Sensitivity of the Second Order Homogenized Elasticity Tensor to Topological Microstructural Changes

The multiscale elasticity model of solids with singular geometrical perturbations of microstructure is considered for the purposes, e.g., of optimum design. The homogenized linear elasticity tensors of first and second orders are considered in the framework of periodic Sobolev spaces. In particular, the sensitivity analysis of second order homogenized elasticity tensor to topological microstructural changes is performed. The derivation of the proposed sensitivities relies on the concept of topological derivative applied within a multiscale constitutive model. The microstructure is topologically perturbed by the nucleation of a small circular inclusion that allows for deriving the sensitivity in its closed form with the help of appropriate adjoint states. The resulting topological derivative is given by a sixth order tensor field over the microstructural domain, which measures how the second order homogenized elasticity tensor changes when a small circular inclusion is introduced at the microscopic level. As a result, the topological derivatives of functionals for multiscale models can be obtained and used in numerical methods of shape and topology optimization of microstructures, including synthesis and optimal design of metamaterials by taking into account the second order mechanical effects. The analysis is performed in two spatial dimensions however the results are valid in three spatial dimensions as well.

Topology Optimization of Microstructures to Reduce Structural Oscillations

Vibration suppression in a field of frequencies and the creation of band gaps which do not allow the propagation of the waves is studied in this paper by means of microstructures designed through topology optimization. Topology optimization is formulated in frequency domain. Band gap design is based on Floquet-Bloch theory and genetic optimization. The result is appealing in view of modern 3D printing techniques.

Architectural Morphogenesis Through Topology Optimization

This chapter illustrates the main approach for a generative use of structural optimization in architecture. Structural optimization is very typical of sectors like mechanical, automotive engineering, while in architecture it is a less used approach that however could give new possibilities to performative design. Topology Optimization, one of its most developed sub-methods, is based on the idea of optimization of material densities within a given design domain, along with least material used and wasted energy. In the text is provided a description of TO methods and the principles of their utilization. The process of topology optimization of microstructures of cellular materials is represented and illustrated, emphasizing the all-important criteria and parameters for structural design. A specific example is given of the research at ACTLAB, ACB Dept, Politecnico di Milano, of performative design with lattice cellular solid structures for architecture.