Optimization of composite structures: A shape and topology sensitivity analysis

Abstract : This thesis is devoted to the study of two main problems, namely the optimal design of multi-layered composite laminates and the topological sensitivity analysis in anisotropic elastostatics. Concerning the composite design, we consider minimal weight structures subjected to stiffness and buckling constraints, where the design variables are the shape/topology of each ply and the stacking sequence. Indeed, the composite laminate is made up of a collection of fiber reinforced orthotropic plies whose main axes can take four different orientations: 0º,90º,45º,-45º. The way these orientations are arranged within the composite defines the stacking sequence. The physical behavior of the multi-layered laminate is governed by the system of linearized von Kármán equations for plates. In order to optimize both design variables, we rely on a decomposition technique which aggregates the constraints into one unique constraint called margin function. Thanks to this approach, a rigorous equivalent bi-level optimization problem is established. The latter problem is made up of a lower level represented by the combinatorial optimization of the stacking sequence and a higher level represented by the shape/topology optimization of each ply. We propose for the stacking sequence optimization an outer approximation method which iteratively solves a set of mixed integer linear problems associated to the evaluation of the constraint margin function. For the shape/topology optimization of each ply, we lean on the level set method for the description of the interfaces and the Hadamard method for boundary variations by means of the computation of the shape gradient. An aeronautic test case is exhibited for different constraints, namely compliance, reserve factor and first buckling load. The second main problem of this thesis deals with the topological derivative of cost functionals that depend on the stress and the displacement (assuming a linearly elastic material behavior) in a general 2D and 3D anisotropic setting, where both the background and the inhomogeneity may have arbitrary anisotropic elastic properties. A small-inhomogeneity expansion of the cost function is mathematically justified for a wide class of displacement and stress-based cost functionals having smooth densities and computational procedures are then discussed. Several 2D and 3D numerical examples are presented, in particular demonstrating the proposed formulation of the topological derivative on practical cases involving anisotropic elasticity and non-quadratic cost functionals. Independently of the foregoing subjects, we treat additionally two optimal design problems. First we consider the optimal distribution of several elastic materials in a fixed working domain with either a sharp or a smooth interface. In order to optimize both the geometry and topology of the mixture, we rely on the level set method and the signed distance function for the description of the interfaces between the different phases. Secondly, in the framework of efficient power complements to aircraft engines, we seek to come up with the optimal micro-structure of micro-tubular fuel cells via an inverse homogenization technique which maximizes the contact surface subjected to a pressure drop and a permeability constraint. The optimal periodic design (fluid/solid) emerges from the application of a shape gradient algorithm coupled to a level-set method for the geometrical description of the corresponding cell problem.
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Submitted on : Friday, February 24, 2017 - 3:46:26 PM
Last modification on : Wednesday, March 27, 2019 - 4:08:31 PM
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Gabriel Delgado. Optimization of composite structures: A shape and topology sensitivity analysis. Optimization and Control [math.OC]. Ecole Polytechnique X, 2014. English. ⟨pastel-01005520v2⟩

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