Rooms with a view: A novel approach to iterated multidimensional wave conversion

Ray tracing in the presence of linear mode conversion leads to a ‘splitting’ of an incoming ray into two outgoing rays. When the rays are confined to a cavity, the rays can re-enter the conversion region many times, leading to iterated conversion. In this paper, we present new methods for the analysis of this problem. These involve a shift from local to global methods of analysis, and a shift in emphasis from the study of ray evolution in the dispersion surface to the study of the iterated dynamics of rays crossing the conversion surface. The analytical methods are quite general and can be applied in phase spaces of arbitrary dimension. In two spatial dimensions, (x, y), i.e. with a four-dimensional ray space, (x, y, k_x, k_y), rays are confined to three-dimensional regions called rooms, with one room for each wave type. In these rooms the rays do not cross, but when they intersect the conversion surface a family of converted rays is produced in the other room. The use of rooms allows a full view of the phase space dynamics of the iterated conversion of ray families. A simple two-dimensional model, inspired by the Budden resonance model, is presented as an example of these ideas.     

Propagation of Surface Waves at High Frequencies

An asymptotic theory is presented for the analysis of surfacewave propagation at high frequencies. The theory is developedfor scalar surface waves satisfying an impedance boundary conditionon a surface, which may be curved and, whose impedance may bevariable. A surface eikonal equation is derived for the phaseof the surface wave field, and it is shown that the wave fieldpropagates over the surface along the surface rays, which arethe characteristics of the surface eikonal equation. The wavefield in space is found by solving certain eikonal and transportequations with the aid of complex rays. The theory is then appliedto several examples: axial waves on a circular cylinder, sphericallysymmetric waves on a sphere, waves on a circular cone with avariable impedance, and waves on the plane boundary of an inhomogeneousmedium. In each case it is found that the asymptotic expansionof the exact solution agrees with the asymptotic solution.     

Geometry of the copositive and completely positive cones

The copositive cone, and its dual the completely positive cone, have useful applications in optimisation, however telling if a general matrix is in the copositive cone is a co-NP-complete problem. In this paper we analyse some of the geometry of these cones. We discuss a way of representing all the maximal faces of the copositive cone along with a simple equation for the dimension of each one. In doing this we show that the copositive cone has faces which are isomorphic to positive semidefinite cones. We also look at some maximal faces of the completely positive cone and find their dimensions. Additionally we consider extreme rays of the copositive and completely positive cones and show that every extreme ray of the completely positive cone is also an exposed ray, but the copositive cone has extreme rays which are not exposed rays.     

A simple method for spherical indentation of an elastic thin layer with surface tension bonded to a rigid substrate

An alternative method is proposed to solve the spherical indentation problem of an elastic thin layer with surface tension bonded to a rigid substrate. Based on the Kerr model, we establish a simple modified governing equation incorporating the surface tension effects for describing the relationship between the pressure and downward deflection of the impressed surface of the layer. This modified governing equation holds both inside and outside the contact zone, making it possible to analyze the whole layer by a unified differential equation. Numerical results are presented for the contact pressure inside the contact zone, the surface deflection of the elastic layer and the load-contact zone width relation to illustrate the present method. The validity and accuracy of the present method are demonstrated by comparing our results with those available in the existing literature.     

Capturing the S in segregation: a simple model of flowing granular mixtures in rotating drums

Flowing granular mixtures in rotating cylindrical drums arise in numerous industrial settings and are of great technological significance worldwide. To date, the development of a robust mathematical model for this process remains an open research problem. However, simple mathematical models may be developed that capture some of the underlying mechanisms, including segregation. The key lies in the analysis of the flowing surface layer whose profile is characteristically S-shaped: past models have shown that capturing this characteristic is an essential ingredient in predicting segregation. Using secondary to first-year undergraduate level mathematics, an analysis is presented of the flowing surface layer, which leads to an S-shaped profile. This problem presents an ideal case study for teaching applied mathematics and modelling. Interesting simulations that aid the visualization of this problem can be downloaded from the web, and simple classroom demonstrations of this process can be assembled with no specialized equipment.     

Potential of a simple layer for a torus with density depending on the distance from a tangent plane to the center

Using a bulk potential, we construct a one-dimensional toroidal layer with corresponding density for the mass distribution. We determine the density of a simple layer, and also find the potential of a simple layer distributed along the surface of an elliptical torus of revolution with density depending on distance as measured along the normal between a tangent plane and the center of the torus.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 29, pp. 85–87, 1989.     

Acoustics of a Splitter Plate

According to geometrical acoustics, zones of silence can occurwhen sound waves traverse flows. These zones are not often observedin practice. It is suggested that one reason for this in jetflows is that the sound scattered from the end of the jet-pipepenetrates the zone of silence to an appreciable extent. Theconjecture is confirmed by considering a line source, oscillatingharmonically in time, in the presence of a semi-infinite plateabove which is a shear layer in which the velocity increaseslinearly over a finite distance and then remains constant. Theedge wave is found to be dominant in the zone of silence andto be of the same order of magnitude as the refracted fieldoutside. Refracted rays produced by complex rays in the fluiddo not seem to be as important as the edge rays. The results are developed so as to allow quantitative investigationof more general configurations.     

Asymptotic behavior of grafting rays

In this paper we study the convergence behavior of grafting rays to the Thurston boundary of Teichmüller space. When the grafting is done along a weighted system of simple closed curves or along a maximal uniquely ergodic lamination this behavior is the same as for Teichmüller geodesics and lines of minima. We also show that the rays grafted along a weighted system of simple closed curves are at bounded distance from Teichmüller geodesics.     

