Showing papers on "Eulerian path published in 2018"

Broadening of Modeled Cloud Droplet Spectra Using Bin Microphysics in an Eulerian Spatial Domain

Abstract: This study investigates droplet size distribution (DSD) characteristics from condensational growth and transport in Eulerian dynamical models with bin microphysics. A hierarchy of modeling ...

An effective criterion for Eulerian multizeta values in positive characteristic

Abstract: Characteristic p multizeta values were initially studied by Thakur, who defined them as analogues of classical multiple zeta values of Euler. In the present paper we establish an effective criterion for Eulerian multizeta values, which characterizes when a multizeta value is a rational multiple of a power of the Carlitz period. The resulting “t-motivic”algorithm can tell whether any given multizeta value is Eulerian or not. We also prove that if ζA(s1, . . . , sr) is Eulerian, then ζA(s2, . . . , sr) has to be Eulerian. When r = 2, this was conjectured (and later on conjectured for arbitrary r) by Lara Rodriguez and Thakur for the zeta-like case from numerical data. Our methods apply equally well to values of Carlitz multiple polylogarithms at algebraic points and zeta-like multizeta values.

Global regularity for 1D Eulerian dynamics with singular interaction forces

Abstract: The Euler--Poisson-alignment (EPA) system appears in mathematical biology and is used to model, in a hydrodynamic limit, a set of agents interacting through mutual attraction/repulsion as well as a...

Lagrangian and Eulerian Description of Bed Load Transport

Abstract: Sediment particles transported as bed load undergo alternating periods of motion and rest, particularly at weak flow intensity. Bed load transport can be investigated by either following the motion of individual particles (Lagrangian approach) or observing the phenomenon at prescribed locations (Eulerian approach). In this paper, the Lagrangian and Eulerian descriptions are merged into a unifying framework that includes definitions for quantities used to describe the kinematics of particle motion, as well as the relationships among these quantities. The alternation of motion and rest is represented by two complementary descriptions: (i) proportion of motion, indicating either the relative time spent in motion by an individual particle or the relative number of moving particles; (ii) persistence of motion, indicating the extent to which the process consists of relatively few long periods of motion or of many short ones. The framework only involves first moments of the key quantities. The conceptual developments are tested against results from an experiment with weak bed load transport, demonstrating the soundness of the approach. From an operational point of view, a Lagrangian observation is difficult to perform, since the particle motion is usually investigated for finite spatial domains (e.g., a measurement window within a laboratory or natural reach). Strategies to overcome such limitations are described, suggesting the possibility of obtaining unbiased mean values for Lagrangian descriptors. The proposed framework can be used in any study aimed at parameterizing the kinematic properties of bed load particles as functions of the hydrodynamic conditions.

A new Eulerian model for viscous and heat conducting compressible flows

Abstract: In this article, a suite of physically inconsistent properties of the Navier–Stokes equations, associated with the lack of mass diffusion and the definition of velocity, is presented. We show that these inconsistencies are consequences of the Lagrangian derivation that models viscous stresses rather than diffusion. A new model for compressible and diffusive (viscous and heat conducting) flows of an ideal gas, is derived in a purely Eulerian framework. We propose that these equations supersede the Navier–Stokes equations. A few numerical experiments demonstrate some differences and similarities between the new system and the Navier–Stokes equations.

Effective Shape Optimization of Laplace Eigenvalue Problems Using Domain Expressions of Eulerian Derivatives

Abstract: We consider to solve numerically the shape optimization models with Dirichlet Laplace eigenvalues. Both volume-constrained and volume unconstrained formulations of the model problems are presented. Different from the literature using boundary-type Eulerian derivatives in shape gradient descent methods, we advocate to use the more general volume expressions of Eulerian derivatives. We present two shape gradient descent algorithms based on the volume expressions. Numerical examples are presented to show the more effectiveness of the algorithms than those based on the boundary expressions.

Batch-Parallel Euler Tour Trees.

Abstract: The dynamic trees problem is to maintain a forest undergoing edge insertions and deletions while supporting queries for information such as connectivity. There are many existing data structures for this problem, but few of them are capable of exploiting parallelism in the batch-setting, in which large batches of edges are inserted or deleted from the forest at once. In this paper, we demonstrate that the Euler tour tree, an existing sequential dynamic trees data structure, can be parallelized in the batch setting. For a batch of $k$ updates over a forest of $n$ vertices, our parallel Euler tour trees perform $O(k \log (1 + n/k))$ expected work with $O(\log n)$ depth with high probability. Our work bound is asymptotically optimal, and we improve on the depth bound achieved by Acar et al. for the batch-parallel dynamic trees problem. The main building block for parallelizing Euler tour trees is a batch-parallel skip list data structure, which we believe may be of independent interest. Euler tour trees require a sequence data structure capable of joins and splits. Sequentially, balanced binary trees are used, but they are difficult to join or split in parallel. We show that skip lists, on the other hand, support batches of joins or splits of size $k$ over $n$ elements with $O(k \log (1 + n/k))$ work in expectation and $O(\log n)$ depth with high probability. We also achieve the same efficiency bounds for augmented skip lists, which allows us to augment our Euler tour trees to support subtree queries. Our data structures achieve between 67--96x self-relative speedup on 72 cores with hyper-threading on large batch sizes. Our data structures also outperform the fastest existing sequential dynamic trees data structures empirically.

