university of Jaffna, sri Lanka. Harvey Mudd College, Claremont, CA. ABSTRACT: The impedance boundary condition has been used with great profit.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 24, NO. 6, NOVEMBER 1988
2503
THE IMPEDANCE BOUNDARY CONDITION IN THE BOUNDARY ELEMENT - VECTOR POTENTIAL FORMULATION
S. Ratnajeevan H. Hoole Srisivane Subramaniam' and Department of Mathematics & Statistics Department ofEngineering university of Jaffna, sri Lanka Harvey Mudd College, Claremont, CA
ABSTRACT: The impedance boundary condition has been used with great profit to eliminate large regions from the solution. It has been used with boundary element formulations solving for the elecmc or magnetic field and vector potential finite element formulations. In the boundary element formulation, the reduction of the order of singularitiesis convenient, if not critical. This paper offers a vector potential, boundary element formulation with the impedance boundary condition. The formulation has a lower order of singularity in relation to the magnetic field intensity and has computational simplicity in relation to the finite element implementation and offers great advantages in open boundary problems. Some important lessons are offered for those from a finite element background using boundary elements
I H,
p'u
I
Fig.1: Penetration into Infinite &a which is:
INTRODUCTION
G = Kg(ar)42%)
The solution of eddy current problems is costly because of the rapid variation of the field quantities in conducting parts. Means are usually sought to enhance efficiency. The impedance condition offers a way to reduce the solution domain by eliminating conducting parts with no externally imposed currents[ldl. Because the fields vanish within a few millimeters of the surfaces of many devices, the behavior is I-dimensionaland,r e f d n g to Fig.1, the diffusion equation [V.
where 6 is 1 at r = 0 and 0 elsewhere, a2 = jopo and KO is the modified ~ - r funchon ~ of the second kind of order [11,121. using G ~second * identity:
II~A V ~ G- G V ~ A d~ I = I[A&/an - G aA/an ids A = - II ~ G d~ J + I[A =/an - ~ a ~ ~ a a i d s
(1)
(2)
e-p
Eqs. 6 and 7 have been used before as boundary conditions to eliminate the conducting part from the solution domain [MI. In soking for the magnetic f=l& it is for us to choose the field descriptor (such as A, E, H)and the formulation ( which may be f ~ t elements e M boundary elements ). This paper addnxses the boundary element formulation where the explicit solution is confined to boundaries. In choosing the descriptor, care has to be taken in examining the singularities [8]. The singularities arise.in our having to determine the effect of a source at its own location. Since E=-joA at the impedance bomdary and H= p-lVxA, it is clear that E and Aare one order higher than H in the radius vector r. Thus the mathematics in solving for E or A is m m easily worked out. Our success in integrahg out thes singulaririesis critical to ability to solve these problems. Moreova, the impedance boundary condition assumes greater importance with boundary elements where inhomogeneities are clumsy and costly to handle. With the impedance boundary condition, some of the inhomogeneous regions may be eliminated. The tie-up between the impedance boundary condition and the boundaryelement method assumesequal importance in open boundary problems. The formulation in terms of E has been described elsewhere 181. In this paper we describe the solution of A while using the impedance boundary condition. OUT
(10) (1 1)
'Ihus dividing the boundary into elements over each of which A is assumed to vary according to a trial function, A may be evaluated through eq.11 at the inteapolation points and the resulting matrix equation solved for A or aA/an, whichever is unknown at the boundary. The singularities arise in our having to evaluate the integrals G(r) = Kdm) and
(3) (4) ,
(9)
&/an = Kl(a r) Cos ~ / 2
(12)
where y is the angle between the normal and the radius vector and K1 is the modified Bessel's function of the second kind of order one. The procedures for evaluating these integrals are described elsewhere [9,10,13,14]. In such evaluation, it is useful to know that &(ar) tends to -In ar as ar tends to zero. This result is m n t upon seaing a = 0 in eq.(8) which r e d w s the equation to that for static field problems; we do know that the Green's function for static problems is -
lnerm.
