Publikationen

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kinds of systems. It extends the proper orthogonal decomposition (POD) by introducing time-dependent shifts of the snapshot matrix. The approach, called shifted proper orthogonal decomposition (sPOD), features a determination of the multiple transport velocities and a separation of these. One- and two-dimensional test examples reveal the good performance of the sPOD for transport-dominated phenomena and its superiority in comparison to the POD.

In this article, we present a state estimation method for an approximately constant volume combustion process. This is an Unscented Kalman Filter used to estimate quantities which are of interest for the combustion process inside the considered combustion tube, i.e., the pressure, velocity and temperature field. This algorithm relies only on a small number of discrete pressure measurements along the combustion tube. The proposed methods are applied in numerical simulations to demonstrate their effectiveness. In a one-dimensional simulator, the flow field in the tube is described by the one-dimensional Euler equations with chemical source terms. To keep the computational effort on a manageable level, the Kalman Filter is built based on a reduced model. Thus, the model reduction for the Euler equations in combination with chemical kinetics is another focus of this contribution.

We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.

We present a model describing simultaneous heat and mass transfer of an absorbing or desorbing laminar liquid film flowing over a vertical isothermal plate. We start with a formulation which is comparable to established models by using simplifying assumptions such as homogeneous velocity and constant film thickness. In contrast to those, we allow for effects like change in properties and differential heat of solution within the bulk of the film. Additionally, enthalpy transport due to interdiffusion is accounted for. The impact of the considered effects are discussed and compared. The numerical solution is obtained by utilising a Newton–Raphson scheme to solve the finite difference formulation of the governing equations. Since the temperature gradients adjacent to wall and phase boundary are expected to be large, we discretise the equations on an irregular grid. The results of the model agree very well with established analytical models. It is found that the influence of released differential heat of solution within the bulk is relatively small. However, the impact on the temperature distribution is in the same order of magnitude as the one of a change in properties. Moreover, when comparing desorption with absorption under equivalent conditions, the mass transfer rate during absorption is higher than during desorption.

In this paper a moment-matching procedure for descriptor systems is presented. Based on a time-domain notion of moments, parametrized families of reduced order models matching the moments of the original system are derived. An example of exploiting the flexibility of choosing the free matrix parameters is demonstrated by achieving two-sided moment matching, i. e., obtaining reduced order models of order $\nu$ matching $2\nu$ moments. In this context, a connection with the Loewner framework is shown which is a special case of the framework presented in this paper.

We consider the problem of finding an optimal data-driven modal decomposition of flows with multiple convection velocities. To this end, we apply the shifted proper orthogonal decomposition (sPOD) which is a recently proposed mode decomposition technique. It overcomes the poor performance of classical methods like the proper orthogonal de- composition (POD) for a class of transport-dominated phenomena with large gradients. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. We propose a new algorithm for computing an sPOD which carries out a residual minimization in which the main cost arises from solving a nonlin- ear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detona- tion combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.