Inner ballistic nonlinear interval uncertainty optimization method based on affine algorithm

Abstract

The invention discloses an inner ballistic nonlinear interval uncertainty optimization method based on an affine algorithm, and the method comprises the steps: determining a Chebyshev polynomial order and uncertainty design variables, and enabling the uncertainty design variables to be charge mass, gunpowder particle arc thickness, gunpowder particle aperture and gunpowder chamber volume; establishing a classic inner ballistic model of mixed charge on the basis of thermodynamics, and calculating corresponding ballistic parameters, including bullet initial velocity, maximum bullet bottom pressure, muzzle pressure and maximum negative pressure difference; a Chebyshev polynomial agent model is constructed; obtaining an interval between an uncertain objective function and an uncertain constraint function corresponding to the uncertain design variable; converting into an uncertainty objective function and an uncertainty constraint function in an affine form including a central value and a deviation range; the inner trajectory uncertainty optimization problem is converted into a deterministic optimization problem; and an optimization solver NSGA-II is adopted to carry out simulation search, and a Pareto optimal solution set is solved. The method is suitable for high-efficiency solving of the engineering uncertainty optimization problem.

Description

Technical Field

[0001] The invention relates to the field of structural optimization design, and in particular to an interior ballistic nonlinear interval uncertainty optimization method based on an affine algorithm. Background Art

[0002] Interior ballistics is a discipline that studies the motion law of projectiles in the bore and the phenomena associated with shooting. Its performance indicators directly affect the combat capability of artillery and are an important basis for measuring the performance of artillery weapon systems. Research on interior ballistics can provide a theoretical basis for weapon design and ballistic performance analysis. However, in engineering, ammunition inevitably has uncertain factors caused by material properties, processing and manufacturing errors, assembly errors, etc. Deterministic interior ballistic optimization cannot accurately characterize the influence of uncertainty. It is of great significance to carry out uncertainty optimization of interior ballistics. Uncertain optimization theory has been widely used in other engineering optimization fields. Common uncertain optimization methods include stochastic programming, fuzzy programming and interval optimization. In interval optimization, uncertain parameters are described by intervals. It is only necessary to know the upper and lower bounds of the parameters, but not their precise probability distribution or fuzzy membership function. Therefore, it is widely used. Interval uncertainty optimization problems are often accompanied by two-layer nested optimization, in which the outer optimization is used to optimize the design vector, and the inner optimization is used to calculate the interval of uncertain objective functions and constraints. However, two-layer nested optimization problems are inevitably accompanied by a lot of computing time and low computing efficiency. In order to solve this problem, interval uncertainty optimization is combined with the affine algorithm. The affine algorithm is an algorithm for solving linear programming problems, which combines the characteristics of the simplex method and the interior point method. The affine algorithm has a polynomial time complexity in theory, usually has better numerical stability, and the convergence speed is less dependent on the specific structure of the problem. This patent considers the impact of ammunition uncertainty on the internal ballistic performance of artillery, and proposes an internal ballistic uncertainty optimization method based on the affine algorithm. First, the polynomial response surface model is used to obtain the explicit expressions of the design parameters and uncertainty parameters relative to the objective function and constraint function. When establishing the optimization model, the interval uncertainty parameters are rewritten as affine type, and then the affine algorithm is used to directly obtain the upper and lower bounds of the uncertainty objective function and constraint function, thereby avoiding the two-layer nesting problem and greatly improving the optimization efficiency. The proposed method can obtain the influence of the internal ballistic uncertainty parameters on the internal ballistic performance of the artillery, which has important theoretical significance and engineering value for reducing the initial velocity probable error and improving the density of artillery firing. Summary of the invention

[0003] The purpose of the present invention is to provide an interior ballistic nonlinear interval uncertainty optimization method based on an affine algorithm, which can efficiently and accurately handle interval uncertainty optimization problems in engineering optimization.

[0004] The technical solution for implementing the present invention is: a nonlinear interval uncertainty optimization method for interior ballistics based on an affine algorithm, comprising the following steps:

[0005] Step 1: Determine the order of the Chebyshev polynomial and the uncertain design variables, where the uncertain design variables are the charge mass m, the arc thickness of the gunpowder particles 2e 1 , the diameter of the powder particles d and the volume of the chamber w 0 ;

[0006] Step 2: Determine the algorithm parameters of the optimization solver NSGA-II, including: population size, crossover rate, mutation rate, termination criterion, and the number of small intervals t for interval division. j , constraint possibility level λ i ;

[0007] Step 3: Sampling within the uncertainty interval to obtain an uncertainty design vector sample library;

[0008] Step 4: Establish a mixed charge classical interior ballistic model based on thermodynamics, design vector samples based on the uncertainty obtained in step 3, and calculate the corresponding ballistic parameters, including projectile initial velocity, maximum projectile base pressure, muzzle pressure, and maximum negative pressure difference;

[0009] Step 5: According to the trajectory parameters obtained in step 4, calculate the Chebyshev polynomial expansion coefficients and construct a Chebyshev polynomial proxy model;

[0010] Step 6: Using the interval set method, obtain the intervals of the uncertain objective function and the uncertain constraint function corresponding to the uncertain design variables;

