Showing papers on "Eulerian path published in 2021"

Higher order initial conditions for mixed baryon–CDM simulations

Abstract: We present a novel approach to generate higher-order initial conditions (ICs) for cosmological simulations that take into account the distinct evolution of baryons and dark matter. We focus on the numerical implementation and the validation of its performance, based on both collisionless N-body simulations and full hydrodynamic Eulerian and Lagrangian simulations. We improve in various ways over previous approaches that were limited to first-order Lagrangian perturbation theory (LPT). Specifically, we (1) generalize nth-order LPT to multi-fluid systems, allowing 2LPT or 3LPT ICs for two-fluid simulations, (2) employ a novel propagator perturbation theory to set up ICs for Eulerian codes that are fully consistent with 1LPT or 2LPT, (3) demonstrate that our ICs resolve previous problems of two-fluid simulations by using variations in particle masses that eliminate spurious deviations from expected perturbative results, (4) show that the improvements achieved by going to higher-order PT are comparable to those seen for single-fluid ICs, and (5) demonstrate the excellent (i.e., few per cent level) agreement between Eulerian and Lagrangian simulations, once high-quality initial conditions are used. The rigorous development of the underlying perturbation theory is presented in a companion paper. All presented algorithms are implemented in the Monofonic Music-2 package that we make publicly available.

An unfitted Eulerian finite element method for the time-dependent Stokes problem on moving domains

Abstract: We analyse a Eulerian Finite Element method, combining a Eulerian time-stepping scheme applied to the time-dependent Stokes equations using the CutFEM approach with inf-sup stable Taylor-Hood elements for the spatial discretisation. This is based on the method introduced by Lehrenfeld \& Olshanskii [ESAIM: M2AN 53(2):585--614] in the context of a scalar convection-diffusion problems on moving domains, and extended to the non-stationary Stokes problem on moving domains by Burman, Frei \& Massing [arXiv:1910.03054 [math.NA]] using stabilised equal-order elements. The analysis includes the geometrical error made by integrating over approximated levelset domains in the discrete CutFEM setting. The method is implemented and the theoretical results are illustrated using numerical examples.

An optimized Eulerian–Lagrangian method for two-phase flow with coarse particles: Implementation in open-source field operation and manipulation, verification, and validation

Abstract: The solid–liquid two-phase flow with coarse particles is ubiquitous in natural phenomena and engineering practice, which is characterized by coarse particles, high particle concentration, and large particle size distribution. In this work, the numerical models describing two-phase flows are reviewed, which given that the Eulerian–Lagrangian method is applicable in this work. Then, some modified models are proposed for the situation where the conventional Eulerian–Lagrangian method is not applicable to deal with coarse particles. The continuous phase equations of liquid are solved based on the finite volume method. The pressure implicit with splitting of operators algorithm for solving the Navier–Stokes (N–S) equations of the pseudo-single-phase flow, considering phase fraction and momentum exchange source term, is proposed. The discrete coarse particle is tracked in the Lagrangian method. A virtual mass distribution function is proposed for calculating coarse particle volume fraction. A weighted function method relating to the particle size is given for the interpolation between the Eulerian and Lagrangian fields. The barycentric coordinates are introduced into the particle localization. All the modified models are algorithmically implanted in the open-source field operation and manipulation (OpenFOAM) as a new solver named coarse discrete particle method FOAM (CoarseDPMFoam). Subsequently, the applicability of the numerical simulation method is verified by some typical test cases. The proposed numerical simulation method provides new ideas and methods for the mechanism investigation and engineering application of the two-phase flow with coarse particles.

Clustering in Massive Neutrino Cosmologies via Eulerian Perturbation Theory

Abstract: We introduce an Eulerian Perturbation Theory to study the clustering of tracers for cosmologies in the presence of massive neutrinos. Our approach is based on mapping recently-obtained Lagrangian Perturbation Theory results to the Eulerian framework. We add Effective Field Theory counterterms, IR-resummations and a biasing scheme to compute the one-loop redshift-space power spectrum. To assess our predictions, we compare the power spectrum multipoles against synthetic halo catalogues from the Quijote simulations, finding excellent agreement on scales $k\lesssim 0.25 \,h \text{Mpc}^{-1}$. Extending the range of accuracy to higher wave-numbers is possible at the cost of producing an offset in the best-fit linear local bias. We further discuss the implications for the tree-level bispectrum. Finally, calculating loop corrections is computationally costly, hence we derive an accurate approximation wherein we retain only the main features of the kernels, as produced by changes to the growth rate. As a result, we show how FFTLog methods can be used to further accelerate the loop computations with these reduced kernels.

