CN108804859B - Design method of symmetric successive local enumeration Latin hypercube test - Google Patents
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Abstract
The invention discloses a symmetrical successive local enumeration Latin hypercube test design method, and belongs to the technical field of engineering optimization design. The realization method comprises the following steps of dividing a design space into a super-checkerboard, gradually generating sample points by using a local enumeration method according to a minimum distance maximization criterion, simultaneously generating two sample points by using a symmetry technology through enumeration at one time, and saving calculation time under the condition of ensuring the space uniformity and the projection uniformity of the sample points. The invention divides the problem into even problem and odd problem according to the number of sample points and provides two corresponding symmetrical local enumeration methods according to the properties of the even problem and the odd problem. The symmetrical successive local enumeration Latin hypercube test design method is applied to the proxy model, can obviously improve the global optimizing capability and optimizing efficiency of the proxy model optimization design method, is suitable for the engineering design optimization field containing high-precision analysis models, and can effectively improve the engineering design optimizing efficiency and shorten the design period.
Description
Technical Field
The invention relates to a symmetrical successive local enumeration Latin hypercube test design method, and belongs to the technical field of engineering optimization design.
Background
With the continuous development of computer technology, high-precision simulation models are widely applied in engineering design to improve analysis precision and design feasibility. However, the traditional design optimization method usually directly calls a high-precision analysis model to perform design optimization, so that technical bottlenecks of high time consumption, long period, low efficiency and the like exist in practical application. Secondly, modern engineering design optimization problems often involve multiple disciplines, and the disciplines are coupled to each other. Taking optimization of aircraft design as an example, multiple disciplines such as pneumatics, structures, stealths are often involved in the design process, and analysis of a single high-precision model is time-consuming. In order to reduce the computation time in the conventional design method, the design optimization method MBDO based on the surrogate model is widely applied to engineering design. The method aims to construct an approximate model equivalent to a high-precision analysis model by using a mathematical means and replace the high-precision analysis model for design optimization so as to improve the design efficiency.
Design of Experiments (DOE), also known as decimation, is a key technology in MBDO. The quality of the sample points generated by the test design method determines the precision of the proxy model, thereby affecting the performances of the MBDO method such as optimizing capability, efficiency and robustness. It is generally accepted that a good set of sample points should have good spatial uniformity and projection uniformity. To improve the sampling quality and efficiency, McKay (1979) proposed a Latin Hypercube Design (LHD) with good one-dimensional projection properties. Ye (2000) studies a column-pair (CP) algorithm that enables the construction of optimally symmetric LHDs. Bates (2004) describes a method of minimizing potential using PermGA to produce an optimal LHD. Ruichen Jin (2005) studied the optimal experimental design method based on the improved random evolution technology (ESE). Grosso (2009) adopts an Iterative Local Search (ILS) to obtain an optimal Latin hypercube design method with a maximum minimum distance (Maximin). Zhu Huaguang et al (2012) proposed a Latin hypercube test design method based on Successive Local Enumeration (SLE). Unlike the LHD method, which uses the sample quality assessment index as the global objective function, the SLE uses the maximized minimum distance as the local objective function (the minimum distance is the minimum of all distances between the generated point and the existing points that have been generated by the SLE), achieving the effect of improving the spatial uniformity of the sample points. By comparing this approach with the MATLAB function lhsdesign, BinGA, PermGA (Bates, 2004), and TPLHD (Viana, 2010) approaches, it can be found that the sampling quality of SLE has significant advantages.
While others have suggested the idea of symmetric experimental design. Keeny (2000) brings the idea of symmetry into the sampling technique. In his study, the SA algorithm (Morris and Mitchell, 1995) and the CP algorithm (Ye, 2000) were modified to the corresponding symmetric algorithms and compared to the original algorithms. Test results for some numerical and engineering applications showAnd the time consumption is saved by half by adopting a symmetric strategy for calculation. At the same time, some equipartition criteria (d) min 、φ P And CL 2 Criterion) shows that the space uniform distribution of the symmetrically sampled sample points is consistent with that of the original algorithm, which shows that the symmetrical sampling technology can save time, improve efficiency and maintain certain algorithm performance. However, this method is only applicable to the problem of an even number of sample points.
