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1675 IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, N0.2, MARCH 1992 A Three-Dimensional infinite Element for Modeling Open-Boundary Field Problems S. Gratkowski and M. Zi6lkowski Electrical Engineering Department Technical University of Szczecin 70-3 11 Szczecin, Poland Abslmcl - In this paper a three dimensional infinite element is considered which has four nodes and is compatiblewith conven- tional cuboids. It is supposed that the infinite element is being developed for exterior potential problems governed by Laplace's equation. The element is formulated in such a simple way that closed-form expressions for the element matrix are obtained. 1. INTRODUCTION The finite element method is widely used for the solution of electromagneticfield problems. Applications of standard finite elements are usually restricted to finite domains. This paper reports the development of a finite element computer program to solve three dimensional open boundary electromagneticfield problems governed by Laplace's equation. Various methods of analysis for the open boundary problems have been investigated [l-91. Most of them refer to two dimensional problems. The methods may be classified into following types: combination of boundary element and finite element method, matching to analyti- cal solution, mappings, "ballooning," infinite elements. Unfortunately, very often it is not quite obvious to implement these methods into existing classical finite element programs. A very simple and efficient infinite element for two dimensional problems was proposed by Pissanetzky [6]. In the present paper the idea has-been generalized to three dimensions. We present a three dimensional infinite element which has four nodes and is compatible with conventional cuboids. Any decaying function can be used to approximate the solution in the infinite elements and we give explicitly the element matrix for the Laplace equation when the decay function is Urn. The formulation of the element is so simple that closed-form expressions for the element matrix are obtained. Two numerical examples are given. Manuscript received July 7, 1991. 11. THE INFINITE ELEMENT In this method, the whole region is divided into an interior region R, and an exterior region Re. The region Ri of interest where loads, sources and inhomogeneities exist., and the infinite region Re are discretized by ordinary cuboids and infinite elements respectively. The infinite element forms a frustum of a rectangular pyramid as shown in Fig.1. The element is attached to an appropriate three dimen- sional finite element. The elements are compatible over the rectangular contact area "1-2-3-4," where they touch each other. Four edges of the infinite element extending to infinity lie on lines radiating from the origin. d Fig. 1. The infinite element. 0018-9464/92$03.00 0 1992 JEEE

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Page 1: A three-dimensional infinite element for modeling open-boundary field problems

1675 IEEE TRANSACTIONS ON MAGNETICS, VOL. 28, N0.2, MARCH 1992

A Three-Dimensional infinite Element for Modeling Open-Boundary Field Problems

S. Gratkowski and M. Zi6lkowski Electrical Engineering Department Technical University of Szczecin

70-3 11 Szczecin, Poland

Abslmcl - In this paper a three dimensional infinite element is considered which has four nodes and is compatible with conven- tional cuboids. It is supposed that the infinite element is being developed for exterior potential problems governed by Laplace's equation. The element is formulated in such a simple way that closed-form expressions for the element matrix are obtained.

1. INTRODUCTION

The finite element method is widely used for the solution of electromagnetic field problems. Applications of standard finite elements are usually restricted to finite domains. This paper reports the development of a finite element computer program to solve three dimensional open boundary electromagnetic field problems governed by Laplace's equation.

Various methods of analysis for the open boundary problems have been investigated [l-91. Most of them refer to two dimensional problems. The methods may be classified into following types: combination of boundary element and finite element method, matching to analyti- cal solution, mappings, "ballooning," infinite elements. Unfortunately, very often it is not quite obvious to implement these methods into existing classical finite element programs.

A very simple and efficient infinite element for two dimensional problems was proposed by Pissanetzky [6] . In the present paper the idea has-been generalized to three dimensions. We present a three dimensional infinite element which has four nodes and is compatible with conventional cuboids. Any decaying function can be used to approximate the solution in the infinite elements and we give explicitly the element matrix for the Laplace equation when the decay function is Urn. The formulation of the element is so simple that closed-form expressions for the element matrix are obtained. Two numerical examples are given.

Manuscript received July 7, 1991.