[0011] Step 7: Using an affine algorithm, convert the uncertainty objective function and uncertainty constraint function obtained in step 6 into an affine form of uncertainty objective function and uncertainty constraint function including a center value and a deviation range;

[0012] Step 8: Use interval order relation ≤ cw The uncertain objective function obtained in step 7 is converted into a deterministic objective function, and the uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method;

[0013] Step 9: Perform interval economic evaluation on each individual uncertain design vector sample in step 3 to obtain an indicator for evaluating the overall parameter error level, quantify the error level of the design variables under uncertain conditions, and combine step 8 to obtain a deterministic objective function and a deterministic constraint function, so that the interior ballistic uncertainty optimization problem is transformed into a deterministic optimization problem;

[0014] Step 10: Use the optimization solver NSGA-II to simulate and search the deterministic optimization problem obtained in step 9 to solve the Pareto optimal solution set. If the algorithm termination condition is met, the nonlinear interval uncertainty optimization of the interior ballistics is completed. Otherwise, it means that the sample size of the uncertainty design vector is insufficient, and return to step 3.

[0015] Further, in step 1, the order of the Chebyshev polynomial is determined to be 3rd order.

[0016] Furthermore, in step 3, sampling is performed within the uncertainty interval to obtain an uncertainty design vector sample library. The specific method is as follows:

[0017] The vector U∈U of the 4-dimensional interval uncertain design variables I =[U L ,U R ] is written as follows:

[0018]

[0019] Where: superscripts I, L, R, c, and w represent the interval, the left boundary of the interval, the right boundary of the interval, the midpoint of the interval, and the radius of the interval, respectively; sampling is performed within the uncertainty interval to obtain an uncertainty design vector sample library.

[0020] Furthermore, in step 4, a mixed charge classical interior ballistic model based on thermodynamics is established, and the corresponding ballistic parameters are calculated according to the uncertainty design vector sample obtained in step 3, including the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference, where:

[0021] The classical interior ballistic model of mixed charge based on thermodynamics is:

[0022]

[0023] In the formula, χ i is the shape characteristic of the i-th type of gunpowder, λ ι , μ ι is the powder type parameter of the i-th type of gunpowder, Z i is the relative thickness of the i-th type of gunpowder combustion, u 1i is the burning rate coefficient of the i-th type of propellant, p is the propellant gas pressure, e 1i is half the arc thickness of the i-th type of gunpowder, m is the mass of the projectile, ω i is the charge mass of the i-th type of gunpowder, e i is the arc thickness of the i-th type of gunpowder, S is the equivalent cross-sectional area of ​​the barrel, is the secondary work coefficient, l 0 is the volume reduction length of the chamber, Φ iis the combustion percentage of the i-th type of gunpowder, l φ f is the reduced diameter length of the free volume of the chamber, i is the gunpowder power of the i-th type of gunpowder, α i is the residual capacity of the i-th type of gunpowder, l and v are the projectile travel and speed respectively, Δ i is the gunpowder filling density of the i-th type of gunpowder, θ is the insulation coefficient, and n is a constant.

[0024] Further, in step 5, according to the ballistic parameters obtained in step 4, the Chebyshev polynomial expansion coefficients are calculated to construct a Chebyshev polynomial proxy model. The specific method is as follows:

[0025] Step 5.1: Consider the uncertain design variables as charge mass m, powder particle arc thickness 2e 1 , the diameter of the powder particles d and the volume of the chamber w 0 , the dimension of uncertain variables is 4, which is a unified writing format. The four uncertain variables are represented by x 1 , x 2 , x 3 , x 4 It means that the 4-dimensional Chebyshev polynomial expansion is regarded as the tensor product of 4 1-dimensional Chebyshev polynomials, that is:

[0026]

[0027] In the formula, x i is the ith interval variable, is a one-dimensional Chebyshev polynomial, n i The Chebyshev polynomial representing the ith interval variable is an nth-degree algebraic polynomial;

[0028] Step 5.2: The initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference are related to the uncertain design variable x. 1 , x 2 , x 3 , x 4 A continuous function, for ease of writing, (x 1 ,x 2 ,x 3 ,x 4 ) is abbreviated as (x). For the initial velocity of the projectile, assuming that the initial velocity function of the projectile f(x) is a 4-dimensional continuous function, then its 4-dimensional Chebyshev polynomial expansion is:

[0029]

[0030] In the formula, p is the subscript i 1 ,...,i 4 The number of 0s in is a 4-dimensional Chebyshev polynomial, are the corresponding polynomial coefficients, It is also a 4th-order tensor. Using the orthogonality of Chebyshev polynomials, the polynomial coefficients are calculated as follows:

[0031]

[0032] The Gauss-Chebyshev integral formula is used for each one-dimensional integral in equation (5), and the corresponding integral result is:

[0033]

[0034] In the formula, h=n+1 is the integration order, is the i-th integral interpolation point of the j-th interval variable, which is also a tensor product:

[0035]

[0036] In the formula, x i is each one-dimensional Chebyshev polynomial interpolation point, that is, the zero point of the Chebyshev polynomial, is the Crocker inner product symbol, so the total number of interpolation points of the Chebyshev polynomial of 4-dimensional variables is N 4 for:

[0037] N 4 =(n+1) 4 (8)

[0038] Step 5.3: According to formulas (4)-(6) in step 5.2, the four-dimensional Chebyshev polynomial proxy functions of the projectile initial velocity, maximum base pressure, muzzle pressure and maximum negative pressure difference are calculated in sequence.

