基于样本映射与动态Kriging的飞行器离散连续优化方法

doi:10.7527/S1000-6893.2023.28726

Abstract

Abstract:

To deal with the problems of high computational cost and poor global convergence that often exist in discretecontinuous mixed optimization of complex flight vehicle systems， a Sample Mapping and Dynamic Kriging based Discrete-Continuous Mixed Optimization method （SMDK-DC） is proposed. In this method， time-consuming simulation model is replaced by Kriging surrogate model to reduce computational expenses. A sample point mapping mechanism based on generalized Manhattan distance criterion is also proposed to efficiently generate uniformly-distributed real sample points in continuous-discrete domain. Expected improvement criteria is combined with significant sampling space to identify new sample points，update Kriging continuously and dynamically， and guide the rapid convergence of the discrete-continuous optimization process. Benchmark cases show that compared with international methods such as SOMI and NOMAD， SMDK-DC has significant advantages in global convergence and robustness. Finally， SMDK-DC is used for solving a multidisciplinary design optimization problem of solid rocket motor. The method， on the premise of satisfying all the constraints of the combustion chamber and internal ballistic discipline， leads to a total impulse increase of at least 12. 92%， and the optimization yield is 1. 71% higher than that of SOMI， which verifying the effectiveness and engineering practicability of SMDK-DC.

Key words: Kriging, discrete-continuous mixed optimization, approximate optimization, expected improvement, significant sampling space

CLC Number:

V221

Haoda LI, Teng LONG, Renhe SHI, Nianhui YE. Kriging?based mixed?integer optimization method using sample mapping mechanism for flight vehicle design[J]. Acta Aeronautica et Astronautica Sinica, 2024, 45(3): 228726-228726.

Figures/Tables 18

Fig.1

Fig.2

Table 1

Concepts involved in building/updating Kriging surrogates

名词	符号	内涵
样本点种子集	$X s e e d$	由Maximin拉丁超方试验设计方法在连续空间内生成的点集，该点集中的样本点尚未映射至连续-离散空间内
备选样本点集	$X c a n d i d a t e$	由样本点种子的连续设计变量值以及各离散变量的可能取值集合确定的样本点集，该集合中的所有样本点均位于连续-离散空间内
真实样本点集	$X r e a l$	用于构建/更新Kriging代理模型的样本点集，该集合中的所有样本点均位于连续-离散空间内
初始点集	$X i n i t$	由序列二次规划方法求解近似优化问题的不同初始点组成的点集，该集合中的所有点均位于连续空间内
伪最优解种子集	$X p s e u d o$	由序列二次规划方法求解近似优化问题所得的各局部最优解组成的点集，该集合中的所有点均位于连续空间内
期望改善度准则辨识的新增样本点	$x E I$	由期望改善度准则辨识的新增样本点，位于连续-离散空间内
基于重点采样空间辨识的新增样本点集	$X S S S$	基于重点采样空间方法辨识的新增样本点集，该集合中的所有点均位于连续-离散空间内

Table 1

Fig.3

Fig.4

Table 2

Table 3

Parameter configuration of SMDK⁃DC

参数	取值
$N d o e$	$5 n v$
$m$	20
$N f e m a x$	200
$n L$	1

Table 3

Table 4

Table 5

Fig.5

Fig.6

Fig.7

Table 6

Definition of design variables

设计变量	符号	设计变量	符号
燃烧室直径	$D c o m b$	翼型体倾角	$α f i n$
理论总冲	$I t h$	喷管喉部半径	$R t h r o a t$
内腔前段半径	$R f r o n t$	喷管扩张比	$ε$
内腔中段半径	$R c o r e$	喷管收敛半角	$α n o z$
内腔后段半径	$R r e a r$	喷管扩张半角	$β n o z$
翼型体长度	$L f i n$	翼型体个数（离散设计变量）	$N f i n$
翼型体高度	$H f i n$	翼型体个数（离散设计变量）	$N f i n$

Table 6

Table 7

Comparison of initial and optimized objective values of solid rocket motor

目标函数	初始方案	SMDK-DC优化方案	SOMI优化方案
发动机总冲 $I S R M$ /（N·s）	$1.329 1 × 107$	$1.500 8 × 107$	$1.475 6 × 107$

