Multi-scale optimisation of variable-stiffness composites for thermal cloak

Abstract

Variable-stiffness composites (VSCs) can be efficiently designed, through multi-scale optimisation, to obtain thermal cloaks that can steer the heat flux to conceal the presence of an obstacle. Specifically, a general class of VSC structures characterised by variable fibres volume fraction, thickness and orthotropy orientation is considered in this work. The theoretical/numerical framework relies on the use of the polar formalism to describe the anisotropic thermal conductivity tensor of the VSC at the macroscopic scale, and on a general numerical homogenisation method to set the link between the design variables and the physical responses defined at different scales. In this context, the goal is to determine the optimal distribution of the fibres volume fraction, the orientation of the main orthotropy axis and the thickness of the VSC structure in order to design an efficient thermal cloak. The design variables fields are represented through non-uniform rational basis spline (NURBS) entities in order to achieve solutions that are compliant with standard computer-aided design software. Moreover, some properties of the NURBS entities, such as the local support property, are conveniently exploited to formally derive the gradient of the physical responses (and hence to speed-up the optimisation process), to automatically satisfy some manufacturability constraints (e.g., the continuity of the fibres-path) and to obtain mesh-independent solutions. The general nature of the proposed approach enables the concurrent optimisation of geometrical and physical variables at multiple scales, thus allowing the identification of optimised solutions that overcome the inherent limitations of the analytical solutions. The effectiveness of this approach is tested on benchmark problems taken from the literature. This entails an investigation into the impact of various factors, including the initial guess, the microscopic configuration of the constitutive phases of the composite material, the boundary conditions, the local thickness, the size and shape of the design region on the optimised solution.