Abstract
This article proposes a new surrogate-based multidisciplinary design optimization algorithm. The main idea is to replace each disciplinary solver involved in a non-linear multidisciplinary analysis by Gaussian process surrogate models. Although very natural, this approach creates difficulties as the non-linearity of the multidisciplinary analysis leads to a non-Gaussian model of the objective function. However, in order to follow the path of classical Bayesian optimization such as the efficient global optimization algorithm, a dedicated model of the non-Gaussian random objective function is proposed. Then, an Expected Improvement criterion is proposed to enrich the disciplinary Gaussian processes in an iterative procedure that we call efficient global multidisciplinary design optimization (EGMDO). Such an adaptive approach allows to focus the computational budget on areas of the design space relevant only with respect to the optimization problem. The obtained reduction of the number of solvers evaluations is illustrated on a classical MDO test case and on an engineering test case.
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Funding
This work was partially supported by the French National Research Agency (ANR) through the ReBReD project under grant ANR-16-CE10-0002 and by a ONERA internal project MUFIN dedicated to multi-fidelity. Part of the research presented in this paper has been performed in the framework of the AGILE 4.0 project (Towards Cyber-physical Collaborative Aircraft Development) and has received funding from the European Union Horizon 2020 Programme under grant agreement no 815122.
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Replication of results
This section gives numerical details on the implementation of Algorithm 1 and Algorithm 2. All the numerical results have been obtained using Python 2.7.15 and the packages openTURNS 1.12 and numpy 1.15.4. In the following, the default values of the algorithm provided by these packages are used except if mentioned otherwise.
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The initial disciplinary DoE, denoted by \(\text {DoE}_{f_{i}}\) and the initial DoE for uncertainty quantification, denoted by DoEUQ are sampled by Latin hypercube sampling (LHS) using the class LHSExperiment of openTURNS assuming uniform probability distributions between the lower and upper bounds.
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For the creation of the Gaussian processes (disciplinary GPs and KL-GP), all the input samples are centered and reduced. The GPs are constructed using the class KrigingAlgorithm of openTURNS, with constant trend (ConstantBasisFactory in openTURNS) and squared exponential correlation function (SquaredExponential in openTURNS). Optimization of the hyperparameters is performed by maximum likelihood. Range for the optimization is set to [0.3, 100] for each hyperparameter (this choice avoids highly oscillating GP).
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The PCE are computed using the class FunctionalChaosAlg orithm of openTURNS. The polynomial basis is obtained by tenzorisation of the 1-D hermite polynomial basis. Coefficients of the PCE are computed by ordinary least square (LeastSquares Strategy) in openTURNS, using a sample of size 100, sampled from an independent multinormal probability distribution of dimension nd (as detailed in Section 4).
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Mean and covariance matrix of the PCE vector are obtained using the FunctionalChaosRandomVector of openTURNS. eigenvalues and eigenvectors of the covariance matrix are computed with the command eig from the Python toolbox numpy.linalg. The number of modes kept in the KL decomposition is defined such as the cumulative sum of the M highest eigenvalues is strictly higher than 1 − 10− 6.
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As detailed in Dubreuil et al. (2018), the optimization of the EI (17) is performed using openTURNS class OptimizationProblem, with the COBYLA method with the parameter RhoBeg set to 0.5. It should be noted that all the inputs are normalized with a linear transformation from their respective range to [0, 1] for the resolution of the optimization problem. Finally 20 multiple starting points sampled uniformly are used to increase the chance of finding the global optimum of the EI.
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Resolution of the MDA for a given realization of the disciplinary GP (see Section 3.2) is performed by the non-linear Jacobi method. The convergence condition is reached when the mean relative change in the coupling variables between two successive iterations is less than 10− 6. Note that the same algorithm and the same convergence condition are applied when the MDA is solved on the mean value of the disciplinary surrogate for enrichment purposes (see Section 3.4.2).
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Appendix
Appendix
XDSM (eXtended Design Structure Matrix (Lambe and Martins 2012)) of the EGMDO process
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Dubreuil, S., Bartoli, N., Gogu, C. et al. Towards an efficient global multidisciplinary design optimization algorithm. Struct Multidisc Optim 62, 1739–1765 (2020). https://doi.org/10.1007/s00158-020-02514-6
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DOI: https://doi.org/10.1007/s00158-020-02514-6