Solving Directed Laplacian Systems in Nearly-Linear Time through Sparse LU Factorizations

Abstract: In this paper, we show how to solve directed Laplacian systems in nearly-linear time. Given a linear system in an n × n Eulerian directed Laplacian with m nonzero entries, we show how to compute an e-approximate solution in time O(m log^O(1) (n) log (1/e)). Through reductions from [Cohen et al. FOCS'16], this gives the first nearly-linear time algorithms for computing e-approximate solutions to row or column diagonally dominant linear systems (including arbitrary directed Laplacians) and computing e-approximations to various properties of random walks on directed graphs, including stationary distributions, personalized PageRank vectors, hitting times, and escape probabilities. These bounds improve upon the recent almost-linear algorithms of [Cohen et al. STOC'17], which gave an algorithm to solve Eulerian Laplacian systems in time O((m+n2^O(√ log n log log n))log^O(1)(n e^-1)). To achieve our results, we provide a structural result that we believe is of independent interest. We show that Eulerian Laplacians (and therefore the Laplacians of all strongly connected directed graphs) have sparse approximate LU-factorizations. That is, for every such directed Laplacian there are lower upper triangular matrices each with at most O(n) nonzero entries such that there product spectrally approximates the directed Laplacian in an appropriate norm. This claim can be viewed as an analog of recent work on sparse Cholesky factorizations of Laplacians of undirected graphs. We show how to construct such factorizations in nearly-linear time and prove that once constructed they yield nearly-linear time algorithms for solving directed Laplacian systems.

A zonal hybrid approach coupling FNPT with OpenFOAM for modelling wave-structure interactions with action of current

Abstract: This paper presents a hybrid numerical approach, which combines a two-phase Navier- Stokes model (NS) and the fully nonlinear potential theory (FNPT), for modelling wave-structure interaction. The former governs the computational domain near the structure, where the viscous and turbulent effects are significant, and is solved by OpenFOAM/InterDyMFoam which utilising the finite volume method (FVM) with a Volume of Fluid (VOF) for the phase identification. The latter covers the rest of the domain, where the fluid may be considered as incompressible, inviscid and irrotational, and solved by using the Quasi Arbitrary Lagrangian- Eulerian finite element method (QALE-FEM). These two models are weakly coupled using a zonal (spatially hierarchical) approach. Considering the inconsistence of the solutions at the boundaries between two different sub-domains governed by two fundamentally different models, a relaxation (transitional) zone is introduced, where the velocity, pressure and surface elevations are taken as the weighted summation of the solutions by two models. In order to tackle the challenges associated and maximise the computational efficiency, further developments of the QALE-FEM have been made. These include the derivation of an arbitrary Lagrangian- Eulerian FNPT and application of a robust gradient calculation scheme for estimating the velocity. The present hybrid model is applied to the numerical simulation of a fixed horizontal cylinder subjected to a unidirectional wave with or without following current. The convergence property, the optimisation of the relaxation zone, the accuracy and the computational efficiency are discussed. Although the idea of the weakly coupling using the zonal approach is not new, the present hybrid model is the first one to couple the QALE-FEM with OpenFOAM solver and/or to be applied to numerical simulate the wave-structure interaction with presence of current.

Probing turbulent superstructures in Rayleigh-B\'{e}nard convection by Lagrangian trajectory clusters.

Abstract: We analyze large-scale patterns in three-dimensional turbulent convection in a horizontally extended square convection cell by Lagrangian particle trajectories calculated in direct numerical simulations. A simulation run at a Prandtl number Pr $=0.7$, a Rayleigh number Ra $=10^5$, and an aspect ratio $\Gamma=16$ is therefore considered. These large-scale structures, which are denoted as turbulent superstructures of convection, are detected by the spectrum of the graph Laplacian matrix. Our investigation, which follows Hadjighasem {\it et al.}, Phys. Rev. E {\bf 93}, 063107 (2016), builds a weighted and undirected graph from the trajectory points of Lagrangian particles. Weights at the edges of the graph are determined by a mean dynamical distance between different particle trajectories. It is demonstrated that the resulting trajectory clusters, which are obtained by a subsequent $k$-means clustering, coincide with the superstructures in the Eulerian frame of reference. Furthermore, the characteristic times $\tau^L$ and lengths $\lambda_U^L$ of the superstructures in the Lagrangian frame of reference agree very well with their Eulerian counterparts, $\tau$ and $\lambda_U$, respectively. This trajectory-based clustering is found to work for times $t\lesssim \tau\approx\tau^L$. Longer time periods $t\gtrsim \tau^L$ require a change of the analysis method to a density-based trajectory clustering by means of time-averaged Lagrangian pseudo-trajectories, which is applied in this context for the first time. A small coherent subset of the pseudo-trajectories is obtained in this way consisting of those Lagrangian particles that are trapped for long times in the core of the superstructure circulation rolls and are thus not subject to ongoing turbulent dispersion.

Efficient time stepping for the multiplicative Maxwell fluid including the Mooney‐Rivlin hyperelasticity

Abstract: Summary A popular version of the finite-strain Maxwell fluid is considered, which is based on the multiplicative decomposition of the deformation gradient tensor. The model combines Newtonian viscosity with hyperelasticity of Mooney-Rivlin type; it is a special case of the viscoplasticty model proposed by Simo and Miehe (1992). A simple, efficient and robust implicit time stepping procedure is suggested. Lagrangian and Eulerian versions of the algorithm are available, with equivalent properties. The numerical scheme is iteration free, unconditionally stable and first order accurate. It exactly preserves the inelastic incompressibility, symmetry, positive definiteness of the internal variables, and w-invariance. The accuracy of the stress computations is tested using a series of numerical simulations involving a non-proportional loading and large strain increments. In terms of accuracy, the proposed algorithm is equivalent to the modified Euler backward method with exact inelastic incompressibility; the proposed method is also equivalent to the classical integration method based on exponential mapping. Since the new method is iteration free, it is more robust and computationally efficient. The algorithm is implemented into MSC.MARC and a series of initial boundary value problems is solved in order to demonstrate the usability of the numerical procedures. This article is protected by copyright. All rights reserved.