Interior ballistic nonlinear interval uncertainty optimization method based on affine algorithm

Abstract

The application discloses an interior ballistic nonlinear interval uncertainty optimization method based on an affine algorithm, determines Chebyshev polynomial order and uncertainty design variables, wherein the uncertainty design variables are charge mass, powder grain arc thickness, powder grain aperture and charge chamber volume; a mixed charge classical interior ballistic model based on thermodynamics is established, corresponding ballistic parameters are calculated, including projectile initial velocity, maximum projectile bottom pressure, muzzle pressure and maximum negative pressure difference; a Chebyshev polynomial proxy model is constructed; intervals of uncertainty objective functions and uncertainty constraint functions corresponding to the uncertainty design variables are obtained; the uncertainty objective functions and the uncertainty constraint functions are converted into affine forms including central values and deviation ranges; the interior ballistic uncertainty optimization problem is converted into a deterministic optimization problem; and an optimization solver NSGA-II is used for analog search to solve a Pareto optimal solution set. The method is suitable for efficient solution of engineering uncertainty optimization problems.

Description

Technical Field

[0001] This invention relates to the field of structural optimization design, and in particular to an optimization method for nonlinear interval uncertainty in internal ballistics based on affine algorithms. Background Technology

[0002] Internal ballistics is a discipline that studies the motion of projectiles within the barrel and the accompanying firing phenomena. Its performance indicators directly affect the combat capability of artillery and are an important basis for evaluating the performance of artillery weapon systems. Research in internal ballistics can provide a theoretical foundation for weapon design and ballistic performance analysis. However, in engineering, ammunition inevitably contains uncertainties caused by material properties, manufacturing errors, assembly errors, etc. Deterministic internal ballistic optimization cannot accurately characterize the impact of uncertainties, making uncertain internal ballistic optimization of great significance. Uncertainty optimization theory has been widely applied in other engineering optimization fields. Common uncertainty optimization methods include stochastic programming, fuzzy programming, and interval optimization. In interval optimization, uncertain parameters are described using intervals. Only the upper and lower bounds of the parameters need to be known, without needing their precise probability distribution or fuzzy membership function, hence its widespread use. Interval uncertainty optimization problems often involve two nested optimization layers. The outer layer optimization is used to find the optimal design vector, while the inner layer optimization is used to calculate the interval of the uncertain objective function and constraints. However, two nested optimization problems inevitably involve a large amount of computation time and low computational efficiency. To address this drawback, this paper combines interval uncertainty optimization with affine algorithms. Affine algorithms are used to solve linear programming problems, combining the characteristics of the simplex method and the interior point method. Theoretically, affine algorithms have polynomial time complexity, generally exhibit better numerical stability, and their 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 affine algorithms. First, a polynomial response surface model is used to obtain explicit expressions of the design parameters and uncertainty parameters relative to the objective function and constraint functions. When establishing the optimization model, the interval uncertainty parameters are rewritten in affine form. Then, the upper and lower bounds of the uncertainty objective function and constraint functions are directly obtained using the affine algorithm, thus avoiding two levels of nesting and greatly improving optimization efficiency. The proposed method can obtain the impact of internal ballistic uncertainty parameters on the internal ballistic performance of artillery, which has significant theoretical and engineering value for reducing initial velocity probabilistic errors and improving artillery firing density. Summary of the Invention

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

[0004] The technical solution for implementing this invention is: a method for optimizing uncertainty in the nonlinear interval of internal ballistics based on an affine algorithm, comprising the following steps:

[0005] Step 1: Determine the order of the Chebyshev polynomial and the uncertainty design variables, where the uncertainty design variables are the charge mass m, the arc thickness of the gunpowder particle 2e1, the aperture of the gunpowder particle d, and the chamber volume w0.

[0006] Step 2: Determine the algorithm parameters of the NSGA-II optimization solver, including: population size, crossover rate, mutation rate, termination criterion, and the number of subintervals t for interval partitioning. j Constraint Possibility Level λ i ;

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

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

[0009] Step 5: Based on the ballistic parameters obtained in Step 4, calculate the Chebyshev polynomial expansion coefficients and construct the Chebyshev polynomial proxy model.

