Philipp Schulze

The Shifted Proper Orthogonal Decomposition: A Mode Decomposition for Multiple Transport Phenomena

Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kind 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 {it 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.

Data-driven interpolation of dynamical systems with delay

Abstract 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.

Data-driven structured realization

Abstract We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form C ˜ ( ∑ k = 1 K h k ( s ) A ˜ k ) − 1 B ˜ where h 1 , h 2 , . . . , h K are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to a more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.

A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows

Abstract We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. xa0g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.

Model Reduction for a Pulsed Detonation Combuster via Shifted Proper Orthogonal Decomposition

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 decomposition (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 nonlinear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detonation 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.

The shifted proper orthogonal decomposition

DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamics