We present the approach for computer simulation of magnetic field in a gap between two cylindrical permanent magnets connected by a yoke using FlexPDE software. Proposed approach shows that this implementation allows using program code written "almost natural mathematical language" for successful simulation in practical goals of physical experiments. Particularly, the provided program code and idea of introduction of an effective backsides curvature of magnets have been used in the series of acoustomagnetic studies, i.e. this publication serves also as an example of "reproducible calculations" ideology providing all actual computational details and code.
When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/3359

Postnikov, E.

Storozhenko, A.

SocioEconomic Challenges Journal (Sumy State University)

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ 
Vol. 5 No 4, 04006(4pp) (2013) Том 5 № 4, 04006(4cc) (2013) 
 
 
The article was reported at the International Conference «Physics and Technology of Nanomaterials and Structures», Kursk, 21-22 November, 2013 
2077-6772/2013/5(4)04006(4) 04006-1  2013 Sumy State University 
Practical Simulation of Magnetic Field of Permanent Magnets System  
to Explore Size of Magnetic Nanoparticles 
 
E. Postnikov1, A. Storozhenko2 
 
1 Department of Theoretical Physics, Kursk State University, Kursk, Russia 
2 Department of Nanotechnology and Engineering Physics, Southwest State University,  
94, 50 Let Oktyabrya Str., 305040 Kursk, Russia  
 
(Received 05 August 2013; revised manuscript received 05 November 2013; published online 10 December 2013) 
 
We present the approach for computer simulation of magnetic field in a gap between two cylindrical 
permanent magnets connected by a yoke using FLEXPDE software. Proposed approach shows that this im-
plementation allows using program code written "almost natural mathematical language" for successful 
simulation in practical goals of physical experiments. Particularly, the provided program code and idea of 
introduction of an effective backsides curvature of magnets have been used in the series of acoustomagnet-
ic studies, i.e. this publication serves also as an example of "reproducible calculations" ideology providing 
all actual computational details and code. 
 
Keywords: FlexPDE, Finite elements method, Computer support of experiments. 
 
 PACS numbers: 07.05.Tp, 75.75.Jn 
 
 
1. INTRODUCTION 
 
Modeling and simulation of a stationary magnetic 
field for various configurations of permanent magnets 
[1] is a one of classical magnetostatic problem keeping 
at the same time actuality up to present. First of all it 
is connected with the development electric motors and 
generators, where permanent magnets are a sufficient 
part, see for example a review corresponding modeling 
approaches in the book [2] and review [3]. 
Another hot topic of a permanent magnet configura-
tion development and a simulation of their fields is low-
field Nuclear Magnetic Resonance (NMR) setups [4, 5]. 
The construction principally close to this, which is also 
widely used for study of ferrofluids on a base of acous-
tomagnetic effect, see for review e.g. [6, 7, 8] and refer-
ences therein. The difference consists of the usage oscil-
lating sound field instead of electromagnetic one. Vari-
ations in concentrations and orientations of magnetic 
particles placed in a strong permanent magnetic field, 
which are induced by a sound wave, produce inductive 
signal detected by an inductor coil. This allow to ex-
plore microscopic size distributions of magnetic nano-
particles, details of sound wave propagation in confined 
liquids, etc. 
However, the details of interactions leading to the 
generations of this inductive signal, sufficiently de-
pends on mutual orientation of oscillating sound wave 
vectors and applied permanent magnetic field. Thus, 
its topology is an important characteristic in acous-
tomagnetic experiments. At the same time, it should be 
pointed out that requirements to the accuracy of mag-
netic field calculations are not so restrictive in the 
macroscopic magnetoacoustic experiments comparing 
with, say, high-effective electrogenerators and NMR 
setups due to averaged character of acoustomagnetic 
inductive signal. Thus, one need a relatively simple 
and straightforward tools, which allow to simulate the 
magnetic field of used system permenant magnets in 
the region overlapping direct measurements area, and 
to concentrate on issues more related to the acous-
tomagnetic details of an experiment. 
For this reason, "on shelf" software seems to be more 
appropriate for usage by physicists in such a situation. 
Thus, the main goal of this work is to present method of 
simulation and corresponding program code for 
FLEXPDE software [9], which allows to evaluate appro-
priate computer simulations, and which was actually 
used for successful acoustomagnetic experiments [10]. 
 
