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Evaluation of Formulations
Table 1 lists the sizes of the three formulations, considering only one loading condition. Note that the CF has the least numbers of variables and constraints. It is obvious that the alternative formulations increase the size of the optimization problem substantially. A major disadvantage of CF is that the equilibrium equations in terms of the displacements must be solved in each iteration. The same process must also be repeated during step size calculation along the search direction. Another disadvantage is that the gradient evaluation requires solution of linear equations which must be done by restarting the analysis program. The sensitivity load vectors must be assembled. This is cumbersome making the implementation of the process more tedious depending on the facilities available in the analysis program. The implementation can be facilitated if the analysis software allows call to the element level calculation directly. Also, the Jacobian and Hessian matrices of the constraints are dense in this formulation, limiting the size of the design problem that can be treated efficiently.Advantages and disadvantages of the two alternative formulations are summarized in Table 2. AF1 is more suitable for problems that do not involve stress constraints. In that case, only the equilibrium constraints in Eq. (19) are included as behavior constraints and the displacement constraints are simple bounds on the variables. In AF2 with forces also as variables, the optimization problem is larger with more variables and constraints. However, the assembly of the global stiffness matrix and its derivatives are avoided, which is the major difference between AF2 and the previous two formulations. Only member-level calculations are required in the constraints in Eqs. (26) and (27). Note that the equilibrium constraints in Eq. (25) are linear and the stress constraints in Eq. (27) are simpler. The gradients of the linear constraints in Eq. (25) are programmed independently and calculated only once in the solution process. Obviously it is not necessary to include member nodal forces as variables in the formulations to obtain explicit expressions for the constraints; however, their inclusion simplifies the form of constraints and Jacobians, and hence the numerical implementations. It is also noted that the alternative formulations require study of the sparsity structure of the Jacobian matrices so as to take full advantage of the sparse matrix operations in SNOPT.
中文译文结构优化的可选公式:对现行的框架结构进行评估
摘要: 基于同时分析和设计的观念，以及为制定结构最适的设计，从而提出二个可选公式,并对结构进行评估。不同的结构变量, 譬如结点位移和杆件强度, 除设计变量的问题之外，我们还应找出最优变量。随着这些公式的运用,平衡等式成为在最优过程中的重要平等约束。客观条件和所有约束条件成为最优变量明确的前提。所以,他们的出处可能很容易的获得并与常规方法进行比较，而特别的灵敏度设计分析规程一定是被用来计算其来源的。由于灵敏度等式还没被完全形成或解决，所以还可以更加容易地使用现有的分析软件以及可选择的公式进行优化。利用连续二次编程的方法来解决一些例题和用来评估公式。用现存的分析程序对可选公式的运用进行解释。并且对公式的利弊进行探讨。我们可以总结出对可选公式的运用得当，能更好的优化框架结构，并有着很好的发展潜力。
CE数据库科目标题:仿真;最优化;制定结构;电脑编程;结构介绍：
为了对支架进行优选设计，2005年由Wang 和Arora提出了三种建立在同时分析和设计(由SAND提出)概念上的可选公式已经开始运用。而作为最优变量，公式中又包括结点的位移，轴力以及所受的压力。在公式中引入更多的变量会改变其特有的形式和他们的原有的形式。根据最优变量,所有公式的作用变得明确。所以,设计灵敏度分析方法是必须以适应结构分析规程且以结构优化为目的的, 不再是当需要时才适应。所有公式运用的相当好将能非常准确的得到最佳解决方案。所得例题的解答会与那些用常规公式获得的解答进行比较。我们会得知在多数情况下可选公式比常规公式的效率要高。并且与用那些常规公式比较起来,用现有的分析软件进行实施显得更加容易。