A Multi-Objective Optimization Method for Uncertain Composite Material Structures

Abstract

This invention relates to a multi-objective interval optimization method for uncertain composite material structures, comprising the following steps: (1) Considering the uncertainty of parameters such as Young's modulus and material density, an interval uncertainty optimization model for composite laminate structures with different material thicknesses under given ply angles and material distributions is established. (2) An interval multi-objective optimization algorithm based on non-probabilistic set theory is used to solve the uncertain composite material structure optimization model. In the solution process, an initial composite material optimization model set is established using the Latin hypercube sampling method; the interval uncertainty analysis calculation of the total mass and first-order vibration frequency of the composite material is performed based on the interval dimension-by-dimensional analysis method; the interval non-dominated sorting is performed based on the dominance probability calculation method and the interval crowding calculation method of non-probabilistic set theory to achieve elite selection of composite material optimization models; iterative optimization is performed until the optimal composite material optimization model set is obtained. This invention will explore a new solution for the structural design optimization of composite materials with uncertainty.

Description

Technical Field

[0001] This invention relates to a multi-objective optimization method for uncertain composite material structures, applicable to multi-objective optimization problems of uncertain composite material structures, and belongs to the fields of composite material structures and multi-objective optimization technology. Background Technology

[0002] With the development of engineering technology, scientists have developed composite laminate structures with good designability to obtain new materials with more comprehensive performance. These structures retain the advantages of the original constituent materials and achieve complementary performance between materials through composite effects. By changing the type of ply materials, ply thickness, ply angle, and ply sequence of the composite laminate structure, different requirements for material strength, elasticity, and stiffness in different structures can be met. Therefore, composite structures are necessary to meet different structural design requirements and preferences. However, during the fabrication of composite laminates, uncertainties inevitably exist, affecting the strength of the composite material due to factors such as material properties, physical properties, geometric measurements, loads, and the environment. Moreover, the limitations of computational resources in the design of composite laminates limit the number of composite structural performance tests, making it difficult to accurately fit the precise probability density or fuzzy membership function between the composite structural performance and the uncertain variables. Among existing uncertainty quantification methods, interval numbers are widely used to solve uncertainty optimization problems because they do not require precise probability density or fuzzy membership functions and require a large amount of sample data. Therefore, the interval uncertainty optimization method can effectively solve the optimization design of uncertain composite material structures, and is beneficial for obtaining the potential fluctuation range of composite material structure performance under the influence of uncertain parameters when there is no precise probability distribution.

[0003] Common interval uncertainty optimization methods transform interval uncertainty optimization problems into deterministic problems through interval order relations or minimum regret criteria. These methods perform well in solving linear interval optimization problems. However, for most composite material structure design optimization problems, the optimization model is nonlinear, or even strongly nonlinear. Therefore, traditional interval optimization methods are insufficient for designing composite material structures. The most common interval optimization method is to first use deterministic optimization algorithms to obtain a deterministic set of optimal composite material structures, and then perform interval analysis on each composite material structure in the set. For the same interval value problem, different mathematical transformation models will result in different deterministic problems. Furthermore, the objectives and constraints of composite material structure optimization are mainly obtained through numerical analysis methods, which are computationally expensive and require specialized academic backgrounds for mathematical model transformation. Therefore, the effectiveness and universality of mathematical transformation models are a major research challenge. Summary of the Invention

[0004] The technical problem solved by this invention is to overcome the above-mentioned problems and provide a multi-objective interval optimization method for uncertain composite material structures. This method has the advantage of being able to directly perform efficient and high-precision multi-objective interval optimization on uncertain composite material structure optimization problems. It does not require complex mathematical transformations. The uncertain composite material structure optimization model participates in the entire optimization process in a black box manner, without the need for complex interval analysis. This provides a general framework for the interval uncertainty optimization of composite laminate structures.

[0005] To overcome the above challenges, this invention proposes a range multi-objective optimization method for uncertain composite material structures.