Some numerical approaches for landslide river blocking: introduction, simulation, and discussion

Abstract: This paper aims to present several approaches for fluid–solid coupling applied in landslide river blocking simulations based on the Abaqus software. These approaches include the coupled Eulerian Lagrangian (CEL) model and the finite element method-smoothed particle hydrodynamics (FEM-SPH) model, which use discrete finite element meshed blocks to simulate landslides, and use continuum models of Eulerian material and SPH material for river simulations, respectively. Another approach is to use the Eulerian model with Eulerian material to simulate landslides and river at the same time. A physical model of an elastic plate subjected to time-dependent water pressure was applied to validate the selected methods in the fluid–solid coupling. The process was recorded and corresponding numerical simulations were performed. The results show that the simulated plate and water behavior is consistent with the physical model in all simulations. Afterwards, the methods were applied to a real-scale rock avalanche river blocking simulation and the processes of sliding mass movement, impulse wave behavior, and the formation of landslide dams were described and analyzed. Applied Eulerian materials and discrete finite element meshed blocks were used to simulate the sliding mass and similar sliding movement characteristics were obtained. However, discrete blocks have shown better potential for simulating more complex situations and processes leading to better mass movement and deposition results. In addition, discrete finite element meshed blocks and other similar discontinuum models can better reflect the structural characteristics of the source rock in the simulation. As for the impulse wave, the use of Eulerian model for simulating the sliding mass will lead to an abnormally high wave height. Whether to consider a water effect and how to consider it affect the shape of the landslide dams and the simulation results show the importance of fluid–solid coupling in the landslide river blocking simulation. Similar and accurate results can be obtained using the Eulerian method and the SPH method for water simulation. However, it is worth noting that the existence of the Eulerian boundary makes the Eulerian method a better option for simulating more complex flow conditions compared to the SPH method. The meshless characteristics of the SPH method make it more computational efficient provided the fluid volume is constant and the final scale of the model is uncertain.

Discrete shallow water equations preserving symmetries and conservation laws

Abstract: The one-dimensional shallow water equations in Eulerian coordinates are considered. Relations between symmetries and conservation laws for the potential form of the equations and symmetries and conservation laws in Eulerian coordinates are shown. An invariant difference scheme for equations in Eulerian coordinates with arbitrary bottom topography is constructed. It possesses all the finite-difference analogs of the conservation laws. Some bottom topographies require moving meshes in Eulerian coordinates, which are stationary meshes in mass Lagrangian coordinates. The developed invariant conservative difference schemes are verified numerically using examples of flow with various bottom topographies.

Clustering in Massive Neutrino Cosmologies via Eulerian Perturbation Theory

Abstract: We introduce an Eulerian Perturbation Theory to study the clustering of tracers for cosmologies in the presence of massive neutrinos. Our approach is based on mapping recently-obtained Lagrangian Perturbation Theory results to the Eulerian framework. We add Effective Field Theory counterterms, IR-resummations and a biasing scheme to compute the one-loop redshift-space power spectrum. To assess our predictions, we compare the power spectrum multipoles against synthetic halo catalogues from the Quijote simulations, finding excellent agreement on scales $k\lesssim 0.25 \,h \text{Mpc}^{-1}$. Extending the range of accuracy to higher wave-numbers is possible at the cost of producing an offset in the best-fit linear local bias. We further discuss the implications for the tree-level bispectrum. Finally, calculating loop corrections is computationally costly, hence we derive an accurate approximation wherein we retain only the main features of the kernels, as produced by changes to the growth rate. As a result, we show how FFTLog methods can be used to further accelerate the loop computations with these reduced kernels.

Königsberg Sightseeing: Eulerian Walks in Temporal Graphs

Abstract: An Eulerian walk (or Eulerian trail) is a walk (resp. trail) that visits every edge of a graph G at least (resp. exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges of Konigsberg problem in 1736. But what if Euler had to take a bus? In a temporal graph \((G,\lambda )\), with \(\lambda : E(G)\rightarrow 2^{[\tau ]}\), an edge \(e\in E(G)\) is available only at the times specified by \(\lambda (e)\subseteq [\tau ]\), in the same way the connections of the public transportation network of a city or of sightseeing tours are available only at scheduled times. In this scenario, even though several translations of Eulerian trails and walks are possible in temporal terms, only a very particular variation has been exploited in the literature, specifically for infinite dynamic networks (Orlin, 1984). In this paper, we deal with temporal walks, local trails, and trails, respectively referring to edge traversal with no constraints, constrained to not repeating the same edge in a single timestamp, and constrained to never repeating the same edge throughout the entire traversal. We show that, if the edges are always available, then deciding whether \((G,\lambda )\) has a temporal walk or trail is polynomial, while deciding whether it has a local trail is \(\textsf {NP}\)-complete even if it has lifetime 2. In contrast, in the general case, solving any of these problems is \(\textsf {NP}\)-complete, even under very strict hypotheses.

