The filtered (and milling) material volume factor can have large areas with intermediate values. This tends to make the optimization problem easier to solve, but the design might rely on areas with unphysical properties due to intermediate values. These areas can then be reduced using a projection operation based on the hyperbolic tangent function (see Ref. 3):

Here θβ and β are the projection point and slope, respectively, and θ is the output material volume factor. This is plotted in Figure 2-2 for θβ = 1/2 and β = 8. Projection can slow down the optimization progress. Therefore, it is turned off by default so that θ = θf.

Volume constraints are common in topology optimization and they are simple to implement using the average material volume factor, θavg: