No abstract

We consider an optimal boundary control problem of the heat equation with convection with bilateral control and state constraints. The goal is to keep the temperature (state) within a desired range while the norm of the boundary controls is minimized. To numerically approximate the unique solution of the problem, a first-order augmented Lagrangian method is utilized. The gradient of the augmented Lagrangian is characterized by the solution of the adjoint equation. To decrease the computational time, we apply a reduced-order approach based on proper orthogonal decomposition (POD). To gain a sufficiently accurate approximation, we propose and compare different manners to compute the snapshots and thus the POD basis. Numerical tests compare the efficiency of the proposed strategies.

The purpose of this work is to enable the use of the Dempster-Shafer evidence theory for uncertainty propagation on computationally expensive automotive crash simulations. This is necessary as the results of these simulations are influenced by multiple possibly uncertain aspects. To avoid negative effects, it is important to detect these factors and their consequences. The challenge when pursuing this effort is the prohibitively high computational cost of the evidence theory. To this end, we present a framework of existing methods that is specifically designed to reduce the necessary number of full model evaluations and parameters. An initial screening removes clearly irrelevant parameters to mitigate the curse of dimensionality. Next, we approximate the full-scale simulation using metamodels to accelerate output generation and thus enable the calculation of global sensitivity indices. These indicate effects of the parameters on the considered output and more profoundly sort out irrelevant parameters. After these steps, the evidence theory can be performed rapidly and feasibly due to fast-responding metamodel and reduced input dimension. It yields bounds for the cumulative distribution function of the considered quantity of interest. We apply the proposed framework to a simplified crash test dummy model. The elementary effects method is used for screening, a kriging metamodel emulates the finite element simulation, and Sobol' sensitivity indices are determined before the evidence theory is applied. The outcome of the framework provide engineers with information about the uncertainties they may face in hardware testing and that should be addressed in future vehicle design.