[0039] Furthermore, in step 6, the interval set method is used to obtain the interval representation f of the uncertain objective function corresponding to the uncertain design variables, namely, the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference. i I (U I ), and the uncertainty constraint function, i.e., the chamber pressure range and the maximum negative pressure upper limit, is represented by the interval g i I (U I ), let the number of objective functions be m and the number of constraint functions be p, then:

[0040]

[0041] Where U I Design variables for uncertainty, fi I (U I ) and g i I (U I ) is the i-th uncertain objective function and uncertain constraint function under the uncertain design variable U I The output of the following, and Denotes the objective function f i In the design variables The values ​​under the left and right endpoints, i.e. the minimum and maximum endpoints, and Represents the constraint function g i The minimum and maximum endpoints of k=1,2,...,r represent the design variables U I is divided into r sample points.

[0042] Furthermore, in step 7, an affine algorithm is used to transform the uncertainty objective function and constraint function represented by the interval obtained in step 6 to obtain the radial type of the deterministic objective function and constraint function. The specific method is:

[0043] Step 7.1. Uncertain design variable, charge mass x 1 , gunpowder particle arc thickness x 2 , gunpowder particle aperture x 3 And the chamber volume x 4 Affine calculation of uncertain variables x i The affine form of

[0044]

[0045] In the formula, x io Design variable x for uncertainty i The central value of the affine type, t is the number of uncertain factors affecting the variable x, ε it is the disturbance term, whose value range is [-1,1], x ij is the noise symbol ε ij The known real coefficients of |ε ij |=1 when affine type x i Get its maximum and minimum values ​​and express them as intervals:

[0046]

[0047] For the uncertain quantity x in the uncertainty optimization problem i It is expressed as:

[0048]

[0049] In the formula, Δx i is the interval radius, i=1,2,3,4.

[0050] Step 7.2: When the objective function is replaced by the symbol d, the objective function is rewritten in affine form:

[0051]

[0052] Where, d 0 is the center value of the affine type, d i are known real coefficients.

[0053] The interval of the objective function at the design variable X:

[0054]

[0055] Step 7.3: Set the constraint function g i (X,U) is replaced by the symbol c, and the constraint function is rewritten in affine form:

[0056]

[0057] In the formula, c 0 is the center value of the affine type, c i are known real coefficients. g i The interval at the design variable X is:

[0058]

[0059] Therefore, the uncertainty problem of artillery interior ballistics can be rewritten as:

[0060]

[0061] Further, step 8, using the interval order relation ≤ cw The uncertain objective function obtained in step 7 is converted into a deterministic objective function. The uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method. The specific method is as follows:

[0062] Step 8.1: Use interval order relation ≤ cw Transform the objective function, where c and w represent the midpoint and radius of the interval respectively, A≤ cw B means that interval B is better than interval A only when the midpoint and radius of interval B are smaller than those of interval A. In this way, the interval uncertainty objective function is transformed into a deterministic dual objective function optimization problem:

[0063]

[0064] Where: fc (U) represents the average performance of the objective function under uncertainty, f w (U) represents the fluctuation range of the objective function under the influence of uncertain factors;

[0065] Step 8.2: According to the six spatial position relationships between the two intervals A and B, the specific value of the interval possibility is calculated using formula (19):

[0066]

[0067] Where: P(A I ≤B I ) or P(A I ≥B I ) represents the probability that interval A is smaller or larger than interval B for two uniformly distributed intervals A and B;

[0068] For inequality constraints Transformed into a deterministic inequality constraint through formula (20):

[0069]

[0070] Where: i is the acceptable level of possibility set by the decision maker for the i-th constraint, representing the degree to which the constraint function needs to be satisfied;

[0071] For the equation First, transform it into the form of formula (21):

[0072]

[0073] Then the two deterministic inequality constraints transformed into equation (22) are solved separately;

[0074]

[0075] Furthermore, in step 9, each individual uncertainty design sample is evaluated for interval economy, and an indicator for evaluating the overall parameter error level is obtained, which is then transformed into a deterministic optimization problem. The specific method is as follows:

[0076] Define a non-negative dimensionless evaluation coefficient ζ to measure the interval economy of all uncertain design variables;

[0077]

[0078] Where: is the interval radius of the uncertain design variable, is the maximum allowable error of the uncertain design variables artificially set in the optimization, and n represents the number of design variables;

[0079] The multi-objective optimization problem of formula (24) is obtained:

[0080]

[0081] Where: is the i-th constraint function Less than or equal to (equal to, greater than or equal to) the i-th interval variable The probability of μ i is a constant;

[0082] The uncertain optimization problem of gun interior ballistics containing four uncertain variables can be written as a deterministic optimization problem as shown in formula (25):

[0083]

[0084] The invention discloses an interior ballistic nonlinear interval uncertainty optimization system based on an affine algorithm, which implements the interior ballistic nonlinear interval uncertainty optimization method based on an affine algorithm to realize the interior ballistic nonlinear interval uncertainty optimization based on an affine algorithm.

