基于随机建模与Bayes推断的结构热传导

李建宇; 付介祥; 郝鑫野; 李广利

doi:10.21656/1000-0887.460055

基于随机建模与Bayes推断的结构热传导

doi: 10.21656/1000-0887.460055

李建宇^{1, 2, ,},
付介祥¹,
郝鑫野¹,
李广利³

1.
天津科技大学机械工程学院, 天津 300457
2.
大连理工大学工业装备结构分析优化与CAE软件全国重点实验室, 辽宁大连 116024
3.
中国科学院力学研究所高温气动国家重点实验室, 北京 100190

基金项目:

国家自然科学基金 12002347

详细信息

通讯作者:
李建宇(1978—)，男，教授，博士(通讯作者. E-mail: lijianyu@tust.edu.cn)

中图分类号: TK124;TP391.9
计量
- 文章访问数: 34
- HTML全文浏览量: 6
- PDF下载量: 13
- 被引次数: 0
出版历程
- 收稿日期: 2025-03-24
- 修回日期: 2025-05-22
- 刊出日期: 2025-08-01

A Digital Twin Modeling Approach for Structural Heat Conduction Analysis Based on Stochastic Modeling and Bayesian Inference

LI Jianyu^{1, 2
, ,},
FU Jiexiang¹,
HAO Xinye¹,
LI Guangli³

1.
School of Mechanical Engineering, Tianjin University of Science & Technology, Tianjin 300457, P.R.China
2.
State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Dalian University of Technology, Dalian, Liaoning 116024, P.R.China
3.
State Key Laboratory of High Temperature Gas Dynamics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, P.R.China

摘要

摘要: 极端热环境条件下结构传热温度场的准确预测是评估装备热-力耦合性能的关键基础.数字孪生(digital twin)技术通过对观测数据与仿真模型的深度融合，可实现温度场的高精度动态重构.然而，考虑观测噪声、模型参数不确定性、边界条件扰动等多源不确定性因素的结构传热温度场预测数字孪生模型目前还不多见.该文基于Bayes推断框架，提出了一种结合随机传热分析的数据与模型融合方法，旨在构建考虑不确定性量化的热传导数字孪生模型.首先，在热传导方程中引入随机扰动热源项，以模拟未被原模型量化表征的不确定性因素；其次，采用随机有限元方法求解随机扰动热传导模型，获得包含物理信息的温度场先验分布；最后，基于Bayes法则，将含噪声的观测数据与模型预测先验分布进行融合，并针对Gauss分布情形推导出温度场后验分布的解析表达式.通过一维和二维热传导算例验证，所提方法不仅能够实现对温度场的高精度预测，还可有效量化预测结果的不确定性.
- 热传导分析 /
- 数字孪生 /
- 随机有限元 /
- Bayes推断 /
- 不确定性量化
Abstract: The accurate prediction of structural heat transfer temperature fields under extreme thermal environments is a critical foundation for evaluating the thermo-mechanical performance of equipment. The digital twin technology enables high-precision dynamic reconstruction of temperature fields through the deep integration of observed data and simulation models. However, digital twin models for predicting structural heat transfer temperature fields with multi-source uncertainties, such as observation noise, model parameter uncertainty, and boundary condition disturbances, are still relatively scarce. Aimed to construct a heat conduction digital twin model with uncertainty quantification, a data-model fusion method combined with stochastic heat conduction analysis was proposed based on the Bayesian inference framework. First, a Gaussian random perturbation heat source term was introduced into the heat conduction equation to simulate uncertainty factors not quantified by the original model. Second, the stochastic heat conduction model was solved with the stochastic finite element method to obtain a prior distribution of the temperature field incorporating physical information. Finally, based on Bayes' theorem, the noisy observation data was fused with the model-predicted prior distribution, and an analytical expression for the posterior distribution of the temperature field was derived for the Gaussian case. The results of 1D and 2D heat conduction examples demonstrate that, the proposed method not only achieves high-precision prediction of the temperature field but also effectively quantifies the uncertainty of the prediction results.
- heat conduction analysis /
- digital twin /
- stochastic finite element /
- Bayesian inference /
- uncertainty quantification

HTML全文

图 1 基于Bayes推理的数据与模型融合的温度场重构过程

Figure 1. Bayesian inference-based fusion of data and model information for temperature field reconstruction

