基于有限元概念的解析解
The Analytical Solutions Based on the Concept of Finite Element Methods
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摘要: 基于有限元概念,本文研究了结构响应对于设计变量的严格解析解,以空间桁架为例,得到了位移显式的解析表达式——一个关于截面设计变量的有理函数,并进行了严格的数学论证;进而推出了有益于结构敏度分析、结构优化的若干结论,针对含有位移显式的优化模型,提出了采用广义几何规划解法的建议;最后计算了若干简例.Abstract: On the basis of the concept of finite element methods, the rigorous analytical solutions of structural response in terms of the design variables are researched in this paper. The spatial trusses are taken as an example for the solution of the analytical expressions of the explicit displacements which are proved mathematically; then some conclusions are reached that are useful to structural sensitivity analysis and optimization. In the third part of the paper, a generalized geometric programming method is sugguested for the optimal model with the explicit displacement. Finally, the analytical solutions of the displacements of three trusses are given as examples.
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[1] Schmit,L.A,and H.Miura,Approximation concepts for efficient structural synthesis.NASA,CR-2556(1976). [2] 钱令希,《工程结构优化设计》,水利电力出版社(1883). [3] 谢邦杰,《线性代数》,人民教育出版社(1978). [4] 隋允康,函数变换下的满应力设计方法。应用力学学报,2 (1987). [5] 隋允康、耿树森,Taylor展式的推广及其在求解数学规划中的应用,工程数学学报,1 (1985). [6] 隋允康、耿树森,广义几何规划二阶缩并的原算法及其在结构优化中的应用,计算结构力学及其应用,1 (1985). [7] 隋允康、由衷.广义几何规划的完全二阶原算法,计算结构力学及其应用,4 (1988). -
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