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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21110%2F20%3A00345847" target="_blank" >RIV/68407700:21110/20:00345847 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21230/20:00345847

Výsledek na webu

<a href="https://mathplus.de/topic-development-lab/tes-winter-2020-21/ma4m/" target="_blank" >https://mathplus.de/topic-development-lab/tes-winter-2020-21/ma4m/</a>

DOI - Digital Object Identifier

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Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

Designing minimum-compliance bending-resistant structures with continuous cross-section parameters has been a challenging task because of its non-convexity. We develop 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 at 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 sucient condition of global $latex epsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. We illustrate these theoretical findings on examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Název v anglickém jazyce

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

Popis výsledku anglicky

Designing minimum-compliance bending-resistant structures with continuous cross-section parameters has been a challenging task because of its non-convexity. We develop 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 at 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 sucient condition of global $latex epsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. We illustrate these theoretical findings on examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Druh

O - Ostatní výsledky

CEP obor

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OECD FORD obor

10102 - Applied mathematics

Projekt

<a href="/cs/project/GX19-26143X" target="_blank" >GX19-26143X: Neperiodické materiály vykazující strukturované deformace: Modulární návrh a výroba</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů