arXiv subjects

Winnifried Wollner

Publications and source records attributed to Winnifried Wollner.

24 records · Page 2Linked to original sources

arXiv2020

Finite Element Methods for One Dimensional Elliptic Distributed Optimal Control Problems with Pointwise Constraints on the Derivative of the State

We investigate $C^1$ finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing $H^{\frac52-ε}$ regularity result for the optimal state to derive $O(h^{\frac12-ε})$ convergence for the approximation of the optimal state in the $H^2$ norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to $H^3$ under appropriate assumptions on the data and obtain $O(h)$ convergence for the approximation of the optimal state in the $H^2$ norm.

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arXiv2019

Optimization with nonstationary, nonlinear monolithic fluid-structure interaction

Within this work, we consider optimization settings for nonlinear, nonstationary fluid-structure interaction. The problem is formulated in a monolithic fashion using the arbitrary Lagrangian-Eulerian framework to set-up the fluid-structure forward problem. In the optimization approach, either optimal control or parameter estimation problems are treated. In the latter, the stiffness of the solid is estimated from given reference values. In the numerical solution, the optimization problem is solved with a gradient-based solution algorithm. The nonlinear subproblems of the FSI forward problem are solved with a Newton method including line search. Specifically, we will formally provide the backward-in-time running adjoint state used for gradient computations. Our algorithmic developments are demonstrated with some numerical examples as for instance extensions of the well-known fluid-structure benchmark settings and a flapping membrane test in a channel flow with elastic walls.

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arXiv2019

Mesh adaptivity for quasi-static phase-field fractures based on a residual-type a posteriori error estimator

In this work, we consider adaptive mesh refinement for a monolithic phase-field description for fractures in brittle materials. Our approach is based on an a posteriori error estimator for the phase-field variational inequality realizing the fracture irreversibility constraint. The key goal is the development of a reliable and efficient residual-type error estimator for the phase-field fracture model in each time-step. Based on this error estimator, error indicators for local mesh adaptivity are extracted. The proposed estimator is based on a technique known for singularly perturbed equations in combination with estimators for variational inequalities. These theoretical developments are used to formulate an adaptive mesh refinement algorithm. For the numerical solution, the fracture irreversibility is imposed using a Lagrange multiplier. The resulting saddle-point system has three unknowns: displacements, phase-field, and a Lagrange multiplier for the crack irreversibility. Several numerical experiments demonstrate our theoretical findings with the newly developed estimators and the corresponding refinement strategy.

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arXiv2019

Quasi-best approximation in optimization with PDE constraints

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square-root of the Tikhonov regularization parameter. Furthermore, if the operators of control-action and observation are compact, this quasi-best-approximation constant becomes independent of the Tikhonov parameter as the meshsize tends to $0$ and we give quantitative relationships between meshsize and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.

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arXiv2019

Multigoal-oriented optimal control problems with nonlinear PDE constraints

In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to the regularized $p$-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated via several numerical examples.

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arXiv2019

A phase-field model for fractures in incompressible solids

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented.

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