OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is a publisher-deposited version published in: http://oatao. Any correspondence concerning this service should be sent to the repository administrator:staff-oatao@inp-toulouse.fr Surrogate modeling approximation using a mixture of experts based on EM joint estimation Dimitri Bettebghor · Nathalie Bartoli · Stéphane Grihon · Joseph Morlier · Manuel Samuelides Abstract An automatic method to combine several local surrogate models is presented. This method is intended to build accurate and smooth approximation of discontinuous functions that are to be used in structural optimization problems. It strongly relies on the Expectation−Maximization (EM) algorithm for Gaussian mixture models (GMM). To the end of regression, the inputs are clustered together with their output values by means of parameter estimation of the joint distribution. A local expert is then built (linear, quadratic, artificial neural network, moving least squares) on each cluster. Lastly, the local experts are combined using the Gaussian mixture model parameters found by the EM algorithm to obtain a global model. This method is tested over both mathematical test cases and an engineering optimization problem from aeronautics and is found to improve the accuracy of the approximation.

A realistic application of advanced structural and multi-objective optimization for the design of a fully assembled aircraft powerplant installation is presented. As opposed to the classical design process of powerplant installation that does not consider the influence of pylon sizing over engine efficiency, we develop in the present a fully integrated approach where both pylon and compressor intercase are designed at once. The main objective is to consider the impact of weight over tip clearance performance criterion and see how these two objectives are antagonistic. In this work, we perform in the same design session tasks traditionally devoted to the airframe manufacturer and aero-engine manufacturer. The overall weight of the assembly is minimized with respect to Specific Fuel Consumption (SFC) criterion. One interesting aspect of the process is that SFC criterion is based on highly proprietary models and its simulation and call within an optimization process is made available through the development of a webservice. One major phenomenon to consider in both pylon and engine design is Fan Blade Off (FBO) event, i.e. the sudden release of a blade. This event causes high impact loads and must be considered carefully in the design. Such a simulation is not an easy task and several nonlinear phenomena must be addressed (e.g. rotordynamics), not to mention the integration of this nonlinear dynamic response in a static structural optimization process. This article describes how the design of the full assembly is performed taking into account both objectives. Such a problem lies in multi-objective optimization field and then we describe the method we use to solve such a problem. The simulation of an FBO post-impact rotor dynamics is also described and we end up with the final results that show the influence of pylon-engine weight sizing over SFC.

In aircraft design, proper tailoring of composite anisotropic characteristics allows to achieve weight saving while maintaining good aeroelastic performance. To further improve the design, dynamic loads and manufacturing constraints should be integrated in the design process. The objective of this paper is to evaluate how the introduction of continuous blending constraints affects the optimum design and the retrieval of the final stacking sequence for a regional aircraft wing. The effect of the blending constraints on the optimum design (1) focuses on static and dynamic loading conditions and identifies the ones driving the optimization and (2) explores the potential weight saving due to the implementation of a manoeuvre load alleviation (MLA) strategy. Results show that while dynamic gust loads can be critical for wing design, in the case of a regional aircraft, their influence is minimal. Nevertheless, MLA strategies can reduce the impact of static loads on the final design in favour of gust loads, underlining the importance of considering such load-cases in the optimisation. In both cases, blending does not strongly affect the load criticality and retrieve a slightly heavier design. Finally, blending constraints confirmed their significant influence on the final discrete design and their capability to produce more manufacturable structures.

An optimization methodology to find concurrently material spatial distribution and material anisotropy repartition is proposed for orthotropic, linear and elastic two dimensional membrane structures. The shape of the structure is parameterized by a density variable that determines the presence or absence of material. The polar method is used to parameterize a general orthotropic material by its elasticity tensor invariants by change of frame. A global structural stiffness maximization problem written as a compliance minimization problem is treated and a volume constraint is applied. The compliance minimization can be put into a double minimization of complementary energy. An extension of the alternate directions algorithm is proposed to solve the double minimization problem. The algorithm iterates between local minimizations in each element of the structure and global minimizations. Thanks to the polar method, the local minimizations are solved explicitly providing analytical solutions. The global minimizations are performed with finite element calculations. The method is shown to be straightforward and efficient. Concurrent optimization of density and anisotropy distribution of a cantilever beam and a bridge are presented.