  We propose a variational principle combining a phase-field functional for
structural topology optimization with a mixed (three-field) Hu-Washizu
functional, then including directly in the formulation equilibrium,
constitutive, and compatibility equations. The resulting mixed variational
functional is then specialized to derive a classical topology optimization
formulation (where the amount of material to be distributed is an \emph{a
priori} assigned quantity acting as a global constraint for the problem) as
well as a novel topology optimization formulation (where the amount of material
to be distributed is minimized, hence with no pre imposed constraint for the
problem).
  Both formulations are numerically solved by implementing a mixed finite
element scheme, with the second approach avoiding the introduction of a global
constraint, hence respecting the convenient local nature of the finite element
discretization. Furthermore, within the proposed approach it is possible to
obtain guidelines for settings proper values of phase-field-related simulation
parameters and, thanks to the combined phase-field and Hu-Washizu rationale, a
monolithic algorithm solution scheme can be easily adopted.
  An insightful and extensive numerical investigation results in a detailed
convergence study and a discussion on the obtained final designs. The numerical
results clearly highlight differences between the two formulations as well as
advantages related to the monolithic solution strategy; numerical
investigations address both two-dimensional and three-dimensional applications.