Disclosure of Invention
In the optimization design based on the proxy model, sample points are often required to be constructed in a design space. However, some existing sampling methods are low in efficiency, and for high-dimensional problems, a large amount of time is needed for sampling, so that the time consumption and the optimization efficiency of the optimization design method are influenced. Aiming at the defect, the invention discloses a symmetric successive local enumeration Latin hypercube experimental design method (A novel algorithm of maximum Latin hypercube design using systematic statistical failure, SYM-SLE) which aims to solve the technical problem that: based on a Latin hypercube test design method of Successive Local Enumeration (SLE), a symmetric sampling method is used, two symmetric sample points are generated by Enumeration at one time, and then the sampling of the designed sample points is realized, and the method has the following advantages: firstly, the calculation time is reduced, and the algorithm efficiency is improved, so that the method is better suitable for engineering application; secondly, by a symmetrical sampling method, the space uniformity of sampling results is ensured, and the performance reduction caused by greatly reducing time consumption is avoided.
The purpose of the invention is realized by the following technical scheme.
The invention discloses a symmetrical successive local enumeration Latin hypercube test design method, which aims to sample any number of sample points in any dimension space, divides the design space into a hypercoarding, uses a local enumeration method to successively generate sample points according to a minimum distance maximization criterion, and simultaneously uses a symmetrical technology to enumerate and generate two sample points at one time, thereby saving about half of the calculation time under the condition of ensuring the space equipartition and projection uniformity of the sample points. The invention divides the problem into even problem and odd problem according to the number of sample points and provides two corresponding symmetrical local enumeration methods according to the properties of the even problem and the odd problem. The symmetrical successive local enumeration Latin hypercube test design method disclosed by the invention is applied to the proxy model, can obviously improve the global optimization capability and the optimization efficiency of the proxy model optimization design method, is suitable for the engineering design optimization field containing a high-precision analysis model, and can effectively improve the engineering design optimization efficiency and shorten the design period.
The invention discloses a symmetrical successive local enumeration Latin hypercube test design method, also called SYM-SLE method, which comprises the following steps:
the method comprises the following steps: for n design sample points in m-dimensional space, when n is an even number, sampling the design sample points by adopting a symmetrical successive local enumeration Latin hypercube test design method, and the implementation steps are as follows:
step 1.1: and dividing a design space according to the number n of the design sample points and the space dimension m.
The design space is divided into a hypercube, each unit cell in the hypercube is a square for a two-dimensional problem, each unit cell is a cube for a three-dimensional problem, and each unit cell is a hypercube hyper-box for a higher-dimensional problem. The sampling problem of finding a set of n design sample points in an m-dimensional space can be viewed as n in the design space m Positioning n design sample points in each hypercube, wherein each design sample point has m coordinate values (x) i1 ,x i2 ,…,x im )∈{1,2,…,m} m (i ═ 1,2, …, n) so that all n design sample points have good performance in space filling and projection performance. According to the number n of design sample points and the space dimension m, the design space is divided into n unit cells, wherein each unit is (n-1) m A size hypercube, said hypercube having (n-1) m And (4) dividing the design space by the unit cell hyper-box.
Step 1.2: a first set of symmetric design sample points is selected in the design space partitioned in step 1.1.
In step (b)Step 1.1 randomly selecting a point from a first cell of the divided design space as a first design sample point, wherein the first sample point is P 1 (1,i 1 ,j 1 ,…,k 1 ),i 1 ,j 1 ,…,k 1 E {1,2, …, n }. Corresponding point of symmetry P n (n,i n ,j n ,…,k n ) Wherein i n ＝n+1-i 1 ,j n ＝n+1-j 1 ,…,k n ＝n+1-k 1 。P 1 ,P 2 That is, as the first set of symmetrical design sample points selected, the sample set is P ═ P 1 ,P n }。
Step 1.3: a second set of symmetric design sample points is selected in the design space partitioned in step 1.1.
