B-DGTO: a new topology optimization approach enabling derivable signed distance feature in density method

Density method and level set method (LSM) stand as the two most prevalent topology optimization approaches. The former boasts strong robustness but suffers from ambiguous boundary geometric information, while the latter describes structural boundaries via implicit functions, enabling accurate high-order boundary information, but suffers from issues of initial guess dependency and incompatibility to standard optimizers. These two approaches have long been isolated with rare mutual compensations. In this paper, we proposed a novel approach that starts from the density field and achieves its transformation to the signed distance function (SDF) by solving design-dependent transient diffusion equation and Poisson equation. The derived SDF maintains geometric equivalence with the original density field. On this foundation, we further developed two B-DGTO (Boundary-fitting Derivable Geodesics-coupled Topology Optimization) frameworks: the density-based B-DGTO and the SDF-based B-DGTO, in supporting the density-level set co-topology optimization. The efficacy of these frameworks is validated through addressing mean curvature constraint and perimeter constraint on the L-brackets and thermal conduction structures. The proposed method provides a systematic framework integrating density method and LSM, holding profound implications for future development.