Hybrid journal (It can contain Open Access articles) ISSN (Print) 0272-4960 - ISSN (Online) 1464-3634 Published by Oxford University Press[411 journals]

Authors:Chen C; Roberts A, Bunder J. Pages: 347 - 379 Abstract: Multiscale modelling aims to systematically construct macroscale models of materials with fine microscale structure. However, macroscale boundary conditions are typically not systematically derived, but rely on heuristic arguments, potentially resulting in a macroscale model which fails to adequately capture the behaviour of the microscale system. We derive the macroscale boundary conditions of the macroscale model for longitudinal wave propagation on a lattice with periodically varying density and elasticity. We model the macroscale dynamics of the microscale Dirichlet, Robin-like, Cauchy-like and mixed boundary value problem. Numerical experiments test the new methodology. Our method of deriving boundary conditions significantly improves the accuracy of the macroscale models. The methodology developed here can be adapted to a wide range of multiscale wave propagation problems. PubDate: Mon, 19 Mar 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy004 Issue No:Vol. 83, No. 3 (2018)

Authors:Sakajo T; Yokoyama T. Pages: 380 - 411 Abstract: A flow of 2D incompressible and inviscid fluid is an example of a Hamiltonian vector field, where its Hamiltonian corresponds to the stream function whose level curves are called streamlines. A 2D Hamiltonian vector field is said to be structurally stable when the topological structure of streamlines is unchanged under any small perturbations of the vector field. In the present paper we show that the streamline topology of every structurally stable Hamiltonian vector field is in one-to-one correspondence with a labelled and directed plane tree and its associated symbolic expression called a regular expression. Consequently, we can characterize all streamline topologies with their corresponding plane trees and regular expressions uniquely. The present theory of tree representations is combinatorial; it brings us a new compression algorithm converting a large amount of streamline plots obtained by laboratory experiments and numerical simulations into a small set of simple symbolic data of regular expressions, which is amenable to a big data analysis for streamline patterns. Conversion to tree structures and their associated regular expressions is easily performed and it is flexibly applicable not only to incompressible flows but also to any physical phenomena described by Hamiltonian vector fields. We also demonstrate how the tree representation is applied to describe variations of streamline topologies for incompressible flows. PubDate: Mon, 26 Mar 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy005 Issue No:Vol. 83, No. 3 (2018)

Authors:Barańska J; Kozitsky Y. Pages: 412 - 435 Abstract: An infinite system of point particles performing random jumps with repulsion in $\mathbb{R}^{d}$ is studied. The states of the system are probability measures $\mu \in \mathcal{P}(\varGamma )$ on the space of particle’s configurations $\varGamma $. Among them there are sub-Poissonian states constituting the set $\mathcal{P}_{\exp}(\varGamma )\subset \mathcal{P}(\varGamma )$. The main result of the paper is the construction of the global in time evolution $\mathcal{P}_{\exp}(\varGamma )\ni \mu _{0} \mapsto \mu _{t}\in \mathcal{P}_{\exp }(\varGamma )$ of states with the help of their correlation functions, which yields that the constructed map is a unique weak solution of the corresponding Fokker–Planck equation. PubDate: Mon, 19 Mar 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy006 Issue No:Vol. 83, No. 3 (2018)

Authors:Gao T; Vanden-Broeck J, Wang Z. Pages: 436 - 450 Abstract: This work is concerned with flexural-gravity solitary waves on water of finite depth. The deformation of the elastic sheet is modelled based on the Cosserat theory of hyperelastic shells satisfying Kirchhoff’s hypotheses. Both steady and unsteady waves are computed numerically for the full Euler equations by using a conformal mapping technique. Complete bifurcation diagrams of solitary waves are presented, and various dynamical experiments, including the evolution of unstable solitary waves and the generation of stable ones, are carried out via direct time-dependent simulations. In particular, we show that generalized solitary waves can also be excited by moving loads on the elastic cover. PubDate: Thu, 22 Mar 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy007 Issue No:Vol. 83, No. 3 (2018)