2. MATHEMATICAL MODEL AND PROGRAM 
CODE 
 
The view of magnetic system, which needs to be 
modeled is presented in the Fig. 1. Two coaxial cylin-
dric magnets are fixed in a yoke forming magnetic cir-
cuit with a relatively wide gap. It should be pointed out 
that the construction can not be disassembled; thus the 
exact length of invisible parts of magnets inside fixing 
metal rings is unknown. The usage of light steel nee-
dles free arranged by magnetic force lines indicated 
also the following facts important for simulation: the 
metal yoke is sufficiently far for ideal, namely, the di-
rection of needles shows that they are strictly directed 
to the permanent magnets even near the surface of 
curved and vertical parts of the yoke; needles are not 
parallel to horizontal parts of permanent magnets even 
near their surface; moreover, needles are directed prac-
tically normally to the cylindrical part around the rears 
of fixing rings near their connection with the curved 
yoke and fast take an inverse slope (i.e. directed to the 
backside of permanent magnet). 
Thus, all these observations indicate that consid-
ered magnetic circuit is far from ideal one, which would 
closing magnetic lines inside of it. Combining with im-
possibility of disassembling the system and measure-
ments of its inner magnetic properties, this situation 
requires to use effective boundary conditions, which 
would lead to reproduction of real structure and inten-
sity of the magnetic field. 
To control the value of magnetic induction, there 
were evaluated its measurements in the gap between 
magnets using Hall magnetometer with a relative error 
1.5 % within the possible interval 1-300 mT. Referent 
 E. POSTNIKOV, A. STOROZHENKO J. NANO- ELECTRON. PHYS. 5, 04006 (2013) 
 
 
04006-2 
points were taken along the symmetry axis between 
magnets from central point upward with the step 5 mm. 
The set of numerical data is provided in the Appendix. 
In this case, the most direct method for the calcula-
tion of magnetic field outside the magnets and yoke 
connecting them is the usage of a scalar magnetic po-
tential  satisfying Laplace equation 2 0   in this 
free space. 
However, the most important part of the problem 
statement is a formulation of proper boundary condi-
tions. First of all, let us note that the yoke in our case 
only perturbs magnetic field of two permanent mag-
nets. At second, the principal region of interest in phys-
ical experiments connected with a usage of this setup, 
is sufficiently wide gap between magnets. Particularly, 
in the works [7, 10], the pipe filled by ferrofluid had 
diameter 16 mm and was placed along the symmetry 
axis between magnets. At the same time, the column of 
ferrofluid had height up to 250 mm that is larger than 
the diameter of magnets. Thus, the magnetic field dis-
tribution in the normal direction to the common axes of 
magnets is an important parameter. 
Thus, we use for simulation cylindrical co-ordinate sys-
tem with axis coinciding with the common axis of both 
magnets. Correspondingly, the boundary condition 
along this axis is 0r  , i.e. command natural 
(psi)  0 in FlexPDE). The area of calculation is chosen 
in such a way that one can take the magnetic potential 
is taken as zero on its boundary and its derivatives (i.e. 
components of magnetic field) are practically equal to zero 
there. Corresponding properties for the values 
Rd  300 mm and Delta  200 mm for radial and longitu-
dinal distances are confirmed by the direct magnetome-
ter measurements. Whence, the boundary conditions on 
the outer boundary of the embedding space are   0 
(the command value(psi)  0 in FlexPDE). 
The most important are boundary conditions on the 
surface of magnet. Ideal solitary cylindric permanent 
magnet has constant magnetization along its axis, 
 
 
 
