热结构不确定性动力学仿真及模型确认方法研究
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摘要
在科学研究中，计算仿真已经成为继理论分析和试验技术之后的第三支柱，虽然计算仿真已经广泛应用于各个工程领域，但是计算仿真的研究和应用仅仅是一个开始。随着复杂工程结构设计对仿真模型精度的要求不断提高，就必须考虑实际存在的各种不确定性，这时仿真模型的真实精度和置信度评估就成为一项重要的研究课题。
模型确认就是从模型用途角度确定一个模型在多大程度上能够精确描述真实物理世界的过程，它不仅仅是一个评定仿真模型准确度的过程，而且是一个通过确认结果提高预测精度的过程。本文以高超声速飞行器典型热结构的仿真计算以及相关动力学分析为基本问题，以模型确认方法为主要研究内容，研究了以下几个方面：
（1）将材料的热传导系数表达为温度的多项式函数，采用遗传算法来识别多项式的系数，从而得到更精确的温度分布，为热结构设计提供指导。同时以美国圣地亚国家实验室提出的热传导模型确认挑战问题为研究对象，阐述了模型确认的贝叶斯框架，介绍了贝叶斯及不确定性量化的基本理论，强调了模型修正在模型确认中的作用，比较了各种模型修正方法的优缺点，将贝叶斯模型修正用于热传导问题的模型确认中，得到了比初始模型更准确的预测结果，研究表明贝叶斯模型修正方法用于模型确认能显著提高预测精度。
（2）针对结构动力学中的非对称阻尼结构，分别采用基于灵敏度分析和遗传算法的修正方法，同时对与刚度矩阵，阻尼矩阵和质量矩阵等相关的结构设计参数进行识别。提出了使用有效模态质量进行结构动力学有限元模型修正的新方法。仿真算例比较了基于有效模态质量灵敏度分析和遗传算法的修正结果，研究表明所提出的方法可以用于结构动力学模型修正中并能够补充频率信息不足的缺陷。
（3）针对小样本模型确认问题，将核密度估计和核主元分析相结合，用于美国圣地亚国家实验室提出的结构动力学模型确认挑战问题的研究中；将置信水平理论和核密度估计相结合，提出了核密度估计中最佳样本方差选择的改进方法，并应用于美国圣地亚国家实验室提出的结构静力学桁架模型确认挑战问题的研究。研究结果表明，将核主元分析、置信水平理等理论结合核密度估计方法处理小样本模型确认问题是非常有效的。
（4）针对热弹耦合梁固有频率的不确定性量化和传递问题，基于欧拉梁的振动方程和傅里叶热传导定律推导了梁的耦合振动方程；采用概率边界方法同时量化随机和认知混合不确定性，并采用双层蒙特卡罗抽样技术求解不确定性的传递；研究了材料参数不确定性对耦合固有频率的影响，研究表明梁的耦合固有频率均值和标准差都为区间参数，并且随着输入参数不确定性的增大而增大。
（5）针对模型形式不确定性量化问题，将最大熵方法用于确定调整参数方法中区间形式模型概率的最优值；将调整参数处理为区间不确定性，提出了模型形式不确定性量化的区间-混合调整参数法。通过单自由度非线性振动系统的频率预测以及二元机翼颤振临界速度的预测实例对区间-混合调整参数方法进行了验证。研究表明区间-混合调整参数法能够处理模型形式不确定性以及参数的随机和认知不确定性共存时的预测问题。
（6）针对C/C复合材料壁板结构，通过瞬态热传导分析得到了各个时刻壁板模型上的温度分布，并分析了各个时刻不同温度场的热模态和热颤振速度；在确定性框架下研究了热结构动力学模型修正，并对修正前后热颤振速度进行了比较；同时考虑随机和认知不确定性研究了热模态和热颤振速度的不确定性分布和边界，并采用裕度与不确定性的量化技术对热颤振速度进行了定量评估和认证。
The computation and simulation has become the third pillar along with theory and experiment inscientific research. Although the computation and simulation has been widely applied to various fieldsof engineering, the third pillar of computation and simulation is just now beginning to be constructed.The true accuracy and confidence of simulation model has become an important topic with the highsimulation requirements in engineering design.
Model validation is the process of determining the degree to which a model is an accuraterepresentation of the real world from the perspective of the intended uses of the model. It is notmerely a process of assessing the accuracy of a simulation model, but also a process to improve thepredictive precision through the model validation results. The typical thermal dynamical structuresimulation analysis for the hypersonic vehicle and the model validation method are studied in thiswork. The main contents are summarized as follows:
(1) The thermal conductivity of the material is expressed as a polynomial function of temperature,and genetic algorithm is used to identify the coefficients of the polynomial in order to get a moreaccurate temperature distribution to provide guidance for the thermal structure design. The Bayesianframework for model validation is achieved to the example of model validation thermal challengeproblem presented in Sandia National Laboratories. The basic theories of Bayesian analysis anduncertainty quantification are introduced and several model updating methods are emphasized andcompared in model validation. Finally, the Bayesian model updating method is applied to modelvalidation thermal challenge problem, and more accurate prediction results are obtained than thosefrom the initial model. The results demonstrate that the model predictive precision can be significantlyimproved when utilizing Bayesian model updating method in model validation.
