The boundary value problem of the solar force-free magnetic field with constant ? and its analytical solution

Solar Physics ◽  
1986 ◽  
Vol 103 (2) ◽  
pp. 317-328 ◽  
Author(s):  
Zhen-Cheng Chen ◽  
Jing-Xiu Wang
Keyword(s):  
Magnetic Field ◽  
Analytical Solution ◽  
Boundary Value Problem ◽  
Boundary Value

scholarly journals Analytical solution of problem of shielding low-frequency magnetic field by thin-walled cylindrical screen in presence of cylinder

Informatics ◽  
2021 ◽  
Vol 18 (3) ◽  
pp. 48-58
Author(s):  
G. Ch. Shushkevich
Keyword(s):  
Magnetic Field ◽  
Analytical Solution ◽  
Boundary Value Problem ◽  
Harmonic Functions ◽  
Boundary Value ◽  
Low Frequency ◽  
Cylindrical Inclusion ◽  
Frequency Magnetic Field ◽  
Cylindrical Screen ◽  
Thin Walled

The analytical solution of boundary value problem describing the process of penetration of low-frequency magnetic field through thin-walled cylindrical screen with cylindrical inclusion is constructed by use of approximate boundary conditions. The source of the field is a thin thread of infinitely small length with an infinitely small cross-section where current circulates. Thread is located in a plane which is perpendicular to axis of cylindrical screen, in outer region with respect to a screen. Initially the potential of initial magnetic field is represented as spherical harmonic functions, then using addition theorems connecting spherical and cylindrical harmonic functions, it became as cylindrical harmonic functions superposition. Secondary potentials of magnetic field are also presented as superposition of cylindrical harmonic functions in three-dimensional space. It is shown that the solution of formulated boundary value problem is reduced to the solution of linear algebraic equations system for coefficients included in the representation of secondary fields. The influence of some aspects of the problem on the value of the screening coefficient of an external magnetic field when passing through a cylindrical copper screen in the presence of a cylindrical inclusion is studied numerically. Calculation results are presented in graphs form. Obtained results can be used to shield technical devices and biological objects against the effects of magnetic fields to provide ecological surrounding of operating electrical installations and devices.

scholarly journals Experiments and Numerical Implementation of a Boundary Value Problem Involving a Magnetorheological Elastomer Layer Subjected to a Nonuniform Magnetic Field

2021 ◽  
Vol 88 (7) ◽  
Author(s):  
Charles Dorn ◽  
Laurence Bodelot ◽  
Kostas Danas
Keyword(s):  
Magnetic Field ◽  
Magnetic Fields ◽  
Boundary Value Problem ◽  
Magnetic Particles ◽  
Boundary Value ◽  
Energetic Efficiency ◽  
Iron Core ◽  
Isotropic Layer ◽  
Magnetorheological Elastomer ◽  
Electromagnetic Coil

Abstract This study investigates experimentally and numerically the response of a magnetorheological elastomer (MRE) layer placed atop an electromagnetic coil. The MRE layer is deflected upon application of a current in the coil, which creates highly nonuniform magnetic fields. Isotropic and transversely isotropic layers (i.e., containing chains of magnetic particles) are tested experimentally, and the isotropic layer exhibits the largest deflection. To enhance the energetic efficiency of the model device, an iron core is introduced inside the electromagnetic coil, thereby leading to an increase in the resulting magnetic field near the center of the MRE layer. In parallel, the boundary value problem —including the MRE layer, the coil, the core (if present) and the surrounding air—is modeled numerically. For this, a magneto-mechanical, vector potential-based variational formulation is implemented in a standard three-dimensional finite element model at finite strains. For the material description, a recently proposed analytical homogenization-guided model is used to analyze the MRE in the “coil-only” configuration. It is then employed to predict the response of the layer in the “coil plus core” configuration, thus circumventing the need for a separate material characterization procedure. The proposed numerical simulation strategy provides a deeper understanding of the underlying complexity of the magnetic fields and of their interaction with the MRE layer. This study also reveals the importance of modeling the entire setup for predicting the response of MRE materials and, as a result, constitutes a step toward designing more efficient MRE-based devices.

scholarly journals The analytical solution of the 3D model with Robin's boundary conditions for 2 peat layers

