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### Summary

The project is concerned with model order 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 with 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.

### Preview: 3rd funding period 2020 – 2024

The main aim of the third funding period is the development of passive and stable port-Hamiltonian reduced order models for the different subsystems of the pulsed detonation combustion (PDC) and the shockless explosion combustion (SEC). The port-Hamiltonian structure allows to reduce the individual subsystems in a modular way. The total system resulting from the connection of the individual reduced order models is again port-Hamiltonian. Further advantages of a port-Hamiltonian formulation are robustness with respect to perturbations due to a physically meaningful formulation, invariance of the reduced order models with respect to interconnection and invariance with respect to space- and time-discretization.

The port-Hamiltonian structure has only been partially included in the formulation of the governing equations for the combustion process. While the port-Hamiltonian structure is well understood for pure flow-equations and reactive flows, this structure for the PDC and SEC under inclusion of the geometry of the testing facilities and for the reduced order models has yet to be incorporated.

A further goal is the modification of the shifted POD to incorporate and preserve port-Hamiltonian structure. Additionally, based on numerical and experimental input- and output-data of the combustion chamber, data-driven model order reduction schemes that preserve port-Hamiltonian structure will be developed.

Further research goals are the analysis of the shifted POD in operator formulation with respect to convergence and the enhancement of the shifted POD with respect to the detection of the transport-velocities and transport-directions, utilizing data-based machine learning techniques.

### 2nd funding period 2016-2020

At first, the identification of dominant modes via the shifted POD was refined. To this end, two new algorithms, which are based on optimization problems, were introduced. 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 other algorithm aims directly for minimizing the residual, i. e., the difference between the original data and the approximation. This method was applied, for instance, to data of the PDC, which are depicted in Fig. 1, left, and which originated 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 is a considerable improvement in comparison to the classical POD which needs ten times as many modes for the same accuracy.

As an alternative approach for the low-dimensional modeling of transport phenomena, structured realizations based on input/output data were investigated. A new realization method that allows to construct low-dimensional surrogate models whose transfer function maintains a special structure was developed, based on frequency data of the transfer function of the original system. This includes, among others, linear delay-systems and systems with higher or fractional order. Furthermore, it was shown that especially delay-systems are well-suited to approximate transport phenomena, and present a natural way for the low-dimensional modeling of such systems. A problem that is still open is the development of realization methods that create low-dimensional surrogate models with port-Hamiltonian structure.

For systems with algebraic constraints (descriptor systems), e. g., boundary conditions, conservation laws, etc., a new port-Hamiltonian description was developed within this project.

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

Based on the modes determined by shifted POD, a new projection method for the creation of low-dimensional reduced order models was developed. The efficiency of this approach was demonstrated for different model classes, for example for the advection-diffusion equation (with periodic and inlet-outlet boundary conditions) and the viscous Burgers‘ equation with periodic boundary conditions, among others.

In the remaining time of the second funding period of the SFB 1029, reduced order models for state estimation in the SEC in A05 and for parameter studies for the PDC in A04 will be developed.

As a first problem, the dynamics of a chemical reactor for different initial temperatures were considered. This presents a big challenge for common model reduction 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, the dependency of the ignition delay from the initial temperature was directly integrated into the model reduction ansatz.

The idea to directly consider the information about the location of the discontinuity (or: the very high gradient) in the approximation ansatz eventually led to the development of the shifted 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. The shifted POD was investigated 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, a port-Hamiltonian formulation for the reactive Navier-Stokes equations was derived, 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, conditions for data-driven modeling that state if a given realization is equivalent to a port-Hamiltonian formulation were established.

## 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.

Black, F., Schulze, P. und Unger, B.: Nonlinear Galerkin Model Reduction for Systems with Multiple Transport Velocities. Preprint 11-2019, Institut für Mathematik, TU Berlin, 2019. http://www.math.tu-berlin.de/preprints/.