IEEE Transactions on Electrical Insulation Vol. 26 No. 3, June 1991 529

Comparative Analysis of Methods for Computing 2-D and 3-D Electric

Fields

H. Steinbigler, D. Haller

Technical University of Munich, Germany

and A. Wolf Siemens AG, Munich, Germany

ABSTRACT On the basis of four HV examples, electrical field calculations were performed using different computational methods. Three examples represent axisymmetric arrangements, one example is three-dimensional. The computations were performed using a boundary element program from the University of Sarajevo, a program based on the charge simulation method, developed at the Technical University of Munich, and the commercial finite element program ANSYS.

1. INTRODUCTION this. The examples presented could serve as benchmark problems for the application of different field calculation

HERE are well-established concepts available for the T calculation of corona inception and spark breakdown voltages in gaseous dielectrics [l-41. The basis for the ap- plication of these concepts is the knowledge of the elec- tric field strength. The remarkable progress of computer technology within the last two decades now opens many interesting possibilities for the numerical calculation of electric fields in HV insulation systems, even for 3-D ar- rangements with complicated boundaries. This numer- ical field calculation can be integrated into concepts of computer-aided design and used for the optimization of HV apparatus.

methods.

2. FIELD CALCULATION METHODS

OR the calculation of 2-D and 3-D electric fields in F HV engineering, different methods are applied. The most important are the finite difference method (FDM), the charge simulation method (CSM), the finite element method (FEM) and the boundary element method (BEM). The principal task in the computation of electrical fields is t o solve the Poisson equation

E (1)

P V.V@=-- There is a number of computer programs available for these calculations, but the application of these programs

discretization. Compromises between accuracy, comput- ing time and amount of memory must be found. I t is the

needs SOme experience, particularly for the proper way Of In case of space-charge-free fields the equation reduces to the Laplace equation

intention of this paper t o give some help with respect to U * U 9 = Q (2)

0018-9367/91/0600-529$1.00 @ 1991IEEE

530 Steinbigler et al.: Methods for Computing 2-0 and 3-D Electric Fields

Whereas in the past 3-D problems could be solved only on mainframe computers, nowadays superminis and work- stations offer the computer power required. However, computing times and the amount of memory to achieve the accuracy required, still play a dominating role. More- over, an important aspect for the acceptance of a program is the ease with which it can be used to describe a prob- lem.

Because of the wide use of FEM programs in a large range of applications and the strength of their pre- and postprocessors, FEM programs are also attractive for elec- tric field calculations. Therefore, the aim of this paper is a comparison of BEM, CSM and FEM considering the modeling effort required as well as the achievable accura- cy. The comparison was performed on various test exam- ples.

2.1 CHARGE SIMULATION M E T H O D

The charge simulation method (CSM) belongs to the family of integral methods for the calculation of electro- magnetic fields. There are two variants of this method: CSM with discrete charges [5] and CSM with area charges [el.

The CSM with discrete charges is based on the principle that the real surface charges on electrodes or dielectric in- terfaces are replaced by a system of point and line charges located outside the field domain. The position and the type of these simulation charges are predetermined. The magnitudes of the charges have to be calculated so that their integrated effect satisfies the boundary conditions exactly a t a selected number of collocation points (point matching). After the determination of these magnitudes by the solution of a linear equation system, it must be checked whether the system of simulation charges fulfills the boundary conditions between the collocation points with sufficient accuracy. Then the potential and the field strength in any point of the field domain can be calculat- ed analytically by the superposition of simple potential and gradient functions. In the case of an axisymmetric field, ring-shaped line charges (ring charges) or straight line charges are mainly used. For the location of the charges outside the field domain, simple empirical rules are applied [7] .

The calculation of the 3-D electric fields by means of discrete simulation charges is restricted to systems with single axisymmetric components arranged arbitrarily in the 3-D s p x e [7 ] , a situation which is very common in HV engineering. For the charge simulation, ring charges with partially constant charge density [8] or continuously

I

I

1

I

I 3

A

Figure 1. (a): Disconnector, (b): electrode areas A-B-C and D-E.

variable charge density [7] are used. In the latter case, the charge density on a ring charge is divided into a constant part and several components with cosine charge distri- bution on the ring with unknown peak values, similar to Fourier analysis [7]. To each ring charge component a collocation point on the boundary is attached where the boundary conditions are satisfied. The resulting linear equation system is solved for the constant parts and the peak values of the ring charge density components.