For the second point, taking into account the projection homogeneity, P 1 And P n Occupation coordinate (1, i) 1 ,j 1 ,…,k 1 ) And (n, n + 1-i) 1 ,n+1-j 1 ,…,n+1-k 1 ) The second point should be located in the cell for the remaining coordinates in the second cell. Enumerating all feasible design sample points in the second unit, also called feasible sample points, and calculating all feasible sample points and P 1 ,P n The distance between the two points is the smaller value, i.e. the characteristic value of the feasible sample point, and the feasible sample point with the largest characteristic value is the second sample point, i.e. the distance between the two points is the second sample point
d 2 ＝max{min[d((2,i 2 ,j 2 ,…,k 2 ),P 1 ),]d((2,i 2 ,j 2 ,…,k 2 ),P n )} (1)
Obtaining a second sample point P 2 (1,i 2 ,j 2 ,…,k 2 ) Wherein the projection uniformity P is taken into account 2 The coordinates cannot coincide with two sample points that have been generated, with the constraint of i 2 ∈{r i |r i ＝1,2,…,n and r i ≠i 1 ,i n }，j 2 ∈{r i |r i ＝1,2,…,n and r i ≠j 1 ,j n }，k 2 ∈{r i |r i ＝1,2,…,n and r i ≠k 1 ,k n }. Corresponding point of symmetry P n-1 ，P 2 ,P n-1 That is, as the second set of symmetrical design sample points is selected, the sample set is P ═ P 1 ,P 2 ,P n-1 ,P n }。
Preferably, all feasible sample points and P are calculated in step 1.3 1 ,P n Using the distance of two norms of feasible sample points, i.e.
Step 1.4: and (4) selecting an s-th group of symmetrical design sample points in the design space divided in the step 1.1.
For the s-th point, the previously generated point P 1 ,P 2 ,…,P s-1 And P n ,P n-1 ,…,P n+2-s Has occupied 2(s-1) × (m-1) columns of space, including { i } in the 2(s-1) × (m-1) columns 1 ,…,i s-1 ,n+1-i 1 ,…,n+1-i s-1 }，{j 1 ,…,j s-1 ,n+1-j 1 ,…,n+1-j s-1 }，{k 1 ,…,k s-1 ,n+1-k 1 ,…,n+1-k s-1 }. The s-th point should be located within the remaining cells in the s-th cell. And calculating the distances between the feasible sample points in the cells and all the generated points before, and taking the value of the minimum distance as the characteristic value of the feasible sample point. Selecting the point with the maximum characteristic value from all feasible sample points as the s point P s (s,i s ,j s ,…,k s ) Correspondingly generating the n +1-s points P n+1-s (n+1-s,n+1-i s ,n+1-j s ,…,n+1-k s )。P s ,P n+1-s And symmetrically designing sample points for the selected s-th group. Thus, the sample set is updated to P ═ P 1 ,P 2 ,…,P s ,P n+1-s ,P n-1 ,P n }。
And 1.5, repeatedly executing the step 1.4 until the (n/2-1) th point is generated according to the number n of the designed sample points.
Step 1.6: for the n/2 th design sample point P n2 In the remaining two cellsLine sample points, calculating the characteristic values of the two feasible sample points, and selecting the largest feasible sample point as P n/2 . Correspondingly generate P n/2+1 Then, a design sample sampling point set P ═ P is obtained 1 ,P 2 ,…,P n/2 ,P n/2+1 ,…,P n-1 ,P n And completing design sample point sampling when n is an even number.
Step two: for n design sample points in m-dimensional space, when n is an odd number, sampling the design sample points by adopting a symmetrical successive local enumeration Latin hypercube test design method, wherein the method comprises the following steps:
step 2.1: when n is an odd number, n +1 is an even number, according to the method of steps 1.1 to 1.6 at (n +1) m Generating n +1 design sample points in the space with the size, and further obtaining a design sample sampling point set P ═ P 1 ,P 2 ,…,P n ,P n+1 And completing sampling of design sample points when n +1 is an even number.
Step 2.2: deleting the (n +1) th sample point P n+1 Simultaneously deleting P in design space n+1 All the lines the sample points occupy and the remaining points are shifted. The method for realizing the movement residual points is that for all other sample points P i (i＝1,2,…,m)＝(p i1 ,p i2 ,…,p im ) If p is ij ＞p n+1,j (j ═ 1,2, …, m), then p ij Should subtract 1 to get the design sample point set P ═ P 1 ,P 2 ,…,P n/2 ,P n/2+1 ,…,P n-1 ,P n And (4) designing sample point sampling when n is odd number.
Step three: the sampling method of the first step and the second step is applied to the field of industrial design optimization including high-precision analysis models, corresponding engineering problems are solved, and the implementation steps are as follows:
step 3.1: and determining the design space dimension m and the number n of design sample points according to the actual problem.
Step 3.2: and according to the parity of the number n of the design sample points, selecting a SYM-SLE method when n in the step one is even number to realize the sampling of the design sample points or selecting a SYM-SLE method when n in the step two is odd number to realize the sampling of the design sample points.
Step 3.3: and (3) constructing a proxy model according to the design sample points obtained in the step (3.2), and calculating the optimal solution of the proxy model, thereby completing engineering design optimization.
