By Peter W. Christensen
This ebook has grown out of lectures and classes given at Linköping collage, Sweden, over a interval of 15 years. It offers an introductory remedy of difficulties and techniques of structural optimization. the 3 simple periods of geometrical - timization difficulties of mechanical buildings, i. e. , measurement, form and topology op- mization, are taken care of. the focal point is on concrete numerical resolution equipment for d- crete and (?nite aspect) discretized linear elastic buildings. the fashion is particular and useful: mathematical proofs are supplied whilst arguments will be saved e- mentary yet are in a different way purely pointed out, whereas implementation info are often supplied. in addition, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented via continually varying―frequently very many― variables, so-called ?rst order tools are significant to the therapy. those equipment are in keeping with sensitivity research, i. e. , on constructing ?rst order derivatives for - jectives and constraints. The classical ?rst order tools 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 regularly used so-called op- mality standards approach is usually of this sort. it might even be famous during this context that 0 order equipment similar to reaction floor equipment, surrogate types, neural n- works, genetic algorithms, and so forth. , basically observe to kinds of difficulties than those taken care of the following and will be provided in different places.
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Additional info for An Introduction to Structural Optimization
A point x ∗ is a local minimum if the objective function g0 only assumes greater or equal values in a surrounding of x ∗ , but may very well assume smaller values elsewhere. Naturally, any global minimum is also a local minimum. e. points for which the gradient of g0 is zero: ⎡ ∂g (x ∗ ) ⎤ 0 ⎢ ∂x1 ⎥ ⎢ ⎥ ∗ .. ⎥ = 0. ∇g0 (x ) = ⎢ ⎢ ⎥ . ⎣ ∂g (x ∗ ) ⎦ 0 ∂xn A stationary point need not be a local minimum, however; it may equally well be a local maximum. For constrained problems, local minima are not even necessarily located at stationary points, as they may be located on the boundary of the feasible set.
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 .
Similarly, if λi = 0, then gi is active: gi (x) = 0. 10) is called a KKT point. For sufficiently regular nonconvex problems, the KKT conditions are necessary, but not sufficient, optimality conditions for (P). That is, local optima are always found among the KKT points, but there may be KKT points that are not local optima. The fact that the KKT conditions cannot be sufficient for optimality is evident by studying the special case of an unconstrained optimization problem, where the KKT points are equivalent to stationary points.
An Introduction to Structural Optimization by Peter W. Christensen