Computational optimization tools for material design of elastic problems using inverse homogenization

JUAN MANUEL PODESTÁ; MENDEZ, CARLOS GUSTAVO; SEBASTIAN TORO; ALFREDO HUESPE; OLIVER, J.

Lisboa

Congreso; 6th International Conference on Engineering Optimization; 2018

Instituto Superior Técnico University of Lisbon - Institute of Mechanical Engineering

The objective of this work is to present new computational optimization toolswhich can be applied to obtain solutions of inverse homogenization problemsrelated to elastic material design satisfying structural requirements. Thespecific problem that is studied in this work consists of determining thematerial micro-architecture, such that the elastic effective properties of thisheterogeneous material copy those of a target elasticity tensor.The spatial distribution of the hard material phase within a given unit cell thatsatisfies the sought requirement is found through a rather conventionaltopology optimization problem [1]. In particular, we use a methodology that isbased on a topology optimization algorithm using topological derivative andthe level set function [2]. And the specific tools that we have developed in thiscomputational context exploit two aspects of this problem: i) the symmetry ofthe target elastic tensor, and ii) the shape of the unit cell in where the optimumtopology is sought [3].Therefore, we present a comprehensive analysis of the connection between thetarget tensor physical features, i.e. its symmetry properties, and the materialdistribution in the micro-structure. According to this idea and assuming periodicmicro-architectures, we analyze several Bravais lattices and the planewallpaper groups in order to study the way in which the symmetries of thesepatterns are reflected in the homogenized elasticity tensor. We analyze theWigner-Seitz cells, i.e. the primitive cells, of the subjacent Bravais lattices thatpreserve all the symmetries and the corresponding implementation of theplane wallpaper groups.Finally, using the connection between physical properties and micro-structuralpatters, we propose a procedure for the inverse problem that selects the mostconvenient external boundary shape of the unit cell in where the topologyoptimization problem is solved, as well as the necessary geometricalsymmetries to be imposed to the material distribution within the unit cell thatguarantee the symmetry of the homogenized elastic tensor of the designedmicro-architecture material.In the search of new extreme material classes, the proposed tools aim tofacilitate the inverse design by obtaining simple micro-architecture solutions. References[1] Bendsoe, M. P., & Sigmund, O. (2013). Topology optimization: theory, methods, and applications. [2] Amstutz, et al. (2010). Topological derivative for multi‐scale linear elasticity models applied to the synthesis of microstructures. IJNME. [3] Podestá J. M. et al. (2018). Material design of elastic structures using Voronoi cells. IJNME, accepted.