By Peter W. Christensen

ISBN-10: 1402086652

ISBN-13: 9781402086656

This booklet has grown out of lectures and classes given at Linköping collage, Sweden, over a interval of 15 years. It provides an introductory therapy of difficulties and techniques of structural optimization. the 3 uncomplicated sessions of geometrical - timization difficulties of mechanical constructions, i. e. , measurement, 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 constructions. the fashion is particular and functional: mathematical proofs are supplied whilst arguments should be stored e- mentary yet are differently merely stated, whereas implementation information are usually supplied. in addition, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented via regularly varying―frequently very many― variables, so-called ?rst order equipment are important to the remedy. those tools are in keeping with sensitivity research, i. e. , on developing ?rst order derivatives for - jectives and constraints. The classical ?rst order tools that we emphasize are CONLIN and MMA, that are in accordance with particular, convex and separable appro- mations. it's going to be remarked that the classical and often used so-called op- mality standards process is additionally of this sort. it could possibly even be famous during this context that 0 order equipment comparable to reaction floor tools, surrogate versions, neural n- works, genetic algorithms, and so forth. , basically observe to types of difficulties than those taken care of right here and will be offered in other places.

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**Extra info for An Introduction to Structural Optimization**

**Example text**

14) are satisfied, so that L has really been minimized for x ∗ . Since we have also maximized ϕ, the x ∗ obtained is the optimum solution of (P)3 . 7 We will solve the two-bar truss problem on page 14 once again, this time using Lagrangian duality. From the primal problem (P)2 on page 45, the La- 52 3 Basics of Convex Programming grangian function becomes 2 8 3 σ0 . 21) We differentiate L and assume that the box constraints are not active: ∂L 2 8 = √ − √ 2 λ = 0, ∂A1 3 3A1 3 ∂L = 1 − 2 λ = 0, ∂A2 A2 which gives √ A1 = 2 λ, A2 = √ 3λ.

Similarly, √ 3F 8 3 − σ0 < F √ + g5 = A2 3A1 A2 − σ0 = g1 ≤ 0, − σ0 = g1 ≤ 0, so λ5 = 0 as well. Thus, only λ1 may be nonzero. 11) 3 + λ1 ⎢ ⎣ 3 ⎦ 0 1 −√ 2 3A2 8F 3F λ1 √ + − σ0 = 0. 11) we get λ1 = A22 = 0. 3 Insertion of this into the first row gives 8A2 2 √ − √ 2 2 = 0, 3 3 3A1 from which we get √ 3 A1 . 12), we finally get A2 = 14F A∗1 = √ 3σ0 and A∗2 = 7F . σ0 Since the problem is convex, cf. 1, we know that this KKT point is also the global minimum of (P)1 , see Fig. 6. e. they were written as g2 ≤ 0, .

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. t. ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3F ⎪ ⎪ ⎪ ⎪ g4 = − − σ0 ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ A2 ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ 3F ⎪ ⎪ ⎪ ⎪ ⎪ g5 = − σ0 ≤ 0 ⎪ ⎪ ⎪ ⎪ ⎪ A2 ⎪ ⎪ ⎩ ⎩ (A1 , A2 ) ∈ X = {(A1 , A2 ) : A1 ≥ 0, A2 ≥ 0}, where we, for convenience, have skipped the arguments in gi (A1 , A2 ), i = 1, . . , 5. Also, the term ρL has been dropped from the objective function since it is only a positive constant, and, hence, does not affect the optimal solution (A∗1 , A∗2 ).

### An Introduction to Structural Optimization by Peter W. Christensen

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