By Peter W. Christensen
Mechanical and structural engineers have constantly strived to make as effective use of fabric as attainable, e.g. via making constructions as gentle as attainable but capable of hold the hundreds subjected to them. long ago, the hunt for extra effective constructions was once a trial-and-error procedure. despite the fact that, within the final 20 years computational instruments in response to optimization concept were constructed that give the chance to discover optimum constructions roughly instantly. as a result of excessive price reductions and function profits that could be accomplished, such instruments are discovering expanding commercial use.
This textbook provides an creation to all 3 sessions 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 remedy is distinctive adequate to let readers to jot down their very own implementations. at the book's homepage, courses should be downloaded that additional facilitate the training of the cloth covered.
The mathematical necessities are saved to a naked minimal, making the e-book compatible for undergraduate, or starting graduate, scholars of mechanical or structural engineering. working towards engineers operating with structural optimization software program could additionally reap the benefits of interpreting this book.
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Additional info for An Introduction to Structural Optimization
The problem is illustrated in Fig. 8. 5λ. L1 (x1 ,λ) L2 (x2 ,λ) Differentiation gives ∂L1 = 2x1 + λ − 6, ∂x1 ∂L2 = 2x2 + λ + 2. 4 Lagrangian Duality 49 Fig. 14), we find the x, denoted x ∗ , that minimizes L for any given λ ≥ 0. 16) λ ∴ x1∗ = 3 − , if 4 ≤ λ ≤ 6 2 ∴ x2∗ = −2, if λ ≥ 2 never satisfied since λ ≥ 0 λ ∴ x2∗ = −1 − , if 0 ≤ λ ≤ 2. 17) 50 3 Basics of Convex Programming Fig. 18) if 0 ≤ λ ≤ 2 if 2 ≤ λ ≤ 4 if 4 ≤ λ ≤ 6 if λ ≥ 6. Note that ϕ is continuously differentiable (ϕ(2) = 0, ϕ(4) = −5, ϕ(6) = −11, ϕ (2) = − 52 , ϕ (4) = − 52 , ϕ (6) = − 72 ).
A) Formulate the problem as a mathematical programming problem. b) Solve the optimization problem by using Lagrangian duality. Chapter 4 Sequential Explicit, Convex Approximations In the previous two chapters we were able to formulate a number of structural optimization problems where both the objective function and all of the constraints were written as explicit functions of the design variables only. For larger problems, however, it is in general practically impossible to obtain such explicit functions.
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. If the problem minx∈X L(x, λ) has exactly one solution for a given λ (a sufficient condition for this is that g0 is strictly convex and X is compact), then ϕ is differentiable at λ, and it holds that ∂ϕ(λ) = gi (x ∗ (λ)), ∂λi i = 1, .