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Off-lattice Boltzmann methods increase the flexibility and applicability of lattice Boltzmann methods by decoupling the discretizations of time, space, and particle velocities. However, the velocity sets that are mostly used in off-lattice Boltzmann simulations were originally tailored to on-lattice Boltzmann methods. In this contribution, we show how the accuracy and efficiency of weakly and fully compressible semi-Lagrangian off-lattice Boltzmann simulations is increased by velocity sets derived from cubature rules, i.e. multivariate quadratures, which have not been produced by the Gauss-product rule. In particular, simulations of 2D shock-vortex interactions indicate that the cubature-derived degree-nine D2Q19 velocity set is capable to replace the Gauss-product rule-derived D2Q25. Likewise, the degree-five velocity sets D3Q13 and D3Q21, as well as a degree-seven D3V27 velocity set were successfully tested for 3D Taylor-Green vortex flows to challenge and surpass the quality of the customary D3Q27 velocity set. In compressible 3D Taylor-Green vortex flows with Mach numbers Ma={0.5;1.0;1.5;2.0} on-lattice simulations with velocity sets D3Q103 and D3V107 showed only limited stability, while the off-lattice degree-nine D3Q45 velocity set accurately reproduced the kinetic energy provided by literature.
This work introduces a semi-Lagrangian lattice Boltzmann (SLLBM) solver for compressible flows (with or without discontinuities). It makes use of a cell-wise representation of the simulation domain and utilizes interpolation polynomials up to fourth order to conduct the streaming step. The SLLBM solver allows for an independent time step size due to the absence of a time integrator and for the use of unusual velocity sets, like a D2Q25, which is constructed by the roots of the fifth-order Hermite polynomial. The properties of the proposed model are shown in diverse example simulations of a Sod shock tube, a two-dimensional Riemann problem and a shock-vortex interaction. It is shown that the cell-based interpolation and the use of Gauss-Lobatto-Chebyshev support points allow for spatially high-order solutions and minimize the mass loss caused by the interpolation. Transformed grids in the shock-vortex interaction show the general applicability to non-uniform grids.
In dieser Arbeit wird eine kompressible Semi-Lagrangesche Lattice-Boltzmann-Methode neu entwickelt und erprobt. Die Lattice-Boltzmann-Methode ist ein Verfahren zur numerischen Strömungssimulation, das auf einer Modellierung von Partikeldichten und deren Interaktion untereinander basiert. In ihrer Ursprungsform ist die Methode jedoch auf schwach kompressible Strömungen mit niedriger Machzahl beschränkt. Wesentliche Nachteile der bisherigen Versuche zur Erweiterung auf supersonische Strömungen sind entweder mangelhafte Stabilität der Verfahren, unpraktikabel große Geschwindigkeitssätze oder die Beschränktheit auf kleine Zeitschrittweiten. Als Alternative zu bisherigen Ansätzen wird in dieser Arbeit ein Semi-Lagrangescher Strömungsschritt eingesetzt. Semi-Lagrangesche Verfahren entkoppeln mittels Interpolation die Orts-, Zeit- und Geschwindigkeitsdiskretisierung der ursprünglichen Lattice-Boltzmann-Methode. Nach der Einleitung wird im zweiten und dritten Kapitel dieser Arbeit zunächst auf die Grundlagen und Prinzipien der Lattice-Boltzmann-Methode eingegangen sowie bisherige Ansätze zur Simulation kompressibler Strömungen aufgeführt. Im Anschluss wird die kompressible Semi-Lagrangesche Lattice-Boltzmann-Methode entwickelt und beschrieben. Die Erweiterung erfolgt im Wesentlichen durch die Verknüpfung der Methode mit geeigneten Gleichgewichtsfunktionen und Geschwindigkeitssätzen. Im vierten Kapitel der Arbeit werden neue Kubatur-basierte Geschwindigkeitssätze entwickelt und getestet, darunter ein D3Q45-Geschwindigkeitssatz zur Berechnung kompressibler Strömungen, der den Rechenaufwand gegenüber konventionellen Geschwindigkeitsdiskretisierungen erheblich verringert. Im fünften Kapitel der Arbeit werden zur Validierung Simulationen von eindimensionalen Stoßrohren, zweidimensionalen Riemann-Problemen und Stoß-Wirbel-Interaktionen durchgeführt. Im Anschluss zeigen Simulationen von dreidimensionalen, kompressiblen Taylor-Green-Wirbeln sowie von wandgebundenen Testfällen die Vorteile der Methode für kompressible Strömungssimulationen. Zu diesem Zweck werden die Überschallströmung um ein zweidimensionales NACA-0012-Profil und um eine dreidimensionale Kugel sowie eine supersonische Kanalströmung untersucht. Dem Simulationsteil folgt eine umfangreiche Diskussion der Semi-Lagrangeschen Lattice-Boltzmann-Methode im Vergleich zu anderen Methoden. Die Vorteile der Methode, wie vergleichsweise große Zeitschrittweiten, körperangepasste Netze und die Stabilität der Methode, werden hier herausgearbeitet.