The engineering design optimization field comprises structural optimization design containing large-scale finite element analysis, pneumatic optimization design containing high-precision fluid mechanics analysis and multidisciplinary design optimization of a complex engineering system, and can effectively improve the engineering design optimization efficiency and shorten the design period.
The field of multidisciplinary design optimization of the complex engineering system comprises the fields of aircrafts, automobiles and ships.
Has the advantages that:
1. the invention discloses a symmetrical successive local enumeration Latin hypercube test design method (SYM-SLE) which combines Successive Local Enumeration (SLE) and a symmetrical sampling method, wherein a new sample point is generated by enumeration, a corresponding symmetrical sample point is generated by symmetry, two sample points are generated by one-time enumeration, the calculation amount is half of that of the SLE method, the calculation time is reduced, and the sampling efficiency is obviously improved, so that the method is better suitable for engineering application.
2. The invention discloses a symmetrical successive local enumeration Latin hypercube test design method (SYM-SLE), which maximizes the minimum distance (the minimum distance is the minimum value of all distances between a generated point and an existing point generated by SLE) through enumeration, and simultaneously adopts different methods for even number problems and odd number problems respectively, so that the generated sampling points meet two expectations of test design, namely space equipartition and projection uniformity. And compared with SLE enumeration, the SLE algorithm effectively relieves the influence of the boundary effect of the SLE algorithm by using a symmetric sampling technology, and further improves the space equipartition of sample points.
3. The invention discloses a symmetrical successive local enumeration Latin hypercube test design method (SYM-SLE), which is suitable for being applied to the field of engineering design optimization including high-precision analysis models, such as structural optimization design including large-scale finite element analysis, pneumatic optimization design including high-precision hydrodynamics analysis, multidisciplinary design optimization of complex engineering systems of aircrafts, automobiles, ships and the like, and can effectively improve the engineering design optimization efficiency and shorten the design period.
Drawings
FIG. 1 is a flow chart of the SYM-SLE method for generating sample points for an even number of sample point problems;
FIG. 2 is a flow chart of a SYM-SLE method for generating sample points for the odd sample point problem;
fig. 3 is a schematic diagram of a sample point generating method of a two-dimensional problem in an embodiment (m is 2, n is 4), where a red point is a finally generated sample point;
fig. 4 is a schematic diagram of a sample point generation method of a two-dimensional problem in an embodiment (m is 2, n is 5), where a red point is a finally generated sample point, and a green point is an initial point;
fig. 5 is a distribution display of SYM-SLE and SLE generating 20 sample points (a1), (b1), (c1) and 50 sample points (a2), (b2), (c2) in two-dimensional space, wherein: FIG. 5(a1), (a2) SYM-SLE results, (b1), (b2) SLE results, (c1), (c2) results for both methods taken together;
FIG. 6 is a two-dimensional projection distribution of SYM-SLE and SLE generating 50 sample points in three-dimensional space, where: fig. 6(a1), (a2), (a3) are SYM-SLE results, (b1), (b2), (b3) are SLE results, (c1), (c2), (c3) are results of both processes taken together;
fig. 7 shows the distribution of SYM-SLE and SLE in two-dimensional space to generate 21 sample points (a1), (b1), (c1) and 51 (a2), (b2), (c2), where: FIG. 7(a1), (a2) SYM-SLE results, (b1), (b2) SLE results, (c1), (c2) results for both methods taken together; (ii) a
FIG. 8 is a comparison of the sampling performance of the SYM-SLE and SLE methods for the case where the initial point is close to the boundary.
Detailed Description
For further explanation of the objects and advantages of the invention, reference should be made to the following detailed description taken in conjunction with the accompanying drawings.
Example 1:
for an even number of sample points, a two-dimensional (m is 2) problem is used below, and a sample point n is 4 as an implementation example to explain the specific implementation process, in which a 2-norm is used to calculate the distance between two sample points. Fig. 3 is a detailed implementation diagram of the sampling problem (m is 2, n is 4) of two-dimensional 4 points of sample points in this embodiment.
The design method of the symmetrical successive local enumeration Latin hypercube test disclosed by the embodiment comprises the following specific implementation steps:
step A: the design space is divided into a super checkerboard (hypercube), as shown in fig. 3, the space is a 4 × 4 checkerboard, each unit is a column, and each unit has four cells.
And B: randomly selecting a point from the first cell of the design space divided in the step A as a first design sample point, wherein the first sample point is P 1 (1,2). Corresponding point of symmetry P 4 (4,3), wherein. P 1 ,P 4 That is, as the first set of symmetrical design sample points selected, the sample set is P ═ P 1 ,P 4 }。
Step C: for the second point, P, taking into account the projection homogeneity 1 And P 4 Occupying coordinates (1,2) and (4,3), the second point should be located in the cell of the remaining coordinates in the second cell, i.e. the second sample point needs to be selected among the remaining two cells (viable sample points) { (2,1), (2,4) }. Enumerating all feasible design sample points in the second unit, also called feasible sample points, and calculating all feasible sample points and P 1 ,P 4 The smaller value of the distance from the two sample points is the feature value of the feasible sample point, and the feasible sample point with the largest feature value is the second sample point: for the first cell (the feasible sample point), its 2-norm distance to the first two sample points is calculated: d [ (2,1), P 1 ]＝1.414，d[(2,1),P 2 ]2.828 due to d [ (2,1), P 1 ]＜d[(2,1),P 2 ]Then the eigenvalue of the first cell (feasible sample point) is eig [ (2,1)]1.414. For the second cell, d [ (2,4), P 1 ]＝2.236，d[(2,4),P 2 ]2.236 due to d [ (2,4), P 1 ]＝d[(2,4),P 2 ]2.236, of the second cellThe characteristic value is eig [ (2,4)]2.236. Comparing the eigenvalue magnitudes of the two cells, with eig [ (2,1)]＜eig[(2,4)]I.e., the third design sample point, is determined as follows: p is 2 (2,4). According to the symmetry property, the third design sample point is P 3 (3,1). Thus, all design sample points are constructed, with a sample set of P ═ P 1 ,P 2 ,P 3 ,P 4 }＝{(1,2),(2,4),(3,1),(4,3)}。
Example 2:
for an odd number of sample points, the following two-dimensional (m is 2) problem, where the sampling point n is 5 is used as an implementation example to illustrate the specific implementation process, and a 2-norm is used to calculate the distance between two sample points. Fig. 4 is a specific implementation diagram of the sampling problem of two-dimensional 5-point sampling of sample points (m is 2, and n is 5) in the present invention.
For generating 5 sample points in the design space, the method for designing the symmetric successive local enumeration latin hypercube disclosed in this embodiment includes the following specific steps:
step A: the SYM-SLE method was used to generate 6 sample points in a 6 × 6 space, as shown in fig. 4(a), which is a 6 × 6 checkerboard with one column per cell and 6 cells per cell. Generating a set of samples as P ═ P 1 ,P 2 ,P 3 ,P 4 ,P 5 ,P 6 }＝{(1,4),(2,1),(3,5),(4,2),(5,6),(6,3)}。
And B, step B: delete last point P 6 (6,3), and deleting P 6 In a row (see fig. 4 (b)). Post-erasure sample point { P } 1 ,P 3 ,P 5 The values in the second dimension (row) are all greater than or equal to 3, and need to be subtracted by 1. This results in 5 sample points (fig. 4(c)), the sample set is:
P＝{P 1 ,P 2 ,P 3 ,P 4 ,P 5 }＝{(1,3),(2,1),(3,4),(4,2),(5,5)}
in addition, fig. 4(d) shows 5-point sample points (m is 2, n is 5) generated by the SLE method, and it can be seen from comparison with fig. 4(c) that the sample points generated by the two methods have different distributions, but d is different min The space uniformity of the two is the same (d) min ＝2.236)。
Example 3: test example
To better illustrate the advantages of the SYM-SLE method in the application of the proxy model design optimization method, Radial Basis Functions (RBFs) were selected and tested on 5 numerical examples. The mathematical model of the examples is shown in table 1. And respectively using SYM-SLE and MATLAB function Lhsdesign (LHD), sampling 100 design sample points for the problems except the fifth problem, sampling 50 design sample points for the other problems, and performing cross validation by using a K-score method, wherein K is 10. Calculating R of each group 2 Three evaluation indexes of RAAE and RMAE are averaged, R 2 Closer to 1, RAAE and RMAE closer to 0, indicating higher global approximation accuracy. The global approximation precision is related to the space equipartition of the sample points, and the better the space equipartition of the sample points is, the higher the global precision of the proxy model is. The above test was carried out 50 times in succession and the resulting R was calculated 2 The average values of RAAE and RMAE are reported in Table 2.