Fig. 1 – The magnet system: radius of cylindrical magnets 
55 mm, height of cylinder 40 mm, where length of visible 
magnetic part 9 mm. The width of gap: 91 mm. Light steel 
needles free hanged by length fibers are arranged along mag-
netic induction lines visualizing them 
surface of magnet. Ideal solitary cylindric permanent 
magnet has constant magnetization along its axis, i.e. 
magnetic are lines normal to its bases and parallel to 
its generator. In our case, the boundary conditions on 
the parallel surfaces divided by the gap are certainly  
  const as well, due to axial symmetry of the system. 
On the other hand, the magnets in the considered 
system are connected by the yoke, i.e. it is magnetic 
circuit. Fig. 1 demonstrates that the lines of magnetic 
field are practically normal to the sides of magnets and 
are directed to them in the backside region. Therefore, 
to simplify the simulation we take   const over all 
boundary of magnetic region, representing additionally 
the backsides as curved surfaces. This gives the second 
set of adjustment parameters. Since the active magnet-
ic regions of the system are smaller than the full yoke 
construction (see Fig. 1) and we are primarily interest-
ed in the simulation of magnetic field inside the gap 
between permanent magnets, we can neglect by the 
decreasing of magnetic potential (by absolute value) 
downto the base of yoke. However, the overall presence 
of the circuit is taken into account introducing reverse 
sign of the potential for the second magnet. 
The resulting program code, which uses features 
described above, is presented below. 
 
COORDINATES 
xcylinder("z","r")  
SELECT  
nominmax  
textsize  25  
contours  40  
vectorgrid  25  
VARIABLES 
psi !Scalar magnetic potential  
DEFINITIONS 
{Loading of experimental data} 
expdata  splinetable('magnetdataZ0.txt')  
{magnets} 
Rm  55 !Radius of the disc  
Hm  40 !Thickness of the disc 
Rd  20 !Length of the curved part 
Pm  6.05e3 !Magnetic potential of the magnet sur-
face {Gap between magnets} 
Gap  91  
{Embedding Space} 
{Distance from the magnet to outer boundary along 
axis}  
Delta  200 
Length  2(Hm + Rd + Delta) + Gap {Length of the 
embedding space}  
Rad  300 {Radius of the embedding space}  
{Definition of a magnetic induction} 
B  – grad(psi)  
INITIAL VALUES 
psi  0  
EQUATIONS 
del2(psi)  0  
BOUNDARIES 
Region 'Empty Space' {The space around magnet} 
!Axis of symmetry 
start(0,0) natural(psi)  0 line to (Delta,0) 
!1st magnet's backside 
value(psi)  Pm 
 PRACTICAL SIMULATION OF MAGNETIC FIELD OF PERMANENT MAGNETS SYSTEM…J. NANO- ELECTRON. PHYS. 5, 04006 (2013) 
 
 
04006-3 
spline to (0.05Rd + Delta,Rm0.2) to 
(0.1Rd + Delta,Rm0.25) 
to (Delta + Rd,Rm) 
!1st magnet's top 
line to (Delta + Rd + Hm,Rm) 
!1st magnet's plane face 
line to (Delta + Rd + Hm,0) 
!Axis of symmetry 
natural(psi)  0 line to (Delta + Rd + Hm + Gap,0)  
!2nd magnet's plane face 
value(psi)  – Pm line to  
(Delta + Rd + Hm + Gap,Rm) 
!2nd magnet's top 
line to (Length – Delta – Rd,Rm) 
!2nd magnet's backside 
spline to (Length – (0.1Rd + Delta),Rm0.25)  
to (Length – Delta – 0.05Rd,Rm0.2) 
to (Length – Delta,0)  
!Axis of symmetry 
natural(psi)  0 line to (Length,0) 
!Outer boundary 
value(psi)  0 
line to (Length,Rad) to (0,Rad) to close  
PLOTS 
vector(B) zoom(0.75Delta,0,Length 1.5Delta, 
1.5Rm)  
as 'Induction of the magnetic field'  
elevation(fit(xcomp(B)),expdata) 
from(Length / 2,0)to(Length / 2,100) as 'Bz '  
contour(xcomp(B)) zoom(0.5Delta,0,Length-
Delta,2Rm)  
as 'Bz' 
contour(ycomp(B)) zoom(0.5Delta,0,Length –
 Delta,2Rm) as 'Br'  
END 
 
