Transactions on Engineering Sciences vol 11, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
The FEMAX package for static and stationary
electric and magnetic fields
I.E. Lager, G. Mur
Faculty ofElectrical Engineering, Laboratory of Electromagnetic
Research, Delft University of Technology, P.O. Box 5031,
NL-2600 GA Delft, The Netherlands
ABSTRACT
A finite element package is described for computing stationary and static
electric and magneticfieldsin inhomogeneous media. A high efficiency and
accuracy are obtained by formulating the problem directly in terms of the
requiredfieldstrength. The medium parameters inside the domain of computation can be both linear and nonlinear and they may be anisotropic,
also. A combination of edge and nodal elements is used for obtaining an
accurate expansion of the field strength throughout the domain of computation. This is done such the conditions on internal surfaces of discontinuity
in medium parameters, as well as along the outer boundary, can be modeled easily. Various post-processing facilities are available for examining the
computed data.
1
INTRODUCTION
Finite element methods for computing stationary and static electromagnetic
fields are usually formulated in terms of (vector) potentials. Potentials have
the advantage that they can be chosen to be continuous across internal surfaces of discontinuity in medium parameters with the consequence that the
very convenient nodal expansion can be used throughout the entire domain
of computation. However, there are the field strengths rather than the potentials that have a physical or practical meaning. Thesefieldstrengths can
only be computed from the potentials by means of a numerical differentiation with an inherent loss of accuracy.
The computation of the electromagnetic field can be carried out more efficiently if the field strength is computed directly. In case of a finite element
formulation this requires a discretization technique that allows the modeling of continuity conditions for the field at interfaces between media with
Transactions on Engineering Sciences vol 11, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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different electric and/or magnetic properties. The well known decoupling of
the electric and magnetic fields in case of the stationary and static states of
the electromagnetic field [1, 2] requires that the stationary magnetic, stationary electric and electrostaticfieldsneed to be examined separately. We
shall combine these three cases in one generic formalism.
The above ideas were implemented in thefiniteelement code FEMAX.
In this code use is made of a combination of consistently linear edge and
nodal expansion functions [3] that together enable the correct and consistent
modeling of the continuity conditions at interfaces inside the domain of
computation.
2
THE GENERIC FORMALISM
The position of an arbitrary point r is specified by its three coordinates
{x,y,z} with respect to a background Cartesian reference frame with origin O and three mutually perpendicular base vectors {t\, 1*2,^3} of unit
length each. The unit vector along the normal to an internal surface of
discontinuity in medium parameters (J) is denoted as i/ and the unit vector
along the outwardly oriented normal on the boundary (dT>) of the domain
of computation (7)) is denoted as n (see Fig. 1).
In order to simplify the computer code, the three respective cases of
stationary and static electric and magneticfieldsare mapped on one generic
formalism. All computations are carried out within the frame of this generic
formalism, while preserving the original physical significance of the field
quantities for the input and output data. The correspondence between the
generic values and the electromagnetic field values is given in Table 1. The
following notation is adopted:
V = V(r) the (generic)fieldstrength,
dVy
n+
F = F(r) the (generic)fluxdensity,
( = £(V\r) a (generic) constitutive parameter;
gimp _ Qimp(r) the (generic) volume
source density of impressed (known) curdT> = dT>v U dT>p
rent,
Q™P = Q%*(r) the (generic) surface
Figure 1. The domain of computation %). source density of impressed (known) current,
pimp _ pimp^ the (generic) impressed (known) volume charge density,
0.imp _ 0.imp^ the (generic) impressed (known) surface charge density,
^ext _ <^ext^ a (generic) known vector function used for prescribing
Dirichlet boundary conditions (the boundary conditions on the subsurface
dVy C dU where tangential components of V are prescribed),
0.ext _ p-ext^ a (generic) known scalar function used for prescribing Neumann boundary conditions (the boundary conditions on the subsurface
3T>p — dV\dVy where normal components of thefluxF are prescribed).
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297
Table 1. Correspondence between generic quantities and stationary and static electromagnetic Held quantities.