11. THE INFINITE ELEMENT

In this method, the whole region is divided into an interior region R, and an exterior region Re. The region Ri of interest where loads, sources and inhomogeneities exist., and the infinite region Re are discretized by ordinary cuboids and infinite elements respectively. The infinite element forms a frustum of a rectangular pyramid as shown in Fig.1. The element is attached to an appropriate three dimen- sional finite element. The elements are compatible over the rectangular contact area "1-2-3-4," where they touch each other. Four edges of the infinite element extending to infinity lie on lines radiating from the origin.

d Fig. 1. The infinite element.

0018-9464/92$03.00 0 1992 JEEE

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For convenience of analysis, we introduce an auxiliary rectangular system ( 5 , ~ ) . The interpolated function V has values VI, V2, V3, V, at nodes "1," "2," "3," and "4," respectively, and varies bilinearly on the rectangular 11 1-2-3-4 : 11

The function V in the infinite element has the form, in terms of the spherical coordinate system (r,e,cp) (x=rsinecoscp, y=rsinOsincp, z=rcose):

where the gradients are taken with respect to x,y,z, and [SIe is of order 4x4. Using (2) and (3), the closed-form expressions for the element matrix can be easily ob- tained:

S:, = a [ &4- 12n( b4+c11) + y + -3(2112b2+241~2+b4+~11-3b~)] ,

SA = a[~l14-12n(b4-cl,)+y+ -3(211ib2 + 2 4 , ~ 2+b4 -d, +3b c)] ,

SG = a [ P114+ 12n(b&+cl,) + y +

- 3(2112b + 2 4 , ~ -bl, -cl, - 3b c)] ,

(tti+l)(VV j+')Vj a C N i V i S i = a[~l14+12n(b4-cl,)+y+ i- 1 1- 1

-3(21,2b2+241~2-b~+~11+3b~)] ,

local coordinates of the node i; f(r) is , \ I

the decay function, and d/(sinecoscp) is the distance from the origin to any point on 1q-2-3-4,11

SA = a[(4n2-2n-2)1,~+3b2&,-6c2112+ -3(2n-l)b~+3d2(4,-2112)] ,

Ni-shape functions.

For the case when f(r)=l/r", n21 and taking into account relations:

SG = a[(2n2-4n-1)1,&-3b2~,-3c21,,+ -9bc-3d2( 4,+112)] ,

SA = a[(4n2-2n-2)11~-6b241+3c2112+ (3) ll = (Y-b)/l, = (d.tgcp-b)/l, Y

5 = @ - C Y 4 = (d ~tgelcoScp-cII~ y

-3(2n- l)clf-3d2(24, -112)] ,

the shaDe functions N: can be exuressed in terms of the 1

rectangular coordinates (x,y,z). The coefficients of the element matrix for the Laplace equation in 3-D are given by:

S& = a[(4n2-2n-2)l1~-6b24,+3c2ll2+ +3(2n-l)~l,-3d~(2~,-1,~)] ,

m (b+l+/d (c+4)X/d Si = a[(2n2-4n-1)1,~-3b2~,-3c21,,+ Si: = J 1 J VNiVN, dzdydx (4) + 9 b ~ - 3 d ~ ( 4 ~ + 1 ~ ~ ) ] ,

d (b-l$/d (C-l+/d

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1677

The solutions are:

(5 ) S i - a[(4n2-2n-2)11~+3b2~l-6c2112+

+3(2n-l)b~+3d2(~,-21,2)] ,

r,-b2+C2+d2, 2 ll2=lI/4, J1=Vl1 ,

It can be shown that the matrix [SIe has the desired properties of symmetry and positive -definiteness.

111. NUMERICAL EXAMPLES

In order to test our three dimensional infinite ele- ment, we have solved two problems for which the exact solutions exist. The test problems are as follows:

Test case A. V(x,y,z)=?

v 2 v - 0 , -00 <x< 00) -00 <y< 00) 0 sz< 00,

0.5 0.5 A.

and the asymptotic behavior at a large distance are: V, - l/r2 and V, - l/r. The integrals (8) and (9) can be expressed by elementary functions.