THE IMPEDANCE BOUNDARY CONDITION Unlie with finite elements, no elaborate mathematics is required to implement the impedance boundary condition of eq.7. As shown in Fig.2, for fmt order interpolation on A, moth order on we have aA& = - Y(A1+ A2)/2. For both A and aA/an fmt order, we merely impose eq.7 at the vertices. There is no contradiction in aA/h at the vertex which has two normals. The contradiction disappears when we recognize that the straight lines actually approximate a smooth boundary. In implementing these conditions we may either use the relationship between A and aA/an to eliminate one of the variables or add the relationship as an additional equation. In our implementation we have opted for the latter course of action because of its programming convenience, although the former approach is more efficient
am,
THE BOUNDARY ELEMENT METHOD For the solution of eq.1 by thc boundary element method, following refs.[9l and [IO], we seek a Green's function G,satisfying
\A 'Visiting at Harvey Mudd College, 1987188.
\A. 49
Fig. 2: Boundary Element Interpolation
0018-9464/88/1100-2503S01.00@1988 IEEE
I I
\
an
~
2504
EXPERIENCE IN USING THE FORMULATION To test the use of the surface impedance with the boundary element method, the simple problem of Fig. 3 was taken up. Here a conductor is in front of an infinite wall of steel with y = 352, computed from its permeability and conductivity at power frequency. The left boundary has an impedance boundary condition and the lower boundary has a Neumann condition, white the other 2 boundarieshave A = 0. The heavy eddy cun’ent effect makes the wall a flux l i e (the high value of y sets A to zero in eq. 7). To see what happens for low 7, it was set to 1 and the problem solved (Fig. 4). Here the strong skin effect is not present and the flux is no longer tangential to the wall. Figs. 5 and 6 give the shielding of a conductor by a finite slab for y = 352 and 1 respectively. In all these problems fmt order A and zeroth order aNan were used.The smooth interior plots and the crudeness of the plots close to the boundary are pointed out It is further remarked, that the flux plots are highly dependent on the interior grid at whose vertices the potentials are computed before plotting.
Fig. 6: Conductor in Front of Slab of steel 7 = 1 First order A, Zeroth order a A / h
\ SOME LESSONS FOR FINITE ELEMENT ANALYSTS In using boundary elements, those coming from a finite elements background make some assumptions that are hue in finite elements, but not in boundary elments. It is worth pointing them out. In boundary elements:
,/
1. A and its normal gradient are not related, unlike in finite elements where the aial function applies to the boundary and the interior. In fact, there is no contradiction in taking A and its normal gradient to be of the same order. First order A requires only that the gradient should be constant in the tangential direction and does not restrict the normal gradient. The exception is at Neumann boundaries, where if A is of uder zero, the tangential derivative of A is zero. The Neumann condition makes the normal derivative zero. Thus the whole vector field is set to zero at the boundary. Fig. 7 shows the problem of Fig. 5 r e p t e d with zeroth order A. The plot is seen to be poor at the Neumann boundary where, in effect. we set the flux density to zero.
Fig. 7: Symmetry to Reduce Geomeby Fig. 5: Conductor in Front of Slab of Steel y = 352
First order A, Zeroth order a A h
Zeroth order A and a A / h
\
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A= 0
-
U Conductor
S O
Shield
Fig. 9 Symmetry to Reduce Unknowns A=O
3 3 2
zeroth order A and a A h
aA
CONCLUSIONS The impedance boundary condition has been used in boundary element vector potential formulations for the fmt time and examples have been presented. First order vector potential has been used. Some important lessons are offered to those coming into boundary elements from a finite elements background. Zeroth order potential is easy to understand and implement, but yields poor flux plots at Neumann boundaries. It has been shown that this problem may be partly overcome by solving in the whole domain and using symmetry not to reduce the solution region, but to map the unknowns on one side on to those on the other.