[0006] The technical solution of this invention is a multi-objective optimization method for uncertain composite material structures. This method considers the uncertainties in parameters such as Young's modulus and material density, and establishes a multi-objective optimization model for composite laminate structures with different ply thicknesses under given ply angles and material distributions. Based on non-probabilistic set theory, the multi-objective optimization method solves the multi-objective optimization problem of uncertain composite material structures, specifically including the following steps:

[0007] Step 1: Based on the uncertainties of Young's modulus and material density parameters, establish an interval uncertainty optimization model for composite laminate structures with different ply thicknesses under given ply angles and material distributions;

[0008] Step 2: An interval multi-objective optimization method based on non-probabilistic set theory is used to solve the interval uncertainty optimization model of the composite laminate structure. This method is then used to optimize the composite laminate structure design, addressing uncertainties based on Young's modulus and material density parameters. Under given ply angles and material distributions, the material thickness of different ply layers is optimized, providing a series of composite laminate structural design schemes with the lowest total mass and highest first-order vibration frequency, thus completing the interval multi-objective optimization design of the composite laminate structure. During the solution process, the Latin hypercube sampling method is used to establish an initial set of composite laminate structural design schemes. The interval dimension-by-dimensional analysis method is then used to further optimize the composite laminate structure. The set of composite laminate structural design schemes is subjected to interval uncertainty analysis to obtain the response intervals of the total mass and first-order vibration frequency for each structural design scheme. Based on the dominance probability calculation method and interval crowding calculation method of non-probability set theory, the response intervals of the total mass and first-order vibration frequency of the composite laminate structural design scheme set are sorted by interval non-dominated order. The structural design scheme set will be sorted in order of interval dominance probability from small to large and interval crowding from large to small. The structural design schemes with higher sorting order will be combined into a new composite laminate structural design scheme set. Iterative optimization is performed until the termination condition is met. The last set of structural design schemes is the optimal composite laminate structural design scheme set.

[0009] Furthermore, the step in step 1 of establishing an interval uncertainty optimization model for composite laminate structures with different ply thicknesses under a given ply angle and material distribution is as follows:

[0010] (1) The range of uncertainty parameters for composite laminate structures is given, referred to as the structural range, as follows:

[0011] a = a C +a R [-1,1]∈a I =[a L ,a U ] = [a C -a R ,a C +a R ],

[0012]

[0013]

[0014] in, and The uncertainty of Young's modulus represents the two different types of materials in a composite material structure; and This represents the material density uncertainty of two different types of materials in a composite material structure; a I and These are the vector and components of the structural interval, respectively; a L and The vector and component representing the lower bound of the structural interval, respectively; a U and The vector and component representing the upper bound of the structural interval, respectively; a C and It is the vector and components of the central value of the structural interval, denoted as: a C =(a L +a U ) / 2 and a R and It is a vector and its components representing the radius of the structural interval, a R =(a U -a L ) / 2 and Reflects the fluctuation range of the structural interval;

[0015] (2) The single-layer thickness of all ply materials in the composite structure is a design variable, using... It means that N D The total number of layers in the composite material; the layup angle of each layer is represented by... Indicates the type of material used for each layer; express;

[0016] Taking the total mass f1 and the first-order vibration frequency f2 of the composite material structure as the objective functions, the interval uncertainty optimization model of the composite laminate structure is expressed as follows:

[0017]

[0018] Among them, F I Let F be the vector of the objective function interval of the composite material structure optimization model. C F R F L and F U These are vectors representing the center value, radius, lower bound, and upper bound of the objective function interval, respectively.

[0019] Furthermore, the specific implementation of the interval multi-objective optimization method based on non-probabilistic set theory in step 2 for solving the interval uncertainty optimization model of composite laminate structure is as follows:

[0020] (1) Determine the maximum number of iterations t max ,use Let N represent the set of all composite material structures in each iteration, where N is the set of all composite material structures in each iteration. z This represents the total number of composite material structures in each iteration.

