无网格VDQ法分析非规则薄板热-振耦合特性

doi:10.7527/S1000-6893.2025.32248

Abstract

Abstract:

A meshless Variational Differential Quadrature （VDQ） method is proposed for thin plates with irregular geometric configurations. By integrating Differential Reproducing Kernel （DRK） interpolation technique and radial basis function approximation， a differential operator suitable for unstructured nodes is constructed. Meanwhile， an adaptive integral operator based on Voronoi diagram theory is developed to achieve numerical integration over irregular computational domains. The effectiveness of this method is validated through systematic investigations of thermally driven vibrations in both homogeneous single-layer plates and Functionally Graded Materials （FGM） with five typical boundary shapes， including square plates， round-corner-defective plates， square-corner-defective plates， circular-hole plates， and square-hole plates. Numerical results demonstrate that compared with traditional grid-based methods， the meshless VDQ method reduces the number of nodes while maintaining engineering accuracy requirements. In thermal-vibration analysis， the maximum deflection magnitudes for both types of plates exhibit a significant decrease trend： intact > round-corner-defective > square-corner-defective > circular-hole > square-hole， revealing the influence mechanism of irregular geometric boundaries on thermal-vibration responses of thin plate structures. This research provides an efficient and reliable numerical tool for thermal-vibration analysis of complex thin plate structures in aerospace， precision machinery， and other engineering fields.

Key words: irregular boundary thin plates, variational differential quadrature (VDQ) method, Voronoi diagram, meshless method, thermal vibration

CLC Number:

V214.4

Ligang WANG, Runze XU, Zhipeng LIU, Yuchao HUANG, Ke WANG. Meshfree VDQ method for thermal-vibration coupling characteristic analysis of irregular thin plates[J]. Acta Aeronautica et Astronautica Sinica, 2026, 47(3): 232248.

Figures/Tables 16

Fig.1

Fig.2

Fig.3

Fig.4

Table 1

Opening parameters of thin plates

开孔结构	中心坐标/m	长度/半径/m
方洞	$a / 2, b / 2$	$2 a / 5$
方角	$0,0$	$a / 5$
圆洞	$a / 2, b / 2$	$a / 5$
圆角	$0,0$	$a / 5$

Table 1

Fig.5

Table 2

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

Table 3

Table 4

Table D1

Material properties of single-layer plate

参数	数值
密度 $ρ$ /（kg·m^-3）	2 370
弹性模量E/Pa	3.22×10¹¹
泊松比 $μ$	0.24
比热容/（J·kg^-1·K^-1）	736.393
热导率K/（W·m^-1·K^-1）	10.120 26
热扩散系数 $α T$ /（m²·s^-1）	7.474 56×10^-6

Table D1

Table D2

Material properties of FMG plate

材料	参数	$P 0$	$P 1$	$P 2$	$P 3$
SUS304	密度 $ρ$ /（kg·m^-3）	8 166	0	0	0
	弹性模量E/Pa	2.010 4×10¹¹	3.079×10^-4	-6.534×10^-7	0
	泊松比 $μ$	0.326 2	-2.002×10^-4	3.797×10^-7	0
	比热容/（J·kg^-1·K^-1）	496.56	-1.151×10^-3	1.636×10^-6	-5.863×10^-10
	热导率K/（W·m^-1·K^-1）	15.379	-1.264×10^-3	2.092×10^-6	-7.223×10^-10
	热扩散系数 $α T$ /（m²·s^-1）	1.233×10^-5	8.086×10^-4	0	0
Si3N4	密度 $ρ$ /（kg·m^-3）	2 370	0	0	0
	弹性模量E/Pa	3.484 3×10¹¹	-3.07×10^-4	2.16×10^-7	-8.946×10^-11
	泊松比 $μ$	0.24	0	0	0
	比热容/（J·kg^-1·K^-1）	555.11	1.016×10^-3	2.92×10^-7	-1.67×10^-10
	热导率K/（W·m^-1·K^-1）	13.723	-1.032×10^-3	5.466×10^-7	-7.876×10^-11
	热扩散系数 $α T$ /（m²·s^-1）	5.872 3×10^-6	9.095×10^-4	0	0

Table D2

References 23

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薄板形状	节点数量
薄板形状	无网格VDQ	有限元
方形板	289	3 003
圆角缺损板	306	2 968
方角缺损板	329	2 891
圆洞板	352	2 759
方洞板	304	2 632

薄板形状	最大挠度值/m
薄板形状	单层板	FGM板
方形板	0.015 0	0.003 1
圆角缺损板	0.013 0	0.002 8
方角缺损板	0.011 0	0.002 6
圆洞板	0.004 5	1.8×10^-9
方洞板	0.003 2	0.7×10^-9

薄板形状	热流密度值/（W·m^-2）
薄板形状	单层板	FGM板
方形板	0.15×10⁴	1.6×10⁴
圆角缺损板	0.15×10⁴	2.0×10⁴
方角缺损板	0.15×10⁴	2.0×10⁴
圆洞板	1.0×10⁴
方洞板	1.6×10⁴

Meshfree VDQ method for thermal-vibration coupling characteristic analysis of irregular thin plates

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References 23

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