Table 7

Table 8

Comparison of initial and optimized design variables of solid rocket motor

设计变量	符号	单位	范围	初始方案	SMDK-DC优化方案	SOMI优化方案
燃烧室直径	$D c o m b$	m	［1.2，1.4］	1.3	1.363 6	1.394 8
理论总冲	$I t h$	N·s	$1.2 × 107, 1.6 × 107$	$1.4 × 107$	$1.599 5 × 107$	$1.562 7 × 107$
内腔前段半径	$R f r o n t$	m	［0.04，0.08］	0.06	0.078 4	0.071 0
内腔中段半径	$R c o r e$	m	［0.10，0.16］	0.13	0.130 0	0.133 2
内腔后段半径	$R r e a r$	m	［0.18，0.24］	0.21	0.183 3	0.211 5
翼型体长度	$L f i n$	m	［0.20，0.60］	0.40	0.402 3	0.388 3
翼型体高度	$H f i n$	m	［0.35，0.55］	0.40	0.480 9	0.493 3
翼型体倾角	$α f i n$	（°）	［30，60］	45	49.851 1	45.212 7
喷管喉部半径	$R t h r o a t$	m	［0.10，0.14］	0.12	0.124 5	0.113 6
喷管扩张比	$ε$		［12，20］	16	12.067 8	13.524 7
喷管收敛半角	$α n o z$	（°）	［45，55］	50	54.133 1	49.417 6
喷管扩张半角	$β n o z$	（°）	［12，17］	15	12.433 2	15.458 4
翼型体个数	$N f i n$		［8，12］	10	9	9

Table 8

Table 9

Comparison of initial and optimized constraints of solid rocket motor

约束条件	符号	单位	范围	初始方案	SMDK-DC优化方案	SOMI优化方案
平均推力	$F m e a n$	kN	$250,300$	233.191 0	250.687 6	261.568 3
最大推力与平均推力偏差	$F m a x - F m e a n$	kN	$≤ 30$	41.907 7	29.598 1	16.021 2
工作时间	$t w o r k$	s	$55,60$	55.341 9	58.479 5	55.603 1
出口平均压强	$p e, m e a n$	Pa	$≥ 20 265$	12 964.746 4	20 545.626 2	20 614.040 9
喉通比	$J$		$≤ 0.65$	0.326 5	0.460 9	0.288 6
燃烧室直径与喷管出口直径差	$D c o m b - 2 R t h r o a t ε$	m	$≥ 0$	0.340 0	0.498 8	0.559 0
药柱装填分数	$φ$		$≥ 0.85$	0.940 8	0.945 2	0.942 9

Table 9

Fig.8

Fig.9

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算例	理论最优解	问题维度	离散变量数量	约束规模
CF	0	8	4	0
Nvs	-43.130 0	10	5	0
Rast12	-12.000 0	12	5	0
Ex	0	5	2	5
Aex	N.A.	5	2	3
G10	7 049.248 0	8	3	6
G7	24.306 2	10	5	8
G1	-15.000 0	13	5	9
SR	-0.999 9	8	4	3
GTCD4	4 354 548.310 0	4	2	1
CPSD1	4.579 5	7	4	9
CPSD2	2.000 0	2	1	2
CPSD3	1.076 5	3	1	3
PVD	5 850.383 7	4	2	4
GPO	-75.134 1	3	1	2