[0021] (2) Based on the Latin hypercube sampling method, the total number N of composite material structures is used as a basis for the sampling method. z The sampling space is layered, and then random samples are taken from each sub-sampling space in turn. After the sample order is shuffled, the samples are combined to finally obtain a set of initial parent composite material structures Z with uniform sampling. t At this point, the iteration count t = 0;

[0022] (3) Based on the simulated binary crossover and polynomial mutation method, the parent composite material structure set Z is analyzed. t Perform crossover and mutation to generate N z A new composite material structure is then combined with the original parent composite material structure to obtain a set of offspring composite material structures.

[0023] (4) Based on the interval-by-dimension analysis method, S t For different composite material structures, interval uncertainty analysis calculations were performed sequentially on the total mass f1 and the first-order vibration frequency f2 of the composite material to obtain the corresponding interval vector F. I The objective function interval F for composite material structures is constructed based on Legendre orthogonal polynomials. I With structural interval a I The response relationship between them is used to fit the structural interval a in the composite material structure. I The hyperplane on each component intercepts F I The polynomial approximation model of the surface formed by the changing trend is used to calculate the fluctuation range f1 of the total mass of the composite material and the first-order vibration frequency f2 corresponding to the extreme points of each component. I and The fluctuation range f1 I and As the final target interval vector of this composite material structure

[0024] (5) The dominance probability calculation method based on non-probability set theory calculates S sequentially. t The dominance probability of each composite material structure in relation to other composite material structures using nonprobabilistic set theory. The target interval vector F1 of the two composite material structures I and If there are no overlapping regions, then if the following conditions are met in the j-th dimension: Then the F1 score on the j-th dimension I Dominate The dominance probability of the nonprobabilistic set theory is 1; if it satisfies the following in the j-th dimension Then F1I dominates in the j-th dimension The dominance probability of the nonprobabilistic set theory is 0; if it satisfies on the j-th dimension, it does not satisfy... Not satisfied When there is an overlapping region, F1 is achieved in the j-th dimension. I Dominate The dominance probability of nonprobabilistic set theory is:

[0025]

[0026] in, and

[0027] Target interval vector of composite material structure based on nonprobabilistic set theory Dominate The total dominance probability is:

[0028] P = p1 * p2 (4)

[0029] Target interval vector of composite material structure Dominate If and only if Dominate The total nonprobabilistic set theory holds when the dominance probability P > α1*α2, where α1 and α2 are thresholds in two dimensions, and α1, α2 ∈ [0.5, 1], if and only if the composite material structure Not S t When dominated by other composite material structures, Consider it a non-dominated solution;

[0030] Similarly, the dominance probability calculation method based on non-probability set theory calculates S sequentially. t The dominance probability of each composite structure with other structures is determined using non-probabilistic set theory. This determines the dominance relationship between composite structures, and based on the non-dominated ranking method, S is... t It is divided into multiple levels, with the first level being the set of all non-dominated solutions; the higher the level of the composite material structure, the more composite material structures it is dominated by, putting it at a disadvantage in elite selection and making it easier to be eliminated.

[0031] (6) The interval crowding calculation method based on nonprobabilistic set theory for S t The congestion of each composite material structure in each layer is calculated sequentially. The congestion of the composite material structure located at the end of each layer is set to the maximum value by default. The congestion of the remaining composite material structures is calculated as the sum of the distances to the two adjacent intervals in the same layer.

[0032] Target interval vectors of two composite material structures and The formula for calculating the interval distance in the j-th dimension is as follows:

[0033]

[0034] Specifically, when the target interval vectors of two composite material structures have no overlapping region, the distance between these two intervals must be greater than 0; when the two interval vectors are connected, the distance is 0; when the two interval vectors have overlapping region, the distance must be less than 0.

