By Peter W. Christensen
This textbook supplies an advent to all 3 sessions 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 equipment. The remedy is certain adequate to allow readers to jot down their very own implementations. at the book's homepage, courses could be downloaded that extra facilitate the training of the cloth coated. The mathematical necessities are saved to a naked minimal, making the publication compatible for undergraduate, or starting graduate, scholars of mechanical or structural engineering. working towards engineers operating with structural optimization software program might additionally make the most of studying this e-book.
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Extra info for An introduction to structural optimization (Solid Mechanics and Its Applications)
If the same function is minimized over the compact set [1, 2], the solution is x ∗ = 1/2. Note that the convexity of the feasible set plays a crucial role here; for example if the strictly convex function x12 + x22 is minimized over the nonconvex, compact set 1 ≤ x12 + x22 ≤ 2, there is an infinite number of global minima, namely all points (x1∗ , x2∗ ) with (x1∗ )2 + (x2∗ )2 = 1. In order to determine whether a continuously differentiable function is convex, we may study its gradient. 1 Let f : S → R, where S is convex and f is continuously differentiable.
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.
A) Formulate the problem as a mathematical programming problem. b) Change variables to nondimensional ones as xi = P /(EAi ), i = 1, 2, and solve the optimization problem by using the KKT conditions. c) Same as b), but solve the optimization problem by using Lagrangian duality instead. 4 The three-bar truss in Fig. 11 is subjected to the force P > 0. One wants to maximize the stiffness of the truss by minimizing its compliance P uy , where uy is the displacement in the y-direction of the node where P is applied.