Abstract

Multiscale topological material design, aiming at obtaining optimal distribution of the material at several scales in structural materials is still a challenge. In this case, the cost function to be minimized is placed at the macro-scale (compliance function), but the design variables (material distribution) lie at both the macro-scale and the micro-scale. The large number of involved design variables and the multi-scale character of the analysis, resulting into a multiplicative cost of the optimization process, often make such approaches prohibitive, even if in 2D cases.

In this work, an integrated approach for multi-scale topological design of structural linear materials is proposed. The approach features the following properties:

• The “topological derivative” is considered the basic mathematical tool to be used for the purposes of determining the sensitivity of the cost function to material removal. In conjunction with a level-set-based “algorithm” it provides a robust and well-founded setting for material distribution optimization.
• The computational cost associated to the multiscale optimization problem is dramatically reduced by resorting to the concept of the online/offline decomposition of the computations. A “Computational Vademecum” containing the micro-scale solution for the topological optimization problem in a RVE for a large number of discrete macroscopic stress-states, is used for solving that problem by simple consultation.
• Coupling of the optimization problem at both scales is solved by a simple iterative “fixedpoint” scheme, which is found to be robust and convergent.
• The proposed technique is enriched by the concept of “manufacturability”, i.e.: obtaining sub-optimal solutions of the original problems displaying homogeneous material over finite sizes domains at the macrostructure: the “structural components”.

The approach is tested by application to some engineering examples, involving minimum compliance design of material and structure topologies, which show the capabilities of the proposed framework.