Generic
formulation
V
F
t
Qimp
or
pimp
^amp
#ext
^.ext
Static
Stationary
electric field electric field
E
E
J
D
a
e
0
0
0
0
-V • J*™P
^r
<yimp
-i/ • J^P |?
n x E^
n x E**
n . J™P
<"
Stationary
magnetic field
H
B
M
jimp
Jf*
(ft*
<C>
n x JET"
_ext
^m
For stationary and staticfields,the generic basic equation, the generic constitutive relation and the generic interface condition read
V x V = Q™P,
F = £V,
v x V \l= Q^.
(1)
(2)
(3)
At the outer boundary dV of the domain D, the generic boundary conditions
n x V = #**' on dT>v
(4)
need to be imposed.
The generic formalism is made complete by adding a set of compatibility relations [4]. In case of the time-domain formulation the compatibility
relations follow implicitly from the field equations. In case of the stationary
and static fields they have to be explicitly made a part of the formulation
in order to ensure the uniqueness of the solution (see also [5]).
In the subdomains of the domain of computation where the electromagnetic field vectors are continuously differentiate functions of the spatial
coordinates the electromagnetic compatibility (divergence) relation
V • F = imp
(5)
applies.
The field vectors are not differentiate with respect to all spatial coordinates at the interfaces between regions with different medium parameters.
In that case Eq. (5) is replaced by
i,.F\*=o*».
(6)
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A third type of compatibility relation is found when studying the behaviour
of the field near the outer boundary of the domain of computation. The
compatibility relation corresponding to Eq. (6) on dVp is obtained as
n • F = -cr«*.
3
(7)
DISCRETIZATION OF THE EQUATIONS IN THE LINEAR
CASE
For topological reasons, the domain of computation to which thefiniteelement method is applied is divided into tetrahedra. The field values are
discretized using a combination of vectorial nodal and edge expansion functions. Nodal expansion functions are used inside homogeneous regions, edge
expansion functions along interfaces between those regions. The minimum
value of the contrast in the medium parameters in neighbouring tetrahedra for which use is made of edge expansion functions is user defined. The
user can prescribe the use of edge expansion functions along (parts of) the
outer boundary. This option is particularly useful in case reentrant corners
are present in the outer boundary, since it reduces the local error at those
locations. The resulting expansion can be written as
V = J>Wp-">,
(8)
i
where Wp^' denotes the edge (E) or nodal (N) expansion used. Note that,
because of employing consistently linear elements, a local approximation
error O(/i^), where h denotes the length of the longest edge of a tetrahedron
used locally, is obtained in the representation of the field strength V.
A weighted residuals method is applied for constructing the system of
linear equations for computing the expansion coefficients [6], The weighted
residuals equations are
Z, v, [ &,, (V x Wf*>) • (V x W?*>) +
(9)
+ Je,,
(V - (f Wf '">)) p»P,
Vj,
where 2%j = D* D *Dj is the cross-section of the span *Dj of the weighting
function wf^
and the span £>, of the expansion function W^
. The
boundary condition (4), the interface compatibility (6) and the boundary
compatibility (7) are implemented by imposing the implied relations between the expansion coefficients, explicitly.
4
COMPUTATION OF NONLINEAR FIELDS
Nonlinear fields are computed by means of the parallel- chord method [7],
At each global iteration the nonlinear medium parameters are updated on
basis of the field strength' values computed at the previous iteration, a
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299
system of linear algebraic equations is built on basis of the set of equations
for the linear case, the system of linear equations is solved and the updated
field strength values' are computed. The iterative process stops when the
relative error at two subsequent global iterations defined as
NONLINEAR ERROR K =
becomes smaller than a prescribed value or when the prescribed maximum
number of iterations is exceeded.
5
THE FEMAX PACKAGE
The FEMAX package consists of the FEMAXSfiniteelement code and
the FEMAXP and FEMAXL codes that can be used for post-processing
the results obtained by means of FEMAXS. The FEMAXS code uses the
SEPRAN finite element package [8] for a number of elementaryfiniteelement tasks like generating the mesh, assembling the system of linear equations from the element matrices and solving the system. The FEMAXP and
FEMAXL post-processors use the graphic routines included in the NAG library [9] and require the GKS interface [10]. The package was written in
FORTRAN-77.
FEMAXS inherits most of the options implemented in the earlier timedomain and frequency- domain versions of the FEMAX package [3]. Some
new features that have been added are:
• A set of integral values can be computed, i.e. surface or line integrals
of the field strength and/or flux density along user-defined (closed)
surfaces and/or lines. These integral values can serve for analysing
the accuracy of the solution.
• In the interior of the domain of computation the user can specify
nonlinear medium parameters. For nonlinear media, the characteristic
£ = £(V,r) or, more general, F = F(V,r) need to be prescribed.
• An MS-DOS post-processor for generating 3D plots is available for
sites where the NAG or GKS libraries cannot be used.
• A set of "M-files" for post-processing the results inside the MATLAB
computing environment [11] have been also written.
6
NUMERICAL RESULTS
In the following, numerical results will be shown for three different problems.
First, for demonstrating the accuracy of the FEMAX package, a test
problem with a known analytical solution, the so called "Charged Brick"
problem, is solved (see Figure 2 a). A "Brick" of electric charge with constant volume density p(x,y, z) = 4%eo [C/nf] is symmetrically placed in the
origine of the reference frame. The medium is assumed to be homogeneous,
linear and isotropic. The electrical permittivity is €Q.
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E [V/m]
Figure 2. The configuration for the " Charged Brick" problem, a) The configuration for
the linear medium, b) The characteristic of the nonlinear medium.
Because of the symetry of the problem it is sufficient to study the configuration in the octant {a,i/,z} > 0. On the planes x = 0, y = 0, z — 0
Neumann boundary conditions of the type n • D = 0 are imposed. Dirichlet
boundary conditions are set elsewhere using the analytical solution. The domain of computation was discretized by means of a relatively coarse mesh
of 10 x 10 x 10 parallelepipeds that are each divided into 6 tetrahedra.
The mesh is increasingly coarser when moving away from the corner of the
"Charged Brick" {aj,2/,z} = 1. Since no discontinuities in the material parameter are present, nodal elements can be used throughout the domain of
computation. With these choices the total number of unknowns amounted
to Nunk = 3993, Nbnd = 693 of them being explicitly specified. The system
of equations was solved using a ICCG iterative solver with an incomplete
LU decomposition preconditioner that allowed obtaining a relative error of
2.23 • 10"" after 24 iterations. The total charged CPU time for solving the
problem on a DEC-AXP 3000 workstation was about 4.5 minutes.
The overall value of the root mean square relative error in the result for
the electric field was 1.34%. In Fig. 4 the plots for the local relative error
in computing the electricfieldstrength along the plane z = 1 are presented.
The integral of the flux density was computed on the surface of the cubes
\x\ < 1.0001, \y\ < 1.0001, \z\ < 1.0001 and \x\ < 1.9999, \y\ < 1.9999,
\z\ < 1.9999. The relative error in computing the surface integrals was
-1.73% and 0.12%, respectively.
Secondly, for demonstrating the accuracy of the FEMAX package in
modeling interfaces between highly contrasting media, the so called "Test
Slot" problem, is solved (see Fig. 3). A single, rectangular slot of an electric
machine is studied. The slot being infinitely extended in the Oz direction
yield the problem to be two-dimensional. Because the symmetry of the
problem it is sufficient to study only one half of the slot.
Transactions on Engineering Sciences vol 11, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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301
y
HB Copper/IT = 1
[
] Iron /XT = 1000
I
I Air/XT = 1
Figure 3. The configuration for the "Test Slot" problem.
The domain of computation was discretized by means of a relatively
coarse mesh of 10 x 10 x 1 parallelepipeds that are divided in 6 tetrahedra.
The mesh is increasingly coarser when moving away from the iron/air interfaces. Edge elements were used along the interfaces while nodal elements
were used elsewhere. With these choices the total number of unknowns
amounted to Nunk = 1668, Nbnd = 130 of them being explicitly specified.
Because of the small number of unknowns, the system of equations was
solved using a direct method, which proved to be faster. The total charged
CPU time for solving the problem was about 45 seconds.
The field distribution throughout the domain of computation is presented in Fig. 5 a. In Fig. 5 b the jump in the normal component of
the field strength along the interfaces AC and CD (see Fig. 3) is depicted.
Note that the value of the contrast in the medium parameters, i.e. 1000, is
accurately reflected in the jump of the normal components. The value of
the line integrals of the field strength was computed along the closed loops
ABGHA and ABCDEFGHA. The relative error in computing the values of
the line integrals was 0.5% and 1.3%, respectively.
Finally, as an example for a nonlinear problem, we include the solution
for the "Charged Brick" problem, in which the linear medium was replaced
with a nonlinear medium that has a relative permittivity e^ — CT(V) shown
in Fig. 2 b. The permittivity was chosen such that the medium is fully
saturated at about 75% of the peak value of the modulus of the electric
field computed in case of the linear "Charged Brick" problem with e^ = 2.
The Dirichlet boundary conditions were set to 0. The same mesh as for
the linear case was used. A number of 11 global iterations were necessary
for obtaining a relative error of 0.0198% (see Eq. (10)). The total charged
CPU time for solving the problem was about 1 hour. The integral of the
flux density was computed on the same surfaces as in the case of the linear
"Charged Brick" and the obtained relative error was -10.5% and -9.27%,
respectively. In Fig. 6 the distribution of the modulus of the electric field
strength and of the relative permeability along the plane z = 1 is presented.
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CONCLUSIONS
We have shown that the FEMAX package is an efficient and accurate code
for computing stationary and static electric and magnetic fields. The freedom of choice between the edge and nodal elements allows an accurate
modeling of the internal surfaces of discontinuity in medium parameters
without excessively increasing the number of unknowns. The use of consistently linear elements yields a local approximation error 0(h?) in the
representation of the field strength. Note that a method based on the use
of vector potentials and linear expansion functions would yield a local approximation error 0(h) in the representation of the magnetic field strength
for about the same computational effort. By mapping the three basic types
of static and stationary electric and magnetic fields on one generic formalism the resulting code is very compact without inducing any restrictions on
the range of problems to be solved. The package is capable of computing
fields in arbitrarily inhomogeneous and (an)isotropic media that may be
nonlinear.
REFERENCES
[1] Haus, H.A. and Melcher, J.R., Electromagnetic Fields and Energy,
Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1989.
[2] Nayfeh, M.H., Brussel, M.K., Electricity and Magnetism, John Willey
& Sons, New York, 1985.
[3] Mur, G., "The FEMAX finite element package for computing threeedimensional time-domain electromagneticfieldsin inhomogeneous media", in Software Applications in Electrical Engineering. Computational
Mechanics Publications, Southampton, 1993.
[4] Lager, I.E., Mur, G., "Compatibility Relations for Time-Domain and
Static Electromagnetic Field Problems" in Applied Computational Electromagnetics Society Journal, Vol. 9, No. 2, pp. 25-29, 1994.
[5] Jiang, B., Wu, J., "The Origin of Spurious Solutions in Computational
Electromagnetics", NASA Technical Memorandum 106921, ICOMP-958, 1995.
[6] Lager, L, Mur, G., "The Finite Element Modeling of Static and Stationary Electric and Magnetic Fields", submitted for publication in IEEE
Transactions on Magnetics.
[7] Ortega, J.M., Rheinboldt, W.C., Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
[8] Segal, A., SEPRAN, Sepra Analysis, User's Manual, Sepra, Leidschendam, The Netherlands, 1984.
Transactions on Engineering Sciences vol 11, © 1996 WIT Press, www.witpress.com, ISSN 1743-3533
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[9] ***, NAGj Graphical Supplement Manual - Mark 2, Numerical Algorithms Group Ltd, Oxford, UK, 1985.
[10] ***, VAX GKS Reference Manual, Volume I, Digital Equipment Corporation, Maynard, Massachusetts, USA, 1987.
[11] * * *, MATLAB. High-Performance Numeric Computation and Visualization Software. Reference Guide, The Math Works, Inc., Natick,
1993.
ERR(Ez) max = 3.98%
Figure 4. Results for the "Charged Brick" problem. The local relative error in computing
the electricfieldstrength for x E [0, 2], y € [0, 2] and z = 1.
Figure 5. Results for the "Test Slot" problem, a) The values of the magnetic field
strength components for x £ [0, 2] and y G [0, 4]. b) Detail representing the jump in the
normal components along the interfaces.
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Figure 6. Results for the nonlinear "Charged Brick" problem. The values of the electric field strength modulus and of the relative electric permittivity for x G [0,2], y G
[0, 2] and z = 1.