Authors:Le Meur H. Pages: 451 - 470 Abstract: The aim of this article is to derive asymptotic models from surface wave equations in the presence of surface tension and viscosity. Using the Navier–Stokes equations with a flat bottom, we derive the viscous 2D Boussinesq system. The assumed scale of transverse variation is larger than the one along the main propagation direction (weak transverse variation). This Boussinesq system is proved to be consistent with the Navier–Stokes equations. This system is only an intermediate result that enables us to derive the Kadomtsev–Petviashvili (KP) equation which is a 2D generalization of the KdV equation. In addition, we get the 1D KdV equation, and lastly the Boussinesq equation. All these equations are derived for general initial conditions either slipping (Euler’s fluid) or sticking (Navier–Stokes fluid) with a given profile in the boundary layer different from the Euler’s one. We discuss whether the Euler’s initial condition is physical. PubDate: Fri, 23 Mar 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy008 Issue No:Vol. 83, No. 3 (2018)

Authors:Deliktas E; Teymur M. Pages: 471 - 495 Abstract: The method of multiple scales is used to examine the slow modulation of a harmonic surface shear horizontal wave over the surface of a non-linear elastic half space covered with two different non-linear elastic layers of uniform thickness. The appropriate non-linear Schrödinger (NLS) equation is derived with coefficients that depend, in a complicated way, on linear and non-linear material parameters of the double-layered half space, the thicknesses of the layers and also the wave number of the waves. This equation reduces to the NLS equation obtained for the case of a single-layered half space when the thickness of one of the layers goes to zero. Finally, the effect of the non-linear properties of the intermediate layer on the existence of solitary waves has been investigated numerically and the results are presented graphically. Also, to reveal the effect of the second layer, the coefficients of the NLS equations obtained for a double-layered half space and for a single-layered half space are compared. It has been observed that the propagation is affected considerably by the existence of a second layer. PubDate: Wed, 11 Apr 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy009 Issue No:Vol. 83, No. 3 (2018)

Authors:Soffer A; Zhao X. Pages: 496 - 513 Abstract: We apply the modulation theory to study the vortex and radiation solution in the 2D nonlinear Schrödinger equation. The full modulation equations which describe the dynamics of the vortex and radiation separately are derived. A general algorithm is proposed to efficiently and accurately find vortices with prescribed values of energy and spin index. The modulation equations are solved by accurate numerical method. Numerical tests and simulations of radiation are given. PubDate: Tue, 08 May 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy016 Issue No:Vol. 83, No. 3 (2018)

Authors:Lazowski A; Shea S. Pages: 514 - 525 Abstract: We introduce a voting procedure that compounds alternative vote (AV) and the method of plurality. For a three-candidate election, we characterize when an AV election can violate monotonicity. We use this characterization to show that the compound procedure is no more likely (and for certain numbers of voters, strictly less likely) to produce an election that can violate monotonicity. We also show that the voting profiles that can violate monotonicity in an AV election are disjoint from those that can violate monotonicity in an election using the compound procedure. Finally, we find sharp lower bounds for the number of voters required for AV and the compound procedure to yield an election that can violate monotonicity. PubDate: Tue, 08 May 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy017 Issue No:Vol. 83, No. 3 (2018)

Authors:Bürger R; Careaga J, Diehl S. Pages: 526 - 552 Abstract: A method is presented for the identification of non-convex flux functions of hyperbolic scalar conservation laws that model sedimentation of solid particles in a liquid. While all previous identification methods are based on data obtained from settling tests in cylindrical vessels, the novel approach is based on the richer solution behaviour produced in a vessel with downward-decreasing cross-sectional area. Except for the initial homogeneous concentration, the data given for the present inverse problem are the location of the descending supernatant-suspension interface as a function of time. The inverse problem is solved by utilizing the construction of solutions of the direct problem by the method of characteristics. In theory, the entire flux function can be estimated from only one batch-settling experiment, and the solution is given by parametric and explicit formulas for the flux function. The method is tested on synthetic data (e.g. generated by numerical simulations with a known flux) and on published experimental data. PubDate: Fri, 04 May 2018 00:00:00 GMT DOI: 10.1093/imamat/hxy018 Issue No:Vol. 83, No. 3 (2018)