This article is concerned with the derivation of a posteriori error estimates for optimization problems subject to an obstacle problem. To circumvent the nondifferentiability inherent to this type of problem, we introduce a sequence of penalized but differentiable problems. We show differentiability of the central path and derive separate a posteriori dual weighted residual estimates for the errors due to penalization, discretization, and iterative solution of the discrete problems. The effectivity of the derived estimates and of the adaptive algorithm is demonstrated on two numerical examples. Introduction.This paper is concerned with the adaptive approximation of optimal control problems governed by the obstacle problem. The construction of the algorithm is based on a regularization approach in combination with an adaptive finite element discretization of the regularized problems. The two errors induced in this way, i.e., the regularization error and the discretization error, are equilibrated by means of suitable error estimators based on the dual weighted residual (DWR) method.Regarding the adaptive approximation of the obstacle problem itself, there is a large amount of contributions regarding a posteriori error estimates available in the literature; see, for instance, [6,30] for dual weighted error estimates and [17,12,32,8,9] for residual type estimates. In particular, we refer to [13], where residual type error estimates for a penalized obstacle problem were derived.In contrast to the solution of the obstacle problem itself, consideration of optimization problems subject to the obstacle problem is complicated by the nondifferentiability of the solution operator of the obstacle problem; see, e.g., [21]. To this end, we consider a sequence of penalized obstacle problems as constraints for our optimization problem. Such an approach is classical and has been investigated by various authors before. We only refer to [3,15,29] and the references therein. For the penalized but differentiable problems, we derive DWR error estimates following the pioneering work of [4]; see also [5,2]. More precisely, we utilize the DWR estimates for control constrained problems proposed in [33]. To simultaneously control the error due to penalization and discretization, we extend the ideas of [36,35] for optimization problems with regularized pointwise state constraints to regularization in the constraining equation; see also the survey [24]. Finally, we include the possibility to balance the former two errors with the error due to the iterative solution of the problems adapting the work of [26,25].

The paper presents a goal-oriented error control based on the dual weighted residual method (DWR) for the finite cell method (FCM), which is characterized by an enclosing domain covering the domain of the problem. The error identity derived by the DWR method allows for a combined treatment of the discretization and quadrature error introduced by the FCM. We present an adaptive strategy with the aim to balance these two error contributions. Its performance is demonstrated for several two-dimensional examples.

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For achieving high material removal rates while grinding free formed surfaces, shape grinding with toroid grinding wheels is favored. The material removal is carried out line by line. The contact area between grinding wheel and workpiece is therefore complex and varying. Without detailed knowledge about the contact area, which is influenced by many factors, the shape grinding process can only be performed sub-optimally. To improve this flexible production process and in order to ensure a suitable process strategy a simulation-tool is being developed. The simulation comprises a geometric-kinematic process simulation and a finite elements simulation. This paper presents basic parts of the investigation, modelling and simulation of the NC-shape grinding process with toroid grinding wheels.

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