Topology optimization of isotropic brittle materials for external-work capacity under quasi-static phase-field fracture

This paper presents a two-dimensional density-based topology optimization framework for isotropic brittle materials under quasi-static phase-field fracture. The objective is to maximize cumulative external work along a prescribed displacement history while accounting for damage evolution. A stepwise coupled sensitivity formulation with causal active-history propagation is derived from the converged displacement and phase-field equations and retains the design dependence of the maximum-history field. The complete fracture analysis is performed first, after which the recorded active branches are propagated and their material contributions are accumulated over the loading sequence. A separate full-history adjoint sensitivity formulation records the same branches during the forward solution and returns the cross-step contributions in reverse load order for directional verification. The formulation is incorporated into a SIMP-based optimization problem with a volume-fraction constraint. Directional finite-difference checks on a cantilever beam and a half-MBB beam show that the full-history adjoint sensitivity is closer to the centered difference than the Li et al. method in the prescribed direction for each case. The numerical examples present the optimized topologies, crack evolution and structural response under the prescribed loading histories.