The results in Table 2 show R in addition to case 3 2 Results of the RMAE and RAAE data show that the overall precision of the proxy model after sampling by using the SYM-SLE method is lower than that of an lhs design (LHD) function, and the other data show that the precision of the RBF model sampled by using the SYM-SLE method is superior to that of the lhs design function.
TABLE 1 mathematical model of numerical example
The results of the SYM-SLE method compared with SLE show that for low-dimensional problems such as cases 1-4, the spatial uniformity of the sample points generated by the SYM-SLE method is slightly worse than that of SLE. For high dimensional problems, as in case 5, the SYM-SLE approach appears superior to SLE.
TABLE 2 verification of numerical example accuracy
Meanwhile, the RBF proxy model obtained in the above experiment is optimized by using a Genetic Algorithm (GA), so as to obtain the corresponding optimal value of the objective function and the time spent for sampling in the process is listed in Table 3. Wherein for each method, the optimized result is the optimal result of 50 trials, and the sampling time is the average of 50 times.
Obviously, for the optimization result, the optimization result of the proxy model obtained by using SYM-SLE and SLE is almost better than the lhdesign function. From the sampling time, the common SYM-SLE sampling time is about 50% of SLE, wherein the SYM-SLE time in case 5 is 55% of SLE time.
TABLE 3 comparison of the results of the three methods optimization with time consumption
Example 4: engineering example
The application of the SYM-SLE method in the design optimization of the high-dimensional complex aircraft is introduced by taking the airfoil aerodynamic design optimization as an example.
Selecting an NACA0012 wing profile as an initial 2-dimensional wing profile, carrying out parametric modeling on the wing profile through a shape function disturbance method, and respectively selecting 5 weight coefficients as design variables, namely x, from the upper wing profile and the lower wing profile ui ,x li (i ═ 1,2,3,4,5), for a total of 10 design variables. The mathematical model of the airfoil aerodynamic optimization problem is as follows:
in the formula, -C L/D Is a negative lift-to-drag ratio, t max The maximum thickness of the airfoil is indicated,
represents the maximum thickness of the reference airfoil, cl being the lift coefficient, cl baseline For reference airfoil lift coefficient, x being the design variable, x lb And x up The design space is lower bound and upper bound, and the value of each dimension of design variable is not more than +/-0.005. After the wing profile parameterized model is established, the computational time-consuming computational fluid mechanics (CF) needs to be calledD) And obtaining the lift-drag ratio of the wing profile by the model.
And respectively sampling 100 design sample points by using SYM-SLE and lhdesign, and performing aerodynamic analysis on the airfoil for 100 times for each sample point according to the flight working condition when the Mach number is 0.63 and the attack angle is 2.5 degrees. Calculating R using a cross-validation method 2 The three evaluation indexes of RAAE and RMAE are subjected to precision verification, and a proxy model is established by using a Kriging method (KRG) for optimization, and the obtained results are shown in Table 4.
TABLE 4 comparison of approximate precisions of airfoil optimization models
In the table R 2 The data of three evaluation indexes of RAAE and RMAE show that the global precision of Kriging (KRG) proxy model sampled by using the SYM-SLE method is better than that of the proxy model using the lhdesign function.
TABLE 5 comparison of aerodynamic airfoil optimization results for SYM-SLE and lhdesign
Table 5 lists the optimal solution found using the Genetic Algorithm (GA) for the KRG proxy model and the corresponding model response values, and the substitution of the optimal solution into the Computational Fluid Dynamics (CFD) model to find the true values in that case. By comparing the KRG model response value with the true value, the response value of the proxy model sampled by using the SYM-SLE is found to be closer to the true value, which shows that compared with the lhdesign function, the SYM-SLE sampling method is more favorable for improving the accuracy of the proxy model, and the optimization result can be more reliable and accurate when the method is applied to the field of industrial optimization.
As can be easily seen from the comparison, the SYM-SLE method can effectively ensure the space equipartition and projection uniformity of all sample points in the experimental design method. The SYM-SLE is applied to the optimization design method based on the proxy model, so that the optimization capability and the optimization efficiency of the optimization design method can be improved, and the robustness of the optimization design method can be enhanced. The SYM-SLE method is suitable for the field of engineering design optimization with huge computation, such as engineering structure optimization design containing large-scale finite element analysis, pneumatic optimization design containing high-precision computational fluid mechanics analysis, and multidisciplinary design optimization of complex engineering systems of aircrafts, automobiles, ships and the like, achieves the expected invention purpose, and can verify the reasonability, the effectiveness and the engineering practicability of the invention.
