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### Summary

The project is concerned with the model reduction of reactive flows. Model reduction techniques are mathematical methods that allow to describe the dynamics of complex, high-dimensional systems by simpler models of lower dimension. However, systems, which are strongly influenced by transport, are usually hard to reduce by classical methods, as for instance, the *proper orthogonal decomposition* (POD) or the *dynamic mode decomposition*. The goal of this project is to extend model reduction approaches in order to be capable of efficiently reducing transport-dominated phenomena. Moreover, these new methods shall be applied to the *pulsed detonation combustion* (PDC) and to the *shockless explosion combustion* (SEC) to support the parameter studies (A04) and the state estimation (A05) of the partner projects.

## 2nd Funding period 2016-2020

In the second period, the goal is to enhance the *shifted POD*, which has been developed in the first period, and to apply it to the problems which are relevant in the CRC, so especially to the SEC and the PDC.

At first, we refined the identification of dominant modes via the shifted POD. To this end, we introduced two new algorithms which are based on optimization problems. One algorithm considers a cost functional which maximizes the largest singular values in the co-moving frames of the different transport directions. Diverse extensions of this optimization problem have the potential to achieve a unique decomposition of multiple transports or to impose additional smoothness requirements on the identified modes. The application of such additional regularizations is currently under investigation.

The other algorithm aims directly for minimizing the residual, i.e., the difference between the original data and the approximation. We applied this method, among others, to the data of the PDC which are depicted in Fig. 1, left, and which originate from a data assimilation as part of the collaboration between A01 and A04. The comparison between the original data, the approximation via shifted POD, and the approximation via POD (with 7 modes, respectively; see Fig. 1) demonstrates that the shifted POD is capable of approximating these complex dynamics comparatively accurately with only 7 modes. This means a considerable improvement in comparison to the classical POD which needs ten times as many modes for the same accuracy.

- Fig. 1: PDC data of the density: Comparison between original data (left), shifted POD approximation with 7 modes (middle), and POD approximation with 7 modes (right)
- © TUB

Another focus of the second period is an energy-based formulation of the full-order as well as of the *reduced-order model *(ROM), using the notion of port-Hamiltonian systems. This allows to guarantee important system properties, as stability and passivity, also in the ROM. Furthermore, the port-Hamiltonian formulation implies that different subsystems (e.g., plena and combustion tubes) can be reduced separately and coupled afterwards, leading to an overall system which is again port-Hamiltonian and, thus, stable and passive. In order to also account for algebraic constraints, as boundary conditions, conservation laws, coupling conditions, etc., we developed a new port-Hamiltonian formulation for descriptor systems, i.e., control systems with algebraic constraints.

As an alternative approach for the low-dimensional modeling of transport phenomena, we investigate structured realizations based on input/output data. We have demonstrated that especially delay systems are well-suited for the approximation of transport phenomena and represent a natural way for low-dimensional modeling of these kinds of phenomena. For this purpose, we investigated how such systems can be created directly from input/output data in the frequency or in the time domain.

Furthermore, we developed novel model reduction methods for systems with inhomogeneous initial conditions as well as for linear switched systems.

Currently, we consider the construction of dynamic ROMs based on shifted POD modes and the application to the SEC. The goal is to obtain dynamic low-dimensional models for the SEC which allow to simulate the SEC process in an efficient way for various parameter configurations.

As a first problem, we considered the dynamics of a chemical reactor for different initial temperatures. This presents a big challenge for common methods since the temperature increases abruptly after the ignition delay time which in turn depends on the initial temperature. In order to still enable an efficient model reduction, we integrated the dependency of the ignition delay from the initial temperature directly into the model reduction ansatz.

The idea to directly consider the information about the location of the discontinuity (or of the very high gradient) in the approximation ansatz eventually led to the development of the shifited POD. This method is based on a low-dimensional modal representation in co-moving frames and, therefore, is a natural way of describing transport phenomena by just a few modes. A special feature of the shifted POD is that it can also represent multiple transport phenomena using just a few modes. This is achieved by an additive and purely data-driven decomposition of the field. We investigated the shifted POD by means of examples of various complexities including the linear wave equation, the crossing of two shocks, and the transport of a vortex pair in two spatial dimensions. These examples demonstrate that the shifted POD is capable of yielding low-dimensional approximations even for non-hyperbolic problems with non-periodic boundary conditions and non-constant transport velocities. Moreover, the identified modes are more accessible for a physical interpretation since the structures in the co-moving frames are usually more distinct than in the lab frame.