Another possibility of charge simulation is the applica-

IEEE Transactions on Electrical Insulation Vol. 26 No. 3, June lQQl 531

Figure 3. Disconnector, submodel.

Figure 2. Disconnector, FE mesh (zoomed).

tion of area charge elements additionally or alternatively to discrete charges [6]. This type of charge is placed di- rectly on electrodes or dielectric interfaces. In this case, the unknowns of the linear equation system are the area charge densities. This charge simulation method can al- so be extended t o 3-D fields with the restriction tha t the electrodes or dielectric boundaries are axisymmet- ric. Similar to the CSM with discrete charges for 3-D fields, cosine charge density components are introduced. In comparison with discrete charges, the potential and gradient functions are more complex and therefore, more computation time is necessary, but on the other hand there is no need for empirical rules for the location of the simulation charges.

2.2 BOUNDARY ELEMENT METHOD

The BEM is, in principle, very similar to the CSM with area elements. As with the CSM, the BEM uses area charge elements to replace the real charges. The BEM does not, however, require tha t the system components possess axial symmetry. The discretization of the real charges is generally achieved using three- or four-noded

boundary elements, which use linear shape functions to approximate the internal charge distribution. To match the outer surface geometry, suitable intermediate points can be added [9]; alternatively, boundary elements having a particular, predefined, form are available e.g. cylinders, cones, spheres and toroids [9]. The evaluation of the re- sulting potential functions of the boundary elements gen- erally requires a numerical integration; only for the outer surfaces of cylinder and sphere elements is a time-saving analytical solution feasible [lo].

2.3 FINITE ELEMENT METHOD

FEM is a numerical computational method for the so- lution of partial differential equations. The method was originally applied in the field of static and dynamic eval- uation of safety-critical mechanical components [ll]. It was later extended to include the study of heat flow, elec- tric and magnetic fields, fluid flow and other field prob- lems.

In the case of electric fields, the field is represented by a number of individual elements, each connected to its neighbors by its corner nodes, thus forming a network of

532 Steinbigler et al.: Methods for Computing 2-D and 3-D Electric Fields

I $ = 2.3 ,Conductive D e f l e c t o r constructed of cubes and tetrahedrons.

\ C o n d u c t o r @ = I

Figure 4. (a): Cable termination, (b): deflector area A-B-C-D.

N nodes. Three-dimensional fields, for example, can be

Within the individual elements, the unknown potential function is approximated by so-called shape functions, generally of low order (depending on the element type). An approximate solution of the exact potential field is then given in the form of an expression whose terms are products of the shape function and the unknown nodal potentials. I t can be shown that the solution of the dif- ferential equation describing the problem (in the electri- cal case, without space charges, the Laplace equation) corresponds to a minimization of the field energy. This leads to a system of algebraic equations, whose solution, under the application of corresponding boundary condi- tions, supplies the required node potentials.

Thus the result of the procedure is a potential distribu- tion over the field in the form of discrete potential values a t the node points of the FEM mesh. The related field strengths a t the element centers are then obtained from the potential gradient. The values of field strength are heavily dependent on the distance between the element centers and the electrode surface, and, therefore, on the element sizes.

In order t o achieve a high mesh density in critical ar- eas, whilst minimizing the total number of elements, the element size can be graduated. An effective technique in terms of both modeling effort required and accuracy of re- sults obtained, is submodeling, whereby a part of the FE model is separated from the remainder and remeshed with an increased mesh density. The submodel can be based on a relatively coarse F E model, allowing higher accura- cy to be achieved in particular areas without significantly increasing the complexity of the whole model. Node po- tentials, extrapolated from the results of the previously analyzed base model, are then applied to the boundaries of the submodel and a second, more exact calculation of this area is performed.

3. COMPARISON C A L C U L AT I 0 N S

N order to compare the computational methods [12], I four examples of HV components were selected, the first three being axisymmetric, the fourth a three-dimensional problem. The computations were carried out using a BEM program from Sarajevo University [9], a CSM pro- gram from Munich Technical University [7], and the FE code ANSYS.

IEEE Transactions on Electrical Insulation

Figure 5. Cable termination, FE mes...

