不确定性温度场和结构的分析方法研究
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摘要
在实际工程结构的分析与设计中,除了要考虑结构的力学行为外,有时还需要考虑结构的热效应(如热变形,热应力)、结构的温度不能超过某一设定值等方面。然而实际工程结构中存在着大量的误差和不确定性,使得结构的物理参数、几何参数以及载荷等具有不确定性,从而导致结构的响应也具有不确定性。因此,研究这些不确定性对结构响应的影响具有重要的工程意义和理论意义。本文对不确定温度场和结构的分析方法进行了系统研究,主要内容如下:
1、稳态随机温度场分析
考虑稳态热传导中的导热系数、换热系数、热流密度、环境温度以及内热源等参数同时具有随机性时,首先,利用Neumann展开Monte-Carlo随机有限元法对温度场的响应问题进行分析,并给出节点温度响应的均值、变异系数和节点温度落在某一区间内的概率计算公式,考察随机变量的变异性大小对节点温度响应的影响。其次,对稳态随机温度场响应的统计特性问题,提出了渐进法与最大熵原理相结合的求解方法,该法利用拉普拉斯多维积分的渐近展开以及函数的泰勒级数展开等方法,求得了节点温度响应的任意阶原点矩的近似解析表达式,之后利用最大熵原理获得节点温度响应的概率密度函数表达式。
2、瞬态随机温度场分析
考虑瞬态热传导中物理参数和边界条件的随机性,利用随机因子法和求解随机变量函数数字特征的代数综合法导出随机瞬态温度场响应的数字特征——均值和方差的拟解析计算表达式,并考察任意参数的随机性对温度场响应的影响。该方法具有只进行一次随机温度场分析便可以获得其响应的数字特征的优点。
3、温度场的非概率凸集合理论模型的摄动数值解法
将结构导热的物理参数、温度场的初始和边界条件等不确定性参数以凸模型加以描述,基于矩阵摄动理论和处理不确定问题的凸集合理论模型的结合,导出有界不确定参数瞬态温度场响应所在集合的上、下界摄动计算公式。
4、具有区间参数的瞬态温度场数值分析
考虑结构瞬态热传导问题的不确定性,将结构各物理参数和温度的初、边值条件均视为区间变量。对具有区间参数的热传导抛物型方程的求解,在空间域上利用有限单元离散,在时间域上利用差分离散,将区间分析和常规的有限元法相结合,建立了求解不确定温度场的基于单元的区间有限元方法。利用矩阵摄动公式求解结构的区间有限元方程,获得了结构瞬态温度场响应的范围。此外,对结构静力区间有限元方程组提出了一种简而易行的解法。该法将含区间变量的整体刚度矩阵在区间变量的中值处进行一阶泰勒展开,并对刚度矩阵展开式进行近似处理,将刚度矩阵的逆矩阵用一系列的Neumann展开级数来表示。为减小区间运算的扩张,利用区间运算的次分配律和相关运算规则,导出不确定结构响应量上、下界的计算式。
5、基于广义密度函数模糊随机温度场分析
考虑热传导中的诸参数同时具有模糊变量和随机变量的混合模型。利用广义密度函数将模糊变量转化为等价随机变量,应用随机变量数字特征的计算式求出了转化后的随机变量的均值和方差,从而原模糊随机温度场问题转换为纯随机温度场问题。利用随机摄动法求解,获得了原模糊随机混合温度场响应的均值和方差。通过算例考察各个不确定性参数的不确定性对温度场响应的影响。
6、基于分解法的最大熵随机有限元法
在分解法的基础之上,对求解随机结构的静力响应问题提出了最大熵分析方法。该法利用单变量分解将多维随机响应函数表述为多个单维随机响应函数的组合形式,将求解随机结构响应统计矩的多维积分表达式转化为单维积分式,对单维积分采用高斯-埃尔米特积分格式求解。在获得结构响应的统计矩之后,利用最大熵原理求得结构响应的概率密度函数解析表达式。
7、随机结构系统固有频率特性分析
考虑随机结构系统固有频率的统计特性,利用单变量分解将高维随机变量特征值函数分解成单维随机变量特征值函数的组合形式,将求解随机结构特征值统计矩的多维积分表达式转化为单维积分式,对单维积分采用8点高斯-埃尔米特积分格式数值求解。在获得随机结构系统固有频率的前四阶原点矩之后,利用最大熵原理求得结构固有频率概率密度函数的解析表达式。
In the process of analysis and design of practical structures, it is necessary to consider the thermal effect(such as thermal distortion, thermal stress, the temperature value smaller than the seted value,and so on) besides considering the structural mechanical behavior. But there are a large errors and uncertainties in the practical structures, which cause the physical parameters, geometrical parameters and loads taking on uncertainties. Consequently, studying the influence of these uncertainties on structural responses has important engineering meaning and academic meaning. The systemic studies on analysis methods of uncertain temperature fields and structures are maded in this dissertation. The main research work can be described as follows.
