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A scalable problem to benchmark robust multidisciplinary design optimization techniques

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Abstract

A scalable problem to benchmark robust multidisciplinary design optimization (RMDO) algorithms is proposed. This allows the user to choose the number of disciplines, the dimensions of the coupling and design variables and the extent of the feasible domain. After a description of the mathematical background, a deterministic version of the scalable problem is defined and the conditions on the existence and uniqueness of the solution are given. Then, this deterministic scalable problem is made uncertain by adding random variables to the coupling equations. Under classical assumptions, the existence and uniqueness of the solution of this RMDO problem is guaranteed. This solution can be easily computed with a quadratic programming algorithm and serves as a reference to assess the performance of RMDO algorithms. This scalable problem has been implemented in the open-source library GEMSEO and tested with two techniques of statistics estimation: Monte-Carlo sampling and Taylor polynomials.

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Notes

  1. \(\forall \textbf{x}\in \mathbb {R}^{{d}}{\setminus }\{0_{{d}}\}, \textbf{x}^\top \textbf{Q}\textbf{x}= \textbf{x}^\top \textbf{Q}_{\textbf{x}_0}\textbf{x}+ \textbf{x}^\top \varvec{\beta }^\top \varvec{\beta }\textbf{x}= \Vert \textbf{x}_0\Vert ^2 + \Vert \varvec{\beta }\textbf{x}\Vert ^2 \ge 0\).

  2. The norm of the residuals divided by the norm of the initial residuals shall be lower than the relative tolerance.

  3. https://www.gemseo.org/.

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Acknowledgements

We wish to acknowledge the PIA framework (CGI, ANR) and the industrial members of the IRT Saint Exupéry project R-Evol: Airbus, Liebherr, Altran Technologies, Capgemini DEMS France, CENAERO and CERFACS for their support, financial funding and own knowledge. We are grateful to Réda Chhaïbi (Institut de Mathématiques de Toulouse) for useful discussions on random matrices. We acknowledge Syver Døving Agdestein for a preliminary work on this topic, during his master internship.

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The authors have not disclosed any funding.

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Authors and Affiliations

  1. Institut de Recherche Technologique Saint Exupéry, 3, rue Tarfaya, 31400, Toulouse, France

    Amine Aziz-Alaoui & Matthias De Lozzo

  2. Institut National des Sciences Appliquées, 135 avenue de Rangueil, 31077, Toulouse, France

    Olivier Roustant

Authors
  1. Amine Aziz-Alaoui
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  2. Olivier Roustant
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  3. Matthias De Lozzo
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Correspondence to Amine Aziz-Alaoui.

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Replication of results

All the details required for the replication of the results presented in this paper are provided in Sects. 3 and 4. The scalable problems (8) and (10) and the QP problems (9), (15) and (16) are available in the open-source library GEMSEO. The MDO problems (8) and (10) can be solved with the packages gemseo and gemseo-umdo respectively, with different MDO formulations and statistics estimators.

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Aziz-Alaoui, A., Roustant, O. & De Lozzo, M. A scalable problem to benchmark robust multidisciplinary design optimization techniques. Optim Eng 25, 941–958 (2024). https://doi.org/10.1007/s11081-023-09830-y

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