This paper is concerned with the topology optimization of structures made of periodically perforated material, where the microscopic periodic cell can be macroscopically modulated and oriented. The main idea is to optimize the homogenized formulation of this problem, which is an easy task of parametric optimization, then to project the optimal microstructure at a desired lengthscale, which is a delicate issue, albeit computationally cheap. The main novelty of our work is, in a plane setting, the conformal treatment of the optimal orientation of the microstructure. In other words, although the periodicity cell has varying parameters and orientation throughout the computational domain, the angles between its members or bars are conserved. The main application of our work is the optimization of so-called lattice materials which are becoming increasingly popular in the context of additive manufacturing. Several numerical examples are presented for compliance minimization in 2-d.
We present a partitioned procedure for fluid-structure interaction problems in which contacts among different deformable bodies can occur. A typical situation is the movement of a thin valve (e.g. the aortic valve) immersed in an incompressible viscous fluid (e.g. the blood). In the proposed strategy the fluid and structure solvers are considered as independent "black-boxes" that exchange forces and displacements; the structure solvers are moreover not supposed to manage contact by themselves. The hypothesis of non-penetration among solid objects defines a non-convex optimization problem. To solve the latter, we use an internal approximation algorithm that is able to directly handle the cases of thin structures and self-contacts. A numerical simulation on an idealized aortic valve is finally realized with the aim of illustrating the proposed scheme. Résumé : Nous présentons un algorithme de couplage partitionné pour des problèmes d'interaction fluide-structure dans lesquels des contacts peuvent se produire entre plusieurs solidesélastiques immergés. La méthode s'applique par exemple aux valves aortiques (qui sont constituées de trois valvules baignées dans un fluide visqueux incompressible). La stratégie proposée considère les solveurs fluide et structure comme des "boites noires" indépendantes. De plus, le solveurs structure n'est pas supposé savoir gérer le contact. La contrainte de non pénétration entre les solides n'est pas convexe. Le problème est résolu de manière itérative en considérant une suite de problèmes avec contrainte convexe. L'algorithme, qui est capable de traiter l'autocontact et les structures minces, est illustré sur une configuration idéalisée de valves aortiques.Mots-clés : interaction fluide-structure, contact, valves cardiaques FSI and multi-body contact. Appliaticon to aortic valves 3
We propose an alternative to the classical post-treatment of the homogenization method for shape optimization. Rather than penalize the material density once the optimal composite shape is obtained (by the homogenization method) in order to produce a workable shape close to the optimal one, we macroscopically project the microstructure of the former through an appropriate procedure that roughly consists in laying the material along the directions of lamination of the composite. We have tested our approach in the framework of compliance minimization in twodimensional elasticity. Numerical results are provided.
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