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 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.
Highlights
Having assigned a design region, a load distribution, and suitable boundary conditions, the goal of structural topology optimization is to identify an optimal distribution of material within the design region
Optimality is reached when the obtained material distribution minimizes a measure of structural compliance and satisfies mechanical equilibrium
Traditional schemes for topology optimization problems generally fix the amount of material to be distributed in the design region, amount which acts as a global constraint in the optimization process
Summary
Having assigned a design region, a load distribution, and suitable boundary conditions, the goal of structural topology optimization is to identify an optimal distribution of material within the design region. Traditional schemes for topology optimization problems generally fix the amount of material to be distributed in the design region, amount which acts as a global constraint in the optimization process This gold-standard strategy is easy to implement, since it adds only a single global equation in the computation; when the evolution of the topology is itself the solution of a finite element computation, the presence of such a global constraint requires to introduce global variables, altering the intrinsic element-based subdivision of the spatial discretization and introducing a coupling term between all elements. The phase-field variable obtained from the solution of the topology evolution equation (which possibly lies outside the admissible bound) is generally projected on the admissible set at each step of the iterative solution procedure – To show the effectiveness of the proposed volume minimization formulation and its finite element mixed implementation both in two-dimensional and three-dimensional applications
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