Solenoidal Functions Research Articles

Numerical Analyses of the Induced Electric Field in a Material Sample Placed Within an Energized Rectangular Cavity

The investigation of the heating mechanism of a material sample in an energized electromagnetic cavity requires the understanding of the interaction of the electromagnetic fields with the material sample in a cavity. The key step for this understanding is to quantify the distribution of the induced electric field by solving an Electric Field Integral Equation (EFIE) or a Magnetic Field Integral Equation (MFIE) inside the material sample placed in the cavity. In this study, a complete set of vector wave functions which include both solenoidal and irrotational functions are employed. EFIE and MFIE are derived based on the expansion of these vector wave functions. Due to the slow convergence of the dyadic Green's function, some schemes were developed to improve the convergence rate of the numerical calculation in order to save computation time. Numerical results of some material samples agree well with theoretical estimations.

Methods of solving certain two-dimensional problems of doppler tomography

In the development of methods of recovering a vector field using radiolocation systems by continuous unmodulated radiation the problem arises of recovering a vector-valued function from information determined by a family of oriented lines. In the two-dimensional case this problem has a nonunique solution in the class of continuous functions of compact support. In the present paper the problem is studied on the set of solenoidal functions and the set of piecewise-constant functions. We give a method of solving the problem and study its stability.

A Nonconforming Finite Element Method for the Stationary Navier--Stokes Equations

Approximations to solutions of the inhomogeneous boundary value problem for the Navier--Stokes equations are constructed via a nonstandard finite element method. The velocity field is approximated using piecewise solenoidal functions that are totally discontinuous across interelement boundaries but which are pointwise divergence free on each element. The pressure is approximated by C0 functions. Optimal rates of convergence results are obtained requiring only local quasi-uniformity assumptions on the meshes.

Problème de Stokes dans ℝ n et espaces de Sobolev avec poids

We present isomorphism results for the Stokes problem in ℝ n using weighted Sobolev spaces and we study the asymptotic behaviour of the solutions. We also study this problem with data in the Hardy space ℋ 1 ( ℝ n ) as well as a few properties of solenoidal functions.

Classifying C1+ structures on dynamical fractals: 1. The moduli space of solenoid functions for Markov maps on train tracks

AbstractSullivan's scaling function provides a complete description of the smooth conjugacy classes of cookie-cutters. However, for smooth conjugacy classes of Markov maps on a train track, such as expanding circle maps and train track mappings induced by pseudo-Anosov systems, the generalisation of the scaling function suffers from a deficiency. It is difficult to characterise the structure of the set of those scaling functions which correspond to smooth mappings. We introduce a new invariant for Markov maps called the solenoid function. We prove that for any prescribed topological structure, there is a one-to-one correspondence between smooth conjugacy classes of smooth Markov maps and pseudo-Hölder solenoid functions. This gives a characterisation of the moduli space for smooth conjugacy classes of smooth Markov maps. For smooth expanding maps of the circle with degree d this moduli space is the space of Hölder continuous functions on the space {0,…, d − 1}ℕ satisfying the matching condition.

The density of solenoidal functions and the convergence of a dual finite element method

The density of solenoidal functions and the convergence of a dual finite element method

Microcomputer controller for a superconducting solenoid

A controller for a superconducting solenoid magnet is described as an example of a microprocessor replacement for hard-wired digital logic in a research laboratory. Its purpose is to take control of the field-changing tasks, and other solenoid status functions such as the superconducting-normal transition. The relatively slow speed of this task is ideally suited to an 8-bit microprocessor, making available a larger data acquisition minicomputer for data processing while the microprocessor operates independently. The minicomputer presents the microcomputer with the desired field value and returns to data processing, while the microprocessor independently measures the present field value and ramps the solenoid power supply to the desired field value, signalling the minicomputer when this has been done.

Net schemes for the navier—stokes equations associated with projection

DIFFERENCE schemes of the projection type for the Navier Stokes equations are considered. In the construction of schemes of the alternating directions type the continuity equation is replaced by a regularized equation. For all the difference schemes considered the convergence of the approximate to the exact solution is proved and estimates of the rate of convergence are given. In [1–4] schemes were considered which reduce at each time step the solution of a Navier- Stokes equation to the solution of quasilinear parabolic equations for the velocity components and to a subsequent projection of the velocity vector obtained in this way onto a space of solenoid functions. This requires a Poisson equation to be solved at each time step. In this paper schemes are proposed combining the projection with the method of alternating directions. Their construction uses methods of regularizing the continuity equation (see [5–7]) and the limitation of non-linearity (see [8, 9]). For all the schemes considered in this paper convergence is proved and estimates are given of the rate of convergence without restrictions on the nature of the dependence of r on h and v, where T is the time step, h is the step along each of the spatial variables, and v is the value of the kinematic viscosity.