[0035] If the composite material structure interval vector The vectors of adjacent intervals at the same level are and but The interval congestion is calculated as follows:

[0036]

[0037] Similarly, the interval crowding calculation method based on nonprobabilistic set theory applies to S. t The interval crowding degree of each composite material structure is calculated sequentially. After the interval crowding degree of all composite material structures has been calculated, the composite material structures at each level are sorted in descending order of interval crowding degree. Individuals with higher interval crowding degree will be given priority in the elite selection process.

[0038] (7) According to the non-dominated sorting level from low to high and the interval crowding degree from large to small, from S t Select N z A new set of paternal composite material structures, composed of elite individuals, Z. t+1 ;

[0039] (8) If the number of iterations t is less than the maximum number of iterations t max Return to (3) to (7), update the parent composite material structure set, and increment the iteration count t by 1; otherwise, complete the optimization process and output the latest parent composite material structure set Z. t This serves as a set of optimal composite laminate structure design schemes.

[0040] The advantages of this invention compared to the prior art are:

[0041] (1) The present invention proposes an interval multi-objective optimization method for uncertain composite material structures, which is applicable to solving interval uncertain multi-objective optimization problems of composite material structures and provides a general solution framework for composite material structure designers.

[0042] (2) This invention considers the uncertainty of parameters such as Young's modulus and material density, and establishes an interval uncertainty optimization model for composite laminate structure with different material thicknesses under given ply angle and material distribution. In the whole optimization process, there is no need to convert the uncertainty optimization problem of composite structure into a deterministic optimization problem, which reduces the difficulty of solving the problem and avoids the complex mathematical transformation process. It has high versatility in solving interval multi-objective problems of uncertain composite structure.

[0043] (3) The interval multi-objective optimization algorithm proposed in this invention can directly perform non-dominated sorting of the target interval vector of composite material structure by using the dominance probability degree and interval crowding degree calculation method based on non-probability set theory. This guides the direction of local search and global search, promotes the process of interval multi-objective optimization of composite material structure, and finally obtains a set of optimal composite material structures with strong diversity and uniform distribution.

[0044] (4) The interval multi-objective optimization algorithm proposed in this invention is based on the non-probability set theory. Based on the Latin hypercube sampling method, a set of uniformly distributed composite material structure sampling points are generated in the early stage of optimization. Compared with the traditional random initialization sampling method, the composite material structure sampling points are more uniformly distributed and will not produce obvious clustering. Moreover, since the samples of each layer are forcibly extracted during the sampling process, the comprehensiveness of the composite material structure design is guaranteed, which can effectively improve the search efficiency in the early stage of the algorithm.

[0045] (5) Existing interval optimization methods for composite material structures neglect the impact of different interval analysis methods on computational resource consumption. Therefore, the interval multi-objective optimization algorithm based on non-probabilistic set theory proposed in this invention improves the algorithm's retrieval capability and, based on the interval dimension-by-dimensional analysis method, can obtain high-precision target interval vector analysis results for composite material structures using only a small number of sample points, significantly reducing computational resource consumption and improving the algorithm's running speed.

[0046] In summary, this invention considers the uncertainties of parameters such as Young's modulus and material density, establishes an interval uncertainty optimization model for composite laminate structures with different ply thicknesses under given ply angles and material distributions, and proposes an interval multi-objective optimization algorithm based on nonprobabilistic set theory. This algorithm does not require mathematical transformation of the uncertainty optimization model and can directly perform efficient and high-precision interval multi-objective optimization on the original problem with limited computing resources, obtaining a set of optimal composite material structures with strong diversity and uniform distribution, which can well meet the design optimization needs of composite material structures. Attached Figure Description

[0047] Figure 1 This is a flowchart of the present invention;

[0048] Figure 2 This is a schematic diagram of a four-layer composite material structure. Detailed Implementation

[0049] The following combination Figure 2 The invention is illustrated in detail by a simple interval multi-objective optimization model.

