A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

Abstract

In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with non-homogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.

Necessary conditions of first-order for an optimal boundary control problem for viscous damage processes in 2D

“…The corresponding optimal control problem then becomes a difficult and unexplored mathematical program with complementarity constraints (MPCC) and it remains open if stationarity conditions can be obtained via a limit passage of the regularized version as considered in this paper. As pointed out in [1] (see [4] for our case) optima of the regularized control problem approximate solutions of the MPCC.…”

“…(ii) Note that in [4] the assumptions for c 1 and c 2 are stated as c 1 ∈ C 1,1 (R) convex and c 2 ∈ C 1,1 (R) concave with c, c ′ 1 , c ′ 2 bounded and c ≥ 0. There, C 1,1 (R) denotes the space of differentiable functions whose derivatives are Lipschitz continuous.…”

“…C(χ)ε(u) + Dε(u t ) · ν = b a.e. on Σ, (4) ∇(χ + χ t ) · ν = 0 a.e. on Σ, (5) u(0) = u 0 , u t (0) = v 0 , χ(0) = χ 0 a.e.…”

Shape Optimization for a Class of Semilinear Variational Inequalities with Applications to Damage Models

“…A weak formulation of (61) and existence of weak solution can be found in [12] with minor adaption. Existence and uniqueness results for strong solutions for the above system with higher-order viscous terms are established in [8]. For the analysis of quasilinear variants of (61) and for rate-independent as well as rate-dependent cases, we refer to [17] and the references therein.…”

“…In this section a time-discrete model to (61) will be investigated in a thermodynamically consistent scheme (in this context it indicates that the time-discrete energy-dissipation inequality is satisfied). A related time-discretisation scheme has been used in [8]. For all k ∈ {1, .…”