Topology Optimization of Structures Made of Discrete Geometric Components With Different Materials

Abstract

We present a new method for the simultaneous topology optimization and material selection of structures made by the union of discrete geometric components, where each component is made of one of multiple available materials. Our approach is based on the geometry projection method, whereby an analytical description of the geometric components is smoothly mapped onto a density field on a fixed analysis grid. In addition to the parameters that dictate the dimensions, position, and orientation of the component, a size variable per available material is ascribed to each component. A size variable value of unity indicates that the component is made of the corresponding material. Moreover, all size variables can be zero, signifying the component is entirely removed from the design. We penalize intermediate values of the size variables via an aggregate constraint in the optimization. We also introduce a mutual material exclusion constraint that ensures that at most one material has a unity size variable in each geometric component. In addition to these constraints, we propose a novel aggregation scheme to perform the union of geometric components with dissimilar materials. These ingredients facilitate treatment of the multi-material case. Our formulation can be readily extended to any number of materials. We demonstrate our method with several numerical examples.

Multi-material topology optimization of lattice structures using geometry projection

“…To calculate the projected density considering the contribution of multiple bars, a p-norm approximation of the maximum function is employed in [35,36]. Unlike these works, in which all bars are made of the same isotropic material, [34] presented a new aggregation scheme to account for the intersection of bars made of different materials by defining an effective density for material i at point p, i.e.,…”

“…To compute the projected density at a point in any of the other regions, we reflect the point with respect to the appropriate symmetry planes so that the reflected point lies on the reference region, and then we perform the geometry projection as usual. This strategy is employed in [34] and is similar to the one introduced in [39]. To perform the reflection with respect to the appropriate symmetry planes, we multiply all the corresponding reflection matrices (we assume all symmetry planes pass through the origin of the unit cell coordinate system).…”

“…Other interpolation schemes (i.e., [30,31]) have also been used to design multi-phase materials with extreme thermal conductivity [32,33]. The reader is referred to [34] for a more detailed description of these approaches.…”

“…This work focuses on the topology optimization of multi-material lattice structures using the geometry projection method. By employing the multi-material interpolation and optimization constraints introduced in [34], the proposed method can readily accommodate any number of materials. It employs inverse homogenization to design the unit cell for extremizing the effective properties of the macrostructure.…”

Adaptive mesh refinement for topology optimization with discrete geometric components

“…To further improve the manufacturability of the optimal design, some geometric constraints have been introduced, such as bounds on the dimensions of the components [3,4], ensuring a minimum separation between geometric components to allow for tool access [6], and ensuring geometric components are fully contained withing irregular, non-convex design regions to avoid impractical cuts [8]. The geometry projection method has also been used for the topology optimization of multimaterial lattices [11] and structures [9].…”