[0050] like Figure 1 As shown, the method of the present invention is specifically implemented as follows:

[0051] Step 1: Give the following... Figure 2 The range of uncertainty parameters for the composite laminate structure shown is referred to simply as the structural range. It is represented as follows:

[0052] a = a C +a R [-1,1]∈a I =[a L ,a U ] = [a C -a R ,a C +a R ],

[0053]

[0054]

[0055] Among them, a I and These are the vector and component of the structural interval, respectively. L and The vector and component representing the lower bound of the structural interval, respectively; a U and The vector and component representing the upper bound of the structural interval, respectively; a C and The vector and components of the structural interval center value can be denoted as: a C =(a L +a U ) / 2 and a R and The vector and components of the structural interval radius can be denoted as: a R =(a U -a L ) / 2 and This reflects the fluctuation range of the structural interval. For example... Figure 2 The four-layer composite structure shown has two ply layers of material type 1, i.e., τ1 = 1 and τ4 = 1. Its Young's modulus uncertainty is a1. I= [4.365e10, 4.635e10], the uncertainty in material density is The first ply angle θ1 is 0°, and the last ply angle θ4 is 90°. The two middle ply materials are both material type 2, i.e., τ2 = 2 and τ3 = 2, and their Young's modulus uncertainty is... Material density uncertainty is The second layer ply angle θ2 is 45°, and the third layer ply angle θ3 is -45°.

[0056] Step 2: Considering the uncertainties in parameters such as Young's modulus and material density, establish as follows: Figure 2 The diagram shows an interval uncertainty optimization model for composite laminate structures with varying ply thicknesses under given ply angles and material distributions. The single-layer thickness of all ply materials in the composite structure is the design variable, denoted by x = (x1, x2, x3, x4). Using the total mass f1 and first-order vibration frequency f2 of the composite structure as the objective functions, the interval uncertainty optimization model for the composite laminate structure can be expressed as follows:

[0057]

[0058] Where x1 is the thickness of the first ply, x2 is the thickness of the second ply, x3 is the thickness of the third ply, and x4 is the thickness of the final ply. The thickness of each ply ranges from 0.01 meters to 0.04 meters, so the total thickness of the ply is 0.08 meters. I Let F be the vector of the objective function interval of the composite material structure optimization model. C F R F L and F U These are vectors representing the center value, radius, lower bound, and upper bound of the objective function interval, respectively.

[0059] Step 3: Solve the interval uncertainty optimization model of the composite laminate structure using an interval multi-objective optimization method based on non-probabilistic set theory. Determine the maximum number of iterations t. max =100, using Let N represent the set of all composite material structures in each iteration, where N is the set of all composite material structures in each iteration. z =10 represents the total number of composite material structures in each iteration.

[0060] Step 4: Based on the Latin hypercube sampling method, the sampling space is layered according to the total number of composite material structures. Then, random sampling is performed sequentially in 10 sub-sampling spaces. After the sample order is shuffled, the samples are combined to obtain a set of uniformly sampled initial parent composite material structures Z. t At this point, the iteration count t = 0.

[0061] Step 5: Based on the simulated binary crossover and polynomial mutation method, analyze the parent composite material structure set Z. t By performing crossover and mutation, 10 new composite material structures are generated, which are then combined with the original parent composite material structures to obtain a set of offspring composite material structures.

[0062] Step 6: Analyze the set of offspring composite material structures S using interval-by-interval dimension-by-dimensional analysis. t For different composite material structures, interval uncertainty analysis calculations were performed sequentially for the total mass f1 and the first-order vibration frequency f2 of the composite material to obtain the corresponding interval vector F. I Based on Legendre orthogonal polynomials, the objective function interval F of composite material structures is constructed. I With structural interval a I The response relationship between them is used to fit the structural interval a in the composite material structure. I The hyperplane on each component intercepts F I The polynomial approximation model of the surface formed by the changing trend is used to calculate the fluctuation range f1 of the total mass of the composite material and the first-order vibration frequency f2 corresponding to the extreme points of each component. I and The fluctuation range f1 I and As the final target interval vector of this composite material structure