The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.
Claims (5)
1. A symmetrical successive local enumeration Latin hypercube test design method, also called SYM-SLE method, is used for aerodynamic design optimization of wing profile, and is characterized in that: comprises the following steps of (a) preparing a solution,
the method comprises the following steps: selecting an initial m-dimensional wing profile, carrying out parametric modeling on the wing profile by a shape function disturbance method, respectively selecting L weight coefficients as design variables from an upper wing profile and a lower wing profile, wherein n is n design sample points, and n is mxL, and a mathematical model of a wing profile pneumatic optimization problem is as follows:
in the formula, -C L/D Is a negative lift-to-drag ratio, t max The maximum thickness of the airfoil is indicated,
represents the maximum thickness of the reference airfoil, cl being the lift coefficient, cl baseline For reference airfoil lift coefficient, x being the design variable, x lb And x up Respectively the lower and upper bounds of the design space, and the values of the design variables of each dimensionNot more than +/-0.005;
step two: determining a dimension space m and design sample points n of a design space, and for n design sample points in the dimension space m:
step 2.1: when n is even number, a symmetrical successive local enumeration Latin hypercube test design method is adopted to realize the sampling of design sample points, the realization steps are as follows,
step 2.1.1: dividing a design space according to the number n of design sample points and the space dimension m;
dividing a design space into a hypercube, wherein each unit cell in the hypercube is a square for a two-dimensional problem, each unit cell is a cube for a three-dimensional problem, and each unit cell is a hypercube for a problem with higher dimensionality; the sampling problem of finding a set of n design sample points in an m-dimensional space can be viewed as n in the design space m Positioning n design sample points in a hypercube, wherein each design sample point has m coordinate values (x) i1 ,x i2 ,…,x im )∈{1,2,…,m} m I is 1,2, …, n, so that all n design sample points have good performance in space filling and projection performance; according to the number n of design sample points and the space dimension m, the design space is divided into n unit cells, wherein each unit is (n-1) m A hypercube of size, said hypercube having (n-1) m The cell high-box realizes the division of the design space;
step 2.1.2: selecting a first group of symmetrical design sample points in the design space divided in the step 1.1;
randomly selecting a point from the first cell of the design space divided in the step 1.1 as a first design sample point, wherein the first sample point is P 1 (1,i 1 ,j 1 ,…,k 1 ),i 1 ,j 1 ,…,k 1 E {1,2, …, n }; corresponding point of symmetry P n (n,i n ,j n ,…,k n ) Wherein i n ＝n+1-i 1 ,j n ＝n+1-j 1 ,…,k n ＝n+1-k 1 ；P 1 ,P n Is to do immediatelyDesigning sample points for the first set of symmetry, wherein the sample set is P ═ P 1 ,P n }；
Step 2.1.3: selecting a second group of symmetrical design sample points in the design space divided in the step 1.1;
for the second point, P, taking into account the projection homogeneity 1 And P n Occupation coordinate (1, i) 1 ,j 1 ,…,k 1 ) And (n, n + 1-i) 1 ,n+1-j 1 ,…,n+1-k 1 ) The second point should be located in the cell for the remaining coordinates in the second cell; enumerating all feasible design sample points in the second unit, also called feasible sample points, and calculating all feasible sample points and P 1 ,P n The distance between two points is the smaller value, i.e. the characteristic value of the feasible sample point, and the feasible sample point with the largest characteristic value is the second sample point, i.e. the distance between two points is the smaller value
d 2 ＝max{min[d((2,i 2 ,j 2 ,…,k 2 ),P 1 ),]d((2,i 2 ,j 2 ,…,k 2 ),P n )} (1)
Obtaining a second sample point P 2 (1,i 2 ,j 2 ,…,k 2 ) Wherein the projection uniformity P is taken into account 2 The coordinates cannot coincide with two sample points that have been generated, with the constraint of i 2 ∈{r i |r i ＝1,2,…,n and r i ≠i 1 ,i n }，j 2 ∈{r i |r i ＝1,2,…,n and r i ≠j 1 ,j n }，k 2 ∈{r i |r i ＝1,2,…,n and r i ≠k 1 ,k n }; corresponding point of symmetry P n-1 ，P 2 ,P n-1 That is, as the second set of symmetrical design sample points is selected, the sample set is P ═ P 1 ,P 2 ,P n-1 ,P n }；
Step 2.1.4: selecting a s-th group of symmetrical design sample points in the design space divided in the step 1.1;
