By Peter W. Christensen

ISBN-10: 1402086652

ISBN-13: 9781402086656

This textbook provides 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 tools. The remedy is designated adequate to permit readers to write down their very own implementations. at the book's homepage, courses will be downloaded that extra facilitate the educational of the fabric lined. The mathematical necessities are stored to a naked minimal, making the ebook compatible for undergraduate, or starting graduate, scholars of mechanical or structural engineering. working towards engineers operating with structural optimization software program could additionally take advantage of analyzing this publication.

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**Extra resources for An introduction to structural optimization (Solid Mechanics and Its Applications)**

**Example text**

14 should be minimized given that the truss should be sufficiently stiff; the maximum nodal displacement Fig. 5 Fig. 5 Exercises 55 Fig. 7 Fig. 8 has to be lower than a prescribed value: max(|u1 |, |u2 |, |u3 |) ≤ u0 , where ui is the displacement vector of node i and u0 > 0 is a given scalar. The truss is subjected to two applied forces. It holds that P >0. The design variables are the cross-sectional areas of the bars: A1 , A2 and A3 . a) Formulate the problem as a mathematical programming problem.

T. 5 ≤ 0 x ∈ X = {x : 0 ≤ x1 ≤ 1, −2 ≤ x2 ≤ 1}. 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 ).

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. , l2 = 2L. 3 How does the solution of the example of Sect. 5 change if the maximum allowable stress in compression is lower than that in tension? 4 Verify the details leading to the solutions (x1∗ , x2∗ ) and (x1∗∗ , x2∗∗ ) in Sect. 4. 5 The stiffness of the two-bar truss subjected to the force P > 0 in Fig. 19 should be maximized by minimizing the displacement u of the free node.

### An introduction to structural optimization (Solid Mechanics and Its Applications) by Peter W. Christensen

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