Read e-book online An introduction to structural optimization (Solid Mechanics PDF

By Peter W. Christensen

ISBN-10: 1402086652

ISBN-13: 9781402086656

This textbook offers an advent to all 3 periods of geometry optimization difficulties of mechanical buildings: sizing, form and topology optimization. the fashion is specific and urban, concentrating on challenge formulations and numerical answer tools. The remedy is exact adequate to permit readers to write down their very own implementations. at the book's homepage, courses might be downloaded that additional facilitate the educational of the fabric lined. The mathematical must haves are saved to a naked minimal, making the publication compatible for undergraduate, or starting graduate, scholars of mechanical or structural engineering. training engineers operating with structural optimization software program may additionally take advantage of analyzing this ebook.

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Additional resources for An introduction to structural optimization (Solid Mechanics and Its Applications)

Example text

6 Let (P) be a convex problem with the set X compact, satisfying Slater’s CQ. Then there exist a λ∗ that solves (D), and an x ∗ ∈ argminx∈X L(x, λ∗ ) that solves (P), where g0 (x ∗ ) = ϕ(λ∗ ). e. 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.

5 Weight Minimization of a Three-Bar Truss Subject to Stress Constraints 27 Fig. 13 Case b). Point B is the solution see Fig. 13. It would appear that the solution is at the intersection A of the σ1 - and σ2 -constraints. However, we must keep in mind that the σ2 -constraint is valid only for x2 > 0. By deleting this constraint, it is evident from the figure, that the point B on the σ1 -constraint curve, where x2 is zero, gives the lowest weight that can be attained. This point is obtained by letting x2 = 0 in the active σ1 -constraint: 2x1 − 4x12 = 0, which gives x1∗ = 1/2 as x1∗ = 0 is not a valid design.

10) directly. In this section, we will therefore describe another method to obtain an optimal solution that will prove more suitable, especially for large-scale structural optimization problems. It may be proven that (P) is equivalent to the following min-max problem: l (PL ) min max L(x, λ) = min max g0 (x) + x∈X λ≥0 x∈X λ≥0 λi gi (x) . i=1 Thus, first the Lagrangian L of (P) is maximized with respect to λ ≥ 0 for a fixed x, and the result is then minimized with respect to x ∈ X . Note that the result of the maximization will be +∞ if some gi (x) > 0, and g0 (x) if all gi (x) ≤ 0, i = 1, .

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An introduction to structural optimization (Solid Mechanics and Its Applications) by Peter W. Christensen


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