Local Multilevel Fast Multipole Algorithm for 3D Electromagnetic Scattering

Transcription

1 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Local Multilevel Fast Multipole Algorithm for 3D Electromagnetic Scattering Jun Hu, Zaiping Nie, Lin Lei, and Jun Wang University of Electronic Science and Technology of China, China Abstract In this paper, a local multilevel fast multipole algorithm (LMLFMA) is proposed to further speed up the efficiency of MLFMA in conjugate gradient (CG) iteration. In the LMLFMA, only local interactions between the subscatters are taken into account. And, the interaction regions in iteration are varying adaptively with iterative current density. With decrease of iterative error, iterative current density tends to real one, the local interaction regions required are diminishing. When the iterative error is less than a critical iteration error, only the interaction between nearby regions at the finest level is considered. Numerical results show that the LMLFMA has good accuracy, and the efficiency can achieve over four times of the efficiency of traditional MLFMA. Introduction Based on multilevel fast multipole algorithm (MLFMA) [1], many large-scale problems can be solved efficiently now. Although MLFMA can solve scattering from object with very large electrical size, the storage and CPU time required are still expensive. To attain a faster solution, many fast algorithms based MLFMA have been developed [2-5]. Fast far field approximation (FAFFA) developed by Chew and Cui reduces the computation complexity of MLFMA greatly, is applied successfully for scattering and radiation [2]. A hybrid method based on MLFMA and adaptive ray propagation technique developed by us has attained faster solution than conventional MLFMA [3]. Although the efficiency of this method is lower than FAFFA, the accuracy is higher. Unfortunately, the two methods are not easily error controllable. A more accurate and better computational property is achieved by combination FAFFA and ray propagation technique in MLFMA [4]. Compared with MLFMA, MLFMA with partly approximation iteration (MLFMA-PAI) developed by us also attains a faster solution [5]. An obvious advantage of the MLFMA-PAI is its easy implementation. Numerical investigations of scattering from conducting sphere with different sizes show that the efficiency of MLFMA-PAI is about 2.6 times of the one of MLFMA. In MLFMA-PAI, a critical iteration error (CIE) is set in CG iteration, which is dependent on the contributions of nearby region and the convergence accuracy required. Before the iterative error reduces to the CIE, all the interactions between the subscatters are evaluated. After the iterative error has satisfied CIE, only the interactions from nearby groups at the finest level are considered. In this paper, we extend the MLFMA-PAI into a local multilevel fast multipole algorithm (LMLFMA). Different from MLFMA-PAI, only the local interactions between the subscatters are taken into account before the iterative error reduces to the CIE in this present method. It further reduces the computational complexity of MLFMA-PAI, and the accuracy is still good. Some typical numerical results given demonstrate the valid and efficiency of the LMLFMA. Multilevel Fast Multipole Algorithm (MLFMA) For sake of simplicity, only electrical field integral equation (EFIE) for 3-D conductive object is considered. It is given by ˆt G(r, r ) J(r )ds = 4πi S kη ˆt E i (r) (1) By applying addition theory, dyadic green function in (1) is rewritten as [1] G(r j, r i ) = ik d 2ˆk(I ik (r ˆkˆk)e jm r ) im α mm (ˆr mm 4π ˆk), r mm > r jm r im (2) where r j, r i is the vector coordinate of field point and source point position respectively, r m, r m coordinate of group center of the field points group, the source points group respectively. is the vector

2 746 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Applying appropriate basis function j i and testing function t j to e.qu.(1), linear algebraic equations is attained finally N A ji a i = b j, j = 1, 2, N (3) It can be evaluated by fast multipole method (FMM) as follows i=1 N A ji a i = m i=1 i G m A ji a i + ik 4π m α mm (ˆk ˆrmm ) d (ˆk 2ˆkVfmj ) i G m ) Vsm i (ˆk a i, j G m (4) In the right side of (4), the first term represents the interactions from nearby regions, the second term represents the interactions from non-nearby regions. Vsm i, α mm and V fmj represents the aggregation, translation, disaggregation term respectively, detailed formula is shown in [6]. The equ.(4) is often rewritten as another representation of matrix-vector multiplication, A a = A near a + U t T V a (5) where the matrix A near, V, T, U is sparse, represents the interaction matrix of nearby region, the matrix of aggragation, translation, disaggregation respectively. MLFMA is the multilevel extension of FMM, the matrix-vector multiplication is implemented in a multilevel multistage fashion, written as A a = A near a + U t NL 1 NL T NL V NL a + U t i T i V i a (6) where V i, T i, U i represents the matrix of aggragation, translation, disaggregation at the ith level respectively, NL is the total level number. In MLFMA, V i, U i (i < NL) is computed by interpolation and anterpolation technique. For N unknowns, the computational complexity and storage requirement are O(NlogN). More details are shown in [6]. Local Multilevel Fast Multipole Algorithm (LMLFMA) In conventional MLFMA, all interactions between nearby regions and non-nearby regions are considered. But in fact, only the interactions between local regions are dominant, especially when the unknown current tends to the real one. Based on this, the matrix-vector multiplication in CG iteration can be evaluated approximately by A a A near a + NL i=2 i=l c U t i T i V i a (7) Where L c is the coarsest level (L c 2), determined by the iterative error. For the case of NL 6, two useful empirical formula about Lc are chosen as follows NL 4 err > err1 L c = NL 2, err2 < err err1 (8) NL err err2 or NL 2 err > err1 L c = NL 1, err2 < err err1 NL err err2 where err is the iterative error, err1, err2 is the error threshold in LMLFMA. Obviously, equ.(8) has better accuracy than equ.(9). After err reaches the CIE, only the interactions from nearby groups at the finest level are considered. (9)

