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A variable-accuracy metamodel-based architecture for global MDO under uncertainty

Abstract

A method for simulation-based multidisciplinary robust design optimization (MRDO) of problems affected by uncertainty is presented. The challenging aspects of simulation-based MRDO are both algorithmic and computational, since the solution of a MRDO problem typically requires simulation-based multidisciplinary analyses (MDA), uncertainty quantification (UQ) and optimization. Herein, the identification of the optimal design is achieved by a variable-accuracy, metamodel-based optimization, following a multidisciplinary feasible (MDF) architecture. The approach encompasses a variable (i) density of the design of experiments for the metamodel training, (ii) sample size for the UQ analysis by quasi Monte Carlo simulation and (iii) tolerance for the multidisciplinary consistency in MDA. The focus is on two-way steady fluid-structure interaction problem, assessed by partitioned solvers for the hydrodynamic and the structural analysis. Two analytical test problems are shown, along with the design of a racing-sailboat keel fin subject to the stochastic variation of the yaw angle. The method is validated versus a standard MDF approach to MRDO, taken as a benchmark and solved by fully coupled MDA, fully converged UQ, without metamodels. The method is evaluated in terms of optimal design performances and number of simulations required to achieve the optimal solution. For the current application, the optimal configuration shows performances very close to the benchmark solution. The convergence analysis to the optimum shows a promising reduction of the computational cost.

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Abbreviations

\(\displaystyle \mathbf {a} \in \mathbb {R}^{n}\) :

multidisciplinary equilibrium solution (\(n\in \mathbb {N}^{+}\))

Δ i :

i-th discipline involved in multidisciplinary analysis

\(\displaystyle f(\mathbf {x}) \in \mathbb {R}\) :

deterministic objective function

\(\displaystyle f(\mathbf {x},\mathbf {a}) \in \mathbb {R}\) :

deterministic multidisciplinary objective function

\(\displaystyle f(\mathbf {x},\mathbf {y}) \in \mathbb {R}\) :

stochastic objective function

\(\displaystyle f(\mathbf {x},\mathbf {y},\mathbf {a}) \in \mathbb {R}\) :

stochastic multidisciplinary objective function

\(\displaystyle \mu (f) \in \mathbb {R}\) :

expected value of the deterministic objective function f

\(\displaystyle N_{{\Delta }} \in \mathbb {N}^{+}\) :

number of disciplinary analysis

\(\displaystyle N_{\text {DV}} \in \mathbb {N}^{+}\) :

number of design variables

\(\displaystyle N_{\text {MDA}} \in \mathbb {N}^{+}\) :

number of iterations for multidisciplinary analysis

\(\displaystyle N_{\text {OS}} \in \mathbb {N}^{+}\) :

number of optimization stages

\(\displaystyle N_{\text {PSO}} \in \mathbb {N}^{+}\) :

number of function evaluations in particle swarm optimization

\(\displaystyle N_{\text {TP}} \in \mathbb {N}^{+}\) :

number of training points for metamodel

\(\displaystyle N_{\text {S}} \in \mathbb {N}^{+}\) :

number of simulations

\(\displaystyle N_{\text {UQ}} \in \mathbb {N}^{+}\) :

number of items for uncertainty quantification

\(\displaystyle p(\mathbf {y}) \in \mathbb {R}\) :

probability density function

\(\displaystyle \tau _{0},\tau _{1} \in \mathbb {R^{+}}\) :

tolerance for the multidisciplinary consistency in MDA

\(\displaystyle \tau _{2} \in \mathbb {R^{+}}\) :

tolerance for the UQ analysis

\(\displaystyle \mathbf {x} \in X \subseteq \mathbb {R}^{N_{\text {DV}}}\) :

deterministic design variables

\(\displaystyle \mathbf {y} \in Y \subseteq \mathbb {R}^{m}\) :

stochastic parameters (\(m\in \mathbb {N}^{+}\))

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Acknowledgments

The present research is supported by the Italian Flagship Project RITMARE, coordinated by the Italian National Research Council and funded by the Italian Ministry of Education, University and Research, within the National Research Program 2011-2013. The authors are grateful to Dr Woei-Min Lin and Dr Ki-Han Kim of the US Navy Office of Naval Research, for their support through NICOP grant N62909-15-1-2016.

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Correspondence to Matteo Diez.

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Leotardi, C., Serani, A., Iemma, U. et al. A variable-accuracy metamodel-based architecture for global MDO under uncertainty. Struct Multidisc Optim 54, 573–593 (2016). https://doi.org/10.1007/s00158-016-1423-4

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  • DOI: https://doi.org/10.1007/s00158-016-1423-4

Keywords

  • Simulation-based design
  • Multidisciplinary robust design optimization (MRDO)
  • Uncertainty quantification (UQ)
  • Metamodel-based optimization
  • Global optimization