Shape design with phase field methods for structural hemivariational inequalities in contact problems

We develop mathematical models for shape design and topology optimization in structural contact problems involving friction between elastic and rigid bodies. The governing mechanical constraint is a nonlinear, non-smooth, and non-convex hemivariational inequality, which provides a more general and realistic description of frictional contact forces than standard variational inequalities, but is also more challenging due to its non-convexity. For energy-type shape functionals, the Eulerian derivative of the hemivariational inequality is derived through rigorous shape sensitivity analysis. The rationality of a regularization approach is justified by asymptotic analysis, and this method is further applied to handle the non-smoothness of general shape functionals in the sensitivity framework. Based on these theoretical results, a numerical boundary variational method is proposed for shape optimization. For topology optimization, three phase-field algorithms are developed: a gradient-flow phase-field method, a phase-field method with second-order regularization of the cost functional, and a phase-field method coupled with topological derivatives. To the best of our knowledge, these approaches are new for shape design in hemivariational inequalities. Various numerical experiments confirm the accuracy and effectiveness of the proposed shape and topology optimization algorithms.

💡 Research Summary

This paper addresses the challenging problem of shape and topology optimization for elastic structures that are in frictional contact with a rigid foundation. The mechanical behavior is modeled by a hemivariational inequality (HVI), which captures non‑smooth and non‑convex friction laws through a Clarke sub‑differential of a locally Lipschitz potential jτ. Compared with classical variational inequalities, HVI provides a more realistic description of friction but introduces serious analytical difficulties: existence and uniqueness of solutions, and the lack of classical differentiability needed for sensitivity analysis.

The authors first establish the mathematical setting. The domain Ω⊂ℝᵈ (d = 2, 3) is bounded with a smooth boundary split into contact, Neumann, Dirichlet and free parts. The elasticity tensor C is symmetric, bounded and uniformly positive definite. The friction potential jτ satisfies measurability, local Lipschitz continuity, linear growth of its sub‑gradient, and a monotonicity condition. Under the condition α_{jτ}<λ₁ m_C (where λ₁ is the smallest eigenvalue of an associated trace eigenproblem) they prove existence and uniqueness of the weak solution u∈V₁ of the HVI.

For shape optimization, the objective functional is of energy type: J(Ω)=½ a(u,u)−(f,u)−(g_N,u)+Ĵ(u_τ), where a(·,·) is the elastic energy, and Ĵ integrates the friction potential over the contact boundary. The design variable is the domain Ω itself, constrained by a prescribed volume. Using a perturbation of identity mapping F_t=I+tV, the paper defines the Eulerian derivative dJ(Ω;V) and derives it rigorously via material derivatives and Clarke’s generalized directional derivative. Lemma 3.2 guarantees the continuity of the state solution under domain perturbations, which is essential for the shape sensitivity analysis. For the energy functional, the Eulerian derivative can be expressed without any regularization, thanks to the regularity of the friction potential.

When the objective functional is not of pure energy type, the non‑smooth term Ĵ prevents direct differentiation. The authors introduce a regularization scheme: they replace the locally Lipschitz potential by a smooth approximation jτ^ε and prove, through asymptotic analysis, that as ε→0 the regularized problem converges to the original HVI. This justification allows them to formulate an adjoint problem for the regularized system and to compute the Eulerian derivative of a general cost functional. The resulting shape gradient is then employed in a boundary variational method, where a Newton iteration solves the nonlinear state and adjoint equations, and a line search updates the domain.

The topology optimization part presents three phase‑field algorithms, all built on the regularized HVI. The first algorithm uses an Allen‑Cahn L²‑gradient flow for the phase‑field variable φ, driving the system toward a stationary configuration that minimizes the regularized objective while satisfying a volume constraint. The second algorithm augments the objective with a second‑order regularization term β‖∇φ‖², which smooths the interface and improves numerical stability. The third algorithm couples the phase‑field evolution with a topological derivative, enabling the creation or removal of small holes in a single step, thus accelerating convergence toward topologically optimal designs. All three schemes employ Newton–Krylov solvers and adaptive mesh refinement to handle the strong nonlinearity and the diffuse interface.

Numerical experiments cover 2‑D and 3‑D test cases. In a 2‑D rectangular domain, the shape optimization reduces the elastic energy while respecting the volume constraint, and the resulting contact stresses match the theoretical Coulomb law. Varying the friction law parameters demonstrates the sensitivity of the optimal shape to the friction model. The topology optimization experiments illustrate the formation of voids and material bridges, the effect of the second‑order regularization on interface thickness, and the rapid hole nucleation achieved by the topological‑derivative‑enhanced algorithm. Convergence studies confirm that the algorithms are robust with respect to mesh size, time step, and regularization parameters.

In conclusion, the paper delivers a comprehensive theoretical and computational framework for shape and topology design of structures governed by hemivariational inequalities. By combining rigorous shape sensitivity analysis, justified regularization, and innovative phase‑field strategies (including topological derivatives), it overcomes the non‑smooth, non‑convex nature of frictional contact problems. The work opens avenues for further research on dynamic contact, nonlinear material behavior, and multiphysics extensions.