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## Model Reduction for Transport Phenomena

Figure 3 - The reconstruction using seven POD modes fails to approximate the main structures.
Figure 2 - The shifted POD reconstruction using seven modes.
Figure 1 - The original density data of a detonation combustion chamber with various transports.

Model Reduction

In the design process of flow machines one usually simulates Navier Stokes equations with billions of degrees of freedom. The simulations are numerically expensive, because computational resources scale with the number of degrees of freedom. Therefore, model reduction techniques are desired to reduce the number of relevant parameters, which describe the system. This is helpful in design processes, where reduced models allow a rapid evaluation for many different parameters.

Mode-Based Model Reduction

Most commonly, reduced models are constructed with the proper orthogonal decomposition (POD). The POD is basically a singular value decomposition applied to selected snapshots of the flow. This decomposes the original dynamics q(x,t) in spatial modes φi(x) with time varying amplitudes αi(t):

$\dpi{300}&space;\small&space;q(x,t)\approx\sum_{i=1}^N&space;\alpha_i(t)\phi_i(x)\qquad(1)$

A reduced model can be constructed from (1) by deriving a dynamical model for the time amplitudes, e.g. by a Galerkin projection of the original system equations. Hence, with the POD ansatz the size of the discretized PDE system ~ O(106) (gripdpoints) is reduced to the number of  the POD modes N ~ O(10). This reduction allows an efficient simulation of the system and enables to optimize and control it.

Research: Mode-Based Model Reduction for Transport Phenomena

Unfortunately, if transports with jumps and kinks dominate the dynamic the POD ansatz eq. (1) performs poorly, since many modes are needed to describe the original system. This fact makes the POD ansatz unfeasible in many interesting flow systems, like propagating flames, moving shocks or traveling acoustic waves. The goal of our research is to improve the ansatz, in order to benefit from the concept in our research.

To this end we follow two different approaches in our research:

• The Shifted POD builds on decomposing the original data into multiple waves with a single, time dependent transport. We have different, data driven formulations to decompose and calculate the modes and amplitudes. Originally we used an iterative  shift and reduce method acting as an approximative filter [1]. An optimization based on the residuum of the approximation is presented in [2]. Currently, we work on optimization on simple singular value based objective functions, first results are in [3]. The method was successfully applied to the effective description of a pulse detonation combustion chamber with its various transports, see Figure 2.
• The Front Transport Reduction addresses model reduction for 2D/3D flows with sharp transported fronts, which change topology. Our basic ansatz assumes that the original flow field q(x,t)=f(φ(x,t)) can be reconstructed by a front shape function f and a level set function φ. The level set function is used to generate a local coordinate, which parametrizes the distance to the front. In this way, we are able to embed the local 1D description of the front for complex 2D front dynamics with merging or splitting fronts, while seeking a low rank description of φ. In [4] we could successfully apply this ansatz to a 2D propagating flame with a moving front.
Figure 4 - Comparison of POD with FTR method for a combustion system.

### Publications

1. 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.DOI: 10.1137/17M1140571
2. 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.DOI: 10.1007/978-3-319-98177-2_17
3. Reiss, J.. Model reduction for convective problems: formulation and application. IFAC-PapersOnLine, 51(2): 186–189, 2018. DOI:10.1016/j.ifacol.2018.03.032
4. Krah, Philipp, Mario Sroka, and Julius Reiss. "Model Order Reduction of Combustion Processes with Complex Front Dynamics." arXiv preprint arXiv:1912.03004 (2019).