[0085] Compared with the prior art, the present invention has the following significant advantages:

[0086] (1) The novel interval uncertainty optimization algorithm established in the present invention can efficiently handle nonlinear interval uncertainty optimization problems. Compared with the traditional interval uncertainty optimization method, this method adopts affine type to describe interval variables, and then uses affine algorithm to obtain the upper and lower bounds of uncertainty objective function and constraint function, converting the uncertainty optimization problem into a single-layer deterministic optimization problem, avoiding nested optimization, and greatly improving the computational efficiency.

[0087] (2) The novel interval uncertainty optimization algorithm established by the present invention has good universality and is particularly suitable for two types of engineering uncertainty optimization problems, complex optimization problems with multiple objective functions and multiple constraint functions, and engineering optimization problems that must use proxy models.

[0088] (3) The present invention introduces the response surface method to construct an explicit polynomial proxy model of the interior ballistic performance index, which solves the problem that the affine algorithm cannot directly process the interior ballistic model in the form of a group of ordinary differential equations, overcomes the difficulty of numerical differential calculation in engineering problems without clear mathematical expressions, and has the advantages of good continuity, high calculation accuracy, parallel calculation, and small amount of calculation.

[0089] (4) The novel interval uncertainty optimization algorithm established by the present invention has an index for evaluating the overall parameter error level. The obtained uncertainty design variables can meet the design requirements within the maximum allowable error level and have good interval economy. BRIEF DESCRIPTION OF THE DRAWINGS

[0090] Figure 1 It is a flow chart of the nonlinear interval uncertainty optimization method of interior ballistics based on affine algorithm of the present invention;

[0091] Figure 2 This is the optimization iterative process based on the genetic algorithm in Example 1 of the present invention;

[0092] Figure 3 is the Pareto frontier of the internal ballistic optimization result of Example 1 of the present invention;

[0093] Figure 4 is the interval of the initial velocity of the projectile after optimization in Example 1 of the present invention;

[0094] Figure 5 is the pressure difference after optimization in Example 1 of the present invention;

[0095] Figure 6 It is the range of the bullet bottom pressure after optimization in Example 1 of the present invention. DETAILED DESCRIPTION

[0096] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0097] Combination Figure 1 The specific steps of establishing the nonlinear interval uncertainty optimization method of interior ballistics based on the affine algorithm described in this embodiment are as follows:

[0098] Step 1: Determine the design space of the Chebyshev polynomial proxy model based on the range of the design variables of the optimization problem. The uncertain design variables are charge mass m, gunpowder particle arc thickness 2e 1 , the diameter of the powder particles d and the volume of the chamber w 0 ; The order of Chebyshev polynomial is 3.

[0099] Step 2: The algorithm parameters of the optimization solver NSGA-II include: population size, crossover rate, mutation rate, termination criterion (maximum evolutionary generations), and the algorithm parameters of the inner interval calculation include: the number of small intervals divided into intervals t j , constraint possibility level λ i .

[0100] Step 3: Vector U∈U of m-dimensional interval uncertainty design variables I =[U L ,U R ] can be written in the following form:

[0101]

[0102] In the formula, the superscripts I, L, R, c, and w represent the interval, the left boundary of the interval, the right boundary of the interval, the midpoint of the interval, and the radius of the interval, respectively. The experimental design method is used to sample in the uncertainty interval to obtain the uncertainty design vector library.

[0103] Step 4: Use the classical interior ballistics model of mixed charge based on thermodynamics:

[0104]

[0105] In the formula, χ i is the shape characteristic of the i-th type of gunpowder, λ and μ are the powder type parameters of the i-th type of gunpowder, Z i is the relative thickness of gunpowder combustion, u 1i is the burning rate coefficient of the i-th type of propellant, p is the propellant gas pressure, e 1i is half the arc thickness of the i-th type of gunpowder, m is the mass of the projectile, ω i is the charge mass, e i is the powder arc thickness, S is the equivalent cross-sectional area of ​​the barrel, is the secondary work coefficient, l 0 Chamber volume reduction length, ψ i is the percentage of gunpowder combustion, l ψ f is the reduced diameter length of the free volume of the chamber, i is the gunpowder power, α i is the residual capacity, l and v are the projectile travel and velocity respectively, Δ i is the gunpowder filling density, and θ is the insulation coefficient.

[0106] When solving the above nonlinear ordinary differential equations, the Runge-Kutta method is generally used for calculation.

[0107] Step 5: According to the output response obtained in step 4, calculate the Chebyshev polynomial expansion coefficients and construct a Chebyshev polynomial proxy model;

[0108] Step 5.1: The k-dimensional Chebyshev polynomial expansion can be viewed as the tensor product of each one-dimensional Chebyshev polynomial, that is:

[0109]

[0110] Function f(x) is a k-dimensional continuous function, and its k-dimensional Chebyshev polynomial expansion is:

[0111]

[0112] In the formula, p is the subscript i 1 ,...,i k The number of 0s in is the k-dimensional Chebyshev polynomial, is the ith one-dimensional Chebyshev polynomial, are the corresponding polynomial coefficients, It is also a k-order tensor. Using the orthogonality of the Chebyshev polynomial, the polynomial coefficients are calculated as follows:

[0113]

[0114] Step 5.2: Use the Gauss-Chebyshev integral formula for each dimension integral in equation (5), and the corresponding integral result can be obtained as follows:

[0115]

[0116] In the formula, h=n+1 is the integration order, x j are the integral interpolation points. It is worth noting that these interpolation points are also a tensor product:

[0117]

[0118] In the formula, x k is each one-dimensional Chebyshev polynomial interpolation point, that is, the zero point of the Chebyshev polynomial, is the Crocker inner product symbol. Therefore, the total number of interpolation points N k for:

[0119] N k =(n+1) k (8)

[0120] Step 6: Use formula (24) to calculate the interval of the uncertainty objective function and the uncertainty constraint function corresponding to the current uncertain design variables:

[0121]

[0122] Where U I Design variables for uncertainty, f i I (U I ) and g i I (U I ) is the i-th uncertain objective function and uncertain constraint function under the uncertain design variable U I The output of the following, and Denotes the objective function f i In the design variables The values ​​under the left and right endpoints, i.e. the minimum and maximum endpoints, and Represents the constraint function g i The minimum and maximum endpoints of , i=1,2,...,q means there are q objective functions, k=1,2,...,r means the design variable U I is divided into r sample points.

[0123] Step 7: Use the affine algorithm to transform the uncertainty objective function and uncertainty constraint function obtained in step 6 into a dual-objective deterministic optimization problem. The specific method is:

[0124] Step 7.1: The affine form of the uncertainty u is denoted by

[0125]

[0126] In the formula, u o is the center value of the affine type, ε t is the noise symbol, its value range is [-1,1], u i is the noise symbol ε t The known real coefficients of . When |ε i When |=1, the affine type u can achieve its maximum and minimum values ​​and can be expressed as an interval form:

[0127]

[0128] For the uncertainty u in the uncertainty optimization problem i It can be expressed as:

[0129]

[0130] In the formula, u io is the median of the interval, Δu i is the interval radius, ε is the disturbance term, and i is the number of design variables; Step 7.2, the basic affine operation rules are as follows: Assume affine type and

[0131]

[0132] Then we have:

[0133]

[0134] For operations such as division, square root, and exponentiation, the min-rang affine approximation model is used to represent them. Its value range is a and b are the upper and lower limits of the variable value.

[0135] Step 7.3: According to the min-rang affine approximation theory, we can get The affine approximation of

[0136]

[0137] The affine approximation of

[0138]

[0139] The affine approximation of

[0140]

[0141] In the formula, a and b are the upper and lower limits of the variable value.

[0142] It can be seen from the above formula that when applying the min-rang affine approximation theory, the affine type The value interval of must not include 0. Compared with the interval algorithm, although the calculation process of the affine algorithm is slightly more complicated, the affine algorithm does not involve complex operations such as derivation and differentiation. It is all composed of basic mathematical operations. The affine calculation process can be easily completed with the help of symbolic calculation software.

[0143] Rewrite the objective function into affine form:

[0144]

[0145] Where, d 0 is the center value of the affine type, d i is a known real coefficient, ε i is the noise symbol; therefore, the interval of the objective function at the design variable X can be obtained:

[0146]

[0147] Step 7.4: Constrain g i Rewritten in affine form:

[0148]

[0149] In the formula, c 0 is the center value of the affine type, c i is a known real coefficient, ε i is the noise symbol; g i The interval at the design variable X is:

[0150]

[0151] Therefore, the above uncertainty problem can be rewritten as:

[0152]

[0153] Step 8: Use interval order relation ≤ cw The uncertain objective function obtained in step 7 is converted into a deterministic objective function. The uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method. The uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method. The specific method is as follows:

[0154] Step 8.1: Use interval order relation ≤ cw Transform the objective function, where the symbols c and w represent the midpoint and radius of the interval respectively. A≤ cw B means that interval B is better than interval A only when the midpoint and radius of interval B are smaller than those of interval A. In this way, the interval uncertainty objective function is transformed into a deterministic dual objective function optimization problem:

[0155]

[0156] Where: f c (U) represents the average performance of the objective function under uncertainty, f w (U) represents the fluctuation range of the objective function under the influence of uncertain factors, so formula (23) minimizes f w (U) Consider robustness.

[0157] Step 8.2: According to the six spatial position relationships between the two intervals A and B, the model of formula (24) is used to calculate the specific value of the interval possibility:

[0158]

[0159] Where: P(A I ≤B I ) or P(A I ≥B I ) represents the probability that interval A is less than or greater than interval B for two uniformly distributed intervals A and B. For example, the ≤-type inequality constraint It can be transformed into a deterministic inequality constraint through formula (25):

[0160]

[0161] Where: i is the acceptable level of possibility set by the decision maker for the i-th constraint, representing the degree to which the constraint function needs to be satisfied. i The larger the value, the stricter the constraint.

[0162] For the equation It can be first transformed into the form of formula (26):

[0163]

[0164] Then it can be transformed into two deterministic inequality constraints in formula (27) and solved separately.

[0165]

[0166] Step 9: Define a non-negative dimensionless evaluation coefficient ζ to measure the interval economy of all uncertain design variables, so as to achieve the design requirements within the maximum allowable error and thus reduce costs.

[0167]

[0168] Where: is the interval radius of the uncertain design variable, is the maximum allowable error of the uncertain design variables artificially set in the optimization, and n represents the number of design variables. The larger the ζ value, the higher the overall error level and the better the interval economy.