Results of its implementation are shown in  
Figs. 2-3. The obtained configuration has allowed to 
use two properties of the magnetic field configuration: 
i) its horizontal direction along the vertical line of 
symmetry, where the tube filled by ferroliquid was 
placed and satisfactory uniformity within each tube's 
cross-section (the described numerical simulation 
namely allows to explore required field configuration) 
and ii) non-uniformity of the field strength along the 
tube (as well controlling field's configuration within 
regions corresponding filled parts of the tube). Whence, 
this all results in a detailed study of the influence of 
crossed acoustic and magnetic field on acoustomagnetic 
phenomena, see the details in [10]. 
 
 
 
 
 
Fig. 2 – Plots of the calculated magnetic field (in mT). Left 
panel: vector plot of magnetic induction, where lengths of vec-
tors are proportional to the it’s absolute value. Right panel: 
comparison of calculated magnetic field induction (dark (blue 
in color online) curve) and experimental referent one (light-
gray (yellow in color online) curve) along the line normal to 
the symmetry axis and crossing it in the middle point between 
magnets 
 
 
 
 
Fig. 3 – Plot of magnetic induction lines for the axial (left) and 
radial (right) components of magnetic field (in mT) 
 
 
 
 E. POSTNIKOV, A. STOROZHENKO J. NANO- ELECTRON. PHYS. 5, 04006 (2013) 
 
 
04006-4 
3. SUMMARY 
 
In this work we have presented explicitly the pro-
gram code for FLEXPDE software and discussed its de-
velopment for the computer simulation of a magnetic 
field in the gap between two coaxial permanent mag-
nets fixed by a yoke. It has been shown that yoke's in-
fluence can be successfully reproduced by introduction 
of effective curved backsides of magnets. Thus, one 
need only two parameters: this quantity and a value of 
scalar magnetic potential on magnets surface. To ad-
just these parameters, a user need simply collate ex-
perimental data along of one characteristic line (that 
could be easily obtained) and results of simulations. 
Here the value of magnetic induction on the middle 
point of axes primarily determines value of surface 
scalar potential and a shape of curve is influenced by 
the parameter of the effective backside. 
As well, we consider this work in accordance with 
"reproducible computations" ideology [11], i.e. providing 
to readers all possible data, computational approaches 
and code, which was used for obtaining scientific results. 
Particularly, the provided code has been implemented 
for the analysis of experimental investigations and their 
physical interpretations in [10]. 
 
ACKNOWLEDGEMENT 
 
AMS is grateful for funding of this work via the 
Grant of the President of the Russian Federation for 
support of young researchers № 14.124.13.5498-МК 
"The Details of Mechanisms of magnetization disturb-
ance of magnetic fluid in a sound wave". 
 
APPENDIX A 
 
The experimental data on induction of magnetic 
field (in mT) in referent points (in mm along radial 
direction), containing in the text file magnetdataZ0.txt: 
 
r 14  
0 
5 10 20 30 40 50 60 70 80 90 100 110 115  
data  
128 
127.7 127 124.4 119.3 111.75 100.35 87.5  
73.55 60.6 49.35 40.1 32.5 29.75 
 
The structure of the data representation: name of 
the co-ordinate (z), number of referent points (14), the 
string of values of z-co-ordinate in these points; identi-
fier (data), the string of values of magnetic induction in 
above- mentioned referent points. 
 
REFERENCES 
 
1. N. Ida, J.P.A. Bastos, Electromagnetics and Calculation of 
Fields (Springer, 1997). 
2. J.F. Gieras, Permanent Magnet Motor Technology: Design 
and Applications (CRC Press, 2002). 
3. H. Tiegna, Y. Amara, G. Barakat, Math. Comput. Simulat. 
90, 162 (2013). 
4. B. Bliimich, F. Casanova, S. Appelt, Chem. Phys. Lett. 477, 
231 (2009). 
5. M.V. Landeghem, E. Danieli, J. Perlo, B.B. Kimich, 
F. Casanova, J. Magn. Res. 215, 74 (2012).  
6. V. Polunin, N. Ignatenko, V. Zraichenko, J. Magn. Magn 
Mater. 85, 141 (1990).  
7. V.M. Polunin, N.S. Kobelev, P.A. Ryapolov, A.M. Storozhenko, 
I.A. Shabanova, Magetohydrodynamics 46 No 1, 31 (2010). 
8. V. M. Polunin, A. M. Storozhenko, Acoustical Phys. 58, 
180 (2012). 
9. http://www.pdesolutions.com. [Electronic resources] 
10. S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, 
E.B. Postnikov, P.A. Ryapolov, Magnetohydrodynamics 47 
1, 29 (2011). 
11. M. Schwab, N. Karrenbach, J. Claerbout, Comput. Sci. 
Eng. 2, 61 (2000). 
 


Practical Simulation of Magnetic Field of Permanent Magnets System to Explore Size of Magnetic Nanoparticles

Electronic Sumy State University Institutional Repository

JOURNAL OF NANO- AND ELECTRONIC PHYSICS ЖУРНАЛ НАНО- ТА ЕЛЕКТРОННОЇ ФІЗИКИ Vol. 5 No 4, 04006(4pp) (2013) Том 5 № 4, 04006(4cc) (2013)   The article was reported at the International Conference «Physics and Technology of Nanomaterials and Structures», Kursk, 21-22 November, 2013 2077-6772/2013/5(4)04006(4) 04006-1  2013 Sumy State University Practical Simulation of Magnetic Field of Permanent Magnets System  to Explore Size of Magnetic Nanoparticles  E. Postnikov1, A. Storozhenko2  1 Department of Theoretical Physics, Kursk State University, Kursk, Russia 2 Department of Nanotechnology and Engineering Physics, Southwest State University,  94, 50 Let Oktyabrya Str., 305040 Kursk, Russia   (Received 05 August 2013; revised manuscript received 05 November 2013; published online 10 December 2013)  We present the approach for computer simulation of magnetic field in a gap between two cylindrical permanent magnets connected by a yoke using FLEXPDE software. Proposed approach shows that this im-plementation allows using program code written "almost natural mathematical language" for successful simulation in practical goals of physical experiments. Particularly, the provided program code and idea of introduction of an effective backsides curvature of magnets have been used in the series of acoustomagnet-ic studies, i.e. this publication serves also as an example of "reproducible calculations" ideology providing all actual computational details and code.  Keywords: FlexPDE, Finite elements method, Computer support of experiments.   PACS numbers: 07.05.Tp, 75.75.Jn   1. INTRODUCTION  Modeling and simulation of a stationary magnetic field for various configurations of permanent magnets [1] is a one of classical magnetostatic problem keeping at the same time actuality up to present. First of all it is connected with the development electric motors and generators, where permanent magnets are a sufficient part, see for example a review corresponding modeling approaches in the book [2] and review [3]. Another hot topic of a permanent magnet configura-tion development and a simulation of their fields is low-field Nuclear Magnetic Resonance (NMR) setups [4, 5]. The construction principally close to this, which is also widely used for study of ferrofluids on a base of acous-tomagnetic effect, see for review e.g. [6, 7, 8] and refer-ences therein. The difference consists of the usage oscil-lating sound field instead of electromagnetic one. Vari-ations in concentrations and orientations of magnetic particles placed in a strong permanent magnetic field, which are induced by a sound wave, produce inductive signal detected by an inductor coil. This allow to ex-plore microscopic size distributions of magnetic nano-particles, details of sound wave propagation in confined liquids, etc. However, the details of interactions leading to the generations of this inductive signal, sufficiently de-pends on mutual orientation of oscillating sound wave vectors and applied permanent magnetic field. Thus, its topology is an important characteristic in acous-tomagnetic experiments. At the same time, it should be pointed out that requirements to the accuracy of mag-netic field calculations are not so restrictive in the macroscopic magnetoacoustic experiments comparing with, say, high-effective electrogenerators and NMR setups due to averaged character of acoustomagnetic inductive signal. Thus, one need a relatively simple and straightforward tools, which allow to simulate the magnetic field of used system permenant magnets in the region overlapping direct measurements area, and to concentrate on issues more related to the acous-tomagnetic details of an experiment. For this reason, "on shelf" software seems to be more appropriate for usage by physicists in such a situation. Thus, the main goal of this work is to present method of simulation and corresponding program code for FLEXPDE software [9], which allows to evaluate appro-priate computer simulations, and which was actually used for successful acoustomagnetic experiments [10].  2. MATHEMATICAL MODEL AND PROGRAM CODE  The view of magnetic system, which needs to be modeled is presented in the Fig. 1. Two coaxial cylin-dric magnets are fixed in a yoke forming magnetic cir-cuit with a relatively wide gap. It should be pointed out that the construction can not be disassembled; thus the exact length of invisible parts of magnets inside fixing metal rings is unknown. The usage of light steel nee-dles free arranged by magnetic force lines indicated also the following facts important for simulation: the metal yoke is sufficiently far for ideal, namely, the di-rection of needles shows that they are strictly directed to the permanent magnets even near the surface of curved and vertical parts of the yoke; needles are not parallel to horizontal parts of permanent magnets even near their surface; moreover, needles are directed prac-tically normally to the cylindrical part around the rears of fixing rings near their connection with the curved yoke and fast take an inverse slope (i.e. directed to the backside of permanent magnet). Thus, all these observations indicate that consid-ered magnetic circuit is far from ideal one, which would closing magnetic lines inside of it. Combining