533 Vol. 28 No. 3, June la91

Figure 6 . Insulator.

3.1 DISCONNECTOR

The first example is a disconnector for a gas-insulated

substation. The dimensions of this axisymmetric device are shown in Figure la. Results on the basis of BEM and CSM have already been presented [13] and a FEM simulation was carried out for comparison in this paper. This comparison was performed in the area of highest field strength along the electrode sections A-B-C and D- E (Figure lb ) . A rough calculation was first performed using the F E mesh shown in Figure 2, followed by a more accurate submodel calculation of the areas of highest field strength. For the section A-B this submodel is shown in Figure 3.

Figure 7. Insulator, FE mesh (zoomed).

The field strength a t the surface sections A-B-C and D-E are listed in Table 1. Discrepancies between the FEM results and those from BEM and CSM lie < 2%. Differences of > 2% appear only where there is a step in the curvature of the top surface leading to a discontinuity in the field strength.

3.2 CABLE TERMINATION

In the second example, a cable termination for a trans- former input was studied. This component, which com- prises layered dielectrics and which has axial symmetry, is shown schematically in the installed condition in Fig- ure 4a. The path A-B-C-D on the deflector end was chosen for comparison of analysis results (Figure 4b).

At points A, B, C and D there are discontinuities in the curvature of the electrode outer surface. Thus the field

534 Steinbigler et al.: Methods for Computing 2-D and 3-0 Electric Fields

3

Figure 8. 3-D arrangement.

strength, which is inversely proportional to the surface curvature, is also discontinuous a t these points. Since none of the three methods can accurately reproduce such a discontinuity, the differences between them were par- ticularly high a t these points.

In the area of the maximum field strength, the electrode assembly can be approximated by a n idealized cylindri- cal capacitor. A comparison of the FEM results with an analytical solution for the field strength a t the inner conductor of the cylindrical capacitor showed a differ- ence of -0.98%. This result confirms the accuracy of the submodeling technique. The F E model of the cable ter- mination is shown in Figure 5. The difference between these FEM results and those from the BEM and CSM techniques also lay in the region of 2% (Table 2).

3.3 INSULATOR

Further calculations were made for an insulator having axial symmetry. It is similar to tha t described in previous studies [14] and is shown in Figure 6. Of particular in- terest was area A-B-C of the upper electrode (Figure 6). The first calculation with F E shown in Figure 7 was fol- lowed by a submodel calculation of area A-B-C in the same way as it was performed for the disconnector exam- ple. The field strength results in the area A-C are shown in Table 3. The differences between results from FEM and those from BEM and CSM were < 2%.

Table 1. Disconnector (Figure l(a) and Figure l(b)): field strength values E, deviation 61 (between FEM and BEM) and deviation 6 2 (between FEM and CSM).

s E E 61 E 62

FEM BEM CSM

[mml rl/ml [l/m1 [ % I ['/ml [ % I

A 0.00 57.81 7.85 59.28

15.71 62.56 23.56 65.49 31.42 63.18 39.27 54.35 47.12 43.12 72.23 19.49 83.73 15.45

C 95.23 12.27

58.37 59.54 63.07 66.50 64.11 54.83 43.72 19.66 15.66 12.49

-1.0 58.31 -0.4 59.59 -0.8 63.08 -1.5 66.39 -1.5 63.98 -0.9 54.81 -1.4 43.74 -0.9 19.71 -1.3 15.67 -1.8 12.49

-0.9 -0.5 -0.8 -1.4 -1.3 - 0 . 8 -1.4 -1.1 -1.4 -1.8

D 0.00 2.46 4.91 7.37 9.82 12.28 14.37 18.54 22.72 26.89 31.07

E 33.16

28.86 28.97 -0.4 39.87 40.63 -1.9 47.92 48.67 -1.5 52.31 52.82 -1.0 51.51 51.91 -0.8 43.79 43.95 -0.4 37.29 37.11 0.5 30.88 30.89 t0.1 26.69 26.46 0.9 23.11 23.04 0.3 19.96 20.32 -1.8

Discontinuity

29.35 -1.7 40.60 -1.8 48.60 -1.4 52.73 -0.8 51.84 -0.6 44.22 -1.0 37.42 -0.4 30.94 -0.2 26.49 0.8 23.07 0.2 20.33 -1.8

3.4 3-D EXAMPLE

Finally, a 3-dimensional electrode arrangement consist- ing of a sphere ( 4 = 1) and a cylinder (4 = U) was analyzed. The field area is bounded by a surrounding cylinder with potential 4 = 0 (Figure 8). The calcula- tions were performed using only CSM and FEM, For the F E model, eight-noded, isoparametric 3-D brick elements with linear shape functions were used.

