不确定性结构分析及拓扑优化研究
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摘要
本学位论文首先以区间参数和未确知参数结构为研究对象，探索性地研究了当结构参数和外载荷为区间变量或未确知变量时结构的静力、动力响应以及非概率可靠性指标；利用3σ准则建立了区间非概率可靠性指标与概率可靠性指标的转换关系，分析了星载伞状可展开天线展开机构的运动功能可靠性；利用水平集拓扑优化方法，对含有可靠性约束的拓扑优化模型进行优化；最后，对随机温度场的相关方面做了研究，利用拓扑优化的方法对热传导介质的分布开展了优化设计。全文的主要内容如下：
1.基于区间模型的结构静力分析
将结构系统中的不确定性参数用区间数表示，建立系统的区间有限元控制方程。对该方程组的求解分别提出了一种摄动方法和一种基于导数信息的仿射算法。通过这两种方法对区间模型的处理，简化了区间运算，有效控制了区间运算结果的扩张。对比研究了经典可靠性理论与区间非概率可靠性理论，利用3σ准则建立概率可靠性指标与非概率可靠性指标之间的转换关系，构建了含有不确定参数的星载伞状可展开天线展开机构的运动区间分析模型，利用优化方法处理区间运算，通过功能函数得到非概率可靠性指标，利用本文建立的3σ准则方法转换为等效概率可靠性指标，并将此理论应用于伞状天线旋转关节运动功能可靠性的分析。
2.基于未确知信息的结构研究
充分利用客观的不确定性信息，构建了物理参数和载荷同时具有未确知性的系留气球设备挂架结构有限元分析模型，提出了基于未确知理论的系留气球设备挂架结构静力分析方法；利用未确知有理数的运算规则，推导出挂架结构位移响应和单元应力响应的计算表达式。得出结构位移和应力响应取某值的可信度与各参数取值可信度的趋势是一致的结论。此外，构建了物理参数（弹性模量、质量密度）同时具有未确知性的空间板梁组合结构动力特性分析模型，并提出了基于未确知因子法的板梁组合结构动力特性分析方法。在缺乏足够数据或者信息不完整的情况下，用未确知信息表述结构模型参数的不确定性，比用成熟的随机方法具有更高的可信度且方法简易可行。
3.基于Levelset方法的连续体结构拓扑优化
在研究水平集拓扑优化方法的基础上，对具有区间参数的不确定性结构的非概率可靠性约束进行了分析，提出了包含非概率可靠度信息的拟安全系数形式，从而将非概率可靠性约束问题显式化处理，使得优化过程形式简单，收敛速度快，避免了复杂的迭代运算。在疲劳可靠性的剩余强度模型基础上，对元件的概率疲劳和非概率疲劳可靠性进行了研究，与水平集拓扑优化方法相结合，以结构柔度最小为目标函数，以体积和疲劳可靠性要求为约束函数进行拓扑优化，优化前对概率疲劳和非概率疲劳可靠性约束进行显式化处理，使得优化过程形式简单，降
低了优化难度，减少了计算工作量。
4.不确定性温度场分析及拓扑优化将热传导中的物理参数(导热系数、对流换热系数)和初、边界条件(环境温度)等参数考虑为随机变量，对随机稳态、随机瞬态温度场以及随机热应力问题进行了分析，并研究了与之相关的可靠性问题。利用移动渐进拓扑优化方法(Method ofMoving Asymptotes，MMA)，通过导出的结构随机温度场的数字特征，定义了以温度随机变量不超越其临界值的热可靠性，建立了以结构总散热弱度均值最小化为目标函数、以给定热可靠度和体积比为约束函数的结构拓扑优化模型，对热可靠性概率约束函数进行了等价显式化处理，对优化模型求解，通过两个算例表明了文中模型的合理性和方法的有效性。
Firstly, structures with interval parameters or unascertained parameters are taken asresearch objects in this paper. Structural static responses, dynamic characteristics,dynamic responses and non-probability reliability index are derived under theconditions that physical parameters of materials, structural geometric dimensions andapplied loads are all interval, random-interval or unascertained variables. Secondly, thetransformational relation between the probabilistic reliability index andnon-probabilistic reliability index was established by applying the3σ principle, andthen the movement function reliability analysis model of deployment mechanism ofsatellite umbrella antenna was built and solved by optimization method. Then, modelswith reliability constraint are optimized based on the topological optimization methodwith level set. Finally, the stochastic temperature field is researched and the heatconduction distributions are analyzed based on topological optimization method. Themain research works can be described as follows:
