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Research Papers: Design Automation

Statistical Surrogate Formulations for Simulation-Based Design Optimization

[+] Author and Article Information
Bastien Talgorn

GERAD,
Montréal, Québec H3T 2A7, Canada
Department of Mechanical Engineering,
McGill University,
Montréal, Québec H3A 0C3, Canada

Sébastien Le Digabel

GERAD,
Montréal, Quebec H3T 2A7, Canada
Département de Mathématiques
et Génie Industriel,
École Polytechnique de Montréal,
Montréal, Québec H3C 3A7, Canada

Michael Kokkolaras

GERAD,
Montréal, Québec H3T 2A7,Canada
Department of Mechanical Engineering,
McGill University,
Montréal, Québec H3A 0C3, Canada

There may be situations where the properties of the objective function or some of the constraints do not require the construction and use of surrogate models, e.g., if one of these functions is smooth and inexpensive and has an analytical expression.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received February 18, 2014; final manuscript received September 9, 2014; published online December 15, 2014. Assoc. Editor: Gary Wang.

J. Mech. Des 137(2), 021405 (Feb 01, 2015) (18 pages) Paper No: MD-14-1128; doi: 10.1115/1.4028756 History: Received February 18, 2014; Revised September 09, 2014; Online December 15, 2014

Typical challenges of simulation-based design optimization include unavailable gradients and unreliable approximations thereof, expensive function evaluations, numerical noise, multiple local optima, and the failure of the analysis to return a value to the optimizer. One possible remedy to alleviate these issues is to use surrogate models in lieu of the computational models or simulations and derivative-free optimization algorithms. In this work, we use the R dynaTree package to build statistical surrogates of the blackboxes and the direct search method for derivative-free optimization. We present different formulations for the surrogate problem (SP) considered at each search step of the mesh adaptive direct search (MADS) algorithm using a surrogate management framework. The proposed formulations are tested on 20 analytical benchmark problems and two simulation-based multidisciplinary design optimization (MDO) problems. Numerical results confirm that the use of statistical surrogates in MADS improves the efficiency of the optimization algorithm.

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Figures

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Fig. 1

Optimization algorithm

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Fig. 2

dynaTree regression on 24 data points in R

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Fig. 3

Probability of feasibility and uncertainty in feasibility. μ(x) is maximal for x = 4, where c∧(x) = 0. In the neighborhood of x = 7, despite the sharp variation in c, the feasibility is predictable, so μ(x) is small.

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Fig. 4

Data and performance profiles for the set of analytical problems. MADS and Quad are used as a reference; ((SP1-Fσ), λ = 1.0) and ((SP3-EIσ), λ = 0.01) are the formulations with the best and worst mean deviation, respectively. (a): data profile, τ = 10−1; (b): data profile, τ = 10−3; (c): data profile, τ = 10−7; (d): performance profile after 1000np evaluations.

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Fig. 5

Data and performance profiles for the simplified wing MDO problem. MADS and Quad are used as a reference; ((SP3-EIσ), λ = 0.01) and ((SP3-EIσ), λ = 1.0) are the formulations with the best and worst mean deviation, respectively. (a): data profile, τ = 10−1; (b): data profile, τ = 10−3; (c): data profile, τ = 10−7; (d): performance profile after 7,000 evaluations.

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Fig. 6

Data and performance profiles for the aircraft-range MDO problem. MADS and Quad are used as a reference; ((SP5-EFσ), λ = 0.1) and ((SP5-EFIσ), λ = 1.0) are the formulations with the best and worst mean deviation, respectively. (a): Data profile, τ = 10−1; (b): data profile, τ = 10−3; (c): data profile, τ = 10−7; (d): performance profile after 10,000 evaluations.

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Fig. 7

Comparison of different surrogate modeling methods using four test functions (left column: models, right column: absolute errors)

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