[0010] Step 6: Use the interval set method to 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, the uncertainty objective function and uncertainty constraint function obtained in Step 6 are transformed into affine forms of uncertainty objective function and uncertainty constraint function, including the center value and the deviation range;

[0012] Step 8: Apply the interval order relation ≤ cw The uncertain objective function obtained in step 7 is transformed into a deterministic objective function. The uncertain constraint function obtained in step 7 is then transformed 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 index for evaluating the overall parameter error level, quantify the error level of the design variables under uncertain conditions, and combine with Step 8 to obtain the deterministic objective function and deterministic constraint function, thus transforming the internal ballistic uncertainty optimization problem into a deterministic optimization problem.

[0014] Step 10: Use the NSGA-II optimization solver to perform a simulation search on the deterministic optimization problem obtained in Step 9 to solve the Pareto optimal solution set. If the algorithm termination condition is met, the uncertainty optimization of the inner ballistic nonlinear interval is completed; otherwise, it indicates 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 3.

[0016] Further, 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 4-dimensional interval uncertainty design variables I =[U L U R Write it in the following form:

[0018]

[0019] 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; sampling is performed within the uncertain interval to obtain an uncertain design vector sample library.

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

[0021] The classical internal ballistic model of mixed-charge propellants based on thermodynamics is as follows:

[0022]

[0023] In the formula, χ i Let λ be the shape characteristic quantity of the i-th type of gunpowder. ι μ ι Let Z be the propellant parameter of the i-th type of gunpowder. i Let u be the relative thickness of the i-th type of gunpowder during combustion. 1i Let p be the burning rate coefficient of the i-th type of gunpowder, p be the pressure of the gunpowder gas, and e be the burning rate coefficient of the i-th type of gunpowder. 1i The thickness of the propellant arc is half that of the i-th type of propellant, m is the mass of the projectile, and ω is... i Let e ​​be the charge mass of the i-th type of gunpowder. i Let S be the gunpowder arc thickness of the i-th type of gunpowder, and S be the equivalent cross-sectional area of ​​the barrel. Φ is the secondary work coefficient, l0 is the volume of the medicine chamber, and Φ is the diameter of the reduced diameter. i Let l be the percentage of gunpowder combustion for class i.φ For the free volume of the drug chamber, the diameter is reduced, f i Let α be the gunpowder power of the i-th type of gunpowder. i Let be the residual volume of the i-th type of gunpowder, l and v be the projectile's stroke and velocity, respectively, and Δ i Let θ be the gunpowder packing density of the i-th type of gunpowder, θ be the adiabatic coefficient, and n be a constant.

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

[0025] Step 5.1: Considering the uncertainties, the design variables are the charge mass m, the gunpowder particle arc thickness 2e1, the gunpowder particle aperture d, and the chamber volume w0. The dimension of the uncertain variables is 4. For a consistent writing format, the four uncertain variables are represented by x1, x2, x3, and x4 respectively. The expansion of the 4-dimensional Chebyshev polynomial is regarded as the tensor product of four 1-dimensional Chebyshev polynomials, that is:

[0026]

[0027] In the formula, x i Let i be the interval variable. It is a one-dimensional Chebyshev polynomial, n i The Chebyshev polynomial representing the i-th interval variable is an n-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 continuous functions of the uncertain design variables x1, x2, x3, and x4. For ease of writing, (x1, x2, x3, x4) is abbreviated as (x). For the initial velocity of the projectile, let the initial velocity function f(x) be a 4-dimensional continuous function. Then its 4-dimensional Chebyshev polynomial expansion is:

[0029]

[0030] In the formula, p is the number of 0s in the subscripts i1,...,i4. It is a 4-dimensional Chebyshev polynomial. These 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] Applying the Gauss-Chebyshev integral formula to each one-dimensional integral in equation (5), the corresponding integral result is obtained as follows:

[0033]

[0034] In the formula, h = n+1 is the order of integration. These are the i-th integral interpolation points of the j-th interval variable, and these interpolation points are also tensor products:

[0035]

[0036] In the formula, x i These are the interpolation points of each one-dimensional Chebyshev polynomial, i.e., the zeros of the Chebyshev polynomial. Since the Krock inner product symbol is used, the total number of interpolation points N4 of the Chebyshev polynomial for 4-dimensional variables is:

[0037] N4 = (n+1) 4 (8)

[0038] Step 5.3: According to formulas (4)-(6) in step 5.2, calculate the 4-dimensional Chebyshev polynomial surrogate functions for the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference 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 projectile initial velocity, maximum base pressure, muzzle pressure, and maximum negative pressure difference. i I (U I ), and the interval representation g of the uncertain constraint function, namely the chamber pressure range and the upper limit of the maximum negative pressure. i I (U I Let m be the number of objective functions and p be the number of constraint functions, then:

[0040]

[0041] In the formula, U I Design variables for uncertainty, f i I (U I ) and g i I (U I Let be the objective function and constraint function for the i-th uncertainty, given the uncertain design variable U. I The output below, and Describe the objective function f i In design variables The values ​​below the left and right endpoints, i.e., the minimum and maximum values ​​at the endpoints. and Represents the constraint function g i The minimum and maximum endpoints, k = 1, 2, ..., r represent the design variable U. I It is divided into r sample points.

[0042] Furthermore, in step 7, an affine algorithm is used to transform the uncertain objective function and constraint function obtained in step 6 into radial forms of the deterministic objective function and constraint function. The specific method is as follows:

[0043] Step 7.1: Affine calculation of design variables for uncertainties: charge mass x1, gunpowder particle arc thickness x2, gunpowder particle aperture x3, and chamber volume x4. Uncertain variable x i Affine notation

[0044]

[0045] In the formula, x io Design variable x for uncertainty i The central value of the affine form, t is the number of uncertainties affecting variable x, ε it The term is a disturbance term, and its value range is [-1, 1]. ij For noise symbol ε ij The known real coefficients, when |ε ij When |=1, affine type x i Find its maximum and minimum values ​​and express them in interval form:

[0046]

[0047] For the uncertainty x in the uncertainty optimization problem i Represented as:

[0048]

[0049] In the formula, Δx i Let be the radius of the interval, 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] In the formula, d0 is the center value of the affine type, d i The coefficients are known.

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

[0054]

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

[0056]

[0057] In the formula, c0 is the center value of the affine type, c i The real coefficients are known. i The interval for design variable X is:

[0058]

[0059] Therefore, the problem of unpredictable ballistics within artillery can be rewritten as follows:

[0060]

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

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

[0063]

[0064] In the formula: 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;

[0065] Step 8.2: Based on the six spatial relationships between the two intervals A and B, calculate the specific value of the interval probability using equation (19):

[0066]

[0067] In the formula: P(A) I ≤B I ) or P(A I ≥B I () represents the probability that, for two uniformly distributed intervals A and B, interval A is less than or greater than interval B.

[0068] For inequality constraints Equation (20) is transformed into a deterministic inequality constraint:

[0069]

[0070] In the formula: λ i The acceptable probability level set by the decision-maker for the i-th constraint represents the degree to which the constraint function needs to be satisfied.

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

[0072]

[0073] Then, the two deterministic inequality constraints of equation (22) are transformed into solutions respectively;

[0074]

[0075] Furthermore, in step 9, each individual sample of the uncertain design undergoes an interval economic evaluation to obtain an index for evaluating the overall parameter error level, thus transforming it 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] In the formula: The radius of the interval for design variables to account for uncertainty. To optimize the maximum permissible error of the artificially set uncertainties in the design variables, n represents the number of design variables;

[0079] The multi-objective optimization problem obtained is shown in equation (24):

[0080]

[0081] In the formula: It 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, μ i It is a constant;

[0082] The artillery trajectory uncertainty optimization problem containing four uncertain variables is written as the deterministic optimization problem shown in formula (25):

[0083]

[0084] An inner ballistic nonlinear interval uncertainty optimization system based on affine algorithm is provided, which implements the inner ballistic nonlinear interval uncertainty optimization method based on affine algorithm to achieve inner ballistic nonlinear interval uncertainty optimization.

