By Peter W. Christensen
This textbook supplies an creation 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 specific sufficient to let readers to write down their very own implementations. at the book's homepage, courses can be downloaded that extra facilitate the training of the fabric lined. The mathematical must haves are saved to a naked minimal, making the e-book appropriate 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 e-book.
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Additional resources for An introduction to structural optimization (Solid Mechanics and Its Applications)
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. 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 .
0 Fig. 16 Case e). 6 Weight Minimization of a Three-Bar Truss Subjectto a Stiffness Constraint 31 This is the same solution as for case b). The reason that we get the same solution although we have doubled the density of bar 2 is of course that bar 2 is not present in the optimal trusses. e. 1. Since the σ2 -constraint curve is parallel to the iso-merit lines, we conclude that in this case there will be an infinite number of solutions, namely all points on the line √ between A and C in Fig. 1!
The design variables are the crosssectional areas of the bars: A1 and A2 . The truss has to be sufficiently stiff; more precisely, the so-called compliance has to be lower than a specified number: −P ux − P uy ≤ c0 , where (ux , uy ) are the displacements of the free node, and c0 > 0 is a given number. 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.
An introduction to structural optimization (Solid Mechanics and Its Applications) by Peter W. Christensen