Topology optimization methods employing binary (also known as discrete) design variables currently lack mathematical formulations to ensure length scale control in their solutions. This paper proposes and applies a morphology-mimicking filtering scheme to provide a minimum size control (often also referred to as minimum length scale control) in this class of binary designs. The Topology Optimization of Binary Structures (TOBS) method was chosen as the foundational framework for this length scale control study. Thermal and structural compliance scenarios were explored under this approach. Numerical results show that the proposed filter efficiently imposes the desired minimum length scale. The optimized designs were also less dependent on the filtering parameters when compared to designs optimized using standard techniques that employ continuous design variables. 

This article proposes a boundary strip indicator for density-based topology optimization that can be used to estimate thedesign’s surface area (perimeter in 2D) or identify a coating layer. We investigate the theoretical properties of the proposedboundary strip indicator and propose a diferentiable approximation that preserves key properties, such as non-negativity.Finally, we use the boundary strip indicator in a heat conduction design optimization problem for a coated structure. Theresulting designs show a strong dependence on the properties of the coating.

We present an approach to binary topology optimization for coated structures. We study a modified version of the standard minimum heat compliance problem that incorporates a coating layer with different conductivity and cost compared to the base material. Our methodology involves defining the discrete material distribution and employing morphology-mimicking non-linear filter operators. We evaluate the effectiveness of our approach through numerical experiments conducted on a modified version of the classic thermal minimal compliance problem. The results demonstrate that our proposed method generates optimized designs that achieve excellent performance for various coating thicknesses. Moreover, the results highlight the interplay between the relative conductivity and cost of the base material and the coating.

This review focuses on material distribution-based topology optimization methods for boundary-effect-dominated problems. More precisely, it addresses problems where the behavior at or near the boundaries of the domain significantly influences the physics, such as problems involving boundary layers or the skin effect. While traditional topology optimization techniques have been highly successful in idealized settings, boundary-sensitive problems introduce unique challenges. We survey the historical development of relevant ideas, including fictitious-domain methods and filtering techniques, and provide a detailed account of modern approaches for handling boundary effects. Key topics include cascades of filters, multi-field representations, and methods for controlling length scale and interface sharpness. We also review specialized strategies for pressure and thermal loads, as well as recent advances in the design of coated structures and impedance-based modeling of boundary layers. This article aims to provide a comprehensive and structured overview of the field.