(2) Finite element model updating based on sensitivity analysis and genetic algorithm respectivelyare used to identify the parameters coupled with mass, stiffness and damping matrixes simultaneouslyfor unsymmetrical damping system. A new finite element model updating method is presented usingeffective modal mass based on sensitivity analysis and genetic algorithm respectively. The simulationresults show that the two updating method using the effective modal mass which providing moreuseful information and can both be used to dynamic model updating.
(3) The kernel density estimation method combined with kernel principal component analysis issuccessfully used to solve the structural dynamic model validation challenge problem presented bySandia National Laboratories. The confidence level method is introduced and the optimum sample variance is determined using an improved method in kernel density estimation to increase thecredibility of model validation and as a numerical example, the static frame model validationchallenge problem presented by Sandia National Laboratories is chosen. The researches demonstratethat the kernel density estimation combined with kernel principal component analysis and theconfidence level methods are effective approach to solve the model validation problem with smallsamples.
(4) The coupled thermoelastic vibration governing equations are derived based on the differentialequations of Fourier heat conduction and transverse vibrations of Euler beam. Mixed aleatory andepistemic uncertainty quantification is described using p-box solution with double-loop Monte Carlosampling techniques. The distribution of coupled natural frequencies is performed when consideringthe material uncertainty with mixed aleatory and epistemic. The researches demonstrate that the meanand standard deviation of coupled nature frequency of beam are interval, and are both increasing asincreased of the input parameter uncertainties.
(5) Model-form probability belongs to epistemic uncertainty which is usually determined based onexpert opinion or experience but is described by interval uncertainty and its optimal value isdetermined through the maximum entropy approach. A new interval adjustment factor approach ispresented to model-form uncertainty quantification. The new method is validated through a nonlinearsingle degree of freedom vibration system for nature frequency, and the flutter velocity prediction of atwo degrees of freedom airfoil subject to unsteady aerodynamics. The studies demonstrate that thenew interval adjustment factor approach is feasible to model prediction for combination withmodel-form, aleatory and epistemic of parameter uncertainty.
(6) The temperature distribution of C/C composite panel structure is determined through transientheat conduction analysis before thermal modal and thermal flutter analysis in different moments.Thermal structural dynamics model updating is performed in deterministic framework and theprediction for thermal flutter velocity with updated model is performed. Uncertainty quantification forthermal modal and thermal flutter analysis considered mixed aleatory and epistemic uncertaintiesfrom the material of C/C composite. Quantification of margins and uncertainties technology isachieved to quantitative assessment and certification for thermal flutter velocity based on the aboveuncertainty analysis results.
模型确认就是从模型用途角度确定一个模型在多大程度上能够精确描述真实物理世界的过程，它不仅仅是一个评定仿真模型准确度的过程，而且是一个通过确认结果提高预测精度的过程。本文以高超声速飞行器典型热结构的仿真计算以及相关动力学分析为基本问题，以模型确认方法为主要研究内容，研究了以下几个方面：
（1）将材料的热传导系数表达为温度的多项式函数，采用遗传算法来识别多项式的系数，从而得到更精确的温度分布，为热结构设计提供指导。同时以美国圣地亚国家实验室提出的热传导模型确认挑战问题为研究对象，阐述了模型确认的贝叶斯框架，介绍了贝叶斯及不确定性量化的基本理论，强调了模型修正在模型确认中的作用，比较了各种模型修正方法的优缺点，将贝叶斯模型修正用于热传导问题的模型确认中，得到了比初始模型更准确的预测结果，研究表明贝叶斯模型修正方法用于模型确认能显著提高预测精度。
（2）针对结构动力学中的非对称阻尼结构，分别采用基于灵敏度分析和遗传算法的修正方法，同时对与刚度矩阵，阻尼矩阵和质量矩阵等相关的结构设计参数进行识别。提出了使用有效模态质量进行结构动力学有限元模型修正的新方法。仿真算例比较了基于有效模态质量灵敏度分析和遗传算法的修正结果，研究表明所提出的方法可以用于结构动力学模型修正中并能够补充频率信息不足的缺陷。
（3）针对小样本模型确认问题，将核密度估计和核主元分析相结合，用于美国圣地亚国家实验室提出的结构动力学模型确认挑战问题的研究中；将置信水平理论和核密度估计相结合，提出了核密度估计中最佳样本方差选择的改进方法，并应用于美国圣地亚国家实验室提出的结构静力学桁架模型确认挑战问题的研究。研究结果表明，将核主元分析、置信水平理等理论结合核密度估计方法处理小样本模型确认问题是非常有效的。
（4）针对热弹耦合梁固有频率的不确定性量化和传递问题，基于欧拉梁的振动方程和傅里叶热传导定律推导了梁的耦合振动方程；采用概率边界方法同时量化随机和认知混合不确定性，并采用双层蒙特卡罗抽样技术求解不确定性的传递；研究了材料参数不确定性对耦合固有频率的影响，研究表明梁的耦合固有频率均值和标准差都为区间参数，并且随着输入参数不确定性的增大而增大。
（5）针对模型形式不确定性量化问题，将最大熵方法用于确定调整参数方法中区间形式模型概率的最优值；将调整参数处理为区间不确定性，提出了模型形式不确定性量化的区间-混合调整参数法。通过单自由度非线性振动系统的频率预测以及二元机翼颤振临界速度的预测实例对区间-混合调整参数方法进行了验证。研究表明区间-混合调整参数法能够处理模型形式不确定性以及参数的随机和认知不确定性共存时的预测问题。
（6）针对C/C复合材料壁板结构，通过瞬态热传导分析得到了各个时刻壁板模型上的温度分布，并分析了各个时刻不同温度场的热模态和热颤振速度；在确定性框架下研究了热结构动力学模型修正，并对修正前后热颤振速度进行了比较；同时考虑随机和认知不确定性研究了热模态和热颤振速度的不确定性分布和边界，并采用裕度与不确定性的量化技术对热颤振速度进行了定量评估和认证。
The computation and simulation has become the third pillar along with theory and experiment inscientific research. Although the computation and simulation has been widely applied to various fieldsof engineering, the third pillar of computation and simulation is just now beginning to be constructed.The true accuracy and confidence of simulation model has become an important topic with the highsimulation requirements in engineering design.
Model validation is the process of determining the degree to which a model is an accuraterepresentation of the real world from the perspective of the intended uses of the model. It is notmerely a process of assessing the accuracy of a simulation model, but also a process to improve thepredictive precision through the model validation results. The typical thermal dynamical structuresimulation analysis for the hypersonic vehicle and the model validation method are studied in thiswork. The main contents are summarized as follows:
(1) The thermal conductivity of the material is expressed as a polynomial function of temperature,and genetic algorithm is used to identify the coefficients of the polynomial in order to get a moreaccurate temperature distribution to provide guidance for the thermal structure design. The Bayesianframework for model validation is achieved to the example of model validation thermal challengeproblem presented in Sandia National Laboratories. The basic theories of Bayesian analysis anduncertainty quantification are introduced and several model updating methods are emphasized andcompared in model validation. Finally, the Bayesian model updating method is applied to modelvalidation thermal challenge problem, and more accurate prediction results are obtained than thosefrom the initial model. The results demonstrate that the model predictive precision can be significantlyimproved when utilizing Bayesian model updating method in model validation.
(2) Finite element model updating based on sensitivity analysis and genetic algorithm respectivelyare used to identify the parameters coupled with mass, stiffness and damping matrixes simultaneouslyfor unsymmetrical damping system. A new finite element model updating method is presented usingeffective modal mass based on sensitivity analysis and genetic algorithm respectively. The simulationresults show that the two updating method using the effective modal mass which providing moreuseful information and can both be used to dynamic model updating.
(3) The kernel density estimation method combined with kernel principal component analysis issuccessfully used to solve the structural dynamic model validation challenge problem presented bySandia National Laboratories. The confidence level method is introduced and the optimum sample variance is determined using an improved method in kernel density estimation to increase thecredibility of model validation and as a numerical example, the static frame model validationchallenge problem presented by Sandia National Laboratories is chosen. The researches demonstratethat the kernel density estimation combined with kernel principal component analysis and theconfidence level methods are effective approach to solve the model validation problem with smallsamples.
(4) The coupled thermoelastic vibration governing equations are derived based on the differentialequations of Fourier heat conduction and transverse vibrations of Euler beam. Mixed aleatory andepistemic uncertainty quantification is described using p-box solution with double-loop Monte Carlosampling techniques. The distribution of coupled natural frequencies is performed when consideringthe material uncertainty with mixed aleatory and epistemic. The researches demonstrate that the meanand standard deviation of coupled nature frequency of beam are interval, and are both increasing asincreased of the input parameter uncertainties.
(5) Model-form probability belongs to epistemic uncertainty which is usually determined based onexpert opinion or experience but is described by interval uncertainty and its optimal value isdetermined through the maximum entropy approach. A new interval adjustment factor approach ispresented to model-form uncertainty quantification. The new method is validated through a nonlinearsingle degree of freedom vibration system for nature frequency, and the flutter velocity prediction of atwo degrees of freedom airfoil subject to unsteady aerodynamics. The studies demonstrate that thenew interval adjustment factor approach is feasible to model prediction for combination withmodel-form, aleatory and epistemic of parameter uncertainty.
(6) The temperature distribution of C/C composite panel structure is determined through transientheat conduction analysis before thermal modal and thermal flutter analysis in different moments.Thermal structural dynamics model updating is performed in deterministic framework and theprediction for thermal flutter velocity with updated model is performed. Uncertainty quantification forthermal modal and thermal flutter analysis considered mixed aleatory and epistemic uncertaintiesfrom the material of C/C composite. Quantification of margins and uncertainties technology isachieved to quantitative assessment and certification for thermal flutter velocity based on the aboveuncertainty analysis results.
引文
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