[0169] Through the above processing, we can get the multi-objective optimization problem of formula (29):

[0170]

[0171] U∈U I =[U L ,U R ], (29)

[0172] in

[0173] f c (U)=[f R (U)+f L (U)] / 2=(maxf(U)+minf(U)) / 2,

[0174] f w (U)=[f R (U)-f L (U)] / 2=(maxf(U)-minf(U)) / 2,

[0175]

[0176] The uncertain optimization problem of gun interior trajectory can be further rewritten as:

[0177]

[0178] At this point, the uncertainty optimization problem is converted into a deterministic optimization problem, which can be solved using existing mature sequential quadratic programming, genetic algorithms and other intelligent optimization algorithms.

[0179] Step 10: Use the optimization solver NSGA-II to perform simulation search to solve the Pareto optimal solution set. If the algorithm termination condition is met, the algorithm terminates, otherwise return to step 3.

[0180] Example

[0181] Internal ballistics design is a multi-solution problem, so it must include a process of selecting and optimizing a solution. Due to the close constraints between internal ballistic parameters, a single parameter cannot fully and profoundly reflect the comprehensive performance of internal ballistics. From the perspective of ballistic safety, the maximum chamber pressure should be within a reasonable range. From the perspective of artillery performance, the initial velocity of the projectile directly affects the range and power of the artillery, so the initial velocity of the projectile should be kept within a reasonable range during optimization and design.

[0182] Step 1: Determine the Chebyshev polynomial order as 3, and consider four uncertain design variables, namely, charge mass m, gunpowder particle arc thickness 2e 1 , the diameter of the powder particles d and the volume of the chamber w 0 .

[0183] Step 2: Determine the algorithm parameters of the optimization solver NSGA-II, including: population size is 100, crossover rate is 0.95, mutation rate is 0.1, termination criterion is 500, etc. Other parameters are reasonably configured.

[0184] Step 3, sampling within the uncertainty interval to obtain an uncertainty design vector sample library;

[0185] The intervals of uncertain design variables are shown in Table 1.

[0186] Table 1 Intervals of uncertain design variables

[0187]

[0188] Each parameter is uniformly sampled within the interval using the hypercube sampling method. The uncertainty design vector is 256 groups, each containing 4 uncertain variables.

[0189] Step 4: Substitute the uncertainty design vector sample obtained in step 3 into formula (2) to calculate the corresponding 256 sets of ballistic parameters, including projectile initial velocity, maximum base pressure, muzzle pressure, and maximum negative pressure difference.

[0190] Step 5: Taking 256 sets of uncertainty design vectors as independent variables and the corresponding 256 sets of ballistic parameters as dependent variables, the Chebyshev polynomial proxy model of the projectile initial velocity, maximum base pressure, muzzle pressure, and maximum negative pressure difference function is calculated using formulas (4)-(6) based on the 4-dimensional Chebyshev polynomial.

[0191] Step 6: Select the midpoint V of the initial velocity of the projectile el c , Radius V el w and economic evaluation coefficient ζ as the optimization objective function, and from the perspective of gun firing safety, the maximum chamber pressure P is selected. dmax , muzzle pressure P pk And the maximum negative pressure difference -Δp is used as the constraint function. The interval set method is used to obtain the interval representation of the uncertain objective function corresponding to the uncertain design variables and the interval representation of the uncertain constraint function.

[0192] Step 7, using the affine algorithm, transform the uncertainty objective function and constraint function represented by the interval obtained in step 6 to obtain the radial type of the deterministic objective function and constraint function. The radial type of the initial velocity of the projectile is as follows

[0193]

[0194] Step 8: Use the interval order relation ≤ cw The uncertain objective function obtained in step 7 is converted into a deterministic objective function. The uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method.

[0195] Step 9: Each individual uncertain design sample is evaluated for interval economy, and an indicator for evaluating the overall parameter error level is obtained, which is then transformed into a deterministic optimization problem.

[0196] The artillery interior ballistics optimization problem is expressed as follows:

[0197]

[0198] The results are calculated using the optimization solution algorithm determined in step 2.

[0199] from Figure 4 It can be seen that the upper and lower bounds of the optimized projectile initial velocity are greater than the original design, which illustrates the effectiveness of the optimization method. Figure 5 The optimized pressure difference curve is better than the original design in terms of both the maximum negative pressure difference and the overall fluctuation. The smoother pressure difference curve indicates that the launch safety has been improved to a certain extent. Figure 6It can be clearly seen that in order to obtain a larger initial velocity of the projectile, the base pressure of the projectile has increased compared to the initial value. This is also in line with objective laws, and the increase is still within the allowable range.

