Topology Optimization of Multi-Material Lattices for Maximal Bulk Modulus

In this paper, we present a method for multi-material topology optimization of lattice structures for maximum bulk modulus. Unlike ground structure approaches that employ 1-d finite elements such as bars and beams to design periodic lattices, we employ a 3-d representation where each lattice bar is described as a cylinder. To accommodate the 3-d bars, we employ the geometry projection method, whereby a high-level parametric description of the bars is smoothly mapped onto a density field over a fixed analysis grid. In addition to the geometric parameters, we assign a size variable per material to each bar. By imposing suitable constraints in the optimization, we ensure that each bar is either made exclusively of one of a set of a multiple available materials or completely removed from the design. These optimization constraints, together with the material interpolation used in our formulation, make it easy to consider any number of available materials. Another advantage of our method over ground structure approaches with 1-d elements is that the bars in our method need not be connected at all times (i.e., they can 'float' within the design region), which makes it easier to find good designs with relatively few design variables. We illustrate the effectiveness of our method with numerical examples of bulk modulus maximization for two-material lattices with orthotropic symmetry, and for two- and three material lattices with cubic symmetry.