不确定性连续体结构的拓扑优化设计研究
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摘要
在实际工程结构系统中,由于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,考虑这些不确定性因素的情况下进行结构拓扑优化设计,被众多学者提出且越来越受到学术界及工程界的关注和重视。然而迄今所见到的关于连续体结构拓扑优化问题的研究,绝大多数属于确定性模型,即将连续体结构参数及其作用载荷等均视为确定值,没有考虑不确定性因素的影响。显然,确定性模型是无法反映出结构参数或作用载荷的不确定性对结构设计性能的影响。因此,开展不确定性连续体结构拓扑优化设计的研究具有重要的理论意义、学术价值和工程实际背景。本文对不确定性连续体结构的拓扑优化设计进行了研究,主要内容如下:
1、研究了几何参数和物理参数均为随机变量的平面连续体结构在结构总应变能约束下的拓扑优化设计问题。以结构总质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量,以结构总应变能概率可靠性指标为约束条件,构建了随机结构的拓扑优化设计数学模型。利用随机因子法和代数综合法,导出了应变能的均值和均方差的计算表达式,提出了相应的求解策略。
2、在考虑结构几何和物理参数均具有随机性的情况下,建立了以质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量的基于频率概率可靠性约束的平面应力薄板结构的动力特性拓扑优化设计模型。同理,研究了物理参数为随机变量的三维结构的频率拓扑优化设计。利用代数综合法,导出了随机参数结构的动力特性数字特征计算表达式,并采用渐进结构优化法进行求解。
3、考虑结构的物理参数和作用载荷均具有模糊性的平面连续体结构的拓扑优化设计问题,利用信息熵将模糊变量转换为等效的随机变量,构建了随机载荷作用下的随机参数连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,以单元应力可靠性为约束条件,并对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了单元应力数字特征的计算表达式,采用双方向渐进结构优化方法求解。
4、研究结构物理参数和作用载荷均为区间变量的平面连续体结构的拓扑优化设计问题。以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,单元应力(结构应变能)非概率可靠性为约束条件,构建了区间载荷作用下区间参数连续体结构的拓扑优化设计数学模型。基于区间因子法和区间运算规则,导出了单元应力(应变能)的均值和离差的计算表达式。采用双方向渐进结构优化法的求解策略。
5、考虑结构的物理参数和作用载荷具有随机性或区间不确定性,利用3σ准则实现随机变量和区间变量之间的相互转换。利用区间因子法和区间运算规则,导出区间参数结构节点位移、单元应力和单元应变能的均值和离差的计算表达式;利用随机因子法和代数综合法,导出随机参数结构静力响应的均值和均方差,进而解决具有随机-区间参数结构的静力分析问题。
6、根据上面的成果,研究混合参数平面连续体结构的静力拓扑优化问题。首先研究了位移可靠性约束下的模糊-区间参数连续体结构的拓扑优化问题。利用3σ准则和信息熵分别将区间变量和模糊变量近似转换为随机变量,构建了随机参数平面连续体结构的拓扑优化模型;利用代数综合法导出了位移的数字特征的计算表达式;通过单位虚载荷计算位移灵敏度然后采用双方向渐进结构优化方法求解。其次,研究了随机-区间结构的拓扑优化设计问题。利用3σ准则将区间变量近似转换为随机变量,建立了一个以质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量,以满足单元应力可靠性为约束条件的数学模型。
There are a large number of errors and uncertainties in practical engineering structures, which cause the physical parameters, geometrical parameters and loads to be uncertain. Therefore, topological optimization designs with these uncertainties are put forward by numerous scholars, and get more and more attentions in academic and engineering kingdom. However, most of topology optimization designs for continuum structures belong to determinate structural optimization design, that is, all the structural parameters are regarded as determinate without considering influence of uncertainties. Obviously, the deterministic models are unable to reflect the influence of uncertain structural parameters on design performances. Consequently, topology optimization designs of continuum structures with uncertainties have wide engineering background, important theoretical meanings and academic values. Studies on topology optimization design of continuum structures with uncertainty parameters are made in this dissertation. The main research work can be described as follows:
1. Topology optimization design problem for planar continuum structure with randomness of both geometry and physical parameters under the structural total strain energy constraint is studied. Mathematical model of topology optimization design of random structure is established, in which the minimum mean of structural weight is taken as objective function, the structural shape topology information is taken as design variables, and the probabilistic reliability index of structural total strain energy is taken as constraint condition. The computational expressions for mean value and mean variance of the structural strain energy response are presented by the random factor method and algebra method. The corresponding solving method is proposed.
2. Dynamic characteristic topology optimization of continuum structure with randomness of both geometry and physical parameters is researched based on natural frequency probability constraints. Mathematical model of topology optimization design about plain stress thin plate is established, in which the minimum mean of structural weight is taken as objective function, structural shape topology information is taken as design variables, and the probabilistic reliability index of structural natural frequency is taken as constraint condition. Similarly, topology optimization design problem for three dimensional structures with randomness of physical parameters under structural frequency constraints is studied. The numerical characteristics of structure with random parameters are deduced by the algebra method. Meanwhile, the evolutionary structural optimization method is used in the optimization.
3. Topology optimization designs of planar continuum structure with fuzzy parameters under fuzzy loads are discussed. Fuzzy variables are transformed into random variables by information entropy. Mathematical model of topology optimization of continuum structures with random parameters under random loads is established. The minimum mean of structural weight is taken as objective function; the structural shape topology information is taken as design variables; and the probabilistic reliability index of structural natural frequency is taken as constraint condition. The element stress constraints are transformed into equivalent normal constraints. The computational expressions of numerical characteristics of stress responses are presented based on random factor method. Bi-directional evolutionary structural optimization method is used in the optimization.
4. Topology optimization design of planar continuum structure under interval loads is discussed considering interval uncertainties of physical parameters. The topology optimization mathematical models with stress (structural strain energy) constraints of planar continuum structures under interval loads are established. The minimum mean of structural weight is taken as objective function, the structural shape topology information is taken as design variables, and the non-probabilistic reliability index is taken as constraint condition. Computational expressions of numerical characteristics of stress (strain energy) based on interval factor method and interval arithmetic rule are deduced. Bi-directional evolutionary structural optimization method is also used in the optimization.
5. In consideration of structural physical parameters and applied loads being random or interval, random variables and interval variables are transformed into each other by 3σcriteria. For interval parameters structures, the computational expressions of numerical characteristics of node displacement, element stress and strain energy responses are deduced based on the interval factor method. For random parameters structures, the computational expressions of numerical characteristics of element stress, node displacement and strain energy responses are presented based on the random factor method and interval arithmetic rules. Then the static analysis problem of random-interval parameters continuum structures is solved.
6. According to the above results, topology optimization design of planar continuum structure with mixed parameters is studied. Firstly, the optimization problem of fuzzy-interval parameters structures under displacement constraints is discussed. Interval and fuzzy variables are transformed respectively into random ones by the 3σcriteria and information entropy. Mathematical model of topology optimization of continuum structures with random parameters is established. The computational expressions of numerical characteristics of displacement responses are presented based on the algebra method. Displacement sensitivity is obtained by unit void load. Bi-directional evolutionary structural optimization method is used in the optimization. Secondly, the optimization problem of random-interval parameters structures under stress constraints is discussed. Interval variables are transformed into random ones by the 3σcriteria. The topology optimization mathematical model is established, taking the minimum mean of structural weight as objective function, the structural shape topology information as design variables, and the stress probabilistic reliability index as constraint condition.
1、研究了几何参数和物理参数均为随机变量的平面连续体结构在结构总应变能约束下的拓扑优化设计问题。以结构总质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量,以结构总应变能概率可靠性指标为约束条件,构建了随机结构的拓扑优化设计数学模型。利用随机因子法和代数综合法,导出了应变能的均值和均方差的计算表达式,提出了相应的求解策略。
2、在考虑结构几何和物理参数均具有随机性的情况下,建立了以质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量的基于频率概率可靠性约束的平面应力薄板结构的动力特性拓扑优化设计模型。同理,研究了物理参数为随机变量的三维结构的频率拓扑优化设计。利用代数综合法,导出了随机参数结构的动力特性数字特征计算表达式,并采用渐进结构优化法进行求解。
3、考虑结构的物理参数和作用载荷均具有模糊性的平面连续体结构的拓扑优化设计问题,利用信息熵将模糊变量转换为等效的随机变量,构建了随机载荷作用下的随机参数连续体结构的拓扑优化设计数学模型,以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,以单元应力可靠性为约束条件,并对应力可靠性约束进行了等价显式化处理。基于随机因子法,利用代数综合法导出了单元应力数字特征的计算表达式,采用双方向渐进结构优化方法求解。
4、研究结构物理参数和作用载荷均为区间变量的平面连续体结构的拓扑优化设计问题。以结构的形状拓扑信息为设计变量,结构总质量均值极小化为目标函数,单元应力(结构应变能)非概率可靠性为约束条件,构建了区间载荷作用下区间参数连续体结构的拓扑优化设计数学模型。基于区间因子法和区间运算规则,导出了单元应力(应变能)的均值和离差的计算表达式。采用双方向渐进结构优化法的求解策略。
5、考虑结构的物理参数和作用载荷具有随机性或区间不确定性,利用3σ准则实现随机变量和区间变量之间的相互转换。利用区间因子法和区间运算规则,导出区间参数结构节点位移、单元应力和单元应变能的均值和离差的计算表达式;利用随机因子法和代数综合法,导出随机参数结构静力响应的均值和均方差,进而解决具有随机-区间参数结构的静力分析问题。
6、根据上面的成果,研究混合参数平面连续体结构的静力拓扑优化问题。首先研究了位移可靠性约束下的模糊-区间参数连续体结构的拓扑优化问题。利用3σ准则和信息熵分别将区间变量和模糊变量近似转换为随机变量,构建了随机参数平面连续体结构的拓扑优化模型;利用代数综合法导出了位移的数字特征的计算表达式;通过单位虚载荷计算位移灵敏度然后采用双方向渐进结构优化方法求解。其次,研究了随机-区间结构的拓扑优化设计问题。利用3σ准则将区间变量近似转换为随机变量,建立了一个以质量均值极小化为目标函数,以结构的形状拓扑信息为设计变量,以满足单元应力可靠性为约束条件的数学模型。
There are a large number of errors and uncertainties in practical engineering structures, which cause the physical parameters, geometrical parameters and loads to be uncertain. Therefore, topological optimization designs with these uncertainties are put forward by numerous scholars, and get more and more attentions in academic and engineering kingdom. However, most of topology optimization designs for continuum structures belong to determinate structural optimization design, that is, all the structural parameters are regarded as determinate without considering influence of uncertainties. Obviously, the deterministic models are unable to reflect the influence of uncertain structural parameters on design performances. Consequently, topology optimization designs of continuum structures with uncertainties have wide engineering background, important theoretical meanings and academic values. Studies on topology optimization design of continuum structures with uncertainty parameters are made in this dissertation. The main research work can be described as follows:
1. Topology optimization design problem for planar continuum structure with randomness of both geometry and physical parameters under the structural total strain energy constraint is studied. Mathematical model of topology optimization design of random structure is established, in which the minimum mean of structural weight is taken as objective function, the structural shape topology information is taken as design variables, and the probabilistic reliability index of structural total strain energy is taken as constraint condition. The computational expressions for mean value and mean variance of the structural strain energy response are presented by the random factor method and algebra method. The corresponding solving method is proposed.
2. Dynamic characteristic topology optimization of continuum structure with randomness of both geometry and physical parameters is researched based on natural frequency probability constraints. Mathematical model of topology optimization design about plain stress thin plate is established, in which the minimum mean of structural weight is taken as objective function, structural shape topology information is taken as design variables, and the probabilistic reliability index of structural natural frequency is taken as constraint condition. Similarly, topology optimization design problem for three dimensional structures with randomness of physical parameters under structural frequency constraints is studied. The numerical characteristics of structure with random parameters are deduced by the algebra method. Meanwhile, the evolutionary structural optimization method is used in the optimization.
3. Topology optimization designs of planar continuum structure with fuzzy parameters under fuzzy loads are discussed. Fuzzy variables are transformed into random variables by information entropy. Mathematical model of topology optimization of continuum structures with random parameters under random loads is established. The minimum mean of structural weight is taken as objective function; the structural shape topology information is taken as design variables; and the probabilistic reliability index of structural natural frequency is taken as constraint condition. The element stress constraints are transformed into equivalent normal constraints. The computational expressions of numerical characteristics of stress responses are presented based on random factor method. Bi-directional evolutionary structural optimization method is used in the optimization.
4. Topology optimization design of planar continuum structure under interval loads is discussed considering interval uncertainties of physical parameters. The topology optimization mathematical models with stress (structural strain energy) constraints of planar continuum structures under interval loads are established. The minimum mean of structural weight is taken as objective function, the structural shape topology information is taken as design variables, and the non-probabilistic reliability index is taken as constraint condition. Computational expressions of numerical characteristics of stress (strain energy) based on interval factor method and interval arithmetic rule are deduced. Bi-directional evolutionary structural optimization method is also used in the optimization.
5. In consideration of structural physical parameters and applied loads being random or interval, random variables and interval variables are transformed into each other by 3σcriteria. For interval parameters structures, the computational expressions of numerical characteristics of node displacement, element stress and strain energy responses are deduced based on the interval factor method. For random parameters structures, the computational expressions of numerical characteristics of element stress, node displacement and strain energy responses are presented based on the random factor method and interval arithmetic rules. Then the static analysis problem of random-interval parameters continuum structures is solved.
6. According to the above results, topology optimization design of planar continuum structure with mixed parameters is studied. Firstly, the optimization problem of fuzzy-interval parameters structures under displacement constraints is discussed. Interval and fuzzy variables are transformed respectively into random ones by the 3σcriteria and information entropy. Mathematical model of topology optimization of continuum structures with random parameters is established. The computational expressions of numerical characteristics of displacement responses are presented based on the algebra method. Displacement sensitivity is obtained by unit void load. Bi-directional evolutionary structural optimization method is used in the optimization. Secondly, the optimization problem of random-interval parameters structures under stress constraints is discussed. Interval variables are transformed into random ones by the 3σcriteria. The topology optimization mathematical model is established, taking the minimum mean of structural weight as objective function, the structural shape topology information as design variables, and the stress probabilistic reliability index as constraint condition.
引文
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