By Peter W. Christensen
This e-book has grown out of lectures and classes given at Linköping college, Sweden, over a interval of 15 years. It provides an introductory remedy of difficulties and techniques of structural optimization. the 3 uncomplicated sessions of geometrical - timization difficulties of mechanical constructions, i. e. , dimension, form and topology op- mization, are taken care of. the point of interest is on concrete numerical answer tools for d- crete and (?nite point) discretized linear elastic buildings. the fashion is particular and useful: mathematical proofs are supplied while arguments might be saved e- mentary yet are differently in basic terms mentioned, whereas implementation information are usually supplied. furthermore, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented by way of continually varying―frequently very many― variables, so-called ?rst order equipment are crucial to the therapy. those tools are in response to sensitivity research, i. e. , on developing ?rst order derivatives for - jectives and constraints. The classical ?rst order equipment that we emphasize are CONLIN and MMA, that are in line with specific, convex and separable appro- mations. it may be remarked that the classical and often used so-called op- mality standards technique can be of this sort. it can even be famous during this context that 0 order equipment akin to reaction floor tools, surrogate types, neural n- works, genetic algorithms, and so forth. , basically observe to forms of difficulties than those taken care of the following and will be awarded in other places.
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Additional resources for An Introduction to Structural Optimization
6, the optimal solution x1∗ = 1, x2∗ = −1 of (P)3 . In general it is quite tedious to write ϕ as an explicit function of λ as done above. When solving small problems by hand it is usually more efficient to first assume that none of the box constraints is active. It may then turn out that the λ∗ that maximizes ϕ results in a x ∗ that does not satisfy the box constraints. 4 Lagrangian Duality 51 Fig. 9 The dual objective function edges of the box region X , and optimize again with these variables kept fixed.
15, we see that the σ1 - and σ2 -constraints are active at the solution. This point has already been calculated for case c) as A∗1 F = σ0 √ 4+ 2 , 14 Fig. 15 Case d). Point A is the solution A∗2 F = σ0 √ 6 2−4 , 14 30 2 Examples of Optimization of Discrete Parameter Systems which gives the optimal weight F Lρ0 σ0 √ 6+5 2 . 7 C ASE E ) ρ1 = ρ3 = ρ0 , ρ2 = 2ρ0 , σ1max = σ3max = 2σ0 , σ2max = σ0 . t. the constraints in (SO)5b nf , see Fig. 16. The solution point is point B, with the optimal truss lacking bar 2: A∗1 = F , 2σ0 with the optimal weight F Lρ0 .
As a much simpler alternative solution procedure for the two-dimensional optimization problem (SO)6nf at hand, we can simply produce finer and finer zoom plots similar to those in Fig. 18 and read off the solution. Since the optimum thickness of bar 2 is very small, it is interesting to investigate how much heavier the optimum structure would be if bar 2 were removed. With no bar 2, the stiffness constraint reads 1 − 1 ≤ 0, x12 whereas the objective function becomes 11x1 . Thus, with bar 2 removed, the optimal solution is x1∗ = 1 and the corresponding (scaled) weight is 11.
An Introduction to Structural Optimization by Peter W. Christensen