with im-possibility of disassembling the system and measure-ments of its inner magnetic properties, this situation requires to use effective boundary conditions, which would lead to reproduction of real structure and inten-sity of the magnetic field. To control the value of magnetic induction, there were evaluated its measurements in the gap between magnets using Hall magnetometer with a relative error 1.5 % within the possible interval 1-300 mT. Referent  E. POSTNIKOV, A. STOROZHENKO J. NANO- ELECTRON. PHYS. 5, 04006 (2013)   04006-2 points were taken along the symmetry axis between magnets from central point upward with the step 5 mm. The set of numerical data is provided in the Appendix. In this case, the most direct method for the calcula-tion of magnetic field outside the magnets and yoke connecting them is the usage of a scalar magnetic po-tential  satisfying Laplace equation 2 0   in this free space. However, the most important part of the problem statement is a formulation of proper boundary condi-tions. First of all, let us note that the yoke in our case only perturbs magnetic field of two permanent mag-nets. At second, the principal region of interest in phys-ical experiments connected with a usage of this setup, is sufficiently wide gap between magnets. Particularly, in the works [7, 10], the pipe filled by ferrofluid had diameter 16 mm and was placed along the symmetry axis between magnets. At the same time, the column of ferrofluid had height up to 250 mm that is larger than the diameter of magnets. Thus, the magnetic field dis-tribution in the normal direction to the common axes of magnets is an important parameter. Thus, we use for simulation cylindrical co-ordinate sys-tem with axis coinciding with the common axis of both magnets. Correspondingly, the boundary condition along this axis is 0r  , i.e. command natural (psi)  0 in FlexPDE). The area of calculation is chosen in such a way that one can take the magnetic potential is taken as zero on its boundary and its derivatives (i.e. components of magnetic field) are practically equal to zero there. Corresponding properties for the values Rd  300 mm and Delta  200 mm for radial and longitu-dinal distances are confirmed by the direct magnetome-ter measurements. Whence, the boundary conditions on the outer boundary of the embedding space are   0 (the command value(psi)  0 in FlexPDE). The most important are boundary conditions on the surface of magnet. Ideal solitary cylindric permanent magnet has constant magnetization along its axis,    Fig. 1 – The magnet system: radius of cylindrical magnets 55 mm, height of cylinder 40 mm, where length of visible magnetic part 9 mm. The width of gap: 91 mm. Light steel needles free hanged by length fibers are arranged along mag-netic induction lines visualizing them surface of magnet. Ideal solitary cylindric permanent magnet has constant magnetization along its axis, i.e. magnetic are lines normal to its bases and parallel to its generator. In our case, the boundary conditions on the parallel surfaces divided by the gap are certainly    const as well, due to axial symmetry of the system. On the other hand, the magnets in the considered system are connected by the yoke, i.e. it is magnetic circuit. Fig. 1 demonstrates that the lines of magnetic field are practically normal to the sides of magnets and are directed to them in the backside region. Therefore, to simplify the simulation we take   const over all boundary of magnetic region, representing additionally the backsides as curved surfaces. This gives the second set of adjustment parameters. Since the active magnet-ic regions of the system are smaller than the full yoke construction (see Fig. 1) and we are primarily interest-ed in the simulation of magnetic field inside the gap between permanent magnets, we can neglect by the decreasing of magnetic potential (by absolute value) downto the base of yoke. However, the overall presence of the circuit is taken into account introducing reverse sign of the potential for the second magnet. The resulting program code, which uses features described above, is presented below.  COORDINATES xcylinder("z","r")  SELECT  nominmax  textsize  25  contours  40  vectorgrid  25  VARIABLES psi !Scalar magnetic potential  DEFINITIONS {Loading of experimental data} expdata  splinetable('magnetdataZ0.txt')  {magnets} Rm  55 !Radius of the disc  Hm  40 !Thickness of the disc Rd  20 !Length of the curved part Pm  6.05e3 !Magnetic potential of the magnet sur-face {Gap between magnets} Gap  91  {Embedding Space} {Distance from the magnet to outer boundary along axis}  Delta  200 Length  2(Hm + Rd + Delta) + Gap {Length of the embedding space}  Rad  300 {Radius of the embedding space}  {Definition of a magnetic induction} B  – grad(psi)  INITIAL VALUES psi  0  EQUATIONS del2(psi)  0  BOUNDARIES Region 'Empty Space' {The space around magnet} !Axis of symmetry start(0,0) natural(psi)  0 line to (Delta,0) !1st magnet's backside value(psi)  Pm  PRACTICAL SIMULATION OF MAGNETIC FIELD OF PERMANENT MAGNETS SYSTEM…J. NANO- ELECTRON. PHYS. 5, 04006 (2013)   04006-3 spline to (0.05Rd + Delta,Rm0.2) to (0.1Rd + Delta,Rm0.25) to (Delta + Rd,Rm) !1st magnet's top line to (Delta + Rd + Hm,Rm) !1st magnet's plane face line to (Delta + Rd + Hm,0) !Axis of symmetry natural(psi)  