Abstract
We consider optimal control problems with distributed control that involve a time-stepping formulation of dynamic one body contact problems as constraints. We link the continuous and the time-stepping formulation by a nonconforming finite element discretization and derive existence of optimal solutions and strong stationarity conditions. We use this information for a steepest descent type optimization scheme based on the resulting adjoint scheme and implement its numerical application.
Highlights
The following work concerns the optimal control of time discretized, dynamic contact problems of a linearly viscoelastic body and a rigid obstacle in the absence of friction, where a linearized nonpenetration condition is employed
The aims of this work are the following: For the optimal control of dynamic contact problems, we first derive a discontinuous finite element formulation in time that yields the contact implicit Newmark scheme [21] with slight modifications concerning the treatment of external forces and the influence of the control
This paper focuses on optimal control problems with a weak formulation of dynamic, viscoelastic contact problems as side constraints and the following section is dedicated to the presentation of the configuration of interest
Summary
The following work concerns the optimal control of time discretized, dynamic contact problems of a linearly viscoelastic body and a rigid obstacle in the absence of friction, where a linearized nonpenetration condition is employed. This condition is referred to as the Signorini condition, after first being introduced by Signorini in [43] in the static one body context. Contact problems have a multitude of applications in mechanics, engineering and medicine, and are well understood in the static context nowadays They are closely related to obstacle problems and both are modeled through structurally similar, elliptic variational inequalities. In [36], Lions and Stampacchia were the first to show the existence of a generally nonlinear but Lipschitz continuous solution operator to these variational inequalities (cf. [23]) and from Mignot’s work in [37], we know the solution operator to even be directionally differentiable in case
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