[0085] Compared with the prior art, the significant advantages of this invention are:

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

[0087] (2) The novel interval uncertainty optimization algorithm established in this 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 surrogate models.

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

[0089] (4) The novel interval uncertainty optimization algorithm established in this invention has an index for evaluating the overall parameter error level. The obtained uncertainty design variables can meet the design requirements under the maximum allowable error level, and have good interval economy. Attached Figure Description

[0090] Figure 1 This is a flowchart of the inner ballistic nonlinear interval uncertainty optimization method based on affine algorithm of the present invention;

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

[0092] Figure 3 It is the Pareto front of the ballistic optimization results in Embodiment 1 of the present invention;

[0093] Figure 4 This is the range of the projectile's initial velocity after optimization in Embodiment 1 of the present invention;

[0094] Figure 5 This is the optimized pressure difference according to Embodiment 1 of the present invention;

[0095] Figure 6 This is the range of the spring base pressure optimized in Embodiment 1 of the present invention. Detailed Implementation

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

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

[0098] Step 1: Determine the design space of the Chebyshev polynomial surrogate model based on the range of design variables for the optimization problem. The uncertain design variables are the charge mass m, the arc thickness of the gunpowder particle 2e1, the aperture of the gunpowder particle d, and the volume of the chamber w0; the Chebyshev polynomial order is 3.

[0099] Step 2: Optimize the algorithm parameters of the solver NSGA-II, specifically including: population size, crossover rate, mutation rate, and termination criterion (maximum number of generations). The algorithm parameters for inner interval calculation include: the number of subintervals t. j Constraint Possibility Level λ i .

[0100] Step 3: The vector U∈U of m-dimensional interval uncertainty design variables I =[U L U R It 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. An uncertainty design vector library is obtained by sampling within the uncertain interval using experimental design methods.

[0103] Step 4: Employ a classical internal ballistic model for mixed-charge propellants based on thermodynamics:

[0104]

[0105] In the formula, χ i Z represents the shape characteristic quantity of the i-th type of gunpowder, λ and μ are the propellant shape parameters of the i-th type of gunpowder, and Z represents the shape characteristic quantity of the i-th type of gunpowder. i The relative thickness of the gunpowder during combustion, u 1i Let p be the burning rate coefficient of the i-th type of gunpowder, p be the pressure of the gunpowder gas, and e be the burning rate coefficient of the i-th type of gunpowder. 1i The thickness of the propellant arc is half that of the i-th type of propellant, m is the mass of the projectile, and ω is... iFor the quality of the propellant charge, e i S is the thickness of the gunpowder arc, and S is the equivalent cross-sectional area of ​​the barrel. For the minor work coefficient, l0 represents the volume of the medicine chamber with reduced diameter and length, ψ i The percentage of gunpowder combustion, l ψ For the free volume of the drug chamber, the diameter is reduced, f i For gunpowder power, α i For the remainder, l and v are the projectile's stroke and velocity, respectively, Δ i θ is the gunpowder packing density, and θ is the adiabatic coefficient.

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

[0107] Step 5: Based on 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, i.e.:

[0109]

[0110] The 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 i1,...,i k The number of zeros in the middle. It is a k-dimensional Chebyshev polynomial. Let i be the i-th one-dimensional Chebyshev polynomial. These are the corresponding polynomial coefficients. It is also a k-order tensor. Using the orthogonality of Chebyshev polynomials, the polynomial coefficients are calculated as follows:

[0113]

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

[0115]

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

[0117]

[0118] In the formula, x k These are the interpolation points of each one-dimensional Chebyshev polynomial, i.e., the zeros of the Chebyshev polynomial. Let N be the Krock inner product symbol. Therefore, the total number of interpolation points N is... k for:

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

[0120] Step 6: Calculate the interval between the uncertainty objective function and the uncertainty constraint function corresponding to the current uncertain design variable using equation (24):

[0121]

[0122] In the formula, U I Design variables for uncertainty, f i I (U I ) and g i I (U I Let be the objective function and constraint function for the i-th uncertainty, given the uncertain design variable U. I The output below, and Describe the objective function f i In design variables The values ​​below the left and right endpoints, i.e., the minimum and maximum values ​​at the endpoints. and Represents the constraint function g i The minimum and maximum endpoints, i = 1, 2, ..., q, indicate that there are q objective functions, and k = 1, 2, ..., r, represent the design variables U. I It is divided into r sample points.

[0123] Step 7: Transform the uncertain objective function and uncertain constraint function obtained in Step 6 into a bi-objective deterministic optimization problem using an affine algorithm. The specific method is as follows:

[0124] Step 7.1, the affine form of the uncertainty u is denoted as

[0125]

[0126] In the formula, u o ε is the center value of the affine type. t The noise symbol has a value range of [-1, 1], u iFor noise symbol ε t The known real coefficients. When |ε i When | = 1, the affine type u can obtain its maximum and minimum values, which can be represented in interval form:

[0127]

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

[0129]

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

[0131]

[0132] Then we have:

[0133]

[0134] For operations such as division, square root, and exponentiation, a min-rang affine approximation model is used. Let the affine form... Its value range is a and b are the upper and lower limits of the values ​​that the variables can take.

[0135] Step 7.3: According to the min-rang affine approximation theory, we can obtain... The affine approximation is

[0136]

[0137] The affine approximation is

[0138]

[0139] The affine approximation is

[0140]

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

[0142] As can be seen from the above formula, when applying the min-rang affine approximation theory, the affine type... The range of values ​​for must not include 0. Compared to interval algorithms, although the calculation process of affine algorithms is slightly more complicated, affine algorithms do not involve complex operations such as differentiation or derivation. They consist entirely of basic mathematical operations, and the calculation process of affine types can be easily completed with the help of symbolic computation software.

[0143] Rewrite the objective function in affine form:

[0144]

[0145] In the formula, d0 is the center value of the affine type, d i Given real coefficients, ε i The symbol is for noise; therefore, the interval of the objective function at the design variable X can be obtained:

[0146]

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

[0148]

[0149] In the formula, c0 is the center value of the affine type, c i Given real coefficients, ε i For noise symbol; g i The interval for design variable X is:

[0150]

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

[0152]

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

[0154] Step 8.1: Use the interval order relation ≤ cw Perform a transformation of the objective function, where c and w represent the midpoint and radius of the interval, respectively. A≤ cw B indicates that interval B is better than interval A only if both the midpoint and radius of interval B are smaller than those of interval A. In this way, the uncertain objective function is transformed into a deterministic biobjective optimization problem:

[0155]

[0156] In the formula: 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, therefore equation (23) minimizes f w (U) Consider robustness.

[0157] Step 8.2: Based on the six spatial relationships between the two intervals A and B, calculate the specific value of the interval probability using the model of equation (24):

[0158]

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

[0160]

[0161] In the formula: λ i λ represents the acceptable probability level set by the decision-maker for the i-th constraint, indicating 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 first be transformed into the form of equation (26):

[0163]

[0164] Then it can be transformed into solving the two deterministic inequality constraints of equation (27) 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 under the maximum allowable error and thus reduce costs.

[0167]

[0168] In the formula: The radius of the interval for design variables to account for uncertainty. The maximum permissible error of the design variables, which are artificially introduced to optimize the process, is ζ, where ζ represents the number of design variables. A larger ζ value indicates a higher overall error level and better interval economy.

[0169] Through the above processing, the multi-objective optimization problem of equation (29) can be obtained:

[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 artillery trajectory uncertainty optimization problem can be further rewritten as:

[0177]

[0178] Thus, the uncertain optimization problem is transformed into a deterministic optimization problem, which can be solved using existing mature intelligent optimization algorithms such as sequential quadratic programming and genetic algorithms.

[0179] Step 10: Use the NSGA-II optimization solver to perform a simulation search to find 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, thus inevitably involving a process of selecting and optimizing solutions. Because of the close relationships between internal ballistic parameters, a single parameter cannot comprehensively and profoundly reflect the overall performance of the 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 muzzle velocity of the projectile directly affects the range and power of the artillery; therefore, optimization and design should ensure that the muzzle velocity is within a reasonable range.

