The Helmholtz Filter
One can impose a minimum length scale, Rmin, on a domain control variable, θc, using a Helmholtz filter (see Ref. 2):
Here θf is the filtered material volume factor. The mesh element size is taken as the default filter radius, but the filter radius should not be smaller than mesh element size, and you have to pick a fixed value to get mesh-independent designs. The filtered variable is continuous and discretized with linear polynomials. The filter can be turned off, in which case θf = θc holds. Figure 2-1 illustrates the filtering process, and it is taken from the Application Library model Topology Optimization of an MBB Beam.
Figure 2-1: A symmetric topology optimization problem is shown with the control variable to the left and the filtered variable to the right.