In complex, expensive optimization domains we often narrowly focus on finding high performing solutions, instead of expanding our understanding of the domain itself. But what if we could quickly understand the complex behaviors that can emerge in said domains instead? We introduce surrogate-assisted phenotypic niching, a quality diversity algorithm which allows to discover a large, diverse set of behaviors by using computationally expensive phenotypic features. In this work we discover the types of air flow in a 2D fluid dynamics optimization problem. A fast GPU-based fluid dynamics solver is used in conjunction with surrogate models to accurately predict fluid characteristics from the shapes that produce the air flow. We show that these features can be modeled in a data-driven way while sampling to improve performance, rather than explicitly sampling to improve feature models. Our method can reduce the need to run an infeasibly large set of simulations while still being able to design a large diversity of air flows and the shapes that cause them. Discovering diversity of behaviors helps engineers to better understand expensive domains and their solutions.
The lattice Boltzmann method (LBM) is an efficient simulation technique for computational fluid mechanics and beyond. It is based on a simple stream-and-collide algorithm on Cartesian grids, which is easily compatible with modern machine learning architectures. While it is becoming increasingly clear that deep learning can provide a decisive stimulus for classical simulation techniques, recent studies have not addressed possible connections between machine learning and LBM. Here, we introduce Lettuce, a PyTorch-based LBM code with a threefold aim. Lettuce enables GPU accelerated calculations with minimal source code, facilitates rapid prototyping of LBM models, and enables integrating LBM simulations with PyTorch's deep learning and automatic differentiation facility. As a proof of concept for combining machine learning with the LBM, a neural collision model is developed, trained on a doubly periodic shear layer and then transferred to a different flow, a decaying turbulence. We also exemplify the added benefit of PyTorch's automatic differentiation framework in flow control and optimization. To this end, the spectrum of a forced isotropic turbulence is maintained without further constraining the velocity field.
Quality diversity algorithms can be used to efficiently create a diverse set of solutions to inform engineers' intuition. But quality diversity is not efficient in very expensive problems, needing 100.000s of evaluations. Even with the assistance of surrogate models, quality diversity needs 100s or even 1000s of evaluations, which can make it use infeasible. In this study we try to tackle this problem by using a pre-optimization strategy on a lower-dimensional optimization problem and then map the solutions to a higher-dimensional case. For a use case to design buildings that minimize wind nuisance, we show that we can predict flow features around 3D buildings from 2D flow features around building footprints. For a diverse set of building designs, by sampling the space of 2D footprints with a quality diversity algorithm, a predictive model can be trained that is more accurate than when trained on a set of footprints that were selected with a space-filling algorithm like the Sobol sequence. Simulating only 16 buildings in 3D, a set of 1024 building designs with low predicted wind nuisance is created. We show that we can produce better machine learning models by producing training data with quality diversity instead of using common sampling techniques. The method can bootstrap generative design in a computationally expensive 3D domain and allow engineers to sweep the design space, understanding wind nuisance in early design phases.