Electromagnetic Field Modeling Using Edge Finite Elements

A. Noreika, P. Tarvydas

Department of Electronics Engineering, Kaunas University of Tehnology Student str. 50, LT-3031 Kaunas, Lithuania, E-mail: alius.noreika@ktu.lt

ABSTRACT: Edge finite elements are typically used in many modern applications, where modeling of distribution of electromagnetic fields is required. These elements differ from nodal finite elements in the fact that they provide continuity of tangential components of the field variables across element interfaces while allowing discontinuity in normal components. This, in turn, eliminates the so-called spurious solutions in eigenvalue analysis in cavities. Comparison of edge and nodal finite elements is presented by analyzing their advantages and disadvantages. Form functions used to describe rectangular finite element in 3D space are presented. Properties of the edge finite element SOLID117, which is implemented in the software package ANSYS/Emag (ANSYS Inc., www.ansys.com) and intended for modeling of low-frequency electromagnetic fields, are discussed here. Generalized electromagnetic field modeling algorithm using edge finite elements was created.

1 Introduction Although numerical application of finite element

method [1,2] is easy and straightforward and it produces accurate results, several problems have been identified when the ordinary nodal based finite elements are employed to compute vector electromagnetic fields. These problems are: long computation time; large amount of memory; satisfaction of the appropriate boundary conditions at material and conducting interfaces; difficulty in treating the conducting and dielectric edges and corners due to the field singularities associated with these structures and etc.

In order to solve these problems, a novel approach has been developed: the approach uses so-called vector finite elements or edge finite elements which assign degrees of freedom to the edges rather than to the nodes of the finite elements. The edge finite element is geometrically the same as its nodal counterpart. The nodal element has one shape function associated with each of the nodes, the edge element has one shape function for each of the edges.

Edge finite element method has several advantages in 3-D electromagnetic field computation compare to the nodal finite element technique. Edge finite elements satisfy continuity of only tangential or normal field components across interfaces between two adjacent finite elements. This property can be used when modelling electromagnetic field, since edge finite element approximation does not impose any additional constraints

on the approximated field apart from those prescribed by the nature of the field itself.

Overall performance of the software, based on edge finite element method, is also higher and requires less amount of memory compare to the software, which implements the conventional nodal finite element method. This fact is important when analyzing complex 3-D electromagnetic field distributions.

2 Fundamental equations for edge elements Maxwells equations for the mathematical models of typical electromagnetic domains without electric charges and current sources are written as (1)(4):

tEHrot

, (1)

tHErot

, (2)

0Hdiv , (3) 0Ediv , (4)

where H magnetic field, B magnetic flux density, E electric field, D electric flux density, permittivity, permeability.

From equations (1) and (2) the vector Helmholtz's equations (5)(7) can be written:

HkHrotrot 2 , (5) EkErotrot 2 , (6) 22 k , (7) where angular frequency.

Equations (5) and (6) are used to calculate modes of magnetic and electric fields. Modes are calculated as eigenvalue problems considering the nondivergent conditions indicated in (3) or (4). In this case when edge finite elements are used to formulate the problem, the nondivergent condition is fulfilled directly.

The formulation of finite element domain using cubic edge elements is derived from ordinary isoparametric elements. The shape function iN is written in the local coordinates ,, as (8):

2008 International Biennial Baltic Electronics Conference (BEC2008) Tallinn, Estonia, October 6-8, 2008

978-1-4244-2060-5/08/$25.00 2008 IEEE

iiiiN

11181 , (8)

where iii ,, is the local coordinates (Fig. 1).

(a) (b)

Fig. 1. Linear cubic element: (a) global coordinate, (b) local coordinate The electric field vector E inside of the element is defined by the shape functions, expressed using (9):

,

,

,

8

1

8

1

8

1,

ii

ii

ii

i

i

ENE

ENE

ENE

(9)

where i

E , iE , iE the components of electric field

E on the nodes of finite element. It is considered that directional components on two nodes which form one edge (Fig. 2) are equal. This condition is depicted in equations (10)(12).