3 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Although the coarsest level is also set in FAFFA-MLFMA, the present method is different from the former. Firstly, the L c increases adaptively with the iterative error decreases. Secondly, no the interactions between far regions are evaluated. Obviously, the LMLFMA requires much less computational complexity than FAFFA- MLFMA. On the other hand, the implementation of LMLFMA is straight. Numerical Results Three typical numerical results are given to demonstrate the validity and efficiency of LMLFMA. The results calculated by MLFMA-PAI are given for comparison. All results are evaluated on DEC workstation XP1000 with 1Gb memory. Because the accuracy and efficiency of MLFMA-PAI have been validated in [5], it is not repeated here. In MLFMA-PAI, tol (convergence error) and CIE (critical iterative error) is set as 0.01 and 0.05 respectively. In LMLFMA, err1 is set as 0.6, err2 is set as 0.1. All errors are in the meaning of relative residual error. Figure 1: Bistatic Normalized RCS of conducting sphere by MLFMA-PAI and LMLFMA. Figure 2: Bistatic RCS of conducting cube by MLFMA, MLFMA-PAI and LMLFMA, VV-pol. The first example is bistatic normalized RCS results of conducting sphere with ka = 50.0, shown in Fig. 1. For sake of clear curves, the result of Mie series is not plotted. The number of unknowns is and the total level number NL = 6. Two results calculated by LMLFMA with different L c are given. In LMLFMA1 and LMLFMA2, L c is determined by equ.(8) and (9) respectively. It is shown that the results by LMLFMA1 agree well with the one by MLFMA-PAI. The results calculated by LMLFMA2 agree not well as the one by LMLFMA1, but the CPU time for iteration is only half of the one in LMLFMA1. The RMS error of MLFMA- PAI, LMLFMA1, LMLFMA2 results is 0.57dB, 0.9dB, 2.3dB respectively. This example indicates that L c had better be set as 2 in the initial iteration in order to attain accurate results. The CPU time for iteration in MLFMA-PAI is 9693 seconds, the one in LMLFMA1 and LMLFMA2 is only 5685, 2808 seconds, it will be seconds if traditional MLFMA is applied. So for this example, the efficiency of CG-MLFMA is speed up by factor of 4.7 by LMLFMA, while it still attains a reasonable accuracy.

4 748 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August Fig. 2 shows the bistatic RCS of 3D conducting cube, which size is 18 wavelength. The results by MLFMA and MLFMA-PAI are also given for comparison. The number of unknowns is , NL = 6. In this example, L c is determined by equ.(8). We can see these results agree well with each other, but the CPU time for iteration in MLFMA-PAI is 5215 seconds, the one in LMLFMA is only 1837 seconds. Figure 3: Monostatic RCS of cone-sphere with a gap by MLFMA-PAI and LMLFMA. Figure 4: Geometrical demonstration of cone-sphere with a gap (unit: mm) Finally, the monostatic vertical polarized RCS of cone-sphere with a gap is given to illustrate the efficiency of LMLFMA. The RCS results and its geometrical structure are shown in Fig. 3, Fig. 4 respectively. The L c is chosen as the same in the second example. The frequency is 9GHz. The nmuber of unknowns is To calculate 181 points RCS, the CPU time for iteration in MLFMA-PAI is seconds, the one in LMLFMA is only seconds. It is seen that the two results agree well with each other, also agree with the measurement. The above examples illustrate clearly the present method is especially suitable to solve scattering from 3D object with large electric sizes. Conclusion Compared with previous fast algorithms based MLFMA such as MLFMA-PAI, FAFFA, ARPMLFMA, the present method is a more efficient method. It can further speed up the efficiency of MLFMