By Peter W. Christensen
This booklet 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 simple periods of geometrical - timization difficulties of mechanical constructions, i. e. , dimension, form and topology op- mization, are taken care of. the focal point 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 may be stored e- mentary yet are another way merely brought up, whereas implementation info are usually supplied. additionally, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented by way of constantly varying―frequently very many― variables, so-called ?rst order equipment are imperative to the remedy. those tools are in accordance with sensitivity research, i. e. , on constructing ?rst order derivatives for - jectives and constraints. The classical ?rst order equipment that we emphasize are CONLIN and MMA, that are in response to particular, 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 could actually even be famous during this context that 0 order equipment corresponding to reaction floor equipment, surrogate versions, neural n- works, genetic algorithms, and so forth. , basically observe to sorts of difficulties than those handled right here and will be provided somewhere else.
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Additional info for An Introduction to Structural Optimization
Using the plus sign instead, gives √ 2 ∗ x2 = . 4 Reverting to the original area variables A1 and A2 , cf. 40), the optimal solution is 1 F F 1+ √ , A∗2 = √ , A∗1 = 2σ0 2 2σ0 2 and the corresponding optimum weight is (3A∗1 + A∗2 )ρ0 L = F Lρ0 σ0 3 √ + 2 . 2 C ASE B ) ρ1 = ρ2 = ρ3 = ρ0 , σ1max = σ3max = 2σ0 , σ2max = σ0 . t. 5 Weight Minimization of a Three-Bar Truss Subject to Stress Constraints 27 Fig. 13 Case b). Point B is the solution see Fig. 13. It would appear that the solution is at the intersection A of the σ1 - and σ2 -constraints.
That is, there does not exist any λ2 ≥ 0 and λ3 ≥ 0 such that −∇g0 (x¯ 2 ) = λ2 ∇g2 (x¯ 2 ) + λ3 ∇g3 (x¯ 2 ), and consequently x¯ 2 is not a KKT point. 4 then implies that x¯ 2 is not an optimum. 5 We study the two-bar truss on page 14 again, but this time we will solve the optimization problem by calculating a KKT point instead. t. A1 ⎪ √ ⎪ ⎪ 3F ⎪ ⎪ ⎪ −σ ≤ − ≤ σ0 0 ⎪ ⎪ A ⎪ 2 ⎩ A1 ≥ 0, A2 ≥ 0. 2 The cone spanned by some vectors v 1 , . . , v l is the set of nonnegative linear combinations of these vectors.
1! Here, C is the point with x1 = 1/ 2 and x2 = 0. e. uT u ≤ δ02 . e. bar 3 is 10 times longer than bars 1 and 2. 33). Inserting β = 1/10 into these expressions we get u= FL EA1 (2A1 + 11A2 ) 2A1 + 10A2 10A2 , so that the stiffness constraint may be written as uT u = F 2 L2 (4A21 + 200A22 + 40A1 A2 ) E 2 A21 (2A1 + 11A2 )2 ≤ δ02 . The density of all bars is ρ0 , which gives the objective function W = ρ0 L(11A1 + A2 ). Dimensionless variables are introduced according to xi = Eδ0 Ai , FL i = 1, 2.
An Introduction to Structural Optimization by Peter W. Christensen