By Peter W. Christensen
This textbook supplies an creation 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 equipment. The therapy is distinctive adequate to permit readers to write down their very own implementations. at the book's homepage, courses might be downloaded that additional facilitate the training of the cloth lined. The mathematical must haves are stored to a naked minimal, making the booklet compatible for undergraduate, or starting graduate, scholars of mechanical or structural engineering. working towards engineers operating with structural optimization software program may additionally take advantage of studying this booklet.
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Additional info for An introduction to structural optimization (Solid Mechanics and Its Applications)
26) where ρ1 , ρ2 and ρ3 are the densities of the bars. The design constraints are A1 ≥ 0, A2 ≥ 0. 27) Concerning designs with A1 or A2 equal to zero, it is clearly impossible to have A1 = A3 = 0 since then there is no equilibrium possible as it would imply collapse of the structure under the given external load. On the other hand, A2 = 0 is a valid design. e. |σi | ≤ σimax , i = 1, 2, 3. 28) The equilibrium equation is found by cutting out the free node as shown in Fig. 11. The equilibrium equations in the x- and y-directions become s2 −s1 − √ + F = 0, 2 s2 s3 + √ = 0.
Then f is (strictly) convex if and only if the gradient ∇f is (strictly) monotone. 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 .
Instead, we will rest content with trying to obtain a local minimum. A point x ∗ is a local minimum if the objective function g0 only assumes greater or equal values in a surrounding of x ∗ , but may very well assume smaller values elsewhere. Naturally, any global minimum is also a local minimum. e. points for which the gradient of g0 is zero: ⎡ ∂g (x ∗ ) ⎤ 0 ⎢ ∂x1 ⎥ ⎢ ⎥ ∗ .. ⎥ = 0. ∇g0 (x ) = ⎢ ⎢ ⎥ . ⎣ ∂g (x ∗ ) ⎦ 0 ∂xn A stationary point need not be a local minimum, however; it may equally well be a local maximum.
An introduction to structural optimization (Solid Mechanics and Its Applications) by Peter W. Christensen