[0063] Step 7: Calculate the set S of offspring composite material structures sequentially using the dominance probability calculation method based on non-probabilistic set theory. t The dominance probability of each composite material structure in relation to other composite material structures using nonprobabilistic set theory. Target interval vectors of two composite material structures and There is an overlapping region in the first dimension, therefore in the first dimension Dominate The dominance probability of nonprobabilistic set theory is:

[0064]

[0065] There is no overlapping region in the second dimension, and it satisfies Then in the second dimension Dominate The dominance probability of the nonprobabilistic set theory is 1. Therefore, the target interval vector of the composite material structure based on the nonprobabilistic set theory is... Dominate The total dominance probability is:

[0066] P=p1*p2=0.7998*1=0.7998>0.5*0.5 (10)

[0067] Therefore, the target interval vector of the composite material structure Dominate Similarly, the dominance probability calculation method based on non-probability set theory calculates S sequentially. t The dominance probability of each composite structure with other structures is determined using non-probabilistic set theory, and the dominance relationship between composite structures is judged accordingly. Based on the non-dominated ranking method, S is... t The system is divided into multiple levels. The first level consists of the set of all non-dominated solutions. The higher the level of a composite material structure, the more composite material structures it is dominated by, putting it at a disadvantage in elite selection and making it more likely to be eliminated.

[0068] Step 8: Calculate the interval crowding degree based on the non-probabilistic set theory for the offspring composite material structure set S. t The congestion of each composite material structure in each layer is calculated sequentially. The congestion of the composite material structure located at the end of each layer is set to the maximum value by default. The congestion of the remaining composite material structures is calculated as the sum of the distances to the two adjacent intervals in the same layer.

[0069] Target interval vectors of two composite material structures and The interval distance in the first dimension is The interval distance in the second dimension is therefore, The interval congestion is Similarly, the interval crowding calculation method based on nonprobabilistic set theory is applied to the set S of offspring composite material structures. t The interval crowding degree of each composite material structure is calculated sequentially. After the interval crowding degree of all composite material structures has been calculated, the composite material structures at each level are sorted in descending order of interval crowding degree. Individuals with higher interval crowding degree will be given priority in the elite selection process.

[0070] Step 9: Following the non-dominated sorting hierarchy from low to high and the interval crowding from high to low, start from the child composite material structure set S. t Select N z A new set of paternal composite material structures, composed of elite individuals, Z. t+1 ;

[0071] Step 10: If the number of iterations t is less than the maximum number of iterations t maxReturn to steps 5-9, update the parent composite material structure set, and increment the iteration count t by 1; otherwise, complete the optimization process and output the latest parent composite material structure set Z. t As the optimal composite material structure. The final N z The set of 10 optimal composite material structures is shown in Table 1 below:

[0072] Table 1 Optimal composite material structure in the last iteration

[0073]

[0074] Table 1 shows the final optimal composite material structure set Z when the number of iterations t = 100. 100 The ply thickness x and the corresponding total mass f1 and first-order vibration frequency f2 fluctuation range of all composite material structures in the text. I and Ultimately, a variety of optimal composite material structures were obtained, allowing designers to determine design schemes based on their needs, thus effectively meeting the requirements for composite material structure design optimization.

[0075] The specific implementation methods of the present invention have been described above. Those skilled in the art should understand that these are merely illustrative examples. Various changes or modifications can be made to these implementation methods without departing from the principles and implementation of the present invention. Therefore, the scope of protection of the present invention is defined by the appended claims.

