Keywords: multiscale, optimization, composites, homogenization, laminates, topology.

Multiscale topology optimization addresses the problem of finding optimal material distributions at different but interconnected structural length scales with the objective of optimally designing the structure and its material. Structure and material evolve concurrently for their optimal layouts as a result of updating the density based design variables such that the global compliance is minimized and a global resource volume constraint is satisfied.

In structural applications fibre reinforced polymers are usually stacked in a number of layers, each consisting of strong fibres bonded together by a resin, to form a laminate. Here we present a multi-scale structural optimization model that takes into account the manufacturing process and characteristics of composite laminates. Layered composites structures have gained an ever increasing popularity in the aerospace, automotive and ship industries because of theirtheir very high strength to weight ratio. The focus of the present work is on the composite structures where fibre mats covering larger areas are often used.

In hierarchical topology optimization the structure design domain is usually discretized using a conforming finite element mesh. The macroscopic density based design variables may be associated with each finite element of the model. This design parameterization enables optimal designs with very high mechanical efficiency but very difficult to manufacture due to its geometrical complexity. Alternatively the number of design variables may be reduced by assuming a design parameterization where the design is assumed uniform within larger subdomains of the structure ("design element") that maybe associated with the different structural constituents (e.g. a lamina within a laminate). This is a very effective approach for practical design problems such as laminated composite structures since they are typically made of laminas constructed from fibres within a medium of uniform macro mechanical properties.

The focus of the present work is on the compliance minimization for this type of composite structures. It assumes a mixed set of micro and macro independent design variables, to characterize the distribution of two materials to obtain the optimal composite microstructures at the micro design level as well as the optimal fibre orientation at the macro level.

Three-dimensional examples involving laminated beams and plates are shown to demonstrate the relevance of the proposed design methodology and also to recognize the optimality of this type of bi-material composite structures.

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