Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results

CISM-course on Computational Nonlinear Mechanics

Advanced computational methods in nonlinear mechanics of solids and fluids are dealt with in this volume. Contributions consider large deformations of structures and solids, problems in nonlinear dynamics, aspects of earthquake analysis, coupled problems, convection-dominated phenomena, and compressible and incompressible viscous flows. Selected applications indicate the relevance of the analysis to the demands of industry and science. The contributors are from research institutions well-known for their work in this field

Elements of plasticity : theory and computation

311 p. : ill. ; 24 cm

Numerical analysis and design of industrial superplastic forming

In the aerospace industry, the production of structural shell components with a prescribed thickness distribution by superplastic forming requires the specification of the initial thickness profile in the undeformed sheet. To this purpose, a finite element simulation methodology developed previously for the analysis of superplastic deformation processes along with the optimum monitoring of the forming pressure is extended to the determination of the initial thickness profile. The utilization of the procedure is shown for the pre-contouring of the blank for the industrial production of a satellite tank. The examination of the produced part confirms that the requested thickness distribution is achieved fairly well and thus proves the appropriateness of the proposed computational approach as a tool for the process design in superplastic net-shape forming