方法	指标	CF	Nvs	Rast12	Ex	Aex	G10	G7
SMDK⁃DC	Mean	0.164 3	-42.052 5	-10.649 4	0	-9.498 4	16 231.066 7	76.471 2
	STD	0.071 9	0.073 6	0.751 1	0	0.392 3	3 202.318 6	13.987 8
	Best	0.097 9	-42.081 8	-11.977 4	0	-9.951 3	12 229.000 0	49.865 8
	Worst	0.286 3	-41.826 8	-9.033 9	0	-8.927 1	22 767.000 0	98.740 3
	NF	0	0	0	0	0	0	0
SOMI	Mean	0.564 4	-41.717 3	-10.635 4	0	-9.414 4	18 135.266 6	78.135 9
	STD	0.049 0	0.379 9	0.081 7	0	0.106 9	8.421 6	7.553 5
	Best	0.432 0	-42.022 0	-10.824 4	0	-9.542 1	18 118.000 0	63.079 8
	Worst	0.647 0	-40.607 8	-10.553 1	0	-9.163 2	18 144.000 0	89.537 7
	NF	0	0	0	0	0	0	0
EGO	Mean	0.170 4	-42.081 8	-7.600 3	1.773 8	-6.459 9	21 708.533 3	1 619.445 1
	STD	0.178 5	0	1.015 1	1.983 9	2.064 0	3 287.223 5	679.034 2
	Best	0.012 6	-42.081 8	-9.938 0	0	-9.589 9	15 026.000 0	315.578 8
	Worst	0.529 4	-42.081 8	-6.237 8	6.010 5	-3.579 5	26 888.000 0	2 875.478 1
	NF	0	0	0	0	0	0	0
NOMAD	Mean	0.004 1	-42.081 3	-10.286 4	0	-9.676 2		217.932 1
	STD	0.008 0	0.001 9	1.994 4	0	0.210 7		253.757 2
	Best	0	-42.081 8	-11.979 5	0	-9.920 0		49.504 6
	Worst	0.023 6	-42.074 3	-5.320 4	0	-9.078 5		1 042.719 9
	NF	0	0	0	0	0	30	2

方法	指标	G1	SR	GTCD4	CPSD1	CPSD2	CPSD3	PVD	GPO
SMDK-DC	Mean	-12.978 6	-0.998 8	4 429 874.487 2	6.860 0	2.000 4	1.130 9	6 317.632 9	-75.116 6
	STD	0.822 1	0.001 8	59 326.083 3	0.667 4	0.000 5	0.040 8	170.195 6	0.047 6
	Best	-13.822 5	-0.999 9	4 357 164.732 9	6.127 0	2.000 0	1.090 2	6 000.755 8	-75.134 1
	Worst	-11.226 3	-0.993 1	4 600 230.010 7	8.436 3	2.002 0	1.250 0	6 625.796 8	-74.950 4
	NF	0	0	0	0	0	0	0	0
SOMI	Mean	-12.663 5	-0.988 6	6 083 668.098 0	7.405 2	2.489 9	1.247 9	6 493.021 1	-73.912 4
	STD	1.144 6	0.013 3	399 536.459 3	1.167 5	0.297 3	0.046 8	338.271 0	0.601 9
	Best	-14.906 8	-0.999 5	4 680 695.857 9	5.288 7	2.022 5	1.173 6	6 100.222 3	-74.816 0
	Worst	-11.506 8	-0.954 9	6 302 160.672 0	8.858 2	2.930 9	1.309 3	7 200.120 5	-72.976 9
	NF	0	0	0	0	0	0	0	0
EGO	Mean	-4.890 5	-0.981 0	6 091 630.366 1	7.301 3	2.376 1	1.277 1	10 727.439 3	-70.405 5
	STD	1.718 8	0.012 1	1 113 100.380 2	0.969 4	0.345 5	0.068 0	3 384.766 7	3.497 3
	Best	-8.399 0	-0.997 0	4 446 783.893 2	5.102 2	2.030 7	1.143 0	6 186.233 4	-75.115 9
	Worst	-2.316 3	-0.955 3	7 488 842.392 8	9.052 5	3.189 8	1.350 0	19 035.347 3	-64.310 0
	NF	2	0	0	0	0	0	0	0
NOMAD	Mean	-8.759 6	-0.998 1	5 628 727.136 4	5.542 4	2.015 7	1.180 8	6 345.940 3	-75.115 1
	STD	0	0.001 3	1 821 853.661 6	0.977 4	0.061 0	0.087 7	361.101 6	0.027 1
	Best	-8.759 6	-0.999 8	4 356 376.928 9	4.597 0	2.000 0	1.076 5	6 157.843 0	-75.133 4
	Worst	-8.759 6	-0.997 0	9 126 867.722 6	7.952 3	2.236 1	1.250 0	7 283.177 1	-75.051 8
	NF	28	0	0	0	0	0	2	0

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Kriging?based mixed?integer optimization method using sample mapping mechanism for flight vehicle design

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