Advanced Method of Moments Based on Iterative Equation System Solvers

Transcription

1 Advanced Method of Moments Based on Iterative Equation System Solvers G. Burger, H.-D. Briins and H. Singer Technical University Hamburg-Harburg Department of Theoretical Electrical Engineering Hamburg, Germany Abstract: The treatment of field and scattering problems by means of method of moments (MOM) produces a system of linear equations which has to be solved. For large-scale structures the numerical effort for an LU decomposition of the system matrix becomes unacceptable. Iterative solvers can be an alternative but they often require a sophisticated preconditioner. A simple but powerful preconditioner can be derived from the physical meaning of the blocks in the impedance matrix and is applicable to a wide range of EMC problems. Using iterative methods can also be of significant advantage in another class of problems. INTRODUCTION The method of moments [5] is perhaps the most developed numerical technique to date for solving scattering and radiation problems in EMC. However the method leads to a system of linear equations that can be solved by an LU decomposition of the order 0(n3). For today s technical structures n = lo4 unknowns are not seldom and one gets an unacceptable solution time for an LU decomposition with such a high number of unknowns even on modern computers. In the past, iterative methods, e.g. conjugate gradients [9], have been tried to solve the equation system. But these early methods show poor and irregular convergence as the system matrix is mostly ill conditioned. In the last years more powerful iterative algorithms, e.g. GM- RES or QMR, have been developed with better convergence properties (for an overview, see e.g. [l]). The use of these methods for solving the equation system is at hands, but experience shows that without special techniques, good results (i.e. fast convergence and high accuracy) could only be achieved for relatively simple structures or in 2 dimensions. To get a sufficiently accurate solution for complicated EMC problems in 3 dimensions a high number of p iterations is needed. Each iteration step produces a numerical effort of the order 0(n2) which means that the complete effort will be 0(pn2). If p in comparison to n is not small there will be no benefit from the iterative method compared to the LU decomposition. There are two possibilities in order to reduce the numerical effort. On one hand the effort in the matrix-vector multiplication can be diminished. Lately some promising methods have been developed that follow this line. Some examples are the impedance matrix localization method (IML) [3], the fast multipole method (FMM) [4] or the multilevel matrix decomposition algorithm (MLMDA) [8]. These algorithms are able to reduce the required numerical effort in the matrix-vector multiplication considerably, for instance in the case of MLMDA from L3(n2) to O(n log2 n). On the other hand the overall numerical cost can also be kept low if the number of iterations p for the solution of the equation system (given certain accuracy) is small, say p 5 20 and independent from n. This can be achieved by the development of po- werful but numerically simple preconditioners. When developing such preconditioners one can sometimes adopt results from the numerical solutions of differential equations. A block Jacobi preconditioner can occasionally be motivated by domains consisting of different physical partitions. Geometrically sepa- rated parts of non-galvanically connected scatterers or multizone scatterers shall be mentioned as examples in the EMC area. In these cases the preconditioner can be easily and with little additional memory requirement derived from the given impedance matrix and a preconditioned version of the iterative solver can be used to substitute the common direct solver. Even if there is no simple preconditioner, iterative methods can still be of advantage. E.g. after a very costly preconditioner has been calculated, it can be used on and on for all steps in a frequency loop. We will first introduce the classes of EMC problems that can be well solved by iterative solvers, where one can achieve a substantial reduction of the solving time. The next section outlines the underlying theory, shows how the block structure of the impedance matrix results from the topology of the configuration and how to derive the block Jacobi preconditioner. Here is also explained how this preconditioner is usable in combination with impedance matrix transformation methods to treat shielding problems. Afterwards it will be demonstrated how an iterative solver can be used in a frequency loop. Some examples will illustrate the efficiency of the solver. We would like to point out the significance of the iterative methods in the computational electromagnetics using MOM. The interpretation of the /97/$

