By Peter W. Christensen
This textbook provides an creation 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 resolution tools. The therapy is precise adequate to permit readers to write down their very own implementations. at the book's homepage, courses could be downloaded that extra facilitate the training of the fabric coated. The mathematical must haves are stored to a naked minimal, making the publication 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 e-book.
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Additional resources for An introduction to structural optimization (Solid Mechanics and Its Applications)
N. e. that it may be written as A = LLT , where L is a nonsingular lower triangular matrix. 3 Consider the function f : R2 → R, f (x1 , x2 ) = x12 + x22 . Then ∇f (x1 , x2 ) = 2x1 , 2x2 ∇ 2 f (x1 , x2 ) = 2 0 . 0 2 The Hessian is positive definite by Sylvester’s criterion since 2 > 0 and 2 · 2 − 0 · 0 > 0. 2(ii) implies that f is strictly convex on R2 . 4 The function f : R → R, f (x) = x 4 has the Hessian ∇ 2 f (x) = 12x 2 > 0, x = 0. 2(i), we conclude that f is (at least) convex. 2(ii), we cannot conclude that f is strictly convex.
This serves to illustrate that one should never uncritically accept a solution obtained by performing structural optimization. 7 Exercises 33 Fig. 18 Point A is the solution of problem (SO)6nf possible to avoid an optimal solution with a very thin bar 2 if the minimization of the manufacturing cost had, somehow, been included in the optimization problem. 1 What happens if F < 0 in the example of Sect. 1? 2 If the length of the second bar in the example of Sect. 5 is changed, the optimum topology of the truss changes: the optimum area of the second bar is zero for l2 ≥ L given β = 1, ρi = ρ0 , and σimax = σ0 , i = 1, 2, 3.
2 Convexity 39 Here, a function g : S → Rn is monotone on S if for all x 1 , x 2 ∈ S with x 1 = x 2 it holds that (x 2 − x 1 )T (g(x 2 ) − g(x 1 )) ≥ 0. Similarly, g is strictly monotone on S if strict inequality holds here. This definition is a generalization of the concept of a monotonically increasing function of one variable: g is monotonically increasing if x2 > x1 implies that g(x2 ) ≥ g(x1 ). 2 The function f : R → R, f (x) = x 4 , is strictly convex on R since ∇f (x) = 4x 3 is strictly monotone on R: (x2 − x1 )(x23 − x13 ) = (x2 − x1 )2 (x12 + x1 x2 + x22 ) = (x2 − x1 )2 1 x1 + x2 2 2 3 + x22 > 0, 4 x1 = x2 .