[0182] Step 1: Determine the order of the Chebyshev polynomial to be 3. Consider four uncertain design variables: charge mass m, gunpowder particle arc thickness 2e1, gunpowder particle aperture d, and chamber volume w0.

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

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

[0185] The ranges of the design variables with uncertainties are shown in Table 1.

[0186] Table 1. Ranges of Design Variables for Uncertainty

[0187]

[0188] Each parameter is sampled uniformly within the interval using a hypercube sampling method. The uncertainty design vector consists of 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 the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference.

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

[0191] Step 6, select the midpoint V of the interval of the projectile's initial velocity. el c radius V el w Using the economic evaluation coefficient ζ as the optimization objective function, and considering the safety of artillery firing, the maximum chamber pressure P is selected. dmax Muzzle pressure P pk The maximum negative pressure difference -Δp is used as the constraint function. The interval set method is employed 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 an affine algorithm, transform the uncertain objective function and constraint function obtained in Step 6 into radial patterns of the deterministic objective function and constraint function. The radial pattern of the projectile's initial velocity is as follows:

[0193]

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

[0195] Step 9: For each individual sample in the uncertain design, perform an interval economic evaluation to obtain an index for evaluating the overall parameter error level, thus transforming it into a deterministic optimization problem.

[0196] The problem of optimizing the internal ballistics of artillery is described as follows:

[0197]

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

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

Claims

1. A method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on an affine algorithm, characterized in that: Includes the following steps: Step 1: Determine the order of the Chebyshev polynomial and the uncertainty design variables, where the uncertainty design variables are the charge mass m, the arc thickness of the gunpowder particle 2e1, the aperture of the gunpowder particle d, and the chamber volume w0. Step 2: Determine the algorithm parameters of the NSGA-II optimization solver, including: population size, crossover rate, mutation rate, termination criterion, and the number of subintervals t for interval partitioning. j Constraint Possibility Level λ i ; Step 3: Sample within the uncertainty interval to obtain an uncertainty design vector sample library; Step 4: Establish a classical internal ballistic model of mixed charge based on thermodynamics. Based on the uncertainty design vector sample obtained in Step 3, calculate the corresponding ballistic parameters, including the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference. Step 5: Based on the ballistic parameters obtained in Step 4, calculate the Chebyshev polynomial expansion coefficients and construct the Chebyshev polynomial proxy model. Step 6: Use the interval set method to 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, the uncertainty objective function and uncertainty constraint function obtained in Step 6 are transformed into affine forms of uncertainty objective function and uncertainty constraint function, including the center value and the deviation range; Step 8: Apply the interval order relation ≤ cw The uncertain objective function obtained in step 7 is transformed into a deterministic objective function. The uncertain constraint function obtained in step 7 is then transformed 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 index for evaluating the overall parameter error level, quantify the error level of the design variables under uncertain conditions, and combine with Step 8 to obtain the deterministic objective function and deterministic constraint function, thus transforming the internal ballistic uncertainty optimization problem into a deterministic optimization problem. Step 10: Use the NSGA-II optimization solver to perform a simulation search on the deterministic optimization problem obtained in Step 9 to solve the Pareto optimal solution set. If the algorithm termination condition is met, the uncertainty optimization of the inner ballistic nonlinear interval is completed; otherwise, it indicates that the sample size of the uncertainty design vector is insufficient, and return to Step 3.

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

3. The method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on affine algorithm according to claim 1, characterized in that, Step 3: Sample within the uncertainty interval to obtain an uncertainty design vector sample library. The specific method is as follows: The vector U∈U of 4-dimensional interval uncertainty design variables I =[U L U R Write it in the following form: in 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; sampling is performed within the uncertain interval to obtain an uncertain design vector sample library.

