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 supplies an introductory remedy of difficulties and techniques of structural optimization. the 3 easy sessions 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 tools for d- crete and (?nite point) discretized linear elastic constructions. the fashion is specific and sensible: mathematical proofs are supplied whilst arguments should be stored e- mentary yet are differently merely pointed out, whereas implementation information are usually supplied. additionally, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented by means of consistently 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 equipment 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 often used so-called op- mality standards approach is additionally of this type. it may 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 on. , basically follow to forms of difficulties than those handled right here and will be provided somewhere else.
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Extra resources for An Introduction to Structural Optimization
We illustrate the method for the problem at hand. If the box constraints are not active, L1 and L2 are minimized when ∂L1 /∂x1 = 0 and ∂L2 /∂x2 = 0. 15) gives x1∗ = 3 − λ 2 λ x2∗ = −1 − . 18) gives the dual objective function ϕ(λ) = − λ2 λ + , 2 2 which is maximized for λ∗ = 12 . 20) we then get x1∗ = 11/4 and x2∗ = −5/4. This cannot be the actual solution, however, since x1∗ 1. We therefore put x1∗ = 1 and try to find the optimum x2∗ . Again, ∂L2 /∂x2 = 0 gives x2∗ = −1 − λ/2. The dual objective function is ϕ(λ) = − λ2 3 − λ + 4, 4 2 which is maximized for λ∗ = 0.
1 A function with several local optima Fig. 2 A convex (left) and a nonconvex (right) set [x3 , x4 ] and x5 are stationary points, x1 , (x3 , x4 ) and x5 are local minima, [x3 , x4 ] and x6 are local maxima. Point x2 is neither a local minimum nor a local maximum. Point x1 is the global minimum and x6 is the global maximum. Let us return to the three-bar truss on page 21 and see if there are any local optima that are not global optima for the four different cases studied. In Fig. 12, point A is the unique global minimum.
In Fig. 13, B is the unique global minimum and A is a local minimum. In Fig. 14, both A and B are global minima. In Fig. 15, A is the unique global minimum and B is a local minimum. Finally, in Fig. 16, B is the unique global minimum, and all points on the line from A to C, not including the end point C itself, are local minima with identical objective function values. Although for general problems local minima are not necessarily global minima, there is an important class of problems for which they are: convex problems.