Download e-book for kindle: An introduction to structural optimization (Solid Mechanics by Peter W. Christensen

By Peter W. Christensen

ISBN-10: 1402086652

ISBN-13: 9781402086656

This textbook offers an advent to all 3 sessions of geometry optimization difficulties of mechanical constructions: sizing, form and topology optimization. the fashion is specific and urban, concentrating on challenge formulations and numerical resolution equipment. The remedy is special adequate to permit readers to put in writing their very own implementations. at the book's homepage, courses will be downloaded that additional facilitate the educational of the fabric coated. The mathematical necessities are stored to a naked minimal, making the ebook compatible for undergraduate, or starting graduate, scholars of mechanical or structural engineering. working towards engineers operating with structural optimization software program might additionally reap the benefits of analyzing this publication.

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Extra info for An introduction to structural optimization (Solid Mechanics and Its Applications)

Example text

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 ).

T. x1 > 0, x2 > 0. For each feasible point (x¯1 , x¯2 ) we can find another feasible point (x¯¯ 1 , x¯¯ 2 ) with x¯¯ 1 > x¯1 and x¯¯ 2 > x¯2 such that δ(x¯¯ 1 , x¯¯ 2 ) < δ(x¯1 , x¯2 ), and consequently no minimum exists. t. x2 ≥ x2min > 0. x1 ≥ x1min > 0, Then, if x1min +x2min > W/C1 , no feasible point exists. Naturally, an optimum cannot exist when there are no feasible points. In general it is extremely computationally demanding to determine a global minimum. Instead, we will rest content with trying to obtain a local minimum.

As previously mentioned, local minima are also global minima for convex problems. 1). e. bounded and closed, a solution always exists (this is true for any continuous objective function, not necessarily convex). If the objective function is strictly convex and the feasible set is convex, there exists at most one solution. If, in addition, the feasible set is compact, there exists exactly one solution. For example, if the strictly convex function 1/x is minimized over the closed, but unbounded convex set x ≥ 1, no solution exists.

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

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