Optimální návrh nosníku pomocí stochastického programování

Optimum Beam Design via Stochastic Programming

journal article - other

en

The purpose of the paper is to discuss the applicability of stochastic programming models and methods to civil engineering design problems. In cooperation with experts in civil engineering, the problem concerning an optimal design of beam dimensions has been chosen. The corresponding mathematical model involves an ODE-type constraint, uncertain parameter related to the material characteristics and multiple criteria. As a result, a multi-criteria stochastic nonlinear optimization model is obtained. It has been shown that two-stage stochastic programming offers a promising approach to solving similar problems. A computational scheme for this type of problems is proposed, including discretization methods for random elements and ODE constraint. An approximation is derived to implement the mathematical model and solve it in GAMS. The solution quality is determined by an interval estimate of the optimality gap computed by a Monte Carlo bounding technique. The parametric analysis of a multi-criteria model results in efficient frontier computation. Furthermore, a progressive hedging algorithm is implemented and tested for the selected problem in view of the future possibilities of parallel computing of large engineering problems. Finally, two discretization methods are compared by using GAMS and ANSYS.

V článku je diskutována použitelnost stochastického programování na úlohy inženýrského návrhu. Je zvolen problém týkající se optimálního návrhu rozměrů nosníku. Odpovídající matematický model vede na úlohu vícekriteriálního stochastického nelineárního programování. Je navrženo výpočtové schéma pro tento typ úloh a odvozená aproximovaná úloha je řešena v programu GAMS. Kvalita řešení je určena pomocí metody Monte Carlo. Výsledkem parametrické analýzy vícekriteriálního modelu je konstrukce eficientní hranice. Dále je implemetován algoritmus progressive hedging. Nakonec jsou srovnány dvě diskretizační metody pomocí programů GAMS a ANSYS.

The purpose of the paper is to discuss the applicability of stochastic programming models and methods to civil engineering design problems. In cooperation with experts in civil engineering, the problem concerning an optimal design of beam dimensions has been chosen. The corresponding mathematical model involves an ODE-type constraint, uncertain parameter related to the material characteristics and multiple criteria. As a result, a multi-criteria stochastic nonlinear optimization model is obtained. It has been shown that two-stage stochastic programming offers a promising approach to solving similar problems. A computational scheme for this type of problems is proposed, including discretization methods for random elements and ODE constraint. An approximation is derived to implement the mathematical model and solve it in GAMS. The solution quality is determined by an interval estimate of the optimality gap computed by a Monte Carlo bounding technique. The parametric analysis of a multi-criteria model results in efficient frontier computation. Furthermore, a progressive hedging algorithm is implemented and tested for the selected problem in view of the future possibilities of parallel computing of large engineering problems. Finally, two discretization methods are compared by using GAMS and ANSYS.

optimální inženýrský návrh, stochastické programování, vícekriteriální programování, metoda Monte Carlo, algoritmus progressive hedging

optimum engineering design, stochastic programming, multi-objective programming, Monte Carlo methods, progressive hedging algorithm

2010

14.06.2010

Institute of Information Theory and Automation

Prague

0023-5954

Kybernetika

46

3

571–582

12