2 fields in each volume can be written as PEC structure El = El 11 J1 + El 22 J1 + Einc 1 > 0) Hl =?-llj ?llj Hint 1? (2b) E2 = E2J Hz =?12J2 2 2 PC) (24 The fields have to satisfy boundary conditions on the surfaces. Since S1 should be a perfect electric conductor the tangential component of the electric field strength has to vanish on S1, i.e. Figure 1: Dielectric body next to a perfect electric conductor in a plane wave field results of the examples in relation to the physical world is out of the scope of this paper, even though all examples are real life EMC problems. Finally, some conclusions and recommendations for future work will be presented. INTEGRAL EQUATIONS AND THEIR SOLUTIONS This section will demonstrate that the impedance matrix has a block structure, which results mainly from the topology of the examined configuration. This block structure gives rise to an easy but efficient preconditioner. nlxel(r)=o VrE&. W On the dielectric boundary Ss the tangential components of the electric and magnetic field strength have to be equal on both sides of S s, i.e. nzxel(r)=nzxez(r) VrESz, (3b) nzxhl(r)=nzxhz(r) VrESz. (3c) Here ni denotes the normal vector on surface Si. Utilizing the representation of the fields and the boundary condition one gets the following operator equation system Integral equation and MOM matrix Consider two surfaces Sr and Ss in free space. The open surface Si represents a perfect electric conductor whereas the closed surface Ss encircles a volume V2 which represents a dielectric body, characterized by its material constants EZ,,L~~, o? (Fig. 1). The configuration is excited by an incident field E2,, Hi and we are looking for the scattered field in free space (volume VI) and the transmitted field in the interior of S s (volume Vz). In order to calculate these fields a single layer current distribution Ji and a double layer current distribution Jf, Ji are assumed. In this notation the current Ji flows on surface Si in volume Vj. The electric and magnetic fields of these currents J{ are given by the integral operators 1 &; J{ = --jwp,, J; + I(, -J;)V kj Gj(R) ds, (14 The subscript tan indicates that only the tangential components of the fields are being considered. The method of moments [5] is employed to solve the operator system. For the purpose, the surfaces Si are divided into patches and on every two patches with a common edge a basis and a test function are defined. This produces n basis and n test functions. The particular surface currents J{ on surface Si in volume Vj are expanded to these n basis functions and will be inserted into the operator system. Subsequently the scalar product with each of the n test functions will be worked out. If one uses an ordering of the basis and test function according to the surfaces and volumes, this will transform the operator system into the following linear block equation system with $J< = z 2 J{ x VGj (R) ds (lb) s Si being the Greens function and Icj being the wave number in volume Vj. With these (equivalent electric) surface currents the where 2 is a n x n matrix and I, U are vectors of the length n. Note that the impedance matrix 2 has a block structure, which can be interpreted: The diagonal blocks describe the interaction of structure parts with itself (self interaction) and off-diagonal blocks describe the interaction between different structure parts. A zero block in the impedance matrix illustrates that there is no interaction between the corresponding parts. 237

3 Block Jacobi preconditioning We want to solve (5) by an iterative Krylov subspace method like GMRES or QMR. An important factor in whether or not an iterative method is successful is the preconditioning technique. Typically it involves replacing the original system (5) by the equivalent system ATIZI = AFlU. (6) The transformed system has the same solution as the original system 21 = U, but the spectral properties of its coefficient matrix M-i2 may be more favorable, as they cause a better convergence rate than that of the original system. In devising a preconditioner, one has to find a matrix M that approximates Z in some sense, and can be much easier to solved than 2. The block Jacobi preconditioner will fulfill all these tasks. The block Jacobi preconditioner can be derived by partitioning of the variables. To do that the index set N = 1,..., n is partitioned as N = lji Ni with mutually disjoint subsets Ni. The elements of M are given by 77&j = Zij if i, j are in the same subset, 0 otherwise. From