Mixed variational formulations for structural topology optimization based on the phase-field approach

TL;DR

This paper introduces a novel mixed variational formulation combining phase-field and Hu-Washizu functionals for structural topology optimization, enabling flexible material distribution strategies and efficient numerical solutions in 2D and 3D.

Contribution

It develops a new variational approach integrating phase-field and mixed finite element methods for topology optimization, including a formulation minimizing material without pre-set constraints.

Findings

The approach effectively distinguishes between constrained and unconstrained formulations.

Numerical results demonstrate the method's robustness and applicability to 2D and 3D problems.

The monolithic solution scheme improves computational efficiency and design quality.

Abstract

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.