✭✷✳✶✸✮ ✇❤❡r❡ t❤❡ ✐♥❞❡① s❡ts ❛r❡ ❞❡✜♥❡❞ ❛s ▼♦r❡♦✈❡r✱ ✇❡ ♦❜t❛✐♥✱ ✉s✐♥❣ ✭✸✳✶✮✱ t❤❛t t❤❡r❡ ❡①✐sts ❛ ♣♦s✐t✐✈❡ ❝♦♥st❛♥t C s✉❝❤ t❤❛t ❚❤❡ ❛❜♦✈❡ ❞✐s❝✉ss✐♦♥ s❤♦✇s ✭✸✳✶✽✮❚❤❡r❡❢♦r❡ (u, ϕ) ∈ H

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We propose and analyze a primal-dual active set method for local and nonlocal Allen-Cahn variational inequalities. An existence result for the non-local variational inequality is shown in a formulation involving Lagrange multipliers for local and non-local constraints. Superlinear local convergence is shown by interpreting the approach as a semi-smooth Newton method. Properties of the method are discussed and several numerical simulations demonstrate its efficiency.

Abstract. The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L 2 -and H −1 -gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches. Mathematics Subject Classification (2000). 74P15, 74P05, 74S03, 35K99.