The automatic meshing of volumes with tetrahedrons becomes extremely time consuming for complex shapes. It is, therefore, advantageous first to split the structure in- to so-called regular bodies with six bounding faces, which can then be rapidly meshed.

Due to the double symmetry of the component it was On the Gnly necessary to construct a quarter model.

IEEE Transactions on Electrical Insulation Vol. 26 No . 3, June 1991

Table 2. Cable termination (Figure 4(a) and Figure 4(b)): field strength values E, deviation 61 (between FEM and BEM) and deviation 6 2 (between FEM and CSM).

FEM BEM CSM S E E 61 E 62

1mm1 [ l / m l [ l / m ] [ % I [ ' / m l 1x1

0.00 2 . 3 9 5 . 0 3 7 . 9 4

11 .27

1 3 . 0 3 1 4 . 8 0 1 6 . 5 7 20 .28

Discontinuity A

3 . 3 9 3.38 0 . 3 3 .44 4 . 3 1 4 . 2 8 0 . 7 4 . 3 7

6 . 6 7 6 . 4 8 2 . 9 6 . 6 5

18 .46 1 8 . 9 3 -2 .5 19 .07

26 .27 27 .04 - 2 . 9 26 .97

32 .77 32 .50 0 . 8 3 3 . 6 3

3 4 . 1 8 35 .16 - 2 . 8 3 5 . 3 5 23 .50 2 3 . 6 9 -0 .8 23 .72

- 1 . 5 - 1 . 4

0 . 3 -3 .2

- 2 . 6 -2 .6 - 3 . 3 - 0 . 9

23 .12 2 2 . 4 1 2 2 . 4 3 - 0 . 1 22 .59 - 0 . 8 25 .96 21 .94 22 .02 - 0 . 4 22 .08 -0 .6 2 8 . 7 9 21 .46 2 1 . 4 9 - 0 . 1 21 .59 - 0 . 6 3 1 . 6 3 Discontinuity D

Table 3. Insulator (Figure 6): field strength values E, devi- ation 61 (between FEM and BEM) and deviation 62 (between FEM and CSM).

FEM BEM CSM s E E 61 E 62

[mml [ ' / m ~ [l/ml [XI [ ' / m l [ % I

A 0.00 3 . 1 4 6 . 2 8 9 .42

1 2 . 5 7 1 5 . 7 1

2 2 . 4 4 25 .81 29 .17 32 .54 35 .90

C 39.27

32 .07 3 7 . 1 3 4 0 . 5 0

42 .12 4 1 . 7 1 37 .18

3 1 . 1 4 29 .76 2 8 . 4 6 27 .10 2 5 . 6 5 24 .02

31 .96 3 6 . 9 5

4 0 . 3 6 41 .94 4 1 . 7 5 37 . 08 3 0 . 8 3

29 .40

2 8 . 1 5 2 6 . 8 5 25 .54 24 I 0 3

0 . 3 0 . 5

0 . 3 0 . 4

- 0 . 1 0 . 3

1 . 0 1 . 2

1.1 0 . 9 0 . 4

- 0 . 1

3 2 . 0 1 36 .87 4 0 . 2 6

4 1 . 7 3 41 .44 3 7 . 1 1 30 .86

29 .42 28 .15 26 .88 25 .52 24 .00

0 . 2 0 . 7

0 . 6 0 . 9

0 . 6 0 . 2 0 . 9

1.1 1.1 0 . 8 0 . 5 0 . 1

edges of symmetry the Neumann boundary conditions ( g = 0) were automatically applied.

As a basis for comparison a n equatorial and a meridian line around the outer surface of the sphere were select- ed. For the 3-D case the discrepancies between FEM and

Table 4. 3-D example (Figure 8) sphere equatorial line: field strength values E and deviation 6 between FEM and CSM.