The test problems were solved numerically using finite and infinite elements. The problems were also solved on the truncated mesh for comparison (Dirichlet and Neumann boundary conditions). For symmetry only the positive octant of three dimensional space is used with the homogeneous Neumann boundary conditions at x=O and y=O. The domain Ri here is a cube l x l x l . The region Ri was modeled by 1000 cubic elements employ- ing linear shape functions with a total of 1331 nodes.

For both of the test cases, we compared exact and numerical solutions on the basis of the standard global error criteria:

N (Vi-veiI2

i- 1 1 IxkO.5, lyl<0.5 (6)

v(x,y,z-o)- 0 Ixbo.5, lybo.5 6,- N c v:i

Test case B. V(x,y,z)=? i- 1

V2V-0, --do <x< 00, -00 < Y < 00, 0 <z< 00, In the above, the subscripts "i" and 'lei" refer to the computed values and the exact values obtained from (8) and (9). The summations are carried out over all nodes.

<Io Ixko.5, lylso.5 (7) The numerical results for the two cases are shown in 0 lxbo.5, lybO.5 Table I and in Figs.2 and 3.

Figs. 2 and 3 show distributions of the absolute errors A=IV-Vexactl on the plane x=y within the cube l x l x l for the solution of the test problems by different methods. It can be seen from the Table I and Figs. 2 and 3 that only our infinite elements give sufficient accuracy. The use of the truncated mesh with V=O or aV/an=O on the truncated boundary yields very inaccurate results.

E, a. = const > 0.

5

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1678

A

TABLE I ERRORS FOR THE SOLUTION OF TEST PROBLEMS

6, % 62 6, % 6, 6, % 6, 13.6 0.0256 26.4 0.0496 86.5 0.1629

Infinite Truncated Mesh

Dirichlet I Neumann Elements I I I I I I I

B I 0.6 10.0006 I 71.4 I 0.0662 I - I -

IV. CONCLUSIONS

A very simple three dimensional infinite element was proposed. The formulation of the element is so simple that closed-form expressions for the element matrix were obtained. The element can easily be incorporated into existing finite element codes and its use does not destroy the symmetry and bandedness of resulting system of equations.

'.a 0'

I *O

Fig. 3. Problem B: Absolute error on the plane x=y; (a) Dirichlet b.c., (b) infinite elements.

REFERENCES

[l] C. Antunes, E. M. Freeman, D. Lowther and P. Silvester, "A static ballooning technique for 2-D open boundary problems," J.Appl.Phys., ~01.53, pp. 8360- 8362, November 1982.

[2] R. W. Thatcher, "On the finite element method for unbounded regions," SIAM J. Num.Anal., vol. 15, pp. 466-477, June 1978.

[3] J. Lm and Z. J. Cendes, "Transfinite elements: A highly efficient procedure for modeling open field problems," J.Appl.Phys., v01.61, pp.

[4] 0. C. Zienkiewin, C. Emson and P. Bettes, "A novel boundary infinite element," 1nt.J.Num.Meth. Eng., vol. 19, pp. 393-404, 1983.

[5] Y. Saito, K. Takahashi and S. Hayano, "Finite element solution of open boundary magnetic field problems," IEEE Trans.Magn., vol. 23, pp. 3569-3571, September 1987.

[6] S. Pissanekky, "A Simple infinite element," COMPEL, vol. 3,

[7] S. Gratkowski, "More on a Simple Infinite Element," COMPEL, vol.5,

[SI T. Nakata, N. Takahashi, K. Fujiwara and M. Sakaguchi, "3-D open boundary magnetic field analysis using infinite element based on hybrid finite element method," IEEE Trans. Mag., vol. 26, pp. 368-370, March 1990.

[9] Jhhoff , G.Meunier and J.Sabonnadiere, "Finite element modeling of open boundary problems," IEEE Trans. Magn., vol. 26, pp. 588-591, March 1990.

3913- 3915, April 1987.

pp.107-114, 1984.

pp.191-194, 1986.

Fig. 2. Problem A Absolute error on the plane x=y; (a) Neumann b.c., (b) Dirichlet b.c., (c) infinite elements.