the block structure of the impedance matrix, a natural choice for the partitioning over the different physical domains is first, to order the unknowns in (5) according to the surfaces. Using this ordering scheme for the partitioning, one will get M=[! i $1. Second, the currents on Sa are also ordered by the volumes in which they radiate. If one now applies one more partitioning 2; 00 over the single volumes, the following preconditioner will result M= 0 Z,l Yz i 1 (7) Normally only one matrix-vector multiplication of a vector with M-l is needed within the iterative method. Instead of calculating Mm1 directly, it is advantageous to factorize M by an LU decomposition and then to perform a forward and backward substitution, provided M-l is applied to a vector. As the single blocks in M according to (7) are decoupled and can be factorized independently, an LU decomposition of M is very simple. Consequently the numerical effort for a decomposition of M is reduced substantially compared to the decomposition of 2. To interprete the numerical process physically, only that part of the impedance matrix, which describes the interaction of a structure part with itself, is used in the preconditioner. Such a preconditioner technique can often be applied successfully in the numerical treatment of differential equations with multiple physical domains. It will be demonstrated that this technique is also applicable for those structured matrices from the MOM. The described preconditioner technique can be expanded in a straight forward manner to configurations which contain more than two non-galvanically coupled substructures or configurations with many different physical domains (multizone scatterers). The efficiency increases with growing number of domains. The block Jacobi preconditioning method itself has the advantage of not requiring any additional storage, as the diagonal blocks are already existing in the impedance matrix, moreover they can be easily implemented and adopted for parallel computation. Matrix transformation methods Geometrically thin but electrically thick penetrable material layers (for instance radoms, plastic casings, vaporized metal coats etc.), which influence the radiation coupling, can often be found in technical configurations. If the field distribution inside a layer is not of interest and if the thickness of the layer is very small in comparison with the dimensions of the coated object, it is possible to consider the layer by generalized boundary conditions for a dielectric body. The treatment of shielding problems is an important application: the shielded space is considered to be a dielectric (air), of which the limited surface is covered by a shielding material (see shielded enclosure example below). In a previous work the possibility to consider the layer in a given equation ZI = U subsequently by transformations of the original impedance matrix 2 and the right hand side U has been shown (for details see [2]). These transformations combine the blocks in the impedance matrix in a certain way but prevent the separation of the currents over the surfaces and volumes, i.e. the partitioning of the variables is unchanged, only the elements of some blocks are modified. Proceeding from that it is possible to apply the preconditioner technique described above to the new equation system. E.g. if the surface 5 s of the dielectric body in Fig. 1 is coated, the subsequent transformations will alter (5) to z,l z? 0 1: Z,l - AY; Z; - AY; BY,2 -BY; -BY; Z; + AY; I[1 I; = 122 with A, B being diagonal matrices. The elements aii, bii of A and B depend on the material constants of the layer and the geometry of the coated body. The preconditioner matrix for this equation system will look as shown in the preceding subsection z,l 0 0 M= 0 Z;-AY,1 0 [ 0 0 Zz +AY,2 1 The executed transformations do generally not influence the quality of the preconditioner M. 238

4 FURTHER APPLICATION POSSIBILITIES Use of preconditioners in iterative methods naturally results in an additional numerical effort, as the preconditioner has to be prepared and applied within the iteration. The preparation phase is most expensive when applying the block Jacobi preconditioner, but it has been shown that the initial costs will pay off during the iterations. The initial costs are also justifiable, if the preconditioner can be used for multiple linear systems repeatedly, e.g. in steps of a frequency loop. Within these steps each time an equation system Z(f)l(f) = U(f) has to be set up and solved. If it is now possible to set up a preconditioner M that is independent of the frequency, its utilization is worthwhile even if its initial costs are very high. One