By Peter W. Christensen
This ebook has grown out of lectures and classes given at Linköping college, Sweden, over a interval of 15 years. It provides an introductory therapy of difficulties and strategies of structural optimization. the 3 simple 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 resolution equipment for d- crete and (?nite point) discretized linear elastic buildings. the fashion is particular and useful: mathematical proofs are supplied whilst arguments could be stored e- mentary yet are in a different way simply mentioned, 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 by means of continually varying―frequently very many― variables, so-called ?rst order equipment are crucial to the therapy. those tools are according to sensitivity research, i. e. , on setting up ?rst order derivatives for - jectives and constraints. The classical ?rst order tools that we emphasize are CONLIN and MMA, that are according to specific, convex and separable appro- mations. it may be remarked that the classical and regularly used so-called op- mality standards strategy can be of this type. it might even be famous during this context that 0 order equipment resembling reaction floor equipment, surrogate types, neural n- works, genetic algorithms, and so on. , basically practice to varieties of difficulties than those handled right here and may be awarded somewhere else.
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N}. e. the values xjmin = −∞ and xjmax = +∞, j = 1, . . , n, are allowed. Naturally, if all lower and upper bounds are infinite, there are in effect no box constraints. Of course, optimization problems may equally well be written as maximization problems instead. However, any maximization problem may be reformulated as a minimization problem by noting that max g0 (x) = − min(−g0 (x)). W. Christensen, A. V. e. a point that satisfies all the ¯ ≤ 0, i = 1, . . , l and x¯ ∈ X . Thus, the problem (P) consists of constraints gi (x) ¯ for all feasible points x¯ of (P).
Solve a min–max problem. It should be noted that the constraints in these optimizations are very simple: x ∈ X and λ ≥ 0, respectively. 4 Lagrangian Duality 47 in (P) we have the constraints gi (x) ≤ 0, i = 1, . . , l, that may be very complicated to deal with directly. The problem of maximizing ϕ is not only easy because of the simple constraints, but also because ϕ is always concave. If the problem minx∈X L(x, λ) has exactly one solution for a given λ (a sufficient condition for this is that g0 is strictly convex and X is compact), then ϕ is differentiable at λ, and it holds that ∂ϕ(λ) = gi (x ∗ (λ)), ∂λi i = 1, .
This serves to illustrate that one should never uncritically accept a solution obtained by performing structural optimization. 7 Exercises 33 Fig. 18 Point A is the solution of problem (SO)6nf possible to avoid an optimal solution with a very thin bar 2 if the minimization of the manufacturing cost had, somehow, been included in the optimization problem. 1 What happens if F < 0 in the example of Sect. 1? 2 If the length of the second bar in the example of Sect. 5 is changed, the optimum topology of the truss changes: the optimum area of the second bar is zero for l2 ≥ L given β = 1, ρi = ρ0 , and σimax = σ0 , i = 1, 2, 3.
An Introduction to Structural Optimization by Peter W. Christensen