  This work presents a rigorous mathematical formulation for topology
optimization of a macrostructure undergoing ductile failure. The prediction of
ductile solid materials which exhibit dominant plastic deformation is an
intriguingly challenging task and plays an extremely important role in various
engineering applications. Here, we rely on the phase-field approach to fracture
which is a widely adopted framework for modeling and computing the fracture
failure phenomena in solids. The first objective is to optimize the topology of
the structure in order to minimize its mass, while accounting for structural
damage. To do so, the topological phase transition function (between solid and
void phases) is introduced, thus resulting in an extension of all the governing
equations. Our second objective is to additionally enhance the fracture
resistance of the structure. Accordingly, two different formulations are
proposed. One requires only the residual force vector of the deformation field
as a constraint, while in the second formulation, the residual force vector of
the deformation and phase-field fracture simultaneously have been imposed. An
incremental minimization principles for a class of gradient-type dissipative
materials are used to derive the governing equations. Here, the level-set-based
topology optimization is employed to seek an optimal layout with smooth and
clear boundaries. Sensitivities are derived using the analytical gradient-based
adjoint method to update the level-set surface for both formulations. Here, the
evolution of the level-set surface is realized by the reaction-diffusion
equation to maximize the strain energy of the structure while a certain volume
of design domain is prescribed. Several three-dimensional numerical examples
are presented to substantiate our algorithmic developments.
