New PDF release: An Introduction to Structural Optimization
By Peter W. Christensen
This publication has grown out of lectures and classes given at Linköping collage, Sweden, over a interval of 15 years. It provides an introductory remedy of difficulties and strategies of structural optimization. the 3 easy periods of geometrical - timization difficulties of mechanical buildings, i. e. , measurement, form and topology op- mization, are handled. the focal point is on concrete numerical answer tools for d- crete and (?nite point) discretized linear elastic constructions. the fashion is specific and sensible: mathematical proofs are supplied whilst arguments may be stored e- mentary yet are in a different way merely pointed out, 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 constantly varying―frequently very many― variables, so-called ?rst order tools are primary to the therapy. those tools are in line 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 specific, convex and separable appro- mations. it may be remarked that the classical and often used so-called op- mality standards approach can also be of this type. it could actually even be famous during this context that 0 order tools akin to reaction floor equipment, surrogate versions, neural n- works, genetic algorithms, and so forth. , basically follow to sorts of difficulties than those taken care of the following and will be provided in different places.
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Extra info for An Introduction to Structural Optimization
E. also a global, minimum of a convex optimization problem. 3) 42 3 Basics of Convex Programming where λi , i = 1, . . , l are called Lagrange multipliers. 10) for all j = 1, . . , n and i = 1, . . , l. Partial differentiation of L with respect to the design variables gives ∂L(x, λ) ∂g0 (x) = + ∂xj ∂xj l λi i=1 ∂gi (x) . ∂xj In most texts, box constraints are not treated separately, but are instead included in gi (x) ≤ 0, i = 1, . . , l, by writing xj − xjmax ≤ 0 and xjmin − xj ≤ 0, j = 1, .
We begin by reviewing some fundamental results of mathematical programming, with focus on convex programming. Actually, most problems of structural optimization are in fact nonconvex, but this does not imply that convex programming is of little importance in structural optimization: we will see in Chap. 4 that convex approximations play a very important role in the solution algorithms for nonconvex problems. All theorems are presented without proofs; these may be found in any good book on nonlinear mathematical programming such as Bazaraa, Sherali and Shetty  or Bertsekas .
Min j =1 xj ≤xj ≤xjmax Thus, in order to minimize the Lagrangian for all x ∈ X , we need only to perform n box constrained minimizations of functions of a single variable. This is the reason why Lagrangian duality is so attractive for convex, separable problems. 14) ∂Lj (xj , λ) = 0, ∂xj cf. Fig. 7. Since xj → Lj (xj , λ) is strictly convex, there is a unique solution to this minimization problem. 14) for each j , 1 ≤ j ≤ n, we may obtain the dual objective function. As usual, the dual problem is solved by maximizing the dual objective function for λ ≥ 0.
An Introduction to Structural Optimization by Peter W. Christensen