Undercut and Overhang Angle Control in Topology Optimization: a Density Gradient based Integral Approach

We present an approach for controlling the undercut and the minimal overhang angle in density based topology optimization, which are useful for reducing support structures in additive manufacturing. We cast both the undercut control and the minimal overhang angle control that are inherently constraints on the boundary shape into a domain integral of Heaviside projected density gradient. Such a Heaviside projection based integral of density gradient leads to a single constraint for controlling the undercut or controlling the overhang angle in the optimization. It effectively corresponds to a constraint on the projected perimeter that has undercut or has slope smaller than the prescribed overhang angle. In order to prevent trivial solutions of intermediate density to satisfy the density gradient constraints, a constraint on density grayness is also incorporated into the formulations. Numerical results on MBB beams, cantilever beams, and 2D and 3D heat conduction problems demonstrate the proposed formulations are effective in controlling the undercut and the minimal overhang angle in the optimized designs.