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- Andréasson, Niclas, 1976, et al.
(författare)
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An Introduction to Continuous Optimization, 3rd edition
- 2016
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- This is the third edition of a book that was originally published in 2005. The book is used primarily in the teaching of the Chalmers course TMA947 Nonlinear Optimization.
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- Evgrafov, Anton, 1978
(författare)
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Approximation of topology optimization problems using sizing optimization problems
- 2004
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The present work is devoted to approximation techniques for singular extremal problems arising from optimal design problems in structural and fluid mechanics. The thesis consists of an introductory part and four independent papers, which however are united by the common idea of approximation and the related application areas. In the first half of the thesis we are concerned with finding the optimal topology of truss-like structures. This class of optimal design problems arises when in order to find the optimal truss not only are we allowed to redistribute the material among the structural members (bars), but also to completely remove some parts altering the connectivity (topology) of the structure. The other half of the thesis addresses the question of the optimal topological design of flow domains for Stokes and Navier-Stokes fluids. For flows, optimizing topology means finding the optimal partition of the given design domain into disjoint parts occupied by the fluid and the impenetrable walls, given the in-flow and the out-flow boundaries. In particular, impenetrable walls change the shape and the connectivity of the flow domain. In the first paper we construct an example demonstrating the singular behaviour of truss topology optimization problems including a linearized global buckling (linear elastic stability) constraint. This singularity phenomenon has not been known before and affects the choice of numerical methods that can be applied to the optimization problem. We propose a simple approximation strategy and establish the convergence of globally optimal solutions to perturbed problems towards globally optimal solutions to the original singular problem. In the second paper we are concerned with the construction of finer approximating problems that allow us to reconstruct the local behaviour of a general class of singular truss topology optimization problems, namely to approximate stationary points to the limiting problem with sequences of stationary points to the regular approximating problems. We do so on the classic problem of weight minimization under stress constraints for trusses in unilateral contact with rigid obstacles. In the third paper we extend a design parametrization previously proposed for the topological design of flow domains for Stokes flows to also include the limiting case of porous materials - completely impenetrable walls. We demonstrate that, in general, the resulting design-to-flow mapping is not closed, yet under mild assumptions it is possible to approximate globally optimal minimal-power-dissipation domains using porous materials with diminishing permeability. In the fourth and last paper we consider the optimal design of flow domains for Navier-Stokes flows. We illustrate the discontinuous behaviour of the design-to-flow mapping caused by the topological changes in the design, and propose "minor" changes to the design parametrization and the equations that allow us to rigorously establish the closedness of the design-to-flow mapping. The existence of optimal solutions as well as the convergence of approximation schemes then easily follows from the closedness result.
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| 3. |
- Evgrafov, Anton, 1978
(författare)
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On globally stable singular truss topologies
- 2005
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Ingår i: Structural and Multidisciplinary Optimization. - : Springer Science and Business Media LLC. - 1615-147X .- 1615-1488. ; 29:3, s. 170-177
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Tidskriftsartikel (refereegranskat)abstract
- We consider truss topology optimization problems including a global stability constraint, which guarantees a sufficient elastic stability of the optimal structures. The resulting problem is a nonconvex semi-definite program, for which nonconvex interior point methods are known to show the best performance. We demonstrate that in the framework of topology optimization, the global stability constraint may behave similarly to stress constraints, that is, that some globally optimal solutions are singular and cannot be approximated from the interior of the design domain. This behaviour, which may be called a global stability singularity phenomenon, prevents convergence of interior point methods towards globally optimal solutions. We propose a simple perturbation strategy, which restores the regularity of the design domain. Further, to each perturbed problem interior point methods can be applied.
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| 9. |
- Patriksson, Michael, 1964, et al.
(författare)
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Introduction to Continuous Optimization
- 2013
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Bok (övrigt vetenskapligt/konstnärligt)abstract
- This second edition introduces several areas and items that were not included in the first edition, as well as several corrections. A brief summary of these changes are given next. Chapter 1 includes a discussion on the diet problem, in addition to that on the staff planning problem, in order to very early on introduce linear programming. Figure 1.1 now has the terminating box ``Implementation'', whereas the original one had an infinite loop! Chapter 3 has been enriched by several new results on separating and supporting hyperplanes, and the associated theory of convex cones and their polar sets. Thanks to this study of separating hyperplanes,Theorem 5.17 on the necessity of the Fritz John conditions now has a complete proof. The end of Chapter 5 also includes a summary of the fascinating story of the development of the Karush--Kuhn--Tucker conditions. The sensitivity analysis in linear programming has been expanded with a discussion in Section 10.5.3 on the addition of a variable or a constraint, as well as an introduction to column generation based on the example of the minimum cost multi-commodity network flow problem (Section 10.6). Chapter 11 includes a brief discussion on Gauss--Newton methods for least-squares problems. Chapter 12 has changed its name from ``Optimization over convex sets'' to ``Feasible-direction methods,'' in order to reflect the fact that the scope is now wider---from essentially polyhedral sets to general closed sets (which, however, most often will be assumed to be convex). In particular, we have added new sections on algorithms defined by closed descent maps---an algorithm principle which was devised and analyzed mainly in the 1960s, and which is a quite elegant means to describe iterative methods. We also utilize this principle to contrast established convergent methods (such as the Frank--Wolfe method) and failed attempts (such as the algorithm of Zoutendijk). We have also added a brief discussion on reduced gradient methods, which are relatives to the simplex method; they are---in their original statement---not convergent, but a small adjustment results in a closed descent map and hence a convergent method. Exercises and their solutions are now placed at the end of the book, rather than at the end of each chapter. The first edition from 2005 has been used in teaching of several courses at Chalmers University of Technology and the University of Gothenburg. We wish to thank all the students who have given us remarks on the book. We would also like to thank Dr. Kin Cheong Sou for remarks and corrections on the first edition.
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- Patriksson, Michael, 1964, et al.
(författare)
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On stochastic structural topology optimization
- 2002
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Ingår i: Nonsmooth/Nonconvex Mechanics: With Applications in Engineering, Proceedings of the International Conference in memory of Professor P. D. Panagiotopoulos, 5-6 July 2002, Thessaloniki, Greece. ; , s. 97-104
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Konferensbidrag (refereegranskat)
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