Mathematical Models for Ice Accretion on Conductors using Free Streamline Theory: Part I: Single Conductor

It is the purpose of this paper to develop theoretical modelsof rime ice accretion and snow loading on a single conductor.The ice-accretion kinetics are formulated in terms of separatedor free streamline airflow, assuming that the conductor doesnot twist under the eccentric ice loading on the windward side. The new results for rime ice accretion lead to interesting comparisonsof the predicted rate of axial growth for attached and separatedpotential flow, respectively. A new ice-accretion model is thenproposed for snow loading and here account is taken of the factthat the length of dendritic snowflakes is comparable with thediameter of the conductor.     

A spectrally accurate algorithm for electromagnetic scattering in three dimensions

In this work we develop, implement and analyze a high-order spectrally accurate algorithm for computation of the echo area, and monostatic and bistatic radar cross-section (RCS) of a three dimensional perfectly conducting obstacle through simulation of the time-harmonic electromagnetic waves scattered by the conductor. Our scheme is based on a modified boundary integral formulation (of the Maxwell equations) that is tolerant to basis functions that are not tangential on the conductor surface. We test our algorithm with extensive computational experiments using a variety of three dimensional perfect conductors described in spherical coordinates, including benchmark radar targets such as the metallic NASA almond and ogive. The monostatic RCS measurements for non-convex conductors require hundreds of incident waves (boundary conditions). We demonstrate that the monostatic RCS of small (to medium) sized conductors can be computed using over one thousand incident waves within a few minutes (to a few hours) of CPU time. We compare our results with those obtained using method of moments based industrial standard three dimensional electromagnetic codes CARLOS, CICERO, FE-IE, FERM, and FISC. Finally, we prove the spectrally accurate convergence of our algorithm for computing the surface current, far-field, and RCS values of a class of conductors described globally in spherical coordinates.     

On heat conductors with a stationary hot spot

We consider a heat conductor having initial constant temperature and zero boundary temperature at every time.The hot spot is the point at which temperature attains its maximum at each given time. For convex conductors, if the hot spot does not move in time, we prove symmetry results for planar triangular and quadrangular conductors.Then, we examine the case of a general conductor and, by an asymptotic formula, we prove that, if there is a stationary critical point, not necessarily the hot spot, then the conductor must satisfy a geometric condition. In particular, we show that there is no stationary critical point inside planar non-convex quadrangular conductors. Mathematics Subject Classification (2000) Primary 35K05, 35K20, 35J05; Secondary 35J25, 35B38, 35B40     

Reflection at a Discontinuity in a Transmission Line

An exact expression is obtained for the reflection coefficientwhen a TEM wave, propagating along a co-axial line in a regionwhere both conductors are perfectly conducting, is incidenton a region where the inner conductor consists of dielectriccoated with a resistive film whilst the outer conductor remainsperfectly conducting.     

Mathematical Modeling of the Telluric Field near Extended Conductors

A procedure is proposed for simulating the telluric field near nonhomogeneous hollow cylindrical extended conductors of arbitrary finite length. The procedure allows for the nonhomogeneous conductivity of the conductor with a thin insulating coating. The surrounding medium is layered with three-dimensional nonhomogeneities, which may intersect the conductor. The external telluric field is specified based on actual distributional information.     

Conformal transformation method as applied to finding the current density distribution and induced magnetic field in a strip conductor with a rectangular cut

Given a conductor with a rectangular cut, the problem of determining the direct current distribution and the induced magnetic field in the conductor is solved with applications to the testing of chip conductors. An analytical expression for the current distribution is found assuming that the corners are rounded. The magnetic field induced by the given current is calculated.     

Even icosahedral Galois representations of prime conductor

In this paper, we use a series of targeted Hunter searches to prove that the minimal prime conductor of an even icosahedral Galois representation is 1951. In addition, we give a complete list of all prime conductors less than 10,000 of even icosahedral Galois representations.     

