Grégoire Allaire

Structural optimization under overhang constraints imposed by additive manufacturing technologies

Abstract This article addresses one of the major constraints imposed by additive manufacturing processes on shape optimization problems – that of overhangs, i.e. large regions hanging over void without sufficient support from the lower structure. After revisiting the ‘classical’ geometric criteria used in the literature, based on the angle between the structural boundary and the build direction, we propose a new mechanical constraint functional, which mimics the layer by layer construction process featured by additive manufacturing technologies, and thereby appeals to the physical origin of the difficulties caused by overhangs. This constraint, as well as some variants, is precisely defined; their shape derivatives are computed in the sense of Hadamards method, and numerical strategies are extensively discussed, in two and three space dimensions, to efficiently deal with the appearance of overhang features in the course of shape optimization processes.

Molding Direction Constraints in Structural Optimization via a Level-Set Method

In the framework of structural optimization via a level-set method, we develop an approach to handle the directional molding constraint for cast parts. A novel molding condition is formulated and a penalization method is used to enforce the constraint. A first advantage of our new approach is that it does not require to start from a feasible initialization, but it guarantees the convergence to a castable shape. A second advantage is that our approach can incorporate thickness constraints too. We do not address the optimization of the casting system, which is considered a priori defined. We show several 3d examples of compliance minimization in linearized elasticity under molding and minimal or maximal thickness constraints. We also compare our results with formulations already existing in the literature.

SOLVING LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES WITH GMRES : AN APPLICATION TO AIRCRAFT DESIGN

To create efficient new aerodynamic designs or predict the onset of flutter, the linearised Navier-Stokes equations might be used. In some cases, many right-hand sides must be solved keeping the same matrix. In this paper, techniques which enable to solve several righthand sides at the same time, such as Block GMRes, or reuse pieces of information computed in the previous solves, such as Krylov space recycling, are investigated. They will be tested on both simple and industrial test cases.

DESIGN OF ISOTROPIC MICROSTRUCTURES VIA A TWO-SCALE APPROACH

Architectured materials are promising to reach extreme properties and ultimately address issues related to lightweight or non conventional properties for bulk materials (eg. high specific rigidity, extremal conductivity or auxetism (negative Poisson’s ratio)) [1]. A very efficient way to obtain optimal forms is via inverse homogenization, i.e. using shape and topology optimization techniques in order to achieve target material properties [2]. A great number of publications has been devoted to the design of isotropic materials with extreme properties. Isotropy is usually prescribed via a combination of symmetric planes and penalization techniques, which are quite delicate to handle in an optimization framework. In this work, we present an approach for the design of isotropic multi-materials with extremal conductivity via laminate geometries, consisting in anisotropic phases [3]. More specifically, we design composites with extremal conductivity using rank-1 laminates, composed by two orthotropic phases along parallel layers. The second phase is obtained by a 90-degree rotation of the first one, while their volume fractions are explicitly chosen so that the laminate is isotropic. The orthotropic phases are considered to have their own periodic micro-structure, composed by multiple phases. By optimally distributing the different phases in a periodic cell, we can achieve the Hashin-Shtrikman bounds for the isotropic laminate. We present examples in two dimensions using the level-set method for shape and topology optimization [4]. Alexis Faure, Georgios Michailidis, Rafael Estevez, Guillaume Parry and Grégoire Allaire