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Description
Higher-order partial differential equations play an increasingly important role in structural topology optimization, particularly in level-set, phase-field, and thermodynamically consistent variational formulations arising in advanced applications such as lightweight structural design, multifunctional materials, compliant devices, and additive manufacturing.
In this work, we consider a topology optimization setting coupling a static mechanical model, taken here as linear elasticity for simplicity, with a fourth-order Cahn–Hilliard-type phase-field equation. The resulting problem is strongly nonlinear, nonconvex, and stiff, thus posing significant challenges for robust numerical solution. To address these difficulties, we adopt a variational formulation in which the coupled system is recast as a constrained optimization problem. Time discretization is performed by a fully implicit Euler scheme, while the spatial approximation relies on a mixed finite element formulation with standard finite elements, introducing an auxiliary field to reduce the original fourth-order phase-field equation to a system of two second-order equations. Pointwise constraints are enforced through a simple bound-enforcement strategy, which in turn influences the nonlinearity, convexity, and conditioning of the discrete problem. To further improve robustness, we employ a continuation technique acting on the phase-field free energy. The resulting framework provides a practical and flexible route for the numerical treatment of higher-order PDE-constrained topology optimization problems.