FEM CSM

s E E 6

[mm1 [ l / m l [ ' / m l [%I

0 .00 8 0 . 4 7 80 .45 t 0 . 1 6 . 9 8 1 3 . 9 6 2 0 . 9 4 2 7 . 9 3

3 4 . 9 1

41 . 8 9 48 * 87

55 * 85 6 2 . 8 3

76 .56 6 8 . 7 8 6 3 I 76 6 1 . 4 5

5 9 . 5 3 57 .37

56 .42 5 6 . 8 7

5 6 . 8 5

7 6 . 2 8 6 8 . 95 6 3 . 4 4 6 0 . 0 6

5 8 . 0 4 56 I 84

5 6 . 1 3 5 5 . 7 6 5 5 . 6 4

0 . 4 - 0 . 2

0 . 5 2 . 3

2 . 6 0 . 9

0 . 5 2 . 0 2 . 2

Table 5. 3-D example (Fiure 8 ) sphere meridian line: field strength values E and deviation 6 between FEM and CSM. FEM CSM

S E E 6 [mm1 I ' / ~ I [ ' /RI 1x1

0.00 80 .47 2 .86 8 0 . 6 5 8 .57 7 7 . 0 3

1 4 . 2 8 71 .96 1 9 . 9 9 6 7 . 1 6 2 5 . 7 0 6 4 . 0 1 3 1 . 1 3 6 1 . 1 4 31 .70 6 0 . 9 1 3 7 . 1 3 5 9 . 3 3

42 .84 57 .52 48 .55 5 6 . 9 6 54 .21 5 6 . 9 3 59 .97 5 7 . 2 9 6 2 . 8 3 5 6 . 8 5

8 0 . 4 5

80 .27 7 7 . 1 5

72 .27 6 7 . 5 6

62 .95 6 0 . 9 6 6 0 . 7 3 5 8 . 3 3

5 7 . 6 9 5 6 . 7 1

5 6 . 1 2 5 5 . 6 9

5 5 . 6 4

t0.1 0 . 5

- 0 . 2

- 0 . 4 - 0 . 6

1 . 7 0 . 3 0 . 3 1 . 7

- 0 . 3 0 . 4

1 . 4 2 . 9

2 .2

CSM results were < 3% (Tables 4 and 5).

535

Since the programs are not mounted on the same com- puter hardware, a comparison of computer time require- ments is not available. I t is clear, however, tha t the effort in constructing the FEM mesh is higher than for the CSM mesh, since the entire area between electrodes and system boundaries must be discretized. Hence, the F E model consisted of 5616 elements, while for the charge simula-

536 Steinbigler et al.: Methods for Computing 2-0 and 3-D Electric Fields

tion method 256 elements were sufficient. As a modeling aid, a 3-D CAD system is recommended to allow an easy subdivision of the volume into sections which can be more easily meshed separately.

4. CONCLUSION

HE comparison of the computational methods shows T a good level of agreement between CSM and BEM. For 2-D problems the discrepancy lies in the region of 1% and in the 3-D case 2%. Somewhat larger discrepancies exist between CSM and FEM. In the 2-D case a differ- ence of 2%, and in the 3-D case 3% were observed. The submodeling process provides an easy-to-use procedure for achieving sufficient accuracy. The effort to produce a submodel is minimal. In general the construction of a F E model requires considerable effort since the entire field region must be meshed, while with BEM and CSM only the outer surface of the electrode and the outer layer of the dielectric must be meshed. A 3-D CAD system can be of value in simplifying the meshing of complex and, in particular, 3-D structures.

In practise, a more significant difference between the techniques is that the FEM can only be used with bound- ed fields. CSM and BEM, on the other hand, can also deal with unbounded fields, for example a sphere in space. When describing such fields with the FEM, the system must be approximated by the application of boundary conditions placed a t an appropriate distance outside the area of interest.

FEM programs offer the advantage of a wide applica- tions base, which, in addition to electric field simulation, includes the analysis of mechanical, thermal and magnet- ic systems, as well as certain optimization tasks. Addi- tionally non-linearities can be taken into account. Lastly, the availability of high performance commercial pre- and postprocessors, most of which have interfaces with CAD systems, for both FEM and BEM should be pointed out.

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Manuscript was received on 1 February 1991, in revised form 6 April 1991.