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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21110%2F21%3A00351153" target="_blank" >RIV/68407700:21110/21:00351153 - isvavai.cz</a>

Nalezeny alternativní kódy

RIV/68407700:21230/21:00351153

Výsledek na webu

<a href="https://doi.org/10.1007/s00158-021-02957-5" target="_blank" >https://doi.org/10.1007/s00158-021-02957-5</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/s00158-021-02957-5" target="_blank" >10.1007/s00158-021-02957-5</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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 ε-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.

Název v anglickém jazyce

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

Popis výsledku anglicky

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 ε-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.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

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

2021

Kód důvěrnosti údajů

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

Název periodika

Structural and Multidisciplinary Optimization

ISSN

1615-147X

e-ISSN

1615-1488

Svazek periodika

64

Číslo periodika v rámci svazku

4

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

19

Strana od-do

1963-1981

Kód UT WoS článku

000664407300001

EID výsledku v databázi Scopus

2-s2.0-85108620101