Solution procedures for three-dimensional eddy current problems

The problem under consideration is that of the scattering of time periodic electromagnetic fields by metallic obstacles. A common approximation here is that in which the metal is assumed to have infinite conductivity. The resulting problem, called the perfect conductor problem, involves solving Maxwell's equations in the region exterior to the obstacle with the tangential component of the electric field zero on the obstacle surface. In the interface problem different sets of Maxwell equations must be solved in the obstacle and outside while the tangential components of both electric and magnetic fields are continuous across the obstacle surface. Solution procedures for this problem are given. There is an exact integral equation procedure for the interface problem and an asymptotic procedure for large conductivity. Both are based on a new integral equation procedure for the perfect conductor problem. The asymptotic procedure gives an approximate solution by solving a sequence of problems analogous to the one for perfect conductors.     

Simulating the Electromagnetic Field on the Surface of Extended Conductors Immersed in a Conducting Medium

A new simulation procedure is developed for the electromagnetic field on the surface of infinitely extended hollow cylindrical conductors coated with a dielectric thin layer and buried in a conducting ground. The field source is a linear alternating current transverse to the conductor, which is specified in the upper halfspace (air). A system of two one-dimensional integral equations is constructed describing the behavior of the electromagnetic field in this system. The problem is solved by a fast algorithm that computes the electromagnetic field for various values of the model parameters. Computation results are reported for several values of the main parameters.__________Translated from Prikladnaya Matematika i Informatika, No. 16, pp. 15–26, 2004.     

A continuum theory for first‐order phase transitions based on the balance of structure order

First‐order phase transitions are modelled by a non‐homogeneous, time‐dependent scalar‐valued order parameter or phase field. The time dependence of the order parameter is viewed as arising from a balance law of the structure order. The gross motion is disregarded and hence the body is regarded merely as a heat conductor. Compatibility of the constitutive functions with thermodynamics is exploited by expressing the second law through the classical Clausius–Duhem inequality. First, a model for conductors without memory is set up and the order parameter is shown to satisfy a maximum theorem. Next, heat conductors with memory are considered. Different evolution problems are established through a system of differential equations whose form is related to the manner in which the memory property is represented. Copyright © 2007 John Wiley & Sons, Ltd.     

Second-order effects in the induction of thin discs

The effect of thickness of a conductor is of considerable interestin many areas of application, in particular, in geophysicalprospecting, in ionospheric physics, and in the induction ofoceanic basins. This paper considers the induction of eddy currentsin a circular conductor of finite radius due to an alternatingprimary magnetic field. A perturbation scheme yields the boundaryconditions to be satisfied by the leading- and first-order potentials,which may be solved through a set of integral equations. Itis shown that, when the thickness of the conductor is about40% of the skin depth, the first-order terms become important.Further, for most practical purposes, both the leading- andfirst-order potentials may be derived from distributions ofsurface currents. An example is given for the case when theprimary field is a vertical dipole. The formulation is easilygeneralized to thin conductors of arbitrary shape.     

Direct solution of an optimal layout problem for isotropic heat conductors with a volume fraction constraint

This paper considers the problem of the optimal layout of two isotropic heat conductors on a plane subject to a constraint on the overall volume fraction of the better conductor. The problem is solved initially using the traditionalG _m -closure technique. Later, the utility of the direct relaxation technique introduced in Ref. 1 is illustrated as we apply it to verify the original results.This paper is dedicated with gratitude by the author to his advisor K. A. Lurie. The author wishes to acknowledge discussions with D. Clomenil.     

On the Dynamics of Bundled Conductors in Overhead Transmission Lines

The Energy Balance Principle (EBP) is well established for estimating the vibration levels of wind induced oscillations of single overhead transmission lines. The mathematical model, wherein a conductor is treated as a continuous system, results in a transcendental eigenvalue problem (EVP), which gives numerical difficulties in the case of bundled conductors. In this paper, different approaches for solving transcendental EVP and their relative merits are discussed. A new method named continuous spectrum approach provides a good engineering solution. Results from different approaches are compared. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exact solutions for equilibrium configurations of the surface of a conducting fluid in a nonuniform magnetic field

We study the two-dimensional magnetic shaping problem for the situation where the free surface of a perfectly conducting fluid is deformed by the magnetic field of a system of linear current-carrying conductors. Equilibrium is achieved due to the balance of capillary and magnetic pressures. We obtain exact solutions of the problem using conformal map techniques. These solutions describe a system of two-dimensional dimples that appear on the initially flat surface of a liquid conductor under the action of a nonuniform magnetic field. We consider the case of two symmetrically located dimples in detail.     

