11 Interfakultäre Einrichtungen

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    Identification of linear time-invariant systems with dynamic mode decomposition
    (2022) Heiland, Jan; Unger, Benjamin
    Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.
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    Multivariate motion patterns and applications to rainfall radar data
    (2023) Fischer, Svenja; Oesting, Marco; Schnurr, Alexander
    The classification of movement in space is one of the key tasks in environmental science. Various geospatial data such as rainfall or other weather data, data on animal movement or landslide data require a quantitative analysis of the probable movement in space to obtain information on potential risks, ecological developments or changes in future. Usually, machine-learning tools are applied for this task, as these approaches are able to classify large amounts of data. Yet, machine-learning approaches also have some drawbacks, e.g. the often required large training sets and the fact that the algorithms are often hard to interpret. We propose a classification approach for spatial data based on ordinal patterns. Ordinal patterns have the advantage that they are easily applicable, even to small data sets, are robust in the presence of certain changes in the time series and deliver interpretative results. They therefore do not only offer an alternative to machine-learning in the case of small data sets but might also be used in pre-processing for a meaningful feature selection. In this work, we introduce the basic concept of multivariate ordinal patterns and the corresponding limit theorem. A simulation study based on bootstrap demonstrates the validity of the results. The approach is then applied to two real-life data sets, namely rainfall radar data and the movement of a leopard. Both applications emphasize the meaningfulness of the approach. Clearly, certain patterns related to the atmosphere and environment occur significantly often, indicating a strong dependence of the movement on the environment.
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    FFT-based homogenization at finite strains using composite boxels (ComBo)
    (2022) Keshav, Sanath; Fritzen, Felix; Kabel, Matthias
    Computational homogenization is the gold standard for concurrent multi-scale simulations (e.g., FE2) in scale-bridging applications. Often the simulations are based on experimental and synthetic material microstructures represented by high-resolution 3D image data. The computational complexity of simulations operating on such voxel data is distinct. The inability of voxelized 3D geometries to capture smooth material interfaces accurately, along with the necessity for complexity reduction, has motivated a special local coarse-graining technique called composite voxels (Kabel et al. Comput Methods Appl Mech Eng 294: 168-188, 2015). They condense multiple fine-scale voxels into a single voxel, whose constitutive model is derived from the laminate theory. Our contribution generalizes composite voxels towards composite boxels (ComBo) that are non-equiaxed, a feature that can pay off for materials with a preferred direction such as pseudo-uni-directional fiber composites. A novel image-based normal detection algorithm is devised which (i) allows for boxels in the firsts place and (ii) reduces the error in the phase-averaged stresses by around 30% against the orientation cf. Kabel et al. (Comput Methods Appl Mech Eng 294: 168-188, 2015) even for equiaxed voxels. Further, the use of ComBo for finite strain simulations is studied in detail. An efficient and robust implementation is proposed, featuring an essential selective back-projection algorithm preventing physically inadmissible states. Various examples show the efficiency of ComBo against the original proposal by Kabel et al. (Comput Methods Appl Mech Eng 294: 168-188, 2015) and the proposed algorithmic enhancements for nonlinear mechanical problems. The general usability is emphasized by examining various Fast Fourier Transform (FFT) based solvers, including a detailed description of the Doubly-Fine Material Grid (DFMG) for finite strains. All of the studied schemes benefit from the ComBo discretization.
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    Efficient wildland fire simulation via nonlinear model order reduction
    (2021) Black, Felix; Schulze, Philipp; Unger, Benjamin
    We propose a new hyper-reduction method for a recently introduced nonlinear model reduction framework based on dynamically transformed basis functions and especially well-suited for transport-dominated systems. Furthermore, we discuss applying this new method to a wildland fire model whose dynamics feature traveling combustion waves and local ignition and is thus challenging for classical model reduction schemes based on linear subspaces. The new hyper-reduction framework allows us to construct parameter-dependent reduced-order models (ROMs) with efficient offline/online decomposition. The numerical experiments demonstrate that the ROMs obtained by the novel method outperform those obtained by a classical approach using the proper orthogonal decomposition and the discrete empirical interpolation method in terms of run time and accuracy.
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    Semi-explicit integration of second order for weakly coupled poroelasticity
    (2024) Altmann, R.; Maier, R.; Unger, B.
    We introduce a semi-explicit time-stepping scheme of second order for linear poroelasticity satisfying a weak coupling condition. Here, semi-explicit means that the system, which needs to be solved in each step, decouples and hence improves the computational efficiency. The construction and the convergence proof are based on the connection to a differential equation with two time delays, namely one and two times the step size. Numerical experiments confirm the theoretical results and indicate the applicability to higher-order schemes.