;

,

,

,

4

3

2

1

87

65

43

21

EEE

EEE

EEE

EEE

(10)

;

,

,

,

8

7

6

5

32

76

85

41

EEE

EEE

EEE

EEE

(11)

.

,

,

,

12

11

10

9

84

73

62

51

EEE

EEE

EEE

EEE

(12)

So the field vector E is expressed using formula (13):

.

00000000

00000000

00000000

12

1

8

4

7

3

6

2

5

1

3

2

7

6

8

5

4

1

8

7

6

5

4

3

2

1

12

1

E

E

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

N

ENEi

ii

(13)

So it follows, that in the edge element the nondivergent condition for electric field (expressed by equation (4)) is always satisfied. For the magnetic field H the solution method is formulated in a similar way.

(a) (b) (c)Fig. 2. Edge element: (a) direction, (b) direction, (c) direction

When using Galerkin method, the functional for fundamental equation for electric field (6) is written as (14):

02 dVEkErotrotNG ii . (14)

The Gauss theorem is applied to transform equation (14) and expression (15) is obtained:

.121

2

12

1

jjjiji

jii

dvENNkNrotNrot

dsnErotNG

(15)

The first member of the right side of equation (15) is a boundary integral term. It depends on the boundary conditions.

Global-coordinate equation (15) can be rewritten using local coordinates as it is shown by expression (16):

12

1

1

1

1

1

1

1

2

jjjijii dddJENNkNrotNrotG , (16)

where J is the determinant of Jacobian matrix ! "J . The rotation parameters of equation (16) are transformed to global coordinates by equation (17).

! " .1

##$

%&&'

(

##$

%&&'

(

##$

%&&'

(

kk

kk

kk

zk

yk

xk

NN

NN

NN

J

Nrot

NrotNrot

(17)

The eigenvalue equations for magnetic fields can be formulated using the vector Helmholtz's equation in analogous way.

The boundary integral member of equation (15) should be eliminated for boundary conditions. Boundary conditions of tangential electric fields tE (18) and tangential magnetic fields tH (19) are considered between two different solution domain regions:

,

21 tt EE (18)

,21 ktt HH (19)

where k current density on boundary; indexes 1 and 2 denote different regions of the solution domain.

Since there is no current in the air region, the tangential component of the field vector is continuous. Thus the boundary integral member of equation (l5) is eliminated between these elements.

When the boundary is formed as a contact between perfect conductors, tangential component of electric field

tE is equal to zero. The boundary integral member of

magnetic field vector H can be written as (20):

.0 #

$%

&'(

dSNnE

t

dSNnHrotdSnHrotN

i

ii

(20)

Therefore this member of the equation can be

eliminated since 0tE on this boundary. If we have a non-conductive symmetric boundary,

tH is equal to zero. Analogously, the boundary integral

term of E is expressed as (21):

.0 #

$%

&'(

dSNnHt

dSNnErotdSnErotN

i

ii

(21)

For this reason this member can be eliminated because

tH is equal to zero on the boundary of symmetry.

3 Edge finite element applications There is a variety of programs, which are designed in order to create engineering environment for electromagnetic field modeling. Software package ANSYS/Emag has the most extensive modeling capabilities using edge finite element method.

Fig. 3. 20-node brick edge finite element SOLID117 Table. 1. Shape functions

Nodal shape functions Edge shape functions tsrNI 111

tsrNJ 11 trsNK 1

tsrNL 11 tsrNM 11

tsrNN 1 rstNQ

strNP 1

gradrtsEQ 11 gradstrER 1 gradrtsES 1

gradstrET 11 tgradrsEU 1

rtgradsEV stgradrEW

tgradsrEX 1 gradtrsEY 11 gradtrsEZ 1

stgradtEA rgradtsEB 1

Solid finite element SOLID117 (Fig. 3), implemented

in the ANSYS/Emag program, is typically used in the analysis of low-frequency problems based on edge finite element formulation [3,4]. The corner nodes of element SOLID117 PI ,, are used to describe the geometry, orient the edges, support time integrated electric potential degrees of freedom. The side nodes AQ ,, are used to support the edge-flux degrees of freedom, define the positive orientation of an edge to point from the adjacent (to the edge) corner node with lower node number to the other adjacent node with higher node number. The edge-flux degrees of freedom are used in both magnetostatic and dynamic analyses; the electric potential degrees of freedom are used only for dynamic analysis.

The vector potential A and time integrated electric scalar potential V can be described as

BBQQ EAEAA

, (22)

PPII NVNVV

; (23)

where: BQ AA ,, edge-flux; PI VV ,, time

integrated electric scalar potential; BQ EE ,, vector

edge shape functions (Table 1); PI NN ,, scalar nodal shape functions (Table 1).

Fig. 4. Electronic device modeling algorithm

The global Cartesian coordinates X, Y, Z can be expressed by the master coordinates r, s, t.

PPII XtsrNXtsrNX ,,,,

, (25)

PPII YtsrNYtsrNY ,,,,

, (26)

PPII ZtsrNZtsrNZ ,,,,

; (27)

where: IX , IY , IZ global Cartesian coordinates of the corner nodes; PI NN ,, first order scalar nodal shape functions.

a) b)

Fig. 5. Geometry model (a) and field distribution modelling results (b) of magnetic solenoid actuator

Bridge between mathematical description of the typical electromagnetic problem is illustrated in Fig. 5. Here a model of solenoid actuator (Fig. 5,a) is created using conventional geometry editing tools. Device consists of ferromagnetic core, coil and a plunger. Source of induced magnetic field is defined by setting density of the electric current flowing through the coil

26 /101 mAJ . Relative magnetic permittivity of air and coil is set to 1 . Modeling results are presented in Fig. 5,b.

Since the process of modeling of electromagnetic fields created by various electronic devices using edge finite elements is analogous to the modeling using node finite elements, algorithms presented in [5,6] can be also used when solving electromagnetic field problems based on edge finite element formulation. The generalized algorithm of field modeling is presented in Fig. 4.

4 ConclusionsWhen using edge finite element technique, the number of generated unknown values is larger than using nodal finite element technique for the mesh of the same spacing. When using the edge finite element approximation the higher number of unknowns is compensated by lower connectivity between edges or a sparsity of the rigidity matrix. Therefore in result, the edge finite element method requires less amount of memory and faster analysis can be achieved.

Edge finite element method can be successfully used when solving various electromagnetic field simulation problems, where the application of the nodal finite element method is complicated, e.g. when using nodal-based approximation of a vector variables in finite element computations.

References [1] John L. Volakis, Chatterjee A., Leo C. Kempbel.

Finite Element Method for Electromagnetics. New Jersey: IEEE Press, Hoes Lane, 2005. 360 p.

[2] Bathe Klaus-Jurgen. Finite element procedures. New Jersey: Prentice Hall, Upper Saddle River, 2004. 1038 p.

[3] Theory Reference for ANSYS and ANSYS Workbench. ANSYS Release 11.0. SAS IP, Inc. 2007. 1110 p.

[4] Elements Reference. ANSYS Release 11.0. SAS IP, Inc. 2007. 1532 p.

[5] H. T. Yu, S. L. Ho, M. Q. Hu, H. C. Wong. Edge-Based FEM-BEM for Wide-Band Electromagnetic Computation // IEEE Transactions on Magnetics. April 2006. Vol. 42, No. 4. P. 771774.

[6] Tarvydas P., Markeviius V., Noreika A. Modeling of Asymmetric Electronic Optical System // BEC 2004. Proceedings. Tallinn, 2004. P. 6770.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description >>> setdistillerparams> setpagedevice