1. The analysis for static random temperature field
Considering the randomness of heat conduction coefficient, heat exchange coefficient, heat flux density and environmental temperature, and so on, Firstly, the Neumann expansions Monte-Carlo stochastic finite element method is employed to analyse the temperature response. Giving the computing formulas of the mean、variance and possibility in some interval of node temperature. The effects of the amount of variances of random variables on node temperature responses are considered. Secondly, Asymptotic-Maximum Entropy Principle for solving the statistical characteristics of static random temperature response is presented. In this method, the asymptotic approximation of Laplace multidimensional integrals and functions'Taylor expansions are employed, then the approximate analytical expressions of the arbitrary order original moments of node temperature response are obtained. On the basis of maximum entropy principle, the Probability Density Function(PDF) of nodal temperature response are developed.
2. The analysis for transient random temperature field
Considering the randomness of physical parameters and boundary conditions of transient heat transfer, the quasi-analytical expressions of numerical characteristics (the mean value and variance) of random temperature field response are derived by the random factor method and the algebra synthesis method which is employed to obtain the random variable’s functional moments. The influence of the randomness of each parameter on the temperature field response is investigated. The proposed method has the merit that the numerical characteristics of stochastic temperature field response can be obtained by analyzing the random temperature field just in one time.
3. Perturbed numerical algorithm of nonprobabilistic convex set theoretical models on the temperature field
The uncertain parameters of physical parameters and initial boundary conditions of heat conduction are described by the convex model. The perturbation formulas of the upper and lower bounds of temperature field response with unknown-but-bounded parameters are given via the combination of matrix perturbation theory and the convex set theory model.
4. Numerical analysis for transient temperature field with interval parameters
Considering the uncertainties of the transient heat transfer, the physical parameters and initial boundary conditions are regarded as interval variables. In order to solve parabolic equation of heat conduction with interval parameters, the regions of space are discretized by finite elements and the regions of time are discretized by finite difference. The interval finite element method based on the element is established via the combination of interval analysis and the traditional finite element method. Then the interval finite equation of structure is solved by matrix perturbation formulas, and the range of temperature field response of the structure is obtained. In addition, a simple method for solving the structural static interval finite element equations is presented. In this method, the global stiffness matrix is first order expanded at the middle value of interval variables by Taylor. And the expansion expression of stiffness matrix is dealt approximatively, the inverse matrix of uncertain stiffness matrix is expressed as a series of Neumann expansion series. The full use of sub-distribution law and other arithmetic rules of interval analysis are made to reduce the extension caused by interval analysis. Finally, the computational formulas of the upper and lower boundaries of the uncertain structures responses are developed.
5. The analysis for temperature field with fuzzy-random parameters based on general density function
The mixture modeling of fuzzy and random variables in heat transfer is discussed. The fuzzy variables are transformed into random variables based on general density function method. The mean values and variance values of transformed random variables are obtained using the formulas for solving random variables’numerical characteristics. So the fuzzy-random temperature field is converted to the pure random temperature field, correspondingly. The mean values and variance values of the random temperature field responses are obtained by employing the conventional stochastic perturbation method. The influence of the uncertainty of each parameter on the structural temperature field responses is analyzed through the example.
6. The maximum entropy stochastic finite element method based on the dimension-reduction method
The maximum entropy analysis method for solving the static response of random structures is presented on basis of dimension-reduction method. In this method, the multi-dimensional random response functions are decomposed into the combination of one-dimensional response functions by the univariate dimension-reduction method, so the multi-dimensional integration which is employed to calculate statistical moments of response of stochastic structures is transformed into one-dimensional integration, and the one-dimensional integration is numerically calculated by the Gauss-Hermite integration. After getting the statistical moments of response of structures, the explicit expression of probability density function of structure's response is obtained using the Maximum Entropy Principle.
7. The eigenfrequencies characteristic analysis for random structure systems
Considering the statistical characteristic of eigenfrequencies of random structure systems, the multi-dimensional random eigenvalues functions are decomposed into the combination of one-dimensional random eigenvalues functions by the univariate dimension-reduction method, so the multi-dimensional integrations which are employed to calculate statistical moments of eigenvalues of random structure systems are transformed into one-dimensional integrations, and the one-dimensional integration is calculated by the Gauss-Hermite 8-point numerical integration. Once getting the first four origin moments of eigenfrequencies of stochastic structure systems, the explicit expressions of probability density function of eigenvalues of structures can be obtained employing the Maximum Entropy Principle.