possibility is to use the LU decomposition of the impedance matrix at the frequency fmin as a preconditioner for all other frequencies [6]. But the preconditioner loses its ability to approximate the impedance matrix with increasing distance from the starting frequency, which results in the necessity to more iterations, and the final overall effort (matrix-vector multiplication with impedance matrix and preconditioner) is hardly profitable compared to that of the LU decomposition. In this case an improvement can be gained by using a new LU decomposition at a higher frequency as a preconditioner for all further frequency steps. This process repeats itself when the quality of this last preconditioner becomes unacceptable, namely below a predefined fixed value. Consequently the iteration steps through the frequency domain of the above algorithm are the following: 0. Set f = fmin 1. Compute ZI = U at f 2. Do an LU decomp. of Z and set 111-l = Z-l 3. Increment f 4. While f < frnaz Compute ZI = U at f Solve equation system iteratively using M-l Increment f If no. of iterations exceeds limit goto 1 This algorithm is very suitable if the (not necessarily equidistant) step width is not too high, otherwise there is the probability that the preconditioner loses its quality already for the next frequency. If these requirements are fulfilled, the solving time in a frequency loop can be reduced significantly by this method (see example below). It should also be mentioned that variants of this algorithm can be applied at other parameter studies as well. It can be used for example to examine the effect of different layers of an antenna or the different conductivity for the same geometry. In the algorithm shown above the frequency then has to be adjusted for the respectively examined parameter. EXAMPLES The method was implemented in the field computation program CONCEPT [7] to demonstrate the efficiency of the preconditioner. This program uses a direct solver (a special version of the LU decomposition, which partly takes advantage of zero blocks in the impedance matrix) for solving the equation system. In the following, the arithmetic times of the direct method are compared to those of the iterative methods in conjunction with the preconditioning time. Model of a human head next to a handy Fields and specific absorption rates (SAR) expected in a human body when exposed to radiations produced by a cellular phone is investigated by means of a model. The model used is depicted in Fig. 2. The handy is modeled as a perfect electric conductor and the human model is made up of a dielectric body. The excitation is given by an antenna with power feed of 1 W at 900 MHz. The whole problem produces 2489 unknowns: 425 unknowns for the handy with antenna and 1032 unknowns each for the inner and outer currents of the dielectric. The model is (concerning its topology) equivalent to the one in Fig. 1 and therefore produces an equation system as in (5). Consequently the preconditioner M contains three diagonal blocks with 425, 1032 and 1032 unknowns and the initial cost in the set-up of the preconditioner is to solve these three decoupled matrix systems. On an IBM RS it takes s. The iterative solver (GMRES) requires 10 iterations to get an accurate solution (norm of residual is less then 10e4) in 12.4 s, so one gets a solution in s whereas the direct solver needs s for the LU decomposition of the impedance matrix. It can easily be seen that one has a speedup of 6.4 by this. Shielded enclosure The configuration according to Fig. 3 was the subject of another numerical investigation, The depicted model is placed in free space and consists of an outer completely closed shielded enclosure (2.5 m x 2 m x 4 m) containing much smaller cubes of edges 0.5 m. The internal cubes are bounded by highly conducting sheets and are connected by means of two wires. The walls of the chamber are formed by plane sheets of relatively lowconductive isotropic material ( S/m, et = 3.7) 1.3 mm thick. A special feature of the enclosure is that all edges consist of metal strips of 5 cm wide (also 1.3 mm thickness, high conductivity). A perfect connection between high and low conducting enclosure surface parts is assumed. This example makes use of the above-mentioned method in order to take into account the conducting walls of the shielded enclosure. The internal structure of the shielded enclosure is depicted as a dielectric (air) and the corresponding impedance matrix is set up. Afterwards the layers are considered by transformations with the separation of inner and outer currents being preserved. In contrary to the example above the currents are partitioned into four index sets: currents that flow on the inner 239

