By Peter W. Christensen

ISBN-10: 1402086652

ISBN-13: 9781402086656

This publication has grown out of lectures and classes given at Linköping collage, Sweden, over a interval of 15 years. It supplies an introductory therapy of difficulties and strategies of structural optimization. the 3 uncomplicated periods of geometrical - timization difficulties of mechanical buildings, i. e. , dimension, form and topology op- mization, are taken care of. the point of interest is on concrete numerical answer equipment for d- crete and (?nite aspect) discretized linear elastic constructions. the fashion is specific and functional: mathematical proofs are supplied while arguments could be stored e- mentary yet are another way purely stated, whereas implementation information are often supplied. in addition, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented via consistently varying―frequently very many― variables, so-called ?rst order tools are crucial to the therapy. those equipment are in accordance with sensitivity research, i. e. , on setting up ?rst order derivatives for - jectives and constraints. The classical ?rst order equipment that we emphasize are CONLIN and MMA, that are in accordance with specific, convex and separable appro- mations. it may be remarked that the classical and regularly used so-called op- mality standards approach can also be of this sort. it could possibly even be famous during this context that 0 order tools corresponding to reaction floor equipment, surrogate types, neural n- works, genetic algorithms, and so on. , primarily observe to forms of difficulties than those taken care of right here and may be provided in other places.

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

**Sample text**

F is (strictly) concave if −f is (strictly) convex. The graph of a convex function thus lies on or below the straight line connecting any two points on this graph. For a strictly convex function, the graph lies strictly below the line, cf. Fig. 3. 38 3 Basics of Convex Programming Fig. 1 The function f : R → R, f (x1 ) = x12 is strictly convex, but f : R2 → R, f (x1 , x2 ) = x12 is only convex. The function f : R2 → R, f (x1 , x2 ) = x1 x2 is neither convex nor concave. By applying the definitions of convex sets and functions, one easily obtains the following lemma.

The truss is subjected to two forces P >0. The volume of the truss may not exceed the value V0 . The design variables are the cross-sectional areas of the bars: A1 , A2 and A3 . a) Formulate the problem as a mathematical programming problem. b) Solve the optimization problem by using Lagrangian duality. 7 The weight of the three-bar truss in Fig. 14 should be minimized given that the truss should be sufficiently stiff; the maximum nodal displacement Fig. 5 Fig. 5 Exercises 55 Fig. 7 Fig. 8 has to be lower than a prescribed value: max(|u1 |, |u2 |, |u3 |) ≤ u0 , where ui is the displacement vector of node i and u0 > 0 is a given scalar.

38 3 Basics of Convex Programming Fig. 1 The function f : R → R, f (x1 ) = x12 is strictly convex, but f : R2 → R, f (x1 , x2 ) = x12 is only convex. The function f : R2 → R, f (x1 , x2 ) = x1 x2 is neither convex nor concave. By applying the definitions of convex sets and functions, one easily obtains the following lemma. 1 (i) The set S = {x ∈ X : gi (x) ≤ 0, i = 1, . . , l} is convex if the functions gi : Rn → R, i = 1, . . , l are convex. (ii) Let S be a convex set. If f : S → R and g : S → R are convex and h : S → R is strictly convex, then αf is convex, where α ≥ 0 is an arbitrary scalar, f + g is convex and f + h is strictly convex.

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