Computational Vademecums for Lattice Materials using algebraic PGD

This dissertation is motivated by the concept of materials by design. Focusing on structures, this states that the properties in a mechanical component are not only inherited by its constituent material, but also by the shape in which this one is distributed in space. Although this notion has been developed for thousands of years in architecture, its relevance at a smaller scale has not been conceived until the arrival of additive manufacturing technologies. The materials by design approach is certainly multidisciplinary, from the study of the shape, its representation in CAD, to the final manufacturing by 3D printing. We here assess the first one, by means of numerical simulations. These play an important role, by reinforcing our physical intuition in the resolution of the following question: which is the material structure at a small scale (meso-structure) that features some prescribed properties at a global scale (bulk material). We introduce the material meso-structure as a lattice model with a parametric shape. The mechanical properties arising at the global scale are recovered by solving an equilibrium problem. Naturally, this one acquires the parametric nature of the lattice model. The main difficulty to handle the emerging mechanical properties of the bulk parametrically, is that the computational complexity of numerical simulations increases exponentially with the number of parameters. To overcome this, we resort to the Proper Generalized Decomposition (PGD), which provides explicit parametric solutions of our equilibrium problem. As a major contribution, we obtain the parametric solutions of the algebraic equations arising from different lattice structures using the same PGD framework. In this sense, the algebraic PGD works as a non-intrusive solver, which is not limited to structural problems but in general, any discrete form of a parametrized linear PDE. The parametric mechanical properties of 2D and 3D lattice materials are explicitly represented by the PGD solutions or computational vademecums. In particular, we reproduce the response of orthotropic Poisson's ratios in terms of the design parameters. Extreme negative values are identified, a characteristic that is relevant regarding the outperforming of auxetic (or negative Poisson's ratios) properties compared to conventional materials. Moreover, these computational vademecums could be further exploited to tailor the material design through multi-objective and constraint optimizations, providing an efficient tool to browse the parametric design space. Finally, we extend our parametric analysis using geometrically nonlinear finite elements to compute equilibrium, and the algebraic PGD a posteriori to interpolate their response. This is achieved with very good accuracy, for engineering purposes, at a considerably low number of modes. The nonlinear parametric framework surely broadens the range of applications, and we highlight this in two distinctive situations. First, we demonstrate its capability to describe the loading magnitude as an extra parameter to the material properties behavior. Last but not least, we show its potential to perform buckling analysis of lattice structures, as a function of the geometric parameters and the loading magnitude itself.