Claims

1. A nonlinear interval uncertainty optimization method for interior ballistics based on an affine algorithm, characterized by: The following steps are involved: Step 1, determine the order of Chebyshev polynomial and uncertainty design variables, where the uncertainty design variables are charge mass m, gunpowder particle arc thickness 2e1, gunpowder particle aperture d and chamber volume w0; Step 2: Determine the algorithm parameters of the optimization solver NSGA-II, including: population size, crossover rate, mutation rate, termination criterion, and the number of small intervals t for interval division. j , constraint possibility level λ i ; Step 3: Sampling within the uncertainty interval to obtain an uncertainty design vector sample library; Step 4: Establish a mixed charge classical interior ballistic model based on thermodynamics, design vector samples based on the uncertainty obtained in step 3, and calculate the corresponding ballistic parameters, including projectile initial velocity, maximum projectile base pressure, muzzle pressure, and maximum negative pressure difference; Step 5: According to the trajectory parameters obtained in step 4, calculate the Chebyshev polynomial expansion coefficients and construct a Chebyshev polynomial proxy model; Step 6: Using the interval set method, obtain the intervals of the uncertain objective function and the uncertain constraint function corresponding to the uncertain design variables; Step 7: Using an affine algorithm, convert the uncertainty objective function and uncertainty constraint function obtained in step 6 into an affine form of uncertainty objective function and uncertainty constraint function including a center value and a deviation range; Step 8: Use interval order relation ≤ cw The uncertain objective function obtained in step 7 is converted into a deterministic objective function, and the uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method; Step 9: Perform interval economic evaluation on each individual uncertain design vector sample in step 3 to obtain an indicator for evaluating the overall parameter error level, quantify the error level of the design variables under uncertain conditions, and combine step 8 to obtain a deterministic objective function and a deterministic constraint function, so that the interior ballistic uncertainty optimization problem is transformed into a deterministic optimization problem; Step 10: Use the optimization solver NSGA-II to simulate and search the deterministic optimization problem obtained in step 9 to solve the Pareto optimal solution set. If the algorithm termination condition is met, the nonlinear interval uncertainty optimization of the interior ballistics is completed. Otherwise, it means that the sample size of the uncertainty design vector is insufficient, and return to step 3.

2. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized in that: Step 1: Determine the order of the Chebyshev polynomial to be 3.

3. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized in that: Step 3: Sampling within the uncertainty interval to obtain an uncertainty design vector sample library. The specific method is: The vector U∈U of the 4-dimensional interval uncertain design variables I =[U L ,U R ] is written as follows: in Where: superscripts I, L, R, c, and w represent the interval, the left boundary of the interval, the right boundary of the interval, the midpoint of the interval, and the radius of the interval, respectively; sampling is performed within the uncertainty interval to obtain an uncertainty design vector sample library.

4. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized by: Step 4: Establish a mixed charge classical interior ballistic model based on thermodynamics, design vector samples based on uncertainty obtained in step 3, and calculate the corresponding ballistic parameters, including projectile initial velocity, maximum base pressure, muzzle pressure, and maximum negative pressure difference, where: The classical interior ballistic model of mixed charge based on thermodynamics is: In the formula, χ i is the shape characteristic of the i-th type of gunpowder, λ and μ are the powder type parameters of the i-th type of gunpowder, Z i is the relative thickness of the i-th type of gunpowder combustion, u 1i is the burning rate coefficient of the i-th type of propellant, p is the propellant gas pressure, e 1i is half the arc thickness of the i-th type of gunpowder, m is the mass of the projectile, ω i is the charge mass of the i-th type of gunpowder, e i is the arc thickness of the i-th type of gunpowder, S is the equivalent cross-sectional area of ​​the barrel, is the secondary work coefficient, l0 is the volume reduction length of the chamber, Φ i is the combustion percentage of the i-th type of gunpowder, l Φ f is the free volume reduction length of the chamber, i is the gunpowder power of the i-th type of gunpowder, α i is the residual capacity of the i-th type of gunpowder, l and v are the projectile travel and speed respectively, Δ i is the gunpowder filling density of the i-th type of gunpowder, θ is the insulation coefficient, and n is a constant.

5. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized by: Step 5: Calculate the Chebyshev polynomial expansion coefficients based on the trajectory parameters obtained in step 4 and construct a Chebyshev polynomial proxy model. The specific method is as follows: Step 5.1, consider the uncertain design variables as charge mass m, gunpowder grain arc thickness 2e1, gunpowder grain aperture d and chamber volume w0, the dimension of uncertain variables is 4, for the unified writing format, the four uncertain variables are represented by x1, x2, x3, x4 respectively, and the 4-dimensional Chebyshev polynomial expansion is regarded as the tensor product of four 1-dimensional Chebyshev polynomials, that is: In the formula, x i is the ith interval variable, is a one-dimensional Chebyshev polynomial, n i The Chebyshev polynomial representing the ith interval variable is an nth-degree algebraic polynomial; Step 5.2, the initial velocity of the projectile, the maximum bottom pressure, the muzzle pressure and the maximum negative pressure difference are continuous functions of the uncertain design variables x1, x2, x3, x4. For the convenience of writing, (x1, x2, x3, x4) is abbreviated as (x). For the initial velocity of the projectile, assuming that the initial velocity function of the projectile f(x) is a 4-dimensional continuous function, then its 4-dimensional Chebyshev polynomial expansion is: Where p is the number of 0s in the subscripts i1,...,i4, is a 4-dimensional Chebyshev polynomial, are the corresponding polynomial coefficients, It is also a 4th-order tensor. Using the orthogonality of Chebyshev polynomials, the polynomial coefficients are calculated as follows: The Gauss-Chebyshev integral formula is used for each one-dimensional integral in equation (5), and the corresponding integral result is: In the formula, h=n+1 is the integration order, is the i-th integral interpolation point of the j-th interval variable, which is also a tensor product: In the formula, x i is each one-dimensional Chebyshev polynomial interpolation point, that is, the zero point of the Chebyshev polynomial, is the Crocker inner product symbol, so the total number of interpolation points N4 of the Chebyshev polynomial of 4-dimensional variables is: N4=(n+1) 4 (8) Step 5.3: According to formulas (4)-(6) in step 5.2, the four-dimensional Chebyshev polynomial proxy functions of the projectile initial velocity, maximum base pressure, muzzle pressure and maximum negative pressure difference are calculated in sequence.

6. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized by: Step 6: Use the interval set method to obtain the interval representation f of the uncertain objective function corresponding to the uncertain design variables, namely, the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference. i I (U I ), and the uncertainty constraint function, i.e., the chamber pressure range and the maximum negative pressure upper limit, is represented by the interval g i I (U I ), let the number of objective functions be m and the number of constraint functions be p, then: Where U I Design variables for uncertainty, f i I (U I ) and g i I (U I ) is the i-th uncertain objective function and uncertain constraint function under the uncertain design variable U I The output of the following, and Denotes the objective function f i In the design variables The values ​​under the left and right endpoints, i.e. the minimum and maximum endpoints, and Represents the constraint function g i The minimum and maximum endpoints of k=1,2,...,r represent the design variables U I is divided into r sample points.

7. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized by: Step 7, using the affine algorithm, transform the uncertainty objective function and constraint function represented by the interval obtained in step 6 to obtain the radial type of the deterministic objective function and constraint function. The specific method is: Step 7.1, affine calculation of uncertain design variables, charge mass x1, gunpowder grain arc thickness x2, gunpowder grain aperture x3 and chamber volume x4, uncertain variable x i The affine form of In the formula, x io Design variable x for uncertainty i The central value of the affine type, t is the number of uncertain factors affecting the variable x, ε il is the disturbance term, whose value range is [-1,1], x ij is the noise symbol ε ij The known real coefficients of |ε ij |=1 when affine type x i Get its maximum and minimum values ​​and express them as intervals: For the uncertain quantity x in the uncertainty optimization problem i It is expressed as: In the formula, Δx i is the interval radius, i=1,2,3,4. Step 7.2: When the objective function is replaced by the symbol d, the objective function is rewritten in affine form: In the formula, d0 is the center value of the affine type, d i are known real coefficients. The interval of the objective function at the design variable X: Step 7.3: Set the constraint function g i (X,U) is replaced by the symbol c, and the constraint function is rewritten in affine form: In the formula, c0 is the center value of the affine type, c i are known real coefficients. g i The interval at the design variable X is: Therefore, the uncertainty problem of artillery interior ballistics can be rewritten as:

8. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1 is characterized by: Step 8: Use the interval order relation ≤ cw The uncertain objective function obtained in step 7 is converted into a deterministic objective function. The uncertain constraint function obtained in step 7 is converted into a deterministic constraint function using the interval possibility method. The specific method is as follows: Step 8.1: Use interval order relation ≤ cw Transform the objective function, where c and w represent the midpoint and radius of the interval respectively, A≤ cw B means that interval B is better than interval A only when the midpoint and radius of interval B are smaller than those of interval A. In this way, the interval uncertainty objective function is transformed into a deterministic dual objective function optimization problem: Where: f c (U) represents the average performance of the objective function under uncertainty, f w (U) represents the fluctuation range of the objective function under the influence of uncertain factors; Step 8.2: According to the six spatial position relationships between the two intervals A and B, the specific value of the interval possibility is calculated using formula (19): Where: P(A I ≤B I ) or P(A I ≥B I ) represents the probability that interval A is smaller or larger than interval B for two uniformly distributed intervals A and B; For inequality constraints Transformed into a deterministic inequality constraint through formula (20): Where: i is the acceptable level of possibility set by the decision maker for the i-th constraint, representing the degree to which the constraint function needs to be satisfied; For the equation First, transform it into the form of formula (21): Then the two deterministic inequality constraints transformed into equation (22) are solved separately; 9. The nonlinear interval uncertainty optimization method for interior ballistics based on affine algorithm according to claim 1, characterized in that: Step 9: Each individual uncertainty design sample is evaluated for interval economy, and an indicator for evaluating the overall parameter error level is obtained, which is then transformed into a deterministic optimization problem. The specific method is as follows: Define a non-negative dimensionless evaluation coefficient ζ to measure the interval economy of all uncertain design variables; Where: is the interval radius of the uncertain design variable, is the maximum allowable error of the uncertain design variables artificially set in the optimization, and n represents the number of design variables; The multi-objective optimization problem of formula (24) is obtained: Where: is the i-th constraint function Less than or equal to (equal to, greater than or equal to) the i-th interval variable The probability of i is a constant; The uncertain optimization problem of gun interior ballistics containing four uncertain variables can be written as a deterministic optimization problem as shown in formula (25):

10. A nonlinear interval uncertainty optimization system for interior ballistics based on an affine algorithm, characterized in that: Implement the nonlinear interval uncertainty optimization method of interior ballistics based on affine algorithm as described in any one of claims 1-9 to achieve nonlinear interval uncertainty optimization of interior ballistics based on affine algorithm.

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