Claims

1. A multi-objective optimization method for uncertain composite material structures, characterized in that: The method addresses the range uncertainty optimization of composite laminate structures, and the specific steps are as follows: Step 1: Based on the uncertainties of Young's modulus and material density parameters, establish an interval uncertainty optimization model for composite laminate structures with different ply thicknesses under given ply angles and material distributions; Step 2: The interval multi-objective optimization method based on non-probabilistic set theory is used to solve the interval uncertainty optimization model of composite laminate structure. This interval multi-objective optimization method is used to optimize the design of composite laminate structure, solve the uncertainty based on Young's modulus and material density parameters, optimize the material thickness of different plyes under given ply angle and material distribution, and provide a series of composite laminate structure design schemes with the lowest total mass of composite structure and the highest first-order vibration frequency, thus completing the interval multi-objective optimization design of composite laminate structure. During the solution process, the Latin hypercube sampling method was used to establish an initial set of composite laminate structure design schemes; Based on the interval-by-interval analysis method, interval uncertainty analysis is performed on the design scheme set of composite laminate structures to obtain the response intervals of the total mass and first-order vibration frequency for each structural design scheme. Based on the dominance probability calculation method and interval crowding calculation method of non-probability set theory, the response intervals of the total mass and first-order vibration frequency of the composite laminate structure design scheme set are sorted by interval non-dominated order. The structural design scheme set will be sorted in order of interval dominance probability from small to large and interval crowding from large to small. The structural design schemes with higher sorting order will be combined into a new composite laminate structure design scheme set. Iterative optimization is performed until the termination condition is met. The final structural design scheme set is the optimal composite laminate structure design scheme set.

2. The interval multi-objective optimization method for uncertain composite material structures according to claim 1, characterized in that: The steps in step 1 of establishing an interval uncertainty optimization model for composite laminate structures with different ply thicknesses under given ply angles and material distributions are as follows: (1) The range of uncertainty parameters for composite laminate structures is given, referred to as the structural range, as follows: a=a C +a R [-1,1]∈a I =[a L ,a U ]=[a C -a R ,a C +a R ], in, and The uncertainty of Young's modulus represents the two different types of materials in a composite material structure; and This represents the material density uncertainty of two different types of materials in a composite structure; a I and These are the vector and components of the structural interval, respectively; a L and The vector and component representing the lower bound of the structural interval, respectively; a U and The vector and component representing the upper bound of the structural interval, respectively; a C and It is the vector and components of the central value of the structural interval, denoted as: a C =(a L +a U ) / 2 and a R and It is a vector and its components representing the radius of the structural interval, a R =(a U -a L ) / 2 and Reflects the fluctuation range of the structural interval; (2) The single-layer thickness of all ply materials in the composite structure is a design variable, using... It means that N D The total number of layers in the composite material; the layup angle of each layer is represented by... Indicates the type of material used for each layer; express; Taking the total mass f1 and the first-order vibration frequency f2 of the composite material structure as the objective functions, the interval uncertainty optimization model of the composite laminate structure is expressed as follows: Among them, F I Let F be the vector of the objective function interval of the composite material structure optimization model. C F R F L and F U These are vectors representing the center value, radius, lower bound, and upper bound of the objective function interval, respectively.

