2 Motivation
CFDComputational Fluid Dynamics is an indispensable technique, when aimed to obtain information about aerodynamic properties, such as drag and lift distributions. Modern CFDComputational Fluid Dynamics solvers, such as DLRGerman Aerospace Center’s TAU (Langer, Schwöppe, and Kroll 2014) often solves the RANSReynolds Averaged Navier Stockes equations to obtain one flow-field. Advanced solvers like TAU apply advanced mathematical knowledge to speed up calculations and heavily exploit multiple CPUsComputer Processing Units in an optimized manner. Nevertheless, depending on the size of the object and accuracy demands or in other terms mesh grid size, the computation often is not economically efficient enough. If the object for which a flow field is desired is a full aircraft, then even with a big cluster and making use of symmetry properties of the shape of the airplane, if such exists, the computation of one single flow field can still easily cost one or even multiple months in computation time.
In modern science, there is a trend towards relying on GPUsGraphics Processing Units instead of CPUsComputer Processing Units. Graphic cards possess much more cores than a CPU. However, even with the utilization of GPUsGraphics Processing Units and GPU-optimized CFDComputational Fluid Dynamics solvers, the computation is still very expensive. Not only in time but also in electricity costs. Running calculations on a cluster for multiple months is such expensive that wind tunnel measurements can be considered to be the economically more efficient choice to make. Regarding accuracy, wind tunnel measurements and CFDComputational Fluid Dynamics simulations with state-of-the-art solvers can be considered to be equally useful. When using CFDComputational Fluid Dynamics solvers, there is one more thing to keep in mind. Each outcome is only valid for one single set of input parameters. Within the set of input parameters, the user often is only interested in the impact of one parameter, e.g., the angle of attack. Consequently, wanting to capture the effect of the change of the angle of attack on the flow field, multiple CFDComputational Fluid Dynamics calculations need to be performed, i.e., for each desired angle of attack. Based on the chosen angle of attack the solver might be able to converge faster to a solution. However, the calculation time needs to be added up for each desired angle of attack. In terms of time and energy costs, this could again be more expensive than wind-tunnel measurements. Wind tunnel measurements are difficult to set up, but once a configuration is available, measuring flow field properties with it, in general, is known to be faster and easier than running CFDComputational Fluid Dynamics simulations.
Within the scope of this work, a data-driven tool was developed that allows predictions for dynamic systems. In (Pierzyna 2021) the first version of it showed promising results. However, it was dedicated to the solution of one single dynamical system, i.e., the Lorenz system (Lorenz 1963). Due to the focus on one singular dynamical system, the proposed control-oriented Cluster-based Network Modeling (CNMc) was not verified for other dynamical systems. Hence, one of the major goals of this thesis is to enable CNMccontrol-oriented Cluster-based Network Modeling to be applied to any general dynamical system. For this, it is important to state that because of two main reasons CNMccontrol-oriented Cluster-based Network Modeling was not built upon the first version of CNMccontrol-oriented Cluster-based Network Modeling, but written from scratch. First, since the initial version of CNMccontrol-oriented Cluster-based Network Modeling was designed for only a single dynamic system, extending it to a general CNMccontrol-oriented Cluster-based Network Modeling was considered more time-consuming than starting fresh. Second, not all parts of the initial version of CNMccontrol-oriented Cluster-based Network Modeling could be executed without errors. The current CNMccontrol-oriented Cluster-based Network Modeling is therefore developed in a modular manner, i.e., on the one hand, the implementation of any other dynamical system is straightforward. To exemplify this, 10 different dynamic systems are available by default, so new dynamic systems can be added analogously.
The second important aspect for allowing CNMccontrol-oriented Cluster-based Network Modeling to be utilized in any general dynamical system is the removal of the two limitations. In the first version of CNMccontrol-oriented Cluster-based Network Modeling the behavior of the dynamical systems had to be circular as, e.g., the ears of the Lorenz system (Lorenz 1963) are. Next, its dimensionality must be strictly 3-dimensional. Neither is a general dynamical system is not bound to exhibit a circular motion nor to be 3-dimensional. By removing these two limitations CNMccontrol-oriented Cluster-based Network Modeling can be leveraged on any dynamical system. However, the first version of CNMccontrol-oriented Cluster-based Network Modeling employed Non-negative Matrix Factorization (NMF) as the modal decomposition method. The exploited NMFNon-negative Matrix Factorization algorithm is highly computationally intensive, which makes a universal CNMccontrol-oriented Cluster-based Network Modeling application economically inefficient. Therefore, the current CNMccontrol-oriented Cluster-based Network Modeling has been extended by the option to choose between the NMFNon-negative Matrix Factorization and the newly implemented Singular Value Decomposition (SVD). The aim is not only that CNMccontrol-oriented Cluster-based Network Modeling is returning results within an acceptable timescale, but also to ensure that the quality of the modal decomposition remains at least at an equal level. Proofs for the latter can be found in section 14.
With these modifications, the current CNMccontrol-oriented Cluster-based Network Modeling is now able to be used in any dynamical system within a feasible time frame. The next addressed issue is the B-spline interpolation. It is used in the propagation step of Cluster-based Network Modeling (CNM) (Fernex, Noack, and Semaan 2021) to smooth the predicted trajectory. However, as already noted in (Pierzyna 2021), when the number of the clustering centroids \(K\) is \(K \gtrapprox 15\), the B-spline interpolation embeds oscillations with unacceptable high deviations from the original trajectories. To resolve this problem, the B-spline interpolation is replaced with linear interpolation. By preventing the occurrence of outliers caused by the B-spline interpolation, neither the autocorrelation defined in subsection 4.0.1 nor the predicted trajectories are made impractical . Apart from the main ability of CNMccontrol-oriented Cluster-based Network Modeling a high number of additional features are available, e.g., the entire pipeline of CNMccontrol-oriented Cluster-based Network Modeling with all its parameters can be adjusted via one file (settings.py), an incorporated log file, storing results at desired steps, the ability to execute multiple dynamical models consequentially and activating and disabling each step of CNMccontrol-oriented Cluster-based Network Modeling. The latter is particularly designed for saving computational time. Also, CNMccontrol-oriented Cluster-based Network Modeling comes with its own post-processor. It is optional to generate and save the plots. However, in the case of utilizing this feature, the plots are available as HTML files which, e.g., allow extracting further information about the outcome or rotating and zooming in 3d plots.