In addition, we derived a port-Hamiltonian formulation for the reactive Navier-Stokes equations as a preliminary work for the second period. This structure shall facilitate stable and passive ROMs via structure-preserving discretization and model reduction techniques. Besides, we established conditions for data-driven modeling which state if a given realization is equivalent to a port-Hamiltonian formulation.

## Publications

**Peer-reviewed publications**

Altmann, R. und P. Schulze: A port-Hamiltonian formulation of the Navier–Stokes equations for reactive flows. *Systems & Control Letters*, 100: 51–55, 2017.

Beattie, C., V. Mehrmann, H. Xu und H. Zwart. Linear port-Hamiltonian descriptor systems. *Mathematics of Control, Signals, and Systems*, 30: 17, 2018.

Beattie, C., S. Gugercin und V. Mehrmann. Model reduction for systems with inhomogeneous initial conditions. *Systems & Control Letters*, 99: 99–106, 2017.

Das, P. und V. Mehrmann: Numerical solution of singularly perturbed convection-diffusion-reaction problems with two small parameters. *BIT Numerical Mathematics*, 56(1): 51–76, 2015.

Lemke, M., A. Miedlar, J. Reiss, V. Mehrmann und J. Sesterhenn: Model reduction of reactive processes. In *Active Flow and Combustion Control 2014*, Springer International Publishing, S. 234–262, 2015.

Reiss, J.. Model reduction for convective problems: formulation and application. *IFAC-PapersOnLine*, 51(2): 186–189, 2018.

Reiss, J., P. Schulze, J. Sesterhenn und V. Mehrmann. The shifted proper orthogonal decomposition: A mode decomposition for multiple transport phenomena. *SIAM Journal on Scientific Computing*, 40: 125–131, 2018.

Schäpel, J.-S., S. Wolff, P. Schulze, P. Berndt, R. Klein, V. Mehrmann und R. King. State estimation for reactive Euler equation by Kalman filtering. *CEAS Aeronautical Journal*, 8(2): 261–270, 2017.

Schulze, P., T. C. Ionescu und J. M. A. Scherpen. Families of moment matching-based reduced order models for linear descriptor systems. In *2016 European Control Conference (ECC)*, S. 1964–1969, 2016.

Schulze, P., J. Reiss und V. Mehrmann. Model reduction for a pulsed detonation combuster via shifted proper orthogonal decomposition. In *Active Flow and Combustion Control 2018*, Springer International Publishing, S. 271–286, 2019.

Schulze, P. und B. Unger. Data-driven interpolation of dynamical systems with delay. *Systems & Control Letters*, 97: 125–131, 2016.

Schulze, P. und B. Unger. Model reduction for linear systems with low-rank switching. *SIAM Journal on Control and Optimization*, 56: 4365–4384, 2018.

Schulze, P., B. Unger, C. Beattie und S. Gugercin. Data-driven structured realization, *Linear Algebra and its Applications*, 537: 250–286, 2018.

**Preprints**

Beattie, C., V. Mehrmann und H. Xu. Port-Hamiltonian realizations of linear time invariant systems. Preprint 23–2015, Institut für Mathematik, TU Berlin, 2015. http://www.math.tu-berlin.de/preprints/.

Binder, A., V. Mehrmann, A. Miedlar und P. Schulze. A Matlab toolbox for the regularization of descriptor systems arising from generalized realization procedures. Preprint 24–2015, Institut für Mathematik, TU Berlin, 2016. http://www.math.tu-berlin.de/preprints/.

Fosong, P. Schulze und B. Unger. From Time-Domain Data to Low-Dimensional Structured Models. ArXiv Preprint 1902.05112, 2019. https://arxiv.org.