IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 5. SEPTEMBER 1991

EFFICIENT HYBRID FINITE ELEMENT - BOUNDARY ELEMENT METHOD FOR 3-DIMENSIONAL OPEN-BOUNDARY FIELD PROBLEMS

Chang-Hoi Ahn, Bong-Sik Jeong, and Soo-Young Lee Korea Advanced Institute of Science and Technology

P.O. Box 150 Chongryangni, Seoul 130-650, Korea

Abstract - Highly storage efficient choices of boundary meshes are devised for hybrid finite element - boundary ele- ment analysis in 3-dimensions, and a comptitationally- efficient algorithm to solve the resulting matrix equations is developed. The boundary mesh makes use of cylindrical symmetry to produce block circulant boundary element ma- trices with proper choices of basis functions, and results in up to 64-fold reduction in matrix storage. Also the block cir- culant nature of the boundary element matrices provides a numerically efficient algorithm for solution of the coupled matrix equations. To demonstrate its efficiency this hybrid scheme is applied to a three-dimensional magneto-static tield problem, and proves storage and computational efficiency. With this efficiency the hybrid scheme becomes more advan- tageous than pure finite element method with approximate boundary conditions, and allows to solve large 3-dimensional open boundary field problems in small workstations.

IN TR 0 D U CT IO N

Numerical solutions of open-boundary electromagnetic field problems have drawn special attentions 111. The widely used method of moment (MOM) based on integral equation generates a full matrix which may require prohibitly large computer memory for 3-dimensional problems 121. The finite element method (FEM) and finite difference method (FDM) based on differential equations generate sparse matrices, but requires extra nodes and approximate boundary conditions on fictitious boundary for open boundary value problems 131. Although the boundary element method (BEM), based on t h e boundary integral equation with FEM- like discretization, requires nodes on t h e material boundary only and handles unbounded problems easily, its application is restricted to homogeneous region [4].

Hybrid finite element . boundary element method (FE- BE) forniulation is also introduced and its accuracy was demonstrated [5 ,6] . In this hybrid scheme the FEM is used in the inhomogeneous region, while BEM is applied to the external unbounded region. This hybrid scheme has merits of both methods. The FEM is used in inhomogeneous region to produce a sparse positive-definite matrix, while BEM represents unbounded region accurately with small number of unknowns. However storage requiremelrts of the full BEM matrices prevent the hybrid scheme from popular- ity, especially for large-sized 3-dimensional problems. Also it requires special algorithm to solve the coupled equations efficiently. Recently we chose the FE-BE interface ;is B circle for 2-dimensional problem, and made the BEM ni:itrices cir- culmt for efficient storage 171. In this paper the method is further extended to 3-dimensional problems with the help of block circulant matrices, and a computationally efficient algo- rithm for the coupled equations is also introduced.

n

Fig.1 FEM and BEM regions for hybrid FE-BE solution of open boundary field problems

STORAGE-EFFICIENT HYBRID METHOD

In the hybrid FE-BE method one divides the analysis area into two regions, i.e. closed inhomogeneous region R I and external unbounded homogeneous region R as shown in Fig.]. Inside of the R I FEM is used with Neumann boundary conditions qb on the boundary To to obtain

A U + B qb = f , ( 1 )

where U is the unknown vector, and A and B are N x N and N x M sparse matrices, respectively. Here N and M are number of unknowns in R I (including the boundary To) and number of Neumann boundary conditions on the To, respectively. For the external region R 2 BEM is used to obtain

C U' + D qb = g , ( 2) where ub is boundary portion of the U, and C and D are M x M full matrices. The f and g in Eqs. (1) and (2) denote source terms. Provided collocation scheme were used, the matrix elements become

D,, = I G(lr - r, 1 ) Q,(r) d r . ( 4)

where w and @ denote basis functions for U* and q*, respec- tively, and ci depends upon Hatness of mesh at node i. Eqs. (1) and (2) provide ( N + M ) linear equations for unknown N u i ' s and M yi's.

r,,

00189464/91$01.00 @ 1991 IEEE

4070 IEEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 5, SEF'EMBER 1991

L=8

1

Fig.2 Boundary mesh for 3-dimensional problems. ( a ) side view with meshes on one segment only, (b) top view with meshes on all 8 segments

For 3-dimensional problems the enormous storage requirements of the BEM matrices practically limit calculable number of unknowns. Suppose one divided each axis of a cube into 30 elements, the total number of boundary nodes becomes 30 x 30 x 6 = 5400 and about 200 MBytes of storage are required for 2 BEM matrices in single precision. To reduce the enormous storage requirements of the full BEM matrices one chooses the boundary mesh axi- symmetric with L identical segments as shown i n Fig. 2. Due to the Green's function involved the BEM matrices can be made block circulant, i.e.

where edge or facet basis functions are used and K edges or facets are assutned i n each segment. If i + n K o r , / + I L K exceeds M = K L , one can subtract M from the indiceh. The block circulant nat,ure of these matrices reduces storage requirements to 1 / L . One may further reduce the storage requirements by a half using top-down symmetry. However

axi-symmetric mesh on the FE-BE interface becomes worse i n the triangle qunlity ;It poles as L increases. Thus increas- ing L is ready to suffer computational inaccuracy caused by worse mesh qu:ility. We do not recommend to use L greater than 32, which gives 11.25" for the smallest mesh angle a t

the poles. Therefore, for the above mentioned cubic prob- lem, less t h a n 1 MBytes of memory are enough with 32 seg- ments by including the cube in a sphere or a finite cylinder. With predetinecl meshes on the axi-symmetric boundary the internal region can be tessellated using automatic mesh gen- erator such a s 3-d Delaunay triangulator. At this point the cubic boundary must be maintained throughout the tessella- tion.

COMPUTATIOR'-EFFIClENT HYBRID METHOD

Another serious drawback of the hybrid FE-BE method was lack of efficient algorithm to solve the matrix equations. Combining Eqs. ( 1 ) nnd ( 2 ) one may obtain a single matrix equation for unknown vector I U : q']''. However the resulting matrix is expressed as ;t sparse and positive-definite FE matrix mixed with a full and non-symmetric BE matrix. Generally BE matrix is not symmetric and so not positive- definite. Although the BE matrix may become symmetric by using Galerkin's scheme, there is no guarantee for positive- definiteness. Therefore, i t is no longer positive (semi- )definite, and computationally efficient iterative algorithm such as ICCG can not be used. Instead we eliminate q6 and obtain

A U = B D - ~ ( C U ~ - ~ ) + ~ . ( 6 )

To use ICCG we p u t BD-'CU' term in the right side of Eq.(3) and solve by iteration. Starting with initial guess of U, the right side of Eq.(3) is calculated and new U is obtained by ICCG. This process goes on u n t i l change of U is less than allowed tolerance. Only one incomplete decomposition is required for the ICCG.

At each iteration step BD-'Cub need be calculated. However the block circulant nature of the BEM matrices greatly reduces computation time of the D-' ;is well as BD-ICu', especially for L = 2" with a positive integer n 181. I n fact inversion time of the block circulant matrix is essentially devoted to the L inversion of K x K matrices, and proportional to LK whereas Gauss-Jordan and other direct inversion of the complete L K x L K matrix requests it time proportional to L 3 K K " . Also one may expect increased accu- racy due to that fact that numerical inversion of only K x K matrices are involved instead of L K x LK matrices. For the solution of matrix equation the gain of time is somewhat less than matrix inversion, but still is of order L 2 / 3 . It is worth mentioning t h a t inverse matrix of it block circulant matrix is also block circulant.

TEST RESULTS

To demonstrate its efficiency this hybrid scheme IS

applied to :I 3-dimensional magnetostatic field problem, and its results are compared with analytic solution. A hollow sphere made of permeable medium is located i n uniform magnetic field, and magnetic field perturbation from the sphere is calculated. For the boundary meshes 8, 16 and 32 identical segments are used, and their solution time is

IEFE TRANSACITONS ON MAGNETICS, VOL. 27, NO. 5. SEPTEMBER 1991

t

4 h t t t I

Fig.3 Hollow sphere in uniform magnetic field

Single matrix method

Double loop method

checked and summarized in Table 1. Both single matrix algorithm and the proposed double-loop iterative algorithm are compared, and the latter is proven to be much more computationally efficient, especially for large number of seg- ments. All the calculations are done on a SUN 3/ 1 I O works- tation with floating-point accelerator board.

L = 8 L = 1 6 L = 3 2

379 1,246 2,853

139 313 SI6

Table I Storage requirement and computation time

Storage requirements of BEM matrix D

L = 1 6 L =32

I 1 I I

Computation times [sec]

3.0

2.5

2.0

1.5

1.0

.5

4071

Bz [T I

- Analytic solution L = 8 L = 16 L = 32

- - - - - - - - -_-.-.-_

z 1.1

(4

Bz IT1

2

x 1.1

F i g 4 Calculated and analytic B, . (a) along z-axis, (b) along x-axis.

CONCLUSION

With proper choice of boundary meshes and iterative solution algorithm the hybrid FE-BE method can be made much more efficient in storage and computation time. This numerical efficiency allows u s to solve large-sized 3- dimensional electromagnetic field problems accurately on a small workstation.

In Fig.4 z-component of the calculated magnetic tield is compared with analytic solutions for L = 8, 16, and 32. Even for this simple problem large number of boundary nodes are necessary for accurate calculation, and both the '

storage efficiency and numerical efficiency of our :ilgorithm were essential in order to get accurate results on a small work station .

A cknowledgenient - This work wus supported by Koreu Science and Engineering Foundution.

REFERENCES

[ I ] C.R.I. Emson, "Methods for the solution of opeii- boundary electromagnetic- field pro bleni s," I EE Proc., vol. 135 Pt. A, pp.151-158, 1988.

4072 IEE TRANSACTIONS ON MAGNETICS, VOL. 27, NO. 5, SEPTEMBER 1991

[ 21 R.F. Harrington, Field Computation by Moment Merhoil, New York: Mucmillan, 1968.

131 0. Axelsson and V.A. Barker, Finire Elemenr Solution Boundary Value Problems, New York: Acadeniic Press, 1984.

141 S . Kagami and 1. Fukai, "Application of boundary ele- ment method to electromagnetic field problems," l E E E Trans. Microwave Theory Tech, vol. MTT-32, pp.455-461, 1984.

151 O.C. Zienkiewicz, D.W. Kelly, and P. Bettess, "The coupling of the finite element method and boundary ele- ment procedure," Inter. . I . Numer. Methods En#. . vol.] 1,

[6] S.J. Salon, "The hybrid finite element - boundary ele- ment method in electromagnetics," I E E E Truns. Mngn.,

[7] S.-Y. Lee, "Highly storage efficient hybrid finite element / boundary element method for electromagnetic scatter- ing," Ekcr . Leu. , vol. 25, pp.1273-1274, 1989.

[SI T. De Mazancourt and D. Gerlic, "The inverse of ;i

block-circulant matrix," l E E E Trans. A n r . Prop., v o l .

pp.355-375, 1977.

vol. MAG-21, pp.1829-1834, 1985.

AP-31, pp.808-810, 1983.