Application of response surface methodology to stiffened panel optimization

In a multilevel optimization frame, the use of surrogate models to approximate optimization constraints allows great time saving. Among available metamodelling techniques we chose to use Neural Networks to perform regression of static mechanical criteria, namely buckling and collapse reserve factors of a stiffened panel, which are constraints of our subsystem optimization problem. Due to the highly non linear behaviour of these functions with respect to loading and design variables, we encountered some difficulties to obtain an approximation of sufficient quality on the whole design space. In particular, variations of the approximated function can be very different according to the value of loading variables. We show how a prior knowledge of the influence of the variables allows us to build an efficient Mixture of Expert model, leading to a good approximation of constraints. Optimization benchmark processes are computed to measure time saving, effects on optimum feasibility and objective value due to the use of the surrogate models as constraints. Finally we see that, while efficient, this mixture of expert model could be still improved by some additional learning techniques

Surrogate modeling approximation using a mixture of experts based on EM joint estimation

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

Optimization methodology of composite panels

This paper deals with mass optimisation of composite laminates aeronautical structures. It focuses on the final stage of the process of composite structure design i.e. defining the lay-up evolution all over the panel, this being directly used for manufacturing. Strength criteria (in-plane behaviour) and stability criteria (out-of-plane behaviour) performed with appropriate industrial tools are evaluated in the multi-level optimisation presented hereafter. The methodology consists in five steps. In the first two steps, optimisation is performed with continuous variables and a homogenized material (i.e. with approached out of plane properties). In the third step, a lay-up table is selected (or built) and translated into a "continuous" material (i.e. out of plane stiffnesses expressed as continuous variables of the lay-up thickness). In the fourth step, optimisation is performed with the "continuous" material. In the last step, a genetic optimisation is used to round off at discrete ply thicknesses. This methodology provides a manufacturable result (in terms of ply continuity) that satisfies all stress and stacking sequence constraints

Approche en paramètres de stratification pour l’optimisation biniveau de structures de fuselage composite

On présente ici une méthode d’optimisation biniveau de grandes structures de fuselage composite. Ce schéma biniveau est inspiré de la formulation Quasi Separable Decomposition (QSD)récemment développée par Haftka et Watson. Le comportement membrane et hors-plan des stratifiés est représenté au moyen des paramètres de stratification. La boucle d’optimisation supérieure fait intervenir la redistribution des efforts et a pour contraintes des quantités calculées par des problèmes d’optimisation locale en stabilité où les facteurs critiques de flambage sont approchés par des modèles réduits

An outer approximation bi-level framework for mixed categorical structural optimization problems

In this paper, mixed categorical structural optimization problems are investigated. The aim is to minimize the weight of a truss structure with respect to cross-section areas, materials and cross-section type. The proposed methodology consists of using a bi-level decomposition involving two problems: master and slave. The master problem is formulated as a mixed integer linear problem where the linear constraints are incrementally augmented using outer approximations of the slave problem solution. The slave problem addresses the continuous variables of the optimization problem. The proposed methodology is tested on three different structural optimization test cases with increasing complexity. The comparison to state-of-the-art algorithms emphasizes the efficiency of the proposed methodology in terms of the optimum quality, computation cost, as well as its scalability with respect to the problem dimension. A challenging 120-bar dome truss optimization problem with 90 categorical choices per bar is also tested. The obtained results showed that our method is able to solve efficiently large scale mixed categorical structural optimization problems.Comment: Accepted for publication in Structural and Multidisciplinary Optimization, to appear 202

Modèles réduits en optimisation multiniveau de structures aéronautiques

Le dimensionnement de grandes structures aéronautiques s’appuie sur des heuristiques qui garantissent l’admissibilité de la structure par rapport aux contraintes de tenue mais pas nécessairement l’optimalité en masse. On se propose ici de formaliser le problème d’optimisation de structures en l’incluant dans la catégorie des problèmes d’optimisation multiniveau. On montre comment l’utilisation de modèles réduits peut simplifier et améliorer les méthodes directes. On présente les premiers résultats obtenus sur un cas test pour les différentes méthodes

Fuselage structure optimization

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A bi-level methodology for solving large-scale mixed categorical structural optimization

In this work, large-scale structural optimization problems involving non-ordinal categorical design variables and continuous variables are investigated. The aim is to minimize the weight of a structure with respect to cross-section areas, with materials and stiffening principles selection. First, the problem is formulated using a bi-level decomposition involving master and slave problems. The master problem is given by a first-order-like approximation that helps to drastically reduce the combinatorial explosion raised by the categorical variables. Continuous variables are handled in a slave problem solved using a gradient-based approach, where the categorical variables are driven by the master problem. The proposed algorithm is tested on three different structural optimization test cases. A comparison to state-of-the-art algorithms emphasize efficiency of the proposed algorithm in terms of the optimum quality, the computation cost, and the scaling with respect to the problem dimension