2015 ◽  
Vol 3 ◽  
pp. 186
Author(s):  
Ērika Teirumnieka ◽  
Ilmārs Kangro ◽  
Edmunds Teirumnieks ◽  
Harijs Kalis
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Stationary Problem ◽  
Elliptic Type ◽  
Exponential Spline ◽  
Initial Boundary Value

<p>In this paper we consider averaging methods for solving the 3-D boundary value problem in domain containing 2 layers of the peat block. We consider the metal concentration in the peat blocks. Using experimental data the mathematical model for calculation of concentration of metal in different points in every peat layer is developed. A specific feature of these problems is that it is necessary to solve the 3-D boundary-value problems for elliptic type partial differential equations of second order with piece-wise diffusion coefficients in every direction and peat layers.</p><p>The special parabolic and exponential spline, which interpolation middle integral values of piece-wise smooth function, are considered. With the help of this splines is reduce the problems of mathematical physics in 3-D with piece-wise coefficients to respect one coordinate to problems for system of equations in 2-D. This procedure allows reduce the 3-D problem to a problem of 2-D and 1-D problems and the solution of the approximated problem is obtained analytically.</p><p>The solution of corresponding averaged 2-D initial-boundary value problem is obtained also numerically, using for approach differential equations the discretization in space applying the central differences. The approximation of the 2-D non-stationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerical solution is compared with the analytical solution.</p>

scholarly journals Analytical Solution for the Boundary Value Problem of Elastic Beams Subjected to Distributed Load

2021 ◽  
Vol 17 (1) ◽  
pp. 75-93
Author(s):  
Mustapha Adewale Usman ◽  
Nur Nabilah Afja Mohd Afandi ◽  
Fatai Akangbe Hammed ◽  
Debora Oluwatobi Daniel
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Boundary Value Problem ◽  
Moving Load ◽  
Boundary Value ◽  
Elastic Beam ◽  
Beam Deflection ◽  
Elastic Beams ◽  
Distributed Load ◽  
Order Four

Analytical solution for the boundary value problem (BVP) of elastic beams subjected to distributed load was investigated. Based on the study, dynamic application curves are developed for beam deflection. The partial differential equation of order four were analysed to determine the dynamic response of the elastic beam under consideration and solved analytically. Effects of different parameters such as the mass of the load, the length of the moving load, the distance covered by the moving load, the speed of the moving and the axial force were considered. Result revealed that the values of the deflection with acceleration being considered are higher than the system where acceleration of the moving load is negligible. These obtained results are in agreement with the existing results.

Analytical Solution of Forced Vibration of a Column Subjected to Conservative and Nonconservative Forces

2001 ◽  
Author(s):  
Zhixiang Xu ◽  
Kunisato Seto ◽  
Hideyuki Tamura
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problem ◽  
Integral Transform ◽  
Boundary Value ◽  
Suspension Bridge ◽  
Follower Force ◽  
Analysis Process ◽  
Finite Integral ◽  
Excitation Force ◽  
Integral Transform Technique

Abstract This paper presents analytical results of forced transverse vibration of a column with a mass attached at free-end subjected to a tangential follower force and a transverse distributed excitation force, that is a simplified model of some structures in civil and mechanical engineering, e.g., a column of a suspension bridge, a launched rocket in the atmosphere. Because the tangential follower force is nonconservative, it is very difficult to get the analytical solution of the problem by usually-used analysis methods with which the adjoint boundary value problem can not be directly obtained. However, by applying the finite integral transform technique, we directly obtained the adjoint boundary value problem in the analysis process, and successfully obtained the analytical solution of the column’s vibration excited by the transverse distributed force.

Mathematical model of magnetic field for a sectorial ferromagnetic cylinder

2021 ◽  
Vol 2021 (49) ◽  
pp. 19-25
Author(s):  
R. M. Dzhala ◽  
◽  
V. R. Dzhala ◽  
B. I. Horon ◽  
B. Ya. Verbenets ◽  
...  
Keyword(s):  
Magnetic Field ◽  
Mathematical Model ◽  
Spatial Distribution ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Algebraic Equations ◽  
Secondary Field

The solution of the boundary-value problem of magnetostatics for a circular ferromagnetic cylinder with a longitudinal sectorial cutout is described. The external primary magnetic field is orthogonal to the cylinder and directed at arbitrary azimuth relative to the cutout. A system of algebraic equations for finding the amplitudes of azimuthal expansions of the spatial distribution of the secondary field of the outer and sectorial partial regions of the cylinder is obtained by the method of rearrangement of functions.

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