By Peter W. Christensen
This e-book has grown out of lectures and classes given at Linköping collage, Sweden, over a interval of 15 years. It provides an introductory therapy of difficulties and techniques of structural optimization. the 3 easy periods of geometrical - timization difficulties of mechanical buildings, i. e. , dimension, form and topology op- mization, are handled. the focal point is on concrete numerical resolution tools for d- crete and (?nite point) discretized linear elastic constructions. the fashion is particular and functional: mathematical proofs are supplied whilst arguments should be stored e- mentary yet are differently merely brought up, whereas implementation info are usually supplied. additionally, because the textual content has an emphasis on geometrical layout difficulties, the place the layout is represented via always varying―frequently very many― variables, so-called ?rst order equipment are primary to the therapy. those tools are in keeping with sensitivity research, i. e. , on constructing ?rst order derivatives for - jectives and constraints. The classical ?rst order equipment that we emphasize are CONLIN and MMA, that are in line with particular, convex and separable appro- mations. it may be remarked that the classical and regularly used so-called op- mality standards procedure can be of this sort. it can even be famous during this context that 0 order tools equivalent to reaction floor equipment, surrogate versions, neural n- works, genetic algorithms, and so forth. , primarily follow to sorts of difficulties than those handled right here and will be provided somewhere else.
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Additional resources for An Introduction to Structural Optimization
The volume of the truss may not exceed the value V0 . The design variables are the cross-sectional areas of the bars: A1 , A2 and A3 . a) Formulate the problem as a mathematical programming problem. b) Solve the optimization problem by using Lagrangian duality. 7 The weight of the three-bar truss in Fig. 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.
20) we then get x1∗ = 11/4 and x2∗ = −5/4. This cannot be the actual solution, however, since x1∗ 1. We therefore put x1∗ = 1 and try to find the optimum x2∗ . Again, ∂L2 /∂x2 = 0 gives x2∗ = −1 − λ/2. The dual objective function is ϕ(λ) = − λ2 3 − λ + 4, 4 2 which is maximized for λ∗ = 0. This results in the optimal solution x1∗ = 1 and x2∗ = −1 which is a feasible point. 14) are satisfied, so that L has really been minimized for x ∗ . Since we have also maximized ϕ, the x ∗ obtained is the optimum solution of (P)3 .
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. 34 2 Examples of Optimization of Discrete Parameter Systems Fig. 5 Young’s modulus is E for both bars. The volume of the truss is not allowed to exceed the value V0 .
An Introduction to Structural Optimization by Peter W. Christensen