1. The structural static analysis based on interval models.
By representing the uncertain parameters as interval numbers, the interval finiteelement governed equations of structural systems are built. To solve these equations, aninterval parameter perturbation method and an affine arithmetic with recursivederivative information are proposed, which can simply the interval computing anddecrease the excessive width of interval operations to a certain extent. Moreover, theclassical reliability theory and the interval non-probabilistic reliability theory are studied.By applying the3σ principle, the transformational relation between the probabilisticreliability index and non-probabilistic reliability index is established. The intervaluncertainty analytical model of deployment mechanism of satellite umbrella antenna isbuilt and solved by optimization method. The non-probabilistic reliability index isobtained from the performance function, and then is transformed to equivalentprobabilistic reliability index by the proposed method in this paper. The movementfunction reliability analysis result of rotation joint in umbrella antenna shows the3σprinciple approach is reasonable and effective.
2. Structural analysis based on unascertained theory.
The undetermined but objective information is considered, and the structuralphysical parameters and the applied loads are defined as undetermined variables in the finite element analysis model of the tethered balloon equipment bracket structures. Anda structure analysis method based on the undetermined factor method is proposed. Fromthe operation rules of undetermined information, the computational expressions of theequipment bracket structural displacement response and the element stress response areobtained. It is concluded that the confidence levels of structural responses are in directproportion to the confidence level of each parameter. And also, the dynamiccharacteristics analysis model of space beam-plate composite structures is built, inwhich structural elastic module and mass densities are all considered as unascertainedvariables. And a structure analysis method based on the unascertained factor method isgiven. Compared to the mature random method, the proposed method can obtainreliability result of safer and higher faith degree in the case of inadequate data or
insufficient information; moreover, it is simple and easy to apply.
3. Topology optimization of continuum structures based on the level set methodBased on the research of the topological optimization method with level set, thenon-probability reliability constraint of interval parameters structure is analyzed.Through deducing and transforming the finite element formula, the suppositional safetyfactor form with non-probability reliability information is proposed, by which thenon-probability reliability constraints are dealt with explicitly. This method is simple inform and quick in convergence speed, avoiding complicated iteration operations.Moreover, based on the stress-strength model for the fatigue reliability of structuralelements, the probability and non-probability fatigue reliability of elements are studiedand a new topological optimization method is presented. This method Combines withthe level set method for structural topology optimization, takes the minimized structuralflexibility as objective function, and chooses the volume and fatigue reliabilityrequirement as the constraint functions. The constraints of fatigue reliability areprocessed explicitly before optimization, which simplified the optimization course,
avoided complicated iteration operations and advanced the computing efficiency.
4. Uncertain temperature field analysis and topology optimization.Considering the randomness of physical parameters(such as heat conductioncoefficient) and initial boundary conditions(such as environmental temperature，heatexchange coefficient) and so on. The problems for static random temperature field,transient random temperature field, random thermal stress and correlative reliabilitieswere analyzed. In the thermal analysis of continuum structure, by treating thecoefficient of thermal conductivity, the intensity of internal heat source and theamplitude of distribution function for given boundary temperature as random parameters, the numeric characteristic of structural random temperature field is derived.The thermal reliability definition is proposed, which is that random temperaturevariables can not exceed their critical values. The topology optimization formulation ofstructure is built, in which the minimum mean value of heat dissipation is taken asobjective function, the given thermal reliability and structural volume ratio asconstraints. In the prophase of topology optimization, the probability constraint functionof thermal reliability is transformed into an explicit function, which simplifies thetopology model. The method of moving asymptotes is used to solve this topologyoptimization problem. Some examples indicate the validity and feasibility of theproposed method.
