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

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

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

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

Deep Learning for Additive Manufacturing-driven Topology Optimization

This paper investigates the potential of Deep Learning (DL) for data-driven topology optimization (TO). Unlike the rest of the literature that mainly applies DL to TO from a mechanical perspective, we developed an original approach to integrate mechanical and geometrical constraints simultaneously. Our approach takes as input the mechanical constraints (Boundary conditions, loads configuration, volume fraction) alongside the geometrical ones (total number of elements, minimum overhang, maximum length, minimum thickness) and generates a 2D design complying with these constraints. Thus, it combines the best of both mechanical (CAE) and geometrical design worlds. Conversely, geometrical design constraints are complex, not yet formalized, and contradictory between Additive Manufacturing (AM) processes, applications, and materials. Some are even descriptive, lacking a well-defined mathematical description, or are well-defined but proprietary and inaccessible. Hence, despite the synergy between AM and TO, integrating AM constraints into the TO formulation is still a hurdle. Furthermore, even when their integration is possible, TO’s convergence to a solution is compromised. On the other hand, DL has proven robust in capturing geometrical and spatial correlations. Consequently, our approach solves the previously listed setbacks by aligning DL to serve Design for AM (DfAM); there is no need to identify an analytical formula for a geometrical constraint but simply a sufficient number of examples describing it, and convergence is no longer a blockade when the DL model is trained on converged designs. Our approach tailors the design’s geometrical aspects with great flexibility and creativity. It reconciles design and manufacturing and accelerates the design life cycle of a part. Moreover, it can be easily updated to include additional constraints and can be implemented in the future into CAD software as a lighter and faster generative design module

Approximation of the critical buckling factor for composite panels

This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented

Bilevel optimization of large scale composite structures

Ce travail de thèse s inscrit dans le domaine de l optimisation de structures aéronautiques composites. On cherche à rendre possible le traitement de problèmes de dimensionnement de telles structures, telles que celles rencontrées dans l industrie aéronautique. Ce type de problèmes présente deux aspects bloquants. En premier lieu, la taille des structures et le type de matériaux rendent le problème d optimisation à la fois de très grande taille et de variables mixtes (continues, discrètes). D autre part, le très grand nombre d analyses de stabilité locale (flambage) nécessaires rend le problème d optimisation très difficile à traiter en terme de coût de calculs. On cherche donc à résoudre le premier aspect au travers de schémas d optimisation dits de décomposition qui permettent de décomposer le problème d optimisation initial en une multitude de sous problèmes d optimisations pouvant être résolus en parallèle et dont le couplage est résolu par un problème d optimisation sur un ensemble de variables réduit. L équivalence théorique entre les différents problèmes d optimisation (en termes de minima locaux) est prouvée et on présente et développe un schéma adapté à la fois aux spécificités des composites et aux contraintes industrielles. Le second point est résolu de manière originale par le développement d une stratégie d approximation des contraintes de stabilité. Cette stratégie de mélanges d experts se base sur des outils statistiques avancés et se révèle adaptée au comportement des composites. Les deux principales avancées de ce travail sont validées sur des cas test académiques et sur une structure aéronautique réaliste. Le fil directeur de ce travail est la mécanique des structures composites, néanmoins le caractère pluridisciplinaire du sujet nous a conduit à des incursions vers les domaines des statistiques (apprentissage), de l analyse numérique (étude de l équation aux dérivées partielles relative au flambage) et enfin de l optimisation théorique.This work lies in the field of aerospace composite structures optimization. We are interested in making possible the treatment of large scale optimization problems, as the ones encountered in aerospace design offices. Resolution of such problem needs two main obstacles to be removed. First one lies in the typical large size of problems and the mixed type of design variables: continuous (geometric dimensions, internal loads) and discrete (ply orientations for laminates). Second one lies in the tremendous amount of local stability analyses (buckling) to be performed in one standard optimization. First aspect is solved with the help of decomposition methods that allow breaking up the initial optimization problem in a multitude of optimization sub problems of reduced dimensions. These problems can be solved concurrently, however the internal load redistribution makes them coupled and an upper level optimization problem is needed to solve this coupling. Innovative solutions, both in terms of composite material mechanics representation and in terms of theoretical optimization properties are presented. Second aspect is solved through the development of an innovative approximation scheme, tailored to buckling behaviour specificities, namely mixture of experts. In particular, the piecewise-like behaviour of such functions is considered. This innovative method relies on advanced statistical tools from unsupervised learning (clustering, law mixture). Finally the two main innovations are extensively discussed and tested over academic benchmark. They are eventually combined for a realistic structural optimization problem (fuselage panel) and allowed retrieving the same weight as traditional method with less iterations. Although the main theme is mechanics and structural optimization, the multidisciplinary aspect of the subject included some research questions and answers in statistics field (statistical learning), numerical analysis (buckling partial differential equation) and theoretical optimization.TOULOUSE-ISAE (315552318) / SudocSudocFranceF

Surrogate-based aerodynamics for composite wing box sizing

The main objective of this paper is to propose an aeroelastic optimization approach capable of performing structural sizing optimization considering structural and aerodynamic constraints. The proposed approach uses a detailed FE model of a composite wing with shell elements in order to obtain realistic weight estimation and structural responses. Moreover, a surrogate model based on rigid RANS computations provides a high-fidelity lift and drag coefficient estimation during the optimization as constraints. The use of RANS computations allows the surrogate model to consider all drag components and not the induced drag only. An approximation of the structural displacement is proposed based on modal projection and principle component analysis. Results shows that a polynomial regression of order four is accurate enough to be used as surrogate model of the drag coefficient and of the lift-to-drag ratio. Moreover, it is possible to conclude that improvements in aerodynamic performance comes at the price of a heavier and stiffer structure.</p

Concurrent optimization of material spatial distribution and material anisotropy repartition for two dimensional structures

International audienceAn 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