4. The method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on affine algorithm according to claim 1, characterized in that: Step 4: Establish a classical internal ballistic model for mixed-charge propellants based on thermodynamics. Using the uncertainty design vector samples obtained in Step 3, calculate the corresponding ballistic parameters, including initial projectile velocity, maximum base pressure, muzzle pressure, and maximum negative pressure difference. Where: The classical internal ballistic model of mixed-charge propellants based on thermodynamics is as follows: In the formula, χ i Z represents the shape characteristic quantity of the i-th type of gunpowder, λ and μ are the propellant shape parameters of the i-th type of gunpowder, and Z represents the shape characteristic quantity of the i-th type of gunpowder. i Let u be the relative thickness of the i-th type of gunpowder during combustion. 1i Let p be the burning rate coefficient of the i-th type of gunpowder, p be the pressure of the gunpowder gas, and e be the burning rate coefficient of the i-th type of gunpowder. 1i The thickness of the propellant arc is half that of the i-th type of propellant, m is the mass of the projectile, and ω is... i Let e ​​be the charge mass of the i-th type of gunpowder. i Let S be the gunpowder arc thickness of the i-th type of gunpowder, and S be the equivalent cross-sectional area of ​​the barrel. Φ is the secondary work coefficient, l0 is the volume of the medicine chamber, and Φ is the diameter of the reduced diameter. i Let l be the percentage of gunpowder combustion for class i. Φ For the free volume of the drug chamber, the diameter is reduced, f i Let α be the gunpowder power of the i-th type of gunpowder. i Let be the residual volume of the i-th type of gunpowder, l and v be the projectile's stroke and velocity, respectively, and Δ i Let θ be the gunpowder packing density of the i-th type of gunpowder, θ be the adiabatic coefficient, and n be a constant.

5. The method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on affine algorithm according to claim 1, characterized in that: Step 5: Based on the ballistic parameters obtained in Step 4, calculate the Chebyshev polynomial expansion coefficients and construct the Chebyshev polynomial proxy model. The specific method is as follows: Step 5.1: Considering the uncertainties, the design variables are the charge mass m, the gunpowder particle arc thickness 2e1, the gunpowder particle aperture d, and the chamber volume w0. The dimension of the uncertain variables is 4. For a consistent writing format, the four uncertain variables are represented by x1, x2, x3, and x4 respectively. The expansion of the 4-dimensional Chebyshev polynomial is regarded as the tensor product of four 1-dimensional Chebyshev polynomials, that is: In the formula, x i Let i be the interval variable. It is a one-dimensional Chebyshev polynomial, n i The Chebyshev polynomial representing the i-th interval variable is an n-degree algebraic polynomial. Step 5.2: The initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference are continuous functions of the uncertain design variables x1, x2, x3, and x4. For ease of writing, (x1, x2, x3, x4) is abbreviated as (x). For the initial velocity of the projectile, let the initial velocity function f(x) be a 4-dimensional continuous function. Then its 4-dimensional Chebyshev polynomial expansion is: In the formula, p is the number of 0s in the subscripts i1,...,i4. It is a 4-dimensional Chebyshev polynomial. These 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: Applying the Gauss-Chebyshev integral formula to each one-dimensional integral in equation (5), the corresponding integral result is obtained as follows: In the formula, h = n+1 is the order of integration. These are the i-th integral interpolation points of the j-th interval variable, and these interpolation points are also tensor products: In the formula, x i These are the interpolation points of each one-dimensional Chebyshev polynomial, i.e., the zeros of the Chebyshev polynomial. Since the Krock inner product symbol is used, the total number of interpolation points N4 of the Chebyshev polynomial for 4-dimensional variables is: N4=(n+1) 4 (8) Step 5.3: According to formulas (4)-(6) in step 5.2, calculate the 4-dimensional Chebyshev polynomial surrogate functions for the initial velocity of the projectile, the maximum base pressure, the muzzle pressure, and the maximum negative pressure difference in sequence.

6. The method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on affine algorithm according to claim 1, characterized in that: Step 6: Using the interval set method, obtain the interval representation f of the uncertain objective function corresponding to the uncertain design variables, namely the projectile initial velocity, maximum base pressure, muzzle pressure, and maximum negative pressure difference. i I (U I ), and the interval representation g of the uncertain constraint function, namely the chamber pressure range and the upper limit of the maximum negative pressure. i I (U I Let m be the number of objective functions and p be the number of constraint functions, then: In the formula, U I Design variables for uncertainty, f i I (U I ) and g i I (U I Let be the objective function and constraint function for the i-th uncertainty, given the uncertain design variable U. I The output below, and Describe the objective function f i In design variables The values ​​below the left and right endpoints, i.e., the minimum and maximum values ​​at the endpoints. and Represents the constraint function g i The minimum and maximum endpoints, k = 1, 2, ..., r represent the design variable U. I It is divided into r sample points.

7. The method for optimizing uncertainty in the nonlinear interval of internal ballistics based on an affine algorithm according to claim 1, characterized in that: Step 7: Using an affine algorithm, transform the uncertain objective function and constraint function obtained in Step 6 into radial forms of the deterministic objective function and constraint function. The specific method is as follows: Step 7.1: Affine calculation of design variables for uncertainties: charge mass x1, gunpowder particle arc thickness x2, gunpowder particle aperture x3, and chamber volume x4. Uncertain variable x i Affine notation In the formula, x io Design variable x for uncertainty i The central value of the affine form, t is the number of uncertainties affecting variable x, ε il The term is a disturbance term, and its value range is [-1, 1]. ij For noise symbol ε ij The known real coefficients, when |ε ij When |=1, affine type x i Find its maximum and minimum values ​​and express them in interval form: For the uncertainty x in the uncertainty optimization problem i Represented as: In the formula, Δx i Let i be the radius of the interval, 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 The real coefficients are known. The interval of the objective function at the design variable X: Step 7.3: Set the constraint function g i If (X,U) is replaced by the symbol c, the constraint function can be rewritten in affine form: In the formula, c0 is the center value of the affine type, c i The real coefficients are known; g i The interval for design variable X is: Therefore, the problem of unpredictable ballistics within artillery can be rewritten as follows:

8. The method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on affine algorithm according to claim 1, characterized in that: Step 8, use the interval order relation ≤ cw The uncertain objective function obtained in step 7 is transformed into a deterministic objective function. The uncertain constraint function obtained in step 7 is then transformed into a deterministic constraint function using the interval possibility method. The specific method is as follows: Step 8.1: Use the interval order relation ≤ cw Perform a transformation of the objective function, where c and w represent the midpoint and radius of the interval, respectively, and A ≤ cw B indicates that interval B is better than interval A only when both the midpoint and radius of interval B are smaller than those of interval A. In this way, the uncertain objective function of intervals is transformed into a deterministic biobjective function optimization problem: In the formula: 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: Based on the six spatial relationships between the two intervals A and B, calculate the specific value of the interval probability using equation (19): In the formula: P(A) I ≤B I ) or P(A I ≥B I () represents the probability that, for two uniformly distributed intervals A and B, interval A is less than or greater than interval B. For inequality constraints Equation (20) is transformed into a deterministic inequality constraint: In the formula: λ i The acceptable probability level set by the decision-maker for the i-th constraint represents the degree to which the constraint function needs to be satisfied. For the equation First, transform it into the form of equation (21): Then, the two deterministic inequality constraints of equation (22) are transformed into solutions respectively; 9. The method for optimizing the uncertainty of the nonlinear interval of internal ballistics based on affine algorithm according to claim 1, characterized in that, Step 9: For each individual sample in the uncertain design, perform an interval economic evaluation to obtain an index for evaluating the overall parameter error level, thus transforming it 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; In the formula: The radius of the interval for design variables to account for uncertainty. To optimize the maximum permissible error of the artificially set uncertainties in the design variables, n represents the number of design variables; The multi-objective optimization problem obtained is shown in equation (24): In the formula: It 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, μ i It is a constant; The artillery trajectory uncertainty optimization problem containing four uncertain variables is written as the deterministic optimization problem shown in formula (25):

10. An optimization system for internal ballistic nonlinear interval uncertainty based on an affine algorithm, characterized in that, Implement the inner ballistic nonlinear interval uncertainty optimization method based on affine algorithm as described in any one of claims 1-9 to achieve inner ballistic nonlinear interval uncertainty optimization based on affine algorithm.

Images