Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

Autor: Didier Henrion, Marek Tyburec, Martin Kružík, Jan Zeman
Přispěvatelé: Faculty of Civil Engineering [Prague] (FSV CTU), Czech Technical University in Prague (CTU), Équipe Méthodes et Algorithmes en Commande (LAAS-MAC), Laboratoire d'analyse et d'architecture des systèmes (LAAS), Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées, Faculty of Electrical Engineering [Prague] (FEL CTU), HAL-LAAS, LAAS, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique (Toulouse) (Toulouse INP), Université de Toulouse (UT)-Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)
Jazyk: angličtina
Rok vydání: 2021
Předmět:
Zdroj: 14th World Congress of Structural and Multidisciplinary Optimization
14th World Congress of Structural and Multidisciplinary Optimization, Jun 2021, Boulder, Colorado, United States
14th World Congress of Structural and Multidisciplinary Optimization (WCSMO 14)
14th World Congress of Structural and Multidisciplinary Optimization (WCSMO 14), Jun 2021, Boulder, Colorado, United States
Structural and Multidisciplinary Optimization
Structural and Multidisciplinary Optimization, 2021, 64 (4), pp.1963-1981. ⟨10.1007/s00158-021-02957-5⟩
ISSN: 1615-147X
1615-1488
DOI: 10.1007/s00158-021-02957-5⟩
Popis: The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global $\varepsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.
16 pages, 7 figures
Databáze: OpenAIRE
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