The BACCO simulation project: biased tracers in real space

Abstract: We present an emulator for the two-point clustering of biased tracers in real space. We construct this emulator using neural networks calibrated with more than $400$ cosmological models in a 8-dimensional cosmological parameter space that includes massive neutrinos an dynamical dark energy. The properties of biased tracers are described via a Lagrangian perturbative bias expansion which is advected to Eulerian space using the displacement field of numerical simulations. The cosmology-dependence is captured thanks to a cosmology-rescaling algorithm. We show that our emulator is capable of describing the power spectrum of galaxy formation simulations for a sample mimicking that of a typical Emission-Line survey at $z \sim 1$ with an accuracy of $1-2\%$ up to nonlinear scales $k \sim 0.7 h \mathrm{Mpc}^{-1}$.

Approximation Algorithms for the Bottleneck Asymmetric Traveling Salesman Problem

Abstract: We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n/ log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, Mądry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem.

Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Abstract: In the paper, by virtue of the Faa di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

Eulerian Dynamics in Multidimensions with Radial Symmetry

Abstract: We study the global wellposedness of pressureless Eulerian dynamics in multidimensions, with radially symmetric data Compared with the one-dimensional system, a major difference in multidimensiona

Lagrangian approach for modal analysis of fluid flows

Abstract: Common modal decomposition techniques for flow-field analysis, data-driven modelling and flow control, such as proper orthogonal decomposition and dynamic mode decomposition, are usually performed in an Eulerian (fixed) frame of reference with snapshots from measurements or evolution equations. The Eulerian description poses some difficulties, however, when the domain or the mesh deforms with time as, for example, in fluid–structure interactions. For such cases, we first formulate a Lagrangian modal analysis (LMA) ansatz by a posteriori transforming the Eulerian flow fields into Lagrangian flow maps through an orientation and measure-preserving domain diffeomorphism. The development is then verified for Lagrangian variants of proper orthogonal decomposition and dynamic mode decomposition using direct numerical simulations of two canonical flow configurations at Mach 0.5, i.e. the lid-driven cavity and flow past a cylinder, representing internal and external flows, respectively, at pre- and post-bifurcation Reynolds numbers. The LMA is demonstrated for several situations encompassing unsteady flow without and with boundary and mesh deformation as well as non-uniform base flows that are steady in Eulerian but not in Lagrangian frames. We show that application of LMA to steady non-uniform base flow yields insights into flow stability and post-bifurcation dynamics. LMA naturally leads to Lagrangian coherent flow structures and connections with finite-time Lyapunov exponents. We examine the mathematical link between finite-time Lyapunov exponents and LMA by considering a double-gyre flow pattern. Dynamically important flow features in the Lagrangian sense are recovered by performing LMA with forward and backward (adjoint) time procedures.

Convergence of Eulerian triangulations

Abstract: We prove that properly rescaled large planar Eulerian triangulations converge to the Brownian map. This result requires more than a standard application of the methods that have been used to obtain the convergence of other families of planar maps to the Brownian map, as the natural distance for Eulerian triangulations is a canonical oriented pseudo-distance. To circumvent this difficulty, we adapt the layer decomposition method established by Curien and Le Gall, which yields asymptotic proportionality between three natural distances on planar Eulerian triangulations: the usual graph distance, the canonical oriented pseudo-distance, and the Riemannian metric. This notably gives the first mathematical proof of a convergence to the Brownian map for maps endowed with their Riemannian metric. Along the way, we also construct new models of infinite random maps, as local limits of large planar Eulerian triangulations.

Eulerian and Lagrangian second-order statistics of superfluid He 4 grid turbulence

Abstract: We use particle-tracking velocimetry to study Eulerian and Lagrangian second-order statistics of superfluid $^{4}\mathrm{He}$ grid turbulence. The Eulerian energy spectra at scales larger than the mean distance between quantum vortex lines behave classically with close to Kolmogorov-1941 scaling and are almost isotropic. The Lagrangian second-order structure functions and frequency power spectra, measured at scales comparable with the intervortex distance, demonstrate a sharp transition from nearly classical behavior to a regime dominated by the motion of quantum vortex lines. Employing the homogeneity of the flow, we verify a set of relations that connect various second-order statistical objects that stress different aspects of turbulent behavior, allowing a multifaceted analysis. We use the two-way bridge relations between Eulerian energy spectra and second-order structure functions to reconstruct the energy spectrum from the known second-order velocity structure function and vice versa. The Lagrangian frequency spectrum reconstructed from the measured Eulerian spectrum using the Eulerian-Lagrangian bridge differs from the measured Lagrangian spectrum in the quasiclassical range, which calls for further investigation.