate new individual turns (2). The specific program flow is shown in Figure 2.

实施例2Example 2

在实施例1的基础上，取拱梁直径5m，10组设备均匀分布在拱梁上，每组重20kg，拱梁两端固定，倾角50°，[d]＝0.65cm，[σ]＝300MPa。采用本发明的优化方法与采用有限元优化方法结果对比如表4所示：On the basis of Example 1, take the arch beam with a diameter of 5m, 10 groups of equipment are evenly distributed on the arch beam, each group weighs 20kg, the two ends of the arch beam are fixed, the inclination angle is 50°, [d]=0.65cm, [σ]= 300MPa. Adopt optimization method of the present invention and adopt finite element optimization method result comparison as shown in table 4:

表4优化结果对比Table 4 Comparison of optimization results

由于等边角钢和扁钢的尺寸必需符合国标，因此根据上表取等边角钢边长80mm，厚度4mm，扁钢宽60mm，厚度4mm，此时变形为0.6506cm，重量84.634kg基本满足要求。由上表可以看出，本文优化的结果和有限元优化分析的拱梁最小重量基本相同。用有限元优化迭代30次时间约为四十多分钟，而用本文用实施例1中的样本、BP网络参数和遗传算法参数耗时约为几分钟，优化速度大大提高。Since the dimensions of the equilateral angle steel and flat steel must conform to the national standard, according to the above table, the side length of the equilateral angle steel is 80mm, the thickness is 4mm, the width of the flat steel is 60mm, and the thickness is 4mm. At this time, the deformation is 0.6506cm, and the weight is 84.634kg, which basically meets the requirements. It can be seen from the above table that the optimization results in this paper are basically the same as the minimum weight of the arch beam in the finite element optimization analysis. It takes about 40 minutes to optimize 30 iterations with finite element, but it takes about a few minutes to use the sample, BP network parameters and genetic algorithm parameters in Example 1 in this paper, and the optimization speed is greatly improved.

本发明在均匀采样的基础上，利用BP神经网络计算适应度和约束条件，用遗传算法获得拱梁重量的全局最优解。用此方法优化速度快，精度高，适合工程的实际应用。On the basis of uniform sampling, the invention uses BP neural network to calculate fitness and constraint conditions, and uses genetic algorithm to obtain the global optimum solution of arch beam weight. The optimization speed and precision are high with this method, which is suitable for the practical application of engineering.

以上实施例仅为本发明的实施方式，其描述较为具体和详细，但并不能因此而理解为对本发明专利范围的限制。应当指出的是，对于本领域的普通技术人员来说，在不脱离本发明构思的前提下，还可以做出若干变形和改进，这些都属于本发明的保护范围。因此，本发明专利的保护范围应以所附权利要求为准。The above examples are only implementations of the present invention, and their descriptions are more specific and detailed, but should not be construed as limiting the patent scope of the present invention. It should be pointed out that those skilled in the art can make several modifications and improvements without departing from the concept of the present invention, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the patent for the present invention should be based on the appended claims.

Claims

1. an arch beam optimal design method based on neural network and genetic algorithm, is characterized in that, comprises the following steps:

(1) Establish the arch beam optimization model

For the arch beam with a certain span and load, the optimization criterion is to make the weight the lightest when the stiffness and strength are satisfied;

(2) The process of arch beam optimization

① Acquisition of sample points: Select sample points for training BP neural network, adopt uniform design method to determine sample points, select training samples according to the national standard specifications of equilateral angle steel and flat iron, and conduct finite element analysis;

②Design of BP network: train the BP neural network with a single hidden layer, judge the constraints and determine the fitness in the genetic algorithm by the BP neural network,

③Optimization by genetic algorithm: The global optimal solution of the weight of the arch beam is obtained by the genetic algorithm, and the structure of the arch beam is optimized so that the weight of the arch beam can be minimized under the condition of satisfying the deformation.

2. the arch beam optimal design method based on neural network and genetic algorithm according to claim 1, is characterized in that, in step (1), the mathematical model of optimization is:

\{\begin{matrix} X x = = {[[{B B}_{11},, {B B}_{22},, {t t}_{11},, {t t}_{22}]]}^{T T} \\ min min W W ((X x)) \\ {σ σ}_{max max} \leq \leq [[σ σ]] \\ {d d}_{max max} \leq \leq [[d d]] \\ {X x}_{min min} \leq \leq X x \leq \leq {X x}_{max max} \end{matrix} - - - - - - ((11))

Among them, X is the design variable; B ₁ and t ₁ are the side length and thickness of the medium angle steel of the arch beam; B ₂ and t ₂ are the width and thickness of the flat iron in the arch beam; d _max and [d] are the vertical The maximum displacement and allowable displacement in the vertical direction; σ _max and [σ] are the maximum stress and allowable stress of the arch beam; X _min and X _max are the upper and lower limits of the design variables; W(X) is the weight value calculated by the BP network.

3. the arch beam optimal design method based on neural network and genetic algorithm according to claim 2, is characterized in that, in step (1), when using genetic algorithm to optimize constraint problem, need design penalty function,

The penalty function, that is, the constraint function is designed as:

P (x) = W (x) + r Σ_{i = 1}^{2} (\max {[0, g_{i} (x)]}^{2})

in,

g_{1} (x) = \frac{σ_{\max}}{[σ]} - 1,

g_{2} (x) = \frac{d_{\max}}{[d]} - 1,

r is a positive coefficient.

4. the arch beam optimal design method based on neural network and genetic algorithm according to claim 3, is characterized in that, in step (1), fitness function is: f (X)=C ₀ -P (X),

Among them, C ₀ is a constant that guarantees that f(X) is positive.

5. the arch beam optimal design method based on neural network and genetic algorithm according to claim 4, is characterized in that, in step (2), the design step of BP network is:

According to the arch-beam optimization model, the number of input nodes is determined to be four, that is, B ₁ , t ₁ , B ₂ , and t ₂ , and the output nodes are three, namely, W, σ, and d; with 16 nodes in a single hidden layer, Use samples to train the BP network.

6. the arch beam optimal design method based on neural network and genetic algorithm according to claim 5, is characterized in that, in step (2), after training BP network, carry out the optimization of genetic algorithm, the step of genetic algorithm optimization is : a. Generate the initial population; b. Use the BP neural network to calculate the fitness and constraint values, and output the result if the optimization criteria and constraints are met at the same time, otherwise turn to c; c. Select individuals with high fitness and perform genetic operations to generate The new individual turns to b.