--ifE----
REFERENCES
[I] K.F.Ali, M.T.Ahamed and P.E.Burke," Surface Impedance - BEM Techniques
aA
%-HI
A=O
aA s g l
Fig. 8 Two Approaches tu Using Symmetry a. Reducing Geumetry b. Mapping Unknowns
for Nonlinear TM-Eddy Current Problems", J. of App.Phys., 61, pp.3925-3927, 1987. [2] S.R.H.Hoole,"Surface Impedance Behaviour of Iron at Comers and Slots", M.Sc. Dissertation, University of London, 1977. [3] T.H.Fawzi, M.T.Ahamed and P.E.Burke, "On the use of the Impedance Boundary Conditions in Eddy Current Problems (Invited)," IEEE Trans., MAG-21 ~~1834-1840, 1985. [4] I.D.Maye!goyz, F.M.Abde1-&der and F.P.Emad, "On Penetrations of Electromagnetic Fields in Nonlinear Conducting Ferromagnetic Media", J . of App. Phys., 55, pp.618-629,1984. S.Ratnajeevan H.Hoole and CJ.Carpenter,"Surface Impedance Models for comers and Slots", IEEE Trans.,MAG-21, pp. 1841-1843, 1985. [@ S.Ratnajeevan H.Hoole, Konrad W e e k and N.Ralnasunm G.Hoole. "The Natural Finite Element Formulation of the Impedance Boundary Condition in Shielding Structures", J. of App. Phys., 63,pp. 3022-3024. April 1988. M.V.K.Chari,"Finite Element Solution of the Eddy Current Problem in Magnetic Strtuctures", IEEE Trans., PAS-93,pp.62-72, 1974. [8] Ahmed, M. R., Burke, P. E. and Lavers, J. D., "Singularity and Comer Effects in Boundary Element Model for a Short, Linearly Magnetic, Conducting Cylinder," J . of App. Phys., 63, pp. 3016-3018. April 15, 1988. [9] Schneider, J. M. and Salon, S. J., "A Boundary Integral Formulation of the Eddy Current Problem," IEEE Trans., MAG-16, pp. 1086-1088, 1980. [lo] Salon, S. J. and Schneider, J. M.,"A Hybrid Finite Element-Boundary Integral Formulation of the Eddy Current Problem", IEEE Trans., MAG-18, pp. 461-466, 1982. [ l l ] Abramovitz, M. and Stegun, I. (Eds.), "Handbook of MathematicalFunctions. Graphs and Mathematical Tables", US.Dept. of Commerce, Natl. Bureau of Standards, App. Math. series 55. Washington, D. C., 1964. [12] Lean, Meng H., "Dual Simple-layer Source Formulation for the Two-dimesnional Eddy Current and Skin Effect Problems", J. App. Phys., 51, pp. 3844-3846, 1985. [13] S. Ratnajeevan H. Hoole, "AnIntroduction to the Computer Aided Analysis and Design of ElectromagneticDevices",Elsevier, New York, 1988. [14] Lean, M.H., and Wexler, A., "Accurate Numerical Integration of Singular Boundary Element Kemals over Boundaries with Curvature," Int. J. Num. Meth. Eng., 21, pp. 211-228, 1985.
[a
2. Raising the order of the approximation does not necessarily increase
the matrix size. For zeroth order. with an n-sided polygonal boundary, there are n mid-side nodes. So also for first order. with n-vertex nodes. But accuracy is higher for fmt order and therefore we must attempt to use fmt order. When going born first to second order, of course. the matrix size will be raised as in ffite elements. 3. Reducing the problem domain using symmetry does not give the best e Nor does it enhance ~ccuracy.Fig. 8 shows reduction in work as in f ~ t elements. two ways in which symmetry may be used. In Fig. 8a,as in finite elements, we may reduce the domain. In finite elements, this would roughly halve the matrix size. In boundary elements, assuming an axa square region, the matrix size goes from 4ax4a to 3ax3a. But, on account of this, a Neumann boundary (the symmetry line) is intrcducedand, if zeroth order A is used, the accuracy goes down compared to the solution of the whole problem which has a Dirichlet boundary all round. The altemative, as in Fig. 8b, is to use symmetry to map variables. That is, using the whole boundary, the elements on symmetric parts are made to have the same unknown values. This reduces the matrix size to 2ax2a. a considerable difference considering that the matrices are full. Fig. 9 gives the solution of the same problem as in Fig. 7, using this approach. This technique is used to avoid the Neumann condition which resulted in poor plots. On the other hand, with finite elements. this approach will increase the work since twice as many elements have to be gone through and mapping employed, only to arrive at the same matrix as would result from halving the geometry.While going from zeroth order A to fmt order A in boundary elements does not increase the matrix size, zeroth order A has the advantage that many students and even scientists, can understand it enough b implement it.
ACKNOWLEDGEMENTS: This project was supported in part by the Harvey Mudd College/Southem California Edison Center for Excellence in Electrical Systems.