for the s-th point, the previously generated point P 1 ,P 2 ,…,P s-1 And P n ,P n-1 ,…,P n+2-s 2(s-1) × (m-1) columns in the occupied space, the 2(s-1) × (m-1) columns including { i } i 1 ,…,i s-1 ,n+1-i 1 ,…,n+1-i s-1 }，{j 1 ,…,j s-1 ,n+1-j 1 ,…,n+1-j s-1 }，{k 1 ,…,k s-1 ,n+1-k 1 ,…,n+1-k s-1 }; the s-th point should be located within the remaining cells in the s-th cell; calculating the distances between the feasible sample points in the cells and all the generated points before, and taking the value of the minimum distance as the characteristic value of the feasible sample point; selecting the point with the maximum characteristic value from all feasible sample points as the s-th point P s (s,i s ,j s ,…,k s ) Correspondingly generating the n +1-s points P n+1-s (n+1-s,n+1-i s ,n+1-j s ,…,n+1-k s )；P s ,P n+1-s Symmetrically designing sample points for the selected s-th group; thus, the sample set is updated to P ═ P 1 ,P 2 ,…,P s ,P n+1-s ,P n-1 ,P n }；
Step 2.1.5, repeatedly executing step 1.4 until the (n/2-1) th point is generated according to the number n of the designed sample points;
step 2.1.6: for the n/2 design sample point P n/2 Calculating the characteristic values of the two feasible sample points after the feasible sample points in the two cells are remained, and selecting the largest feasible sample point as P n/2 (ii) a Correspondingly generate P n/2+1 Then, a design sample sampling point set P ═ P is obtained 1 ,P 2 ,…,P n/2 ,P n/2+1 ,…,P n-1 ,P n Finishing sampling sample points when n is an even number;
step 2.2: when n is odd number, a symmetrical successive local enumeration Latin hypercube test design method is adopted to realize the sampling of design sample points, the realization steps are as follows,
step 2.2.1: when n is an odd number, n +1 is an even number, according to the method of steps 1.1 to 1.6 at (n +1) m Generating n +1 design sample points in the space of the size, and further obtaining a design sample sampling point set as P ═ P 1 ,P 2 ,…,P n ,P n+1 }，Namely, when n +1 is an even number, sample point sampling is designed;
step 2.2.2: deleting the (n +1) th sample point P n+1 Simultaneously deleting P in design space n+1 All the lines and columns occupied by the sample points, and moving the rest points; the method for realizing the movement residual points is that for all other sample points P i (i＝1,2,…,m)＝(p i1 ,p i2 ,…,p im ) If p is ij ＞p n+1,j J is 1,2, …, m, then p ij Subtracting 1 to obtain a design sample point set P ═ P 1 ,P 2 ,…,P n/2 ,P n/2+1 ,…,P n-1 ,P n And (4) designing sample point sampling when n is odd number.
2. The design method of symmetric successive local enumeration Latin hypercube design according to claim 1, characterized in that: step three: the design method of the first step and the second step is applied to the field of industrial design optimization containing high-precision analysis models to solve the corresponding engineering problem and is realized by the following steps,
step 3.1: determining a design space dimension m and a design sample point number n according to an actual problem;
step 3.2: according to the parity of the number n of the design sample points, a SYM-SLE method when n in the step one is even number is selected to realize the sampling of the design sample points or a SYM-SLE method when n in the step two is odd number is selected to realize the sampling of the design sample points;
step 3.3: and (3) constructing a proxy model according to the design sample points obtained in the step (3.2), and calculating the optimal solution of the proxy model, thereby completing engineering design optimization.
3. A method according to claim 1 or 2, wherein the method comprises: the three-step industrial design optimization field comprises structural optimization design containing large-scale finite element analysis, pneumatic optimization design containing high-precision fluid mechanics analysis and multidisciplinary design optimization of a complex engineering system, so that the engineering design optimization efficiency can be effectively improved, and the design period can be shortened.
4. The method according to claim 3, wherein the method comprises: the field of multidisciplinary design optimization of the complex engineering system comprises the fields of aircrafts, automobiles and ships.
5. The method of claim 4, wherein the method comprises: all feasible sample points and P are calculated in step 1.3 1 ,P n Using a distance of two norms of feasible sample points, i.e.
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