Computation of Magnetic Fields in Open Traps with Oppositely Directed Fields

The article presents the results of numerical calculation of the magnetic field in open traps. The magnetic field in traps is generated by a system of circular conductors carrying currents in opposite directions. Each pair of conductors creates a magnetic field configuration of so-called cusped geometry (an antiprobkotron). Such configurations may be called traps with oppositely directed fields. Their main shortcoming is the high loss of particles along the magnetic forcelines through the annular slit and the axial opening. These particle losses are reduced by introducing a system of conductors with different configurations of circular loops that carry currents in opposite directions. The magnetic fields of the complex system of current carrying loops are calculated by summing over all pairs of conductors. Computation results are reported for open traps with a cusped magnetic field (antiprobkotrons) for various configurations of conductor pairs.     

More Accurate Skin-Depth Approximations

The penetration of a high-frequency alternating or rotatingmagnetic field into a conductor is considered. The magneticReynolds number is assumed to be small. The standard high-frequency(or skin-depth) approximation is shown to be incorrect in theinterior of the conductor, leading to large relative errors.Two terms of the correct expansion are calculated. It is foundthat a ‘focusing’ effect occurs, in a manner similarto the formation of caustics in geometrical optics. As a result,at some interior points the field is several times larger thanexpected. The behaviour near such singular points is found.The implications for electromagnetically driven fluid flow arebriefly discussed.     

The Stability Theory of Charge Distributions on Conductors

Stability of solution is considered for the charge-density problemof several rigid charged conductors, stability being with respectto the geometrical configuration of the system. Under the analyticalrestrictions usual in potential theory for the surfaces on whichthe charges reside, two theorems are proved establishing thestability: the first for when the potentials of the conductorsare fixed, and the second for when their total charges are fixed.Subsequently, it is shown how the rigidity condition can berelaxed, but a counterexample to stability is described fora situation where the surface analytical conditions are notfulfilled. Finally, an application of the theory is given, inwhich the second of the said theorems is used to close a gapin the theory by the energy method of the electrostatic forceon a conductor.     

An Adaptive Finite Element Method for the H- ψ Formulation of Time-dependent Eddy Current Problems

In this paper, we develop an adaptive finite element method based on reliable and efficient a posteriori error estimates for the H − ψ formulation of eddy current problems with multiply connected conductors. Multiply connected domains are considered by making “cuts”. The competitive performance of the method is demonstrated by an engineering benchmark problem, Team Workshop Problem 7, and a singular problem with analytic solution.W. Zheng was supported in part by China NSF under the grant 10401040.Z. Chen was supported in part by China NSF under the grant 10025102 and 10428105, and by the National Basic Research Project under the grant 2005CB321701.     

Polygonal heat conductors with a stationary hot spot

We consider a convex polygonal heat conductor whose inscribed circle touches every side of the conductor. Initially, the conductor has constant temperature and, at every time, the temperature of its boundary is kept at zero. The hot spot is the point at which temperature attains its maximum at each given time. It is proved that, if the hot spot is stationary, then the conductor must satisfy two geometric conditions. In particular, we prove that these geometric conditions yield some symmetries provided the conductor is either pentagonal or hexagonal. This research was partially supported by Grants-in-Aid for Scientific Research (B) (# 12440042) and (B) (# 15340047) of Japan Society for the Promotion of Science, and by a Grant of the Italian MURST.     

On the Optimal Insulation of Conductors

We coat a conductor with an insulator and equate the effectiveness of this procedure with the rate at which the body dissipates heat when immersed in an ice bath. In the limit, as the thickness and conductivity of the insulator approach zero, the dissipation rate approaches the first eigenvalue of a Robin problem with a coefficient determined by the shape of the insulator. Fixing the mean of the shape function, we search for the shape with the least associated Robin eigenvalue. We offer exact solutions for balls; for general domains, we establish existence and necessary conditions and report on the results of a numerical method.