1.基于区间模型的结构静力分析
将结构系统中的不确定性参数用区间数表示，建立系统的区间有限元控制方程。对该方程组的求解分别提出了一种摄动方法和一种基于导数信息的仿射算法。通过这两种方法对区间模型的处理，简化了区间运算，有效控制了区间运算结果的扩张。对比研究了经典可靠性理论与区间非概率可靠性理论，利用3σ准则建立概率可靠性指标与非概率可靠性指标之间的转换关系，构建了含有不确定参数的星载伞状可展开天线展开机构的运动区间分析模型，利用优化方法处理区间运算，通过功能函数得到非概率可靠性指标，利用本文建立的3σ准则方法转换为等效概率可靠性指标，并将此理论应用于伞状天线旋转关节运动功能可靠性的分析。
2.基于未确知信息的结构研究
充分利用客观的不确定性信息，构建了物理参数和载荷同时具有未确知性的系留气球设备挂架结构有限元分析模型，提出了基于未确知理论的系留气球设备挂架结构静力分析方法；利用未确知有理数的运算规则，推导出挂架结构位移响应和单元应力响应的计算表达式。得出结构位移和应力响应取某值的可信度与各参数取值可信度的趋势是一致的结论。此外，构建了物理参数（弹性模量、质量密度）同时具有未确知性的空间板梁组合结构动力特性分析模型，并提出了基于未确知因子法的板梁组合结构动力特性分析方法。在缺乏足够数据或者信息不完整的情况下，用未确知信息表述结构模型参数的不确定性，比用成熟的随机方法具有更高的可信度且方法简易可行。
3.基于Levelset方法的连续体结构拓扑优化
在研究水平集拓扑优化方法的基础上，对具有区间参数的不确定性结构的非概率可靠性约束进行了分析，提出了包含非概率可靠度信息的拟安全系数形式，从而将非概率可靠性约束问题显式化处理，使得优化过程形式简单，收敛速度快，避免了复杂的迭代运算。在疲劳可靠性的剩余强度模型基础上，对元件的概率疲劳和非概率疲劳可靠性进行了研究，与水平集拓扑优化方法相结合，以结构柔度最小为目标函数，以体积和疲劳可靠性要求为约束函数进行拓扑优化，优化前对概率疲劳和非概率疲劳可靠性约束进行显式化处理，使得优化过程形式简单，降
低了优化难度，减少了计算工作量。
4.不确定性温度场分析及拓扑优化将热传导中的物理参数(导热系数、对流换热系数)和初、边界条件(环境温度)等参数考虑为随机变量，对随机稳态、随机瞬态温度场以及随机热应力问题进行了分析，并研究了与之相关的可靠性问题。利用移动渐进拓扑优化方法(Method ofMoving Asymptotes，MMA)，通过导出的结构随机温度场的数字特征，定义了以温度随机变量不超越其临界值的热可靠性，建立了以结构总散热弱度均值最小化为目标函数、以给定热可靠度和体积比为约束函数的结构拓扑优化模型，对热可靠性概率约束函数进行了等价显式化处理，对优化模型求解，通过两个算例表明了文中模型的合理性和方法的有效性。
Firstly, structures with interval parameters or unascertained parameters are taken asresearch objects in this paper. Structural static responses, dynamic characteristics,dynamic responses and non-probability reliability index are derived under theconditions that physical parameters of materials, structural geometric dimensions andapplied loads are all interval, random-interval or unascertained variables. Secondly, thetransformational relation between the probabilistic reliability index andnon-probabilistic reliability index was established by applying the3σ principle, andthen the movement function reliability analysis model of deployment mechanism ofsatellite umbrella antenna was built and solved by optimization method. Then, modelswith reliability constraint are optimized based on the topological optimization methodwith level set. Finally, the stochastic temperature field is researched and the heatconduction distributions are analyzed based on topological optimization method. Themain research works can be described as follows:
1. The structural static analysis based on interval models.
By representing the uncertain parameters as interval numbers, the interval finiteelement governed equations of structural systems are built. To solve these equations, aninterval parameter perturbation method and an affine arithmetic with recursivederivative information are proposed, which can simply the interval computing anddecrease the excessive width of interval operations to a certain extent. Moreover, theclassical reliability theory and the interval non-probabilistic reliability theory are studied.By applying the3σ principle, the transformational relation between the probabilisticreliability index and non-probabilistic reliability index is established. The intervaluncertainty analytical model of deployment mechanism of satellite umbrella antenna isbuilt and solved by optimization method. The non-probabilistic reliability index isobtained from the performance function, and then is transformed to equivalentprobabilistic reliability index by the proposed method in this paper. The movementfunction reliability analysis result of rotation joint in umbrella antenna shows the3σprinciple approach is reasonable and effective.
2. Structural analysis based on unascertained theory.
The undetermined but objective information is considered, and the structuralphysical parameters and the applied loads are defined as undetermined variables in the finite element analysis model of the tethered balloon equipment bracket structures. Anda structure analysis method based on the undetermined factor method is proposed. Fromthe operation rules of undetermined information, the computational expressions of theequipment bracket structural displacement response and the element stress response areobtained. It is concluded that the confidence levels of structural responses are in directproportion to the confidence level of each parameter. And also, the dynamiccharacteristics analysis model of space beam-plate composite structures is built, inwhich structural elastic module and mass densities are all considered as unascertainedvariables. And a structure analysis method based on the unascertained factor method isgiven. Compared to the mature random method, the proposed method can obtainreliability result of safer and higher faith degree in the case of inadequate data or
insufficient information; moreover, it is simple and easy to apply.
3. Topology optimization of continuum structures based on the level set methodBased on the research of the topological optimization method with level set, thenon-probability reliability constraint of interval parameters structure is analyzed.Through deducing and transforming the finite element formula, the suppositional safetyfactor form with non-probability reliability information is proposed, by which thenon-probability reliability constraints are dealt with explicitly. This method is simple inform and quick in convergence speed, avoiding complicated iteration operations.Moreover, based on the stress-strength model for the fatigue reliability of structuralelements, the probability and non-probability fatigue reliability of elements are studiedand a new topological optimization method is presented. This method Combines withthe level set method for structural topology optimization, takes the minimized structuralflexibility as objective function, and chooses the volume and fatigue reliabilityrequirement as the constraint functions. The constraints of fatigue reliability areprocessed explicitly before optimization, which simplified the optimization course,
avoided complicated iteration operations and advanced the computing efficiency.
4. Uncertain temperature field analysis and topology optimization.Considering the randomness of physical parameters(such as heat conductioncoefficient) and initial boundary conditions(such as environmental temperature，heatexchange coefficient) and so on. The problems for static random temperature field,transient random temperature field, random thermal stress and correlative reliabilitieswere analyzed. In the thermal analysis of continuum structure, by treating thecoefficient of thermal conductivity, the intensity of internal heat source and theamplitude of distribution function for given boundary temperature as random parameters, the numeric characteristic of structural random temperature field is derived.The thermal reliability definition is proposed, which is that random temperaturevariables can not exceed their critical values. The topology optimization formulation ofstructure is built, in which the minimum mean value of heat dissipation is taken asobjective function, the given thermal reliability and structural volume ratio asconstraints. In the prophase of topology optimization, the probability constraint functionof thermal reliability is transformed into an explicit function, which simplifies thetopology model. The method of moving asymptotes is used to solve this topologyoptimization problem. Some examples indicate the validity and feasibility of theproposed method.
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