基于辛叠加方法与迁移学习的功能梯度任意四边形开孔板自由振动问题求解

doi:10.7527/S1000-6893.2025.32453

Abstract

Abstract:

Functionally graded plates with cutouts are common load-bearing structures in engineering， making the study of their dynamic behavior crucial. For plates with regularly shaped cutouts， the symplectic superposition method combined with subdomain decomposition technique can be used to establish a comprehensive analytical solution framework. However， when the research object is extended to plates with arbitrarily shaped cutouts， which have broader applications， the analytical solution faces significant challenges due to the increased complexity of the boundary conditions. While approximate or numerical methods exist， they may suffer from time-consuming computations and strong mesh dependency， often leading to insufficient accuracy in results. A novel solution framework that integrates the symplectic superposition method with the transfer learning technique is established. It leverages the analytically solvable natural frequencies of functionally graded rectangular plates with rectangular cutouts for knowledge transfer， thereby achieving efficient and highly accurate solutions for the free vibration problems of functionally graded plates with arbitrary quadrilateral cutouts. First， a multi-layer perceptron neural network is pre-trained on large-scale analytical solution dataset to extract the complex mapping relationships of geometric parameters between different shapes. Second， the pre-trained model is transferred based on few-shot finite element simulation data， effectively transferring knowledge from the natural frequency dataset of the rectangular plates with rectangular cutouts to the free vibration of plates with arbitrary quadrilateral cutouts. Finally， the accuracy and applicability of the proposed solution framework for predicting the natural frequencies of functionally graded plates with arbitrary quadrilateral cutouts are validated， using metrics such as mean squared error and coefficient of determination. The transfer learning technique is used to leverage small-sample data and existing analytical solutions to efficiently and accurately solve free vibration problems of functionally graded with arbitrary quadrilateral plates cutouts. The proposed framework， adaptable to various working conditions through model parameter adjustments， offers a novel strategy for the mechanical analysis of complex-shaped plates.

Key words: quadrilateral plates with cutouts, free vibration, symplectic superposition method, transfer learning

CLC Number:

V214.3

Chaoyu CHENG, Dian XU, Chengjie GUO, Jinbao LI, Rui LI. Free vibration solution of functionally graded plates with arbitrary quadrilateral cutouts based on symplectic superposition method and transfer learning[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(5): 232453.

Figures/Tables 10

Fig.1

Fig.2

Table 1

First ten-order non-dimensional frequency parameter， ωa2ρch/Dc， of functionally graded rectangular plates with rectangular cutouts

$k$	方法	ωa² $ρ c h / D c$
$k$	方法	模态1	模态2	模态3	模态4	模态5	模态6	模态7	模态8	模态9	模态10
0	本文	23.454	40.171	40.171	71.239	73.338	111.540	111.540	144.480	172.120	200.170
	有限元	23.457	40.180	40.180	71.248	73.365	111.560	111.560	144.500	172.170	200.200
	文献［18］	23.418	39.994	39.994	72.218	72.945	111.572	111.572	144.945	172.870	201.263
	文献［19］	23.422	40.058	40.058	71.157	73.002	111.272	111.272	144.358	171.649	200.014
0.5	本文	19.859	34.015	34.015	60.322	62.099	94.444	94.444	122.340	145.740	169.490
	有限元	19.868	34.029	34.029	60.339	62.132	94.480	94.480	122.380	145.800	169.550
1.0	本文	17.895	30.650	30.650	54.355	55.957	85.103	85.103	110.240	131.320	152.730
	有限元	17.898	30.658	30.658	54.363	55.978	85.123	85.123	110.260	131.370	152.760
10.0	本文	14.938	25.586	25.586	45.374	46.711	71.041	71.041	92.025	109.620	127.490
	有限元	14.940	25.589	25.589	45.374	46.772	71.048	71.048	92.027	109.640	127.500

Table 1

Table 2

Table 3

Sampling spaces and step sizes for functionally graded plates with rectangular cutouts

几何参数	采样空间	采样步长
B/m	$32$
$θ 1$ /（°）	［50，90］	20
a/m	［0.04，2.20］	0.08
b/m	［0.04，2.20］	0.08
$θ 2$ /（°）	［5，90］	5
R	0.5

Table 3

Fig.3

Table 4

Sampling spaces and step sizes for functionally graded plates with arbitrary quadrilateral cutouts

组别	采样类别	B/m	$θ 1$ /（°）	a/m	b/m	$θ 2$ /（°）	R
1~7	采样空间	A×［0.2，1.0］	［50，90］	［0.20，2.20］	a×［0.2，1.0］	［10，90］	［0.3，0.5］
1（12）	采样步长	0.8	40	2.0	0.8	80	0.1
2（24）	采样步长	0.8	40	2.0	0.8	80	0.1
3（48）	采样步长	0.8	40	2.0	0.8	80	0.1
4（96）	采样步长	0.8	40	0.8	0.8	80	0.1
5（144）	采样步长	0.4	40	0.8	0.4	80	0.1
6（216）	采样步长	0.4	40	0.8	0.4	40	0.1
7（324）	采样步长	0.4	20	0.8	0.4	40	0.1

Table 4

Fig.4

Fig.5

Table 5

Partial frequency results of functionally graded plates with arbitrary quadrilateral cutouts

B/m	R	自振频率/Hz
B/m	R	有限元分析	全模型微调	冻结+ 重新训练	无知识迁移
$325$	0.3	0.720 84	0.716 59	0.826 16	0.762 17
	0.3	0.395 85	0.398 16	0.406 10	0.403 10
	0.3	0.510 36	0.510 02	0.507 18	0.509 91
	0.4	0.453 67	0.466 79	0.478 81	0.441 11
	0.4	0.459 03	0.462 43	0.463 67	0.464 25
	0.4	0.708 29	0.705 61	0.753 98	0.803 54
$925$	0.3	0.132 53	0.134 06	0.130 16	0.125 83
	0.3	0.139 41	0.136 93	0.142 93	0.122 19
	0.3	0.128 09	0.126 11	0.134 07	0.147 36
	0.4	0.137 08	0.138 19	0.123 60	0.160 87
	0.4	0.104 96	0.101 31	0.111 44	0.080 67
	0.4	0.137 79	0.136 70	0.141 25	0.134 99

Table 5

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材料	弹性模量/GPa	密度/（kg∙m^-3）	泊松比
金属Al	70	2 707	0.3
陶瓷Al₂O₃	380	3 800	0.3
陶瓷ZrO₂	200	5 700	0.3

Free vibration solution of functionally graded plates with arbitrary quadrilateral cutouts based on symplectic superposition method and transfer learning

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