Resonance scattering on a hole on the boundary surface with finite symmetry group

We consider optimal, in the number of operations, computation schemes for the solution of the problem of resonance scattering on a hole on a boundary surface with a discontinuously acting group. We show that the numerical solution of the diffraction problem on the hole can be represented as a discrete analog of the potential density of a simple layer on the boundary surface.     

A mathematical study of the linear theory for orthotropic elastic simple shells

The theory of simple shells is a surface‐related Cosserat model for thin elastic shells. In this direct approach, each material point is connected with a triad of rigidly rotating directors. This paper presents a study of the governing equations for orthotropic elastic simple shells in the framework of the linearized theory. We establish the uniqueness of classical solutions, without any restrictive assumption on the strain energy function. The continuous dependence of solutions on the body loads and initial data is proved. Also, the existence of weak solutions to the equations of simple shells is proved by means of an inequality of Korn's type established for such directed surfaces. Copyright © 2009 John Wiley & Sons, Ltd.     

An estimate of the heat fluxes to the surface of blunt bodies moving at hypersonic velocity in the atmosphere

The problem of the chemically equilibrium three-dimensional boundary layer on a blunt body which is in motion in the atmosphere is considered. A solution of the system of equations of the boundary layer is found by the method of successive approximations, and simple analytic expressions are written in the first approximation for the surface friction and heat flux coefficients. Formulae are obtained in the final form for estimating the convective heat flux in the neighbourhood of the critical point of spherical blunting.     

An algorithm for determining redundant inequalities and all solutions to convex polyhedra

Given a system of linear inequalities that define a polyhedral cone, we develop a simple technique for finding redundant inequalities. We thus insure that only the cone's extreme rays are calculated. The general solution for the system is developed in terms of the extreme rays. The method leads directly to the general solution for either bounded or unbounded polyhedra.     

An integral equation approximation of the mixed problem for the laplacian in R3

An integral representation for the solutions of the interior and exterior homogeneous mixed problems on a regular bounded open set in R ³ is given in terms of a potential of simple layer on the Dirichlet data and of double layer on the Neumann data. A coercive non-symmetric bilinear form is provided. Singular finite elements containing the first terms of the assymptotic expansion of the singularity are built and error estimates for the surface distributions are given.     

玻壳成型过程诱导残余应力的数值预测

建立了玻壳压制成型固化过程中残余应力预测的数值模拟模型,采用平行平板间玻璃熔体的固化问题来描述成型过程中残余应力形成的机理,并假定材料为热流变简单粘弹性材料．基于板壳理论,将产品视为平板单元的组合,并采用有限元法来求解,这种方法可以象全三维计算一样一层层地计算残余应力,非常适合复杂形状的薄压制成型产品．最后通过实验比较验证了所提出的模型和方法．     

An infinite mapping layer for deep water waves

In order to accurately and efficiently model the propagation of surface gravity waves in infinitely deep domains, we introduce an infinite mapping layer which transforms the unbounded physical domain into a bounded computational domain by means of a mapping function. The method was implemented in the framework of the finite element method, where it can be straightforwardly combined with a perfectly matched layer to model unbounded domains in horizontal direction. Finally, the application of the method is demonstrated by a simple 3D example. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One singularly perturbed Stefan problem describing destruction of the fuel layer in a laser target

A mathematical model of degradation of a deuterium-tritium (D-T) fuel layer located on the interior wall of a spherical shell is proposed. Such a shell with a solid layer frozen on it is a laser target that is employed in controlled thermonuclear fusion. As the laser target is delivered from the cryogenic chamber to the focus of the laser beam, the chamber stays, during a certain time interval, in a warm-gas cloud. During this time interval, the D-T layer degrades, and, in particular, its surface becomes nonideal. The mathematical model is formulated as a Stefan problem for a system of parabolic equations with nonlinear initial-boundary conditions. Small-parameter methods are applied to obtain an analytic solution to this problem, and the time during which the changes in the geometric parameters of the target’s fuel layer do not exceed technologically admissible values is estimated.     

Extending external rays throughout the Julia sets of rational maps

For polynomial maps in the complex plane, the notion of external rays plays an important role in determining the structure of and the dynamics on the Julia set. In this paper we consider an extension of these rays in the case of rational maps of the form Fλ(z) = z ⁿ + λ/z ⁿ where n > 1. As in the case of polynomials, there is an immediate basin of ∞, so we have similar external rays. We show how to extend these rays throughout the Julia set in three specific examples. Our extended rays are simple closed curves in the Riemann sphere that meet the Julia set in a Cantor set of points and also pass through countably many Fatou components. Unlike the external rays, these extended rays cross infinitely many other extended rays in a manner that helps determine the topology of the Julia set.     

A new framework for the solution of DEA models

We provide an alternative framework for solving data envelopment analysis (DEA) models which, in comparison with the standard linear programming (LP) based approach that solves one LP for each decision making unit (DMU), delivers much more information. By projecting out all the variables which are common to all LP runs, we obtain a formula into which we can substitute the inputs and outputs of each DMU in turn in order to obtain its efficiency number and all possible primal and dual optimal solutions. The method of projection, which we use, is Fourier–Motzkin (F–M) elimination. This provides us with the finite number of extreme rays of the elimination cone. These rays give the dual multipliers which can be interpreted as weights which will apply to the inputs and outputs for particular DMUs. As the approach provides all the extreme rays of the cone, multiple sets of weights, when they exist, are explicitly provided. Several applications are presented. It is shown that the output from the F–M method improves on existing methods of (i) establishing the returns to scale status of each DMU, (ii) calculating cross-efficiencies and (iii) dealing with weight flexibility. The method also demonstrates that the same weightings will apply to all DMUs having the same comparators. In addition it is possible to construct the skeleton of the efficient frontier of efficient DMUs. Finally, our experiments clearly indicate that the extra computational burden is not excessive for most practical problems.