0 line to (Delta + Rd + Hm + Gap,0)  !2nd magnet's plane face value(psi)  – Pm line to  (Delta + Rd + Hm + Gap,Rm) !2nd magnet's top line to (Length – Delta – Rd,Rm) !2nd magnet's backside spline to (Length – (0.1Rd + Delta),Rm0.25)  to (Length – Delta – 0.05Rd,Rm0.2) to (Length – Delta,0)  !Axis of symmetry natural(psi)  0 line to (Length,0) !Outer boundary value(psi)  0 line to (Length,Rad) to (0,Rad) to close  PLOTS vector(B) zoom(0.75Delta,0,Length 1.5Delta, 1.5Rm)  as 'Induction of the magnetic field'  elevation(fit(xcomp(B)),expdata) from(Length / 2,0)to(Length / 2,100) as 'Bz '  contour(xcomp(B)) zoom(0.5Delta,0,Length-Delta,2Rm)  as 'Bz' contour(ycomp(B)) zoom(0.5Delta,0,Length – Delta,2Rm) as 'Br'  END  Results of its implementation are shown in  Figs. 2-3. The obtained configuration has allowed to use two properties of the magnetic field configuration: i) its horizontal direction along the vertical line of symmetry, where the tube filled by ferroliquid was placed and satisfactory uniformity within each tube's cross-section (the described numerical simulation namely allows to explore required field configuration) and ii) non-uniformity of the field strength along the tube (as well controlling field's configuration within regions corresponding filled parts of the tube). Whence, this all results in a detailed study of the influence of crossed acoustic and magnetic field on acoustomagnetic phenomena, see the details in [10].      Fig. 2 – Plots of the calculated magnetic field (in mT). Left panel: vector plot of magnetic induction, where lengths of vec-tors are proportional to the it’s absolute value. Right panel: comparison of calculated magnetic field induction (dark (blue in color online) curve) and experimental referent one (light-gray (yellow in color online) curve) along the line normal to the symmetry axis and crossing it in the middle point between magnets     Fig. 3 – Plot of magnetic induction lines for the axial (left) and radial (right) components of magnetic field (in mT)     E. POSTNIKOV, A. STOROZHENKO J. NANO- ELECTRON. PHYS. 5, 04006 (2013)   04006-4 3. SUMMARY  In this work we have presented explicitly the pro-gram code for FLEXPDE software and discussed its de-velopment for the computer simulation of a magnetic field in the gap between two coaxial permanent mag-nets fixed by a yoke. It has been shown that yoke's in-fluence can be successfully reproduced by introduction of effective curved backsides of magnets. Thus, one need only two parameters: this quantity and a value of scalar magnetic potential on magnets surface. To ad-just these parameters, a user need simply collate ex-perimental data along of one characteristic line (that could be easily obtained) and results of simulations. Here the value of magnetic induction on the middle point of axes primarily determines value of surface scalar potential and a shape of curve is influenced by the parameter of the effective backside. As well, we consider this work in accordance with "reproducible computations" ideology [11], i.e. providing to readers all possible data, computational approaches and code, which was used for obtaining scientific results. Particularly, the provided code has been implemented for the analysis of experimental investigations and their physical interpretations in [10].  ACKNOWLEDGEMENT  AMS is grateful for funding of this work via the Grant of the President of the Russian Federation for support of young researchers № 14.124.13.5498-МК "The Details of Mechanisms of magnetization disturb-ance of magnetic fluid in a sound wave".  APPENDIX A  The experimental data on induction of magnetic field (in mT) in referent points (in mm along radial direction), containing in the text file magnetdataZ0.txt:  r 14  0 5 10 20 30 40 50 60 70 80 90 100 110 115  data  128 127.7 127 124.4 119.3 111.75 100.35 87.5  73.55 60.6 49.35 40.1 32.5 29.75  The structure of the data representation: name of the co-ordinate (z), number of referent points (14), the string of values of z-co-ordinate in these points; identi-fier (data), the string of values of magnetic induction in above- mentioned referent points.  REFERENCES  1. N. Ida, J.P.A. Bastos, Electromagnetics and Calculation of Fields (Springer, 1997). 2. J.F. Gieras, Permanent Magnet Motor Technology: Design and Applications (CRC Press, 2002). 3. H. Tiegna, Y. Amara, G. Barakat, Math. Comput. Simulat. 90, 162 (2013). 4. B. Bliimich, F. Casanova, S. Appelt, Chem. Phys. Lett. 477, 231 (2009). 5. M.V. Landeghem, E. Danieli, J. Perlo, B.B. Kimich, F. Casanova, J. Magn. Res. 215, 74 (2012).  6. V. Polunin, N. Ignatenko, V. Zraichenko, J. Magn. Magn Mater. 85, 141 (1990).  7. V.M. Polunin, N.S. Kobelev, P.A. Ryapolov, A.M. Storozhenko, I.A. Shabanova, Magetohydrodynamics 46 No 1, 31 (2010). 8. V. M. Polunin, A. M. Storozhenko, Acoustical Phys. 58, 180 (2012). 9. http://www.pdesolutions.com. [Electronic resources] 10. S.G. Yemelyanov, V.M. Polunin, A.M. Storozhenko, E.B. Postnikov, P.A. Ryapolov, Magnetohydrodynamics 47 1, 29 (2011). 11. M. Schwab, N. Karrenbach, J. Claerbout, Comput. Sci. Eng. 2, 61 (2000).  

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Practical Simulation of Magnetic Field of Permanent Magnets System to Explore Size of Magnetic Nanoparticles

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