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图 2 一维稳态热传导示意图

Figure 2. Schematic diagram of the 1D steady-state heat conduction

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图 3 热导率温度相关和无关两种情形下的一维稳态热传导温度分布图

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 3. The 1D steady-state heat transfer temperature distribution with temperature-dependent and temperature-independent thermal conductivities

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图 4 一维瞬态热传导示意图

Figure 4. Schematic diagram of the 1D transient heat conduction

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图 5 热导率温度相关和无关两种情形下，t=2 s, 4 s, 6 s时刻的一维瞬态热传导温度分布

Figure 5. The 1D transient heat conduction temperature distribution at t=2 s, 4 s, 6 s with temperature-dependent and temperature-independent thermal conductivities

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图 6 观测数据为1个的温度场预测结果

Figure 6. Prediction results of the temperature field with 1 observation

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图 7 观测数据为3个的温度场预测结果

Figure 7. Prediction results of the temperature field with 3 observations

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图 8 不考虑观测数据不确定性与考虑观测数据不确定性的预测结果比对

Figure 8. Comparison of prediction results without observation data uncertainty vs. with observation data uncertainty

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图 9 第一个观测点(节点1)温度预测结果

Figure 9. Temperature prediction results of the 1st observation point (node 1)

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图 10 第二个观测点(节点41)温度预测结果

Figure 10. Temperature prediction results of the 2nd observation point (node 41)

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图 11 非观测点(节点4)温度预测结果

Figure 11. Temperature prediction results of the non-observed point (node 4)

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图 12 二维稳态热传导示意图

Figure 12. Schematic diagram of the 2D steady-state heat conduction

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图 13 二维稳态热传导温度场预测结果示意图

Figure 13. Predicted results of the 2D steady-state heat conduction temperature field

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图 14 x=1处各个节点的先验均值、后验均值、先验分布95%置信区间、后验分布95%置信区间以及真值示意图

Figure 14. The prior mean, posterior mean, prior distribution 95% confidence interval, posterior distribution 95% confidence interval, and true values of each node at x=1

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图 15 二维瞬态热传导示意图

Figure 15. Schematic diagram of the 2D transient heat conduction

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图 16 观测点的温度预测曲线图

Figure 16. Temperature prediction results at observed points

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图 17 非观测点的温度预测曲线图

Figure 17. Temperature prediction results at non-observed points

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表 1 不同数量观测点条件下，x=0.007 5 m处的温度场预测结果

Table 1. Prediction results of the temperature field at x=0.007 5 m under different observations

	1 observation	2 observations	3 observations	4 observations
prior-mean	452.7	452.7	452.7	452.7
prior-STD	47.1	47.1	47.1	47.1
posterior-mean	464.4	463.3	464.5	465.0
posterior-STD	25.0	16.0	11.9	8.9
observed data	465.4	465.4	465.4	465.4

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表 2 四个非观测点的温度值预测结果

Table 2. Temperature prediction results at 4 non-observation locations

	node A₁	node B₁	node C₁	node D₁
prior-mean	190.6	112.4	203.2	176.2
prior-STD	23.9	9.9	12.9	24.4
posterior-mean	160.6	100.3	163.6	144.5
posterior-STD	2.3	1.6	1.9	2.5
observed data	159.9	101.0	164.0	147.2

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表 3 观测点和非观测点温度值预测结果

Table 3. Prediction results of observed and unobserved temperature values

time/s	observed point(2, 0.6)					non-observed point(0, 1)
time/s	prior-mean	prior-TSD	posterior-mean	posterior-TSD	observed value	prior-mean	prior-TSD	posterior-mean	posterior-TSD	observed value
0.5	50.0	4.1	60.3	1.3	63.2	158.3	4.7	168.3	1.9	164.5
1	122.0	4.0	137.1	1.3	143.1	323.3	4.3	337.3	1.9	334.7
1.5	205.4	4.4	225.7	1.2	233.3	459.9	4.9	480.3	1.8	475.7
2	293.9	3.9	319.6	1.4	323.4	579.1	3.9	599.9	1.8	599.2
2.5	385.3	4.3	416.2	1.3	416.7	688.3	4.9	709.9	1.8	712.3
3	478.3	4.7	512.2	1.2	514.4	791.6	5.3	813.5	1.9	819.4
3.5	572.3	3.9	613.3	1.4	613.9	891.5	4.1	921.5	1.8	922.9
4	666.8	3.9	710.5	1.4	712.7	989.5	4.2	1023.4	1.7	1 024.5

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