This contribution presents a theoretical and computational framework for two-scale shape optimisation of nonlinear elastic structures. Particularly, minimum compliance optimisation problems with composite (matrix-inclusion) microstructures subjected to static loads and volume-type design constraints are focused. A homogenisation-based FE$$^2$$ 2 scheme is extended by an enhanced formulation of variational (shape) sensitivity analysis based on Noll’s intrinsic, frame-free formulation of continuum mechanics. The obtained overall two-scale sensitivity information couples shape variations across micro- and macroscopic scales. A numerical example demonstrates the capabilities of the proposed variational sensitivity analysis and the (shape) optimisation framework. The investigations involve a mesh morphing scheme for the design parametrisation at both macro- and microscopic scales.

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The contribution presents the essential steps for the extension of the well‐known FE2 method by elements from variational design sensitivity analysis. The derived sensitivity information for a given structure can be used to suggest improved designs using methods for structural optimisation with objective functions, constraints and design variables on multiple scales. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nowadays profitability and efficiency of practical and industrial applications and therefore of material design is becoming a more important issue. Due to different well-known and established approaches for analysis and simulation of complex heterogeneous materials on multiple scales based on numerical homogenisation techniques, development and production of high performance materials (using 3d printers for example) and as a consequence optimisation and design of materials is reality. The objective is to find optimal structures with optimal material distribution under given constraints related to posed problems and tailoring applications to their special requirements.