New PDF release: An Introduction to Structural Optimization

By Peter W. Christensen

ISBN-10: 1402086652

ISBN-13: 9781402086656

ISBN-10: 1402086660

ISBN-13: 9781402086663

Mechanical and structural engineers have continuously strived to make as effective use of fabric as attainable, e.g. via making constructions as mild as attainable but capable of hold the masses subjected to them. long ago, the hunt for extra effective buildings was once a trial-and-error method. although, within the final 20 years computational instruments according to optimization thought were constructed that give the chance to discover optimum buildings roughly immediately. end result of the excessive expense reductions and function earnings that could be accomplished, such instruments are discovering expanding business use.
This textbook offers an advent to all 3 periods of geometry optimization difficulties of mechanical buildings: sizing, form and topology optimization. the fashion is particular and urban, targeting challenge formulations and numerical answer tools. The remedy is special adequate to permit readers to jot down their very own implementations. at the book's homepage, courses can be downloaded that extra facilitate the training of the fabric covered.

The mathematical must haves are stored to a naked minimal, making the ebook appropriate for undergraduate, or starting graduate, scholars of mechanical or structural engineering. practising engineers operating with structural optimization software program could additionally take advantage of interpreting this book.

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The KKT conditions state that −∇g0 (x) ¯ At point x¯ 1 cone spanned2 by the gradients of the active constraints at a point x. in the figure, −∇g0 (x¯ 1 ) = λ1 ∇g1 (x¯ 1 ) + λ2 ∇g2 (x¯ 1 ), λ1 ≥ 0, λ2 ≥ 0, so x¯ 1 is a KKT point and consequently the optimal solution. Regarding point x¯ 2 , we see that −∇g0 (x¯ 2 ) does not belong to the cone spanned by the active constraints at x¯ 2 . That is, there does not exist any λ2 ≥ 0 and λ3 ≥ 0 such that −∇g0 (x¯ 2 ) = λ2 ∇g2 (x¯ 2 ) + λ3 ∇g3 (x¯ 2 ), and consequently x¯ 2 is not a KKT point.

5 Young’s modulus is E for both bars. The volume of the truss is not allowed to exceed the value V0 . The total length of the bars is h, and bar 1 has length αh, where α is a scalar between αmin and αmax . The cross-sectional areas of the bars are A1 = A and A2 = βA, where β is a scalar. The design variables are α and β. Since α determines the “shape” of the truss, and β the cross-sectional area of bar 2, the problem to be solved is a combined shape and sizing optimization problem. t. ⎪ Ah ⎩ β ≥ 0.

T. the constraints in (SO)5b , nf which is illustrated in Fig. 14. It is not evident from the figure whether the solution is at the intersection A between the σ1 - and σ2 -constraints, or the point B corresponding to a design without bar 2. Point A may be calculated by solving the two equations obtained when equality is satisfied in the σ1 - and σ2 -constraints, which leads to √ √ 4+ 2 6 2−4 ∗ ∗ , x2 = . x1 = 14 14 Point B is x1∗∗ = 1/2, x2∗∗ = 0. It turns out that these two points yield the same value of the objective function, and thus, there are two solutions to this problem!

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An Introduction to Structural Optimization by Peter W. Christensen


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