Penalization
Intermediate design variables do not have a clear physical interpretation so the optimization problem has to be modified if they appear in the optimal design. In the context of volume constraints/objectives, this is typically achieved using penalization. This means that, while θ is used for the volume computation, the penalized material volume factor, θp, is used for the material interpolation:
The SIMP (Solid Isotropic Material with Penalization) and RAMP (Rational Approximation of Material Properties) are used for solid mechanics, while the Darcy interpolation is used in fluid mechanics.