1、稳态随机温度场分析
考虑稳态热传导中的导热系数、换热系数、热流密度、环境温度以及内热源等参数同时具有随机性时,首先,利用Neumann展开Monte-Carlo随机有限元法对温度场的响应问题进行分析,并给出节点温度响应的均值、变异系数和节点温度落在某一区间内的概率计算公式,考察随机变量的变异性大小对节点温度响应的影响。其次,对稳态随机温度场响应的统计特性问题,提出了渐进法与最大熵原理相结合的求解方法,该法利用拉普拉斯多维积分的渐近展开以及函数的泰勒级数展开等方法,求得了节点温度响应的任意阶原点矩的近似解析表达式,之后利用最大熵原理获得节点温度响应的概率密度函数表达式。
2、瞬态随机温度场分析
考虑瞬态热传导中物理参数和边界条件的随机性,利用随机因子法和求解随机变量函数数字特征的代数综合法导出随机瞬态温度场响应的数字特征——均值和方差的拟解析计算表达式,并考察任意参数的随机性对温度场响应的影响。该方法具有只进行一次随机温度场分析便可以获得其响应的数字特征的优点。
3、温度场的非概率凸集合理论模型的摄动数值解法
将结构导热的物理参数、温度场的初始和边界条件等不确定性参数以凸模型加以描述,基于矩阵摄动理论和处理不确定问题的凸集合理论模型的结合,导出有界不确定参数瞬态温度场响应所在集合的上、下界摄动计算公式。
4、具有区间参数的瞬态温度场数值分析
考虑结构瞬态热传导问题的不确定性,将结构各物理参数和温度的初、边值条件均视为区间变量。对具有区间参数的热传导抛物型方程的求解,在空间域上利用有限单元离散,在时间域上利用差分离散,将区间分析和常规的有限元法相结合,建立了求解不确定温度场的基于单元的区间有限元方法。利用矩阵摄动公式求解结构的区间有限元方程,获得了结构瞬态温度场响应的范围。此外,对结构静力区间有限元方程组提出了一种简而易行的解法。该法将含区间变量的整体刚度矩阵在区间变量的中值处进行一阶泰勒展开,并对刚度矩阵展开式进行近似处理,将刚度矩阵的逆矩阵用一系列的Neumann展开级数来表示。为减小区间运算的扩张,利用区间运算的次分配律和相关运算规则,导出不确定结构响应量上、下界的计算式。
5、基于广义密度函数模糊随机温度场分析
考虑热传导中的诸参数同时具有模糊变量和随机变量的混合模型。利用广义密度函数将模糊变量转化为等价随机变量,应用随机变量数字特征的计算式求出了转化后的随机变量的均值和方差,从而原模糊随机温度场问题转换为纯随机温度场问题。利用随机摄动法求解,获得了原模糊随机混合温度场响应的均值和方差。通过算例考察各个不确定性参数的不确定性对温度场响应的影响。
6、基于分解法的最大熵随机有限元法
在分解法的基础之上,对求解随机结构的静力响应问题提出了最大熵分析方法。该法利用单变量分解将多维随机响应函数表述为多个单维随机响应函数的组合形式,将求解随机结构响应统计矩的多维积分表达式转化为单维积分式,对单维积分采用高斯-埃尔米特积分格式求解。在获得结构响应的统计矩之后,利用最大熵原理求得结构响应的概率密度函数解析表达式。
7、随机结构系统固有频率特性分析
考虑随机结构系统固有频率的统计特性,利用单变量分解将高维随机变量特征值函数分解成单维随机变量特征值函数的组合形式,将求解随机结构特征值统计矩的多维积分表达式转化为单维积分式,对单维积分采用8点高斯-埃尔米特积分格式数值求解。在获得随机结构系统固有频率的前四阶原点矩之后,利用最大熵原理求得结构固有频率概率密度函数的解析表达式。
In the process of analysis and design of practical structures, it is necessary to consider the thermal effect(such as thermal distortion, thermal stress, the temperature value smaller than the seted value,and so on) besides considering the structural mechanical behavior. But there are a large errors and uncertainties in the practical structures, which cause the physical parameters, geometrical parameters and loads taking on uncertainties. Consequently, studying the influence of these uncertainties on structural responses has important engineering meaning and academic meaning. The systemic studies on analysis methods of uncertain temperature fields and structures are maded in this dissertation. The main research work can be described as follows.
1. The analysis for static random temperature field
Considering the randomness of heat conduction coefficient, heat exchange coefficient, heat flux density and environmental temperature, and so on, Firstly, the Neumann expansions Monte-Carlo stochastic finite element method is employed to analyse the temperature response. Giving the computing formulas of the mean、variance and possibility in some interval of node temperature. The effects of the amount of variances of random variables on node temperature responses are considered. Secondly, Asymptotic-Maximum Entropy Principle for solving the statistical characteristics of static random temperature response is presented. In this method, the asymptotic approximation of Laplace multidimensional integrals and functions'Taylor expansions are employed, then the approximate analytical expressions of the arbitrary order original moments of node temperature response are obtained. On the basis of maximum entropy principle, the Probability Density Function(PDF) of nodal temperature response are developed.
2. The analysis for transient random temperature field
Considering the randomness of physical parameters and boundary conditions of transient heat transfer, the quasi-analytical expressions of numerical characteristics (the mean value and variance) of random temperature field response are derived by the random factor method and the algebra synthesis method which is employed to obtain the random variable’s functional moments. The influence of the randomness of each parameter on the temperature field response is investigated. The proposed method has the merit that the numerical characteristics of stochastic temperature field response can be obtained by analyzing the random temperature field just in one time.
3. Perturbed numerical algorithm of nonprobabilistic convex set theoretical models on the temperature field
The uncertain parameters of physical parameters and initial boundary conditions of heat conduction are described by the convex model. The perturbation formulas of the upper and lower bounds of temperature field response with unknown-but-bounded parameters are given via the combination of matrix perturbation theory and the convex set theory model.
4. Numerical analysis for transient temperature field with interval parameters
Considering the uncertainties of the transient heat transfer, the physical parameters and initial boundary conditions are regarded as interval variables. In order to solve parabolic equation of heat conduction with interval parameters, the regions of space are discretized by finite elements and the regions of time are discretized by finite difference. The interval finite element method based on the element is established via the combination of interval analysis and the traditional finite element method. Then the interval finite equation of structure is solved by matrix perturbation formulas, and the range of temperature field response of the structure is obtained. In addition, a simple method for solving the structural static interval finite element equations is presented. In this method, the global stiffness matrix is first order expanded at the middle value of interval variables by Taylor. And the expansion expression of stiffness matrix is dealt approximatively, the inverse matrix of uncertain stiffness matrix is expressed as a series of Neumann expansion series. The full use of sub-distribution law and other arithmetic rules of interval analysis are made to reduce the extension caused by interval analysis. Finally, the computational formulas of the upper and lower boundaries of the uncertain structures responses are developed.
5. The analysis for temperature field with fuzzy-random parameters based on general density function
The mixture modeling of fuzzy and random variables in heat transfer is discussed. The fuzzy variables are transformed into random variables based on general density function method. The mean values and variance values of transformed random variables are obtained using the formulas for solving random variables’numerical characteristics. So the fuzzy-random temperature field is converted to the pure random temperature field, correspondingly. The mean values and variance values of the random temperature field responses are obtained by employing the conventional stochastic perturbation method. The influence of the uncertainty of each parameter on the structural temperature field responses is analyzed through the example.
6. The maximum entropy stochastic finite element method based on the dimension-reduction method
The maximum entropy analysis method for solving the static response of random structures is presented on basis of dimension-reduction method. In this method, the multi-dimensional random response functions are decomposed into the combination of one-dimensional response functions by the univariate dimension-reduction method, so the multi-dimensional integration which is employed to calculate statistical moments of response of stochastic structures is transformed into one-dimensional integration, and the one-dimensional integration is numerically calculated by the Gauss-Hermite integration. After getting the statistical moments of response of structures, the explicit expression of probability density function of structure's response is obtained using the Maximum Entropy Principle.
7. The eigenfrequencies characteristic analysis for random structure systems
Considering the statistical characteristic of eigenfrequencies of random structure systems, the multi-dimensional random eigenvalues functions are decomposed into the combination of one-dimensional random eigenvalues functions by the univariate dimension-reduction method, so the multi-dimensional integrations which are employed to calculate statistical moments of eigenvalues of random structure systems are transformed into one-dimensional integrations, and the one-dimensional integration is calculated by the Gauss-Hermite 8-point numerical integration. Once getting the first four origin moments of eigenfrequencies of stochastic structure systems, the explicit expressions of probability density function of eigenvalues of structures can be obtained employing the Maximum Entropy Principle.
引文
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