5 structure, the inner and outer currents of the shielded enclosure and the currents flowing on the metal strips at the edges of the chamber. Hence the impedance matrix with 5348 unknowns is transformed into four diagonal blocks. One of them has 600 unknowns (from internal structure), one has 300 unknowns (from metal strips), and two have 2224 unknowns (internal and external currents on the walls). These four diagonal blocks will be used as the preconditioner. The calculation of the preconditioner needs 1372 s, whereas the calculation of the iterative solver (GMRES) needs 15 iterations which were completed in 257.4s. The direct solver needs s and therefore one gets a speedup of 6.2. Frequency loop The following example will demonstrate the advantage of an iterative solver even when the initial costs of the preconditioner are very high: We have a model of a VW Golf with a top antenna (Fig. 4). The effect of the field will be investigated in a frequency range from 2 MHz up to 100 MHz with an equidistant step width of 2 MHz. Figure 2: Cellular phone next to a model of a human head Figure 4: Model of a VW Golf As described above, in the first place a LU decomposition of the impedance matrix (1789 unknowns) is calculated and used as a preconditioner for the following frequencies. When the number of iterations exceeds the limit of 10, a new decomposition is calculated for the next step and used as the new preconditioner, and so forth. Within these 50 iteration steps from 2 MHz to 100 MHz steps a new decomposition had to be calculated three times, as the number of iterations exceeded the limit of the previous step. The overall solution time amounted to 3155s. If the direct solver is used in each step, we get a solution time of s, meaning a speed-up gain of 5.5. CONCLUSION Figure 3: Completely closed chamber with low conducting walls and internal structure It has been shown that the iterative process can be used to solve the MOM system of equations, as far as an appropriate preconditioner exists. The block Jacobi preconditioner provides a very simple and efficient method. Different physical domains pro- 24.o

6 duce a block structure in the impedance matrix and the blocks used in the preconditioner have the same size as those in the impedance matrix. These classes of problems appear quite often in the area of EMC (see examples above). The efficiency of the process increases with the growing number of different domains. Even if there is only one domain, the iteration process can still somewhat reduce the solution time involved in the frequency loop. The initial costs of the preconditioner might involve factorization of possibly large amount of blocks during the setup phase. It should be attempted in the future to reduce the numerical effort required for preconditioners with big block sizes. An appropriate preconditioner should also be developed for systems where only one domain (e.g. one PEC structure) is present. Lastly, in order to deal with large scale problems more efficiently, the process should be further integrated with numerical methods that would reduce the amount of the computations for matrix-vector multiplications (e.g. the MLMDA). REFERENCES [l] R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, Ch. Romine, and H.van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM Publications, [2] G. Burger, H-D. Brtins, H. Singer, Simulation of thin layers in the method of moments, in Proc. of the 11th Int. Zurich Symp. on EMC, paper 64K3,1995. [3] EX. Canning, Improved impedance matrix localization method IEEE Trans. Antennas and Propagat., vol. 41, no. 5, pp , May [4] R. Coifman, V. Rohklin, and S. Wandzura, The fast multipole methods for the wave equation: A pedestrian prescription, IEEE Antennas and Propagat. Mag., vol. 35, pp. 7-12, June [5] R.F. Harrington, Field Computation by Moment Methods, New York: IEEE Press, reprint [6] G. Hoyler and R. Unbehauen, An efficient algorithm for the treatment of multiple frequencies with the method of moments, in Proc. of the Int. Symp. on Electromagnetic Compatibility EMC 96 Roma, paper J-3, [7] Th. Mader, Berechnung elektromagnetischer Felderscheinungen in abschnittsweise homogenen Medien mit Obeq%ichenstromsimulation, Doct. Thesis, TU Hamburg-Harburg, [8] E. Michielssen, and A. Boag, A multilevel matrix decomposition algorithm for analyzing scattering from large structures, IEEE Trans. Antennas and Propagat., vol. 44, no. 8, pp , Aug [9] T.K. Sarkar, K.R. Siarkiewicz, and R.F. Stratton, Survey of numerical methods for solution of large systems of linear equations for electromagnetic field problems, IEEE Trans. Antennas and Propagat., vol. 29, no. 6, pp , Nov