3. The interval multi-objective optimization method for uncertain composite material structures according to claim 1, characterized in that: The specific implementation of the interval multi-objective optimization method based on non-probabilistic set theory in step 2 to solve the interval uncertainty optimization model of composite laminate structure is as follows: (1) Determine the maximum number of iterations t max ,use Let N represent the set of all composite material structures in each iteration, where N is the set of all composite material structures in each iteration. z This represents the total number of composite material structures in each iteration; (2) Based on the Latin hypercube sampling method, the total number N of composite material structures is used as a basis for the sampling method. z The sampling space is layered, and then random samples are taken from each sub-sampling space in turn. After the sample order is shuffled, the samples are combined to finally obtain a set of initial parent composite material structures Z with uniform sampling. t At this point, the iteration count t = 0; (3) Based on the simulated binary crossover and polynomial mutation method, the parent composite material structure set Z is analyzed. t Perform crossover and mutation to generate N z A new composite material structure is then combined with the original parent composite material structure to obtain a set of offspring composite material structures. (4) Based on the interval-by-dimension analysis method, S t For different composite material structures, interval uncertainty analysis calculations were performed sequentially on the total mass f1 and the first-order vibration frequency f2 of the composite material to obtain the corresponding interval vector F. I The objective function interval F for composite material structures is constructed based on Legendre orthogonal polynomials. I With structural interval a I The response relationship between them is used to fit the structural interval a in the composite material structure. I The hyperplane on each component intercepts F I The polynomial approximation model of the surface formed by the changing trend is used to calculate the fluctuation range f1 of the total mass of the composite material and the first-order vibration frequency f2 corresponding to the extreme points of each component. I and The fluctuation range f1 I and As the final target interval vector of this composite material structure (5) The dominance probability calculation method based on non-probability set theory calculates S sequentially. t The dominance probability of each composite material structure in relation to other composite material structures using nonprobabilistic set theory. Target interval vectors of two composite material structures and If there are no overlapping regions, then if the following conditions are met in the j-th dimension: Then on the j-th dimension Dominate The dominance probability of nonprobabilistic set theory is 1; If it satisfies the following in the j-th dimension Then on the j-th dimension Dominate The dominance probability of nonprobabilistic set theory is 0; If it satisfies the condition in the j-th dimension, then it does not satisfy the condition. Not satisfied When there is an overlapping region, it is in the j-th dimension. Dominate The dominance probability of nonprobabilistic set theory is: in, and Target interval vector of composite material structure based on nonprobabilistic set theory Dominate The total dominance probability is: P = p1 * p2 (4) Target interval vector of composite material structure Dominate If and only if Dominate The total nonprobabilistic set theory holds when the dominance probability P > α1*α2, where α1 and α2 are thresholds in two dimensions, and α1, α2 ∈ [0.5, 1], if and only if the composite material structure Not S t When dominated by other composite material structures, Consider it a non-dominated solution; Similarly, the dominance probability calculation method based on non-probability set theory calculates S sequentially. t The dominance probability of each composite structure with other structures is determined using non-probabilistic set theory. This determines the dominance relationship between composite structures, and based on the non-dominated ranking method, S is... t It is divided into multiple levels, with the first level being the set of all non-dominated solutions; the higher the level of the composite material structure, the more composite material structures it is dominated by, putting it at a disadvantage in elite selection and making it easier to be eliminated. (6) The interval crowding calculation method based on nonprobabilistic set theory for S t The congestion of each composite material structure in each layer is calculated sequentially. The congestion of the composite material structure located at the end of each layer is set to the maximum value by default. The congestion of the remaining composite material structures is calculated as the sum of the distances to the two adjacent intervals in the same layer. Target interval vectors of two composite material structures and The formula for calculating the interval distance in the j-th dimension is as follows: Specifically, when the target interval vectors of two composite material structures have no overlapping region, the distance between these two intervals must be greater than 0; when the two interval vectors are connected, the distance is 0; when the two interval vectors have overlapping region, the distance must be less than 0. If the composite material structure interval vector The vectors of adjacent intervals at the same level are and but The interval congestion is calculated as follows: Similarly, the interval crowding calculation method based on nonprobabilistic set theory applies to S. t The interval crowding degree of each composite material structure is calculated sequentially. After the interval crowding degree of all composite material structures has been calculated, the composite material structures at each level are sorted in descending order of interval crowding degree. Individuals with higher interval crowding degree will be given priority in the elite selection process. (7) According to the non-dominated sorting level from low to high and the interval crowding degree from large to small, from S t Select N z A new set of paternal composite material structures, composed of elite individuals, Z. t+1 ; (8) If the number of iterations t is less than the maximum number of iterations t max Return to (3) to (7), update the parent composite material structure set, and increment the iteration count t by 1; otherwise, complete the optimization process and output the latest parent composite material structure set Z. t This serves as a set of optimal composite laminate structure design schemes.

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