Towards an efficient global multidisciplinary design optimization algorithm

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.

Disciplinary proper orthogonal decomposition and interpolation for the resolution of parameterized multidisciplinary analysis

“…The objective of this section is to use an MDA solver, called DPOD+I, to solve the MDA analysis with an enrichment strategy based on the work presented in Reference 11. In the previous Section 4.2.2, an initial model was trained to obtain a POD basis for each discipline.…”

“…Unfortunately, trajectories of GPs are difficult to simulate when the number of inputs of the GP (design variables and POD coefficients) is high, leading to a representation with a large number of random variables, which is not suitable for the following. One solution proposed in Reference 11 is to use perfectly dependent GPs. The idea is to model the unknown reduced disciplinary solver as a conditioned GP with mean and variance obtained as described in Section 4.1.2 but with a correlation function constant and equal to one.…”

“…Remarks: This simplification in the GP modeling is not detrimental in the evaluation of the interpolation uncertainty as this GP model shares the same variance as the original one, only the covariance is different. It has been shown in Reference 11 that this modeling choice is numerically efficient to quantify the uncertainty due to GP interpolation. Getting a random solution of such surrogate model is inexpensive. Indeed, for a given draw of $$$ {\hat{\xi}}_i^1 $$$ and $$$ {\hat{\xi}}_j^2 $$$, the solution of Equation (20) only requires some calls to the analytical means and variances of the GPs.…”

“…In this paper an approximation of each disciplinary solver is proposed using a POD+I type technique to adapt the strategy developed in Reference 11 to the context of coupled simulations with high‐dimensional coupling variables. A similar approach has been proposed in Reference 24 where only one of the disciplinary solvers was approximated.…”

“…These studies rely on an independent construction of each surrogate model and do not consider the resulting approximation error made by coupling the disciplinary surrogate models. To improve the disciplinary surrogate model construction in an MDO context, it is proposed in Reference 11 to use Gaussian Processes (GPs) to approximate each disciplinary solver. Then, a dedicated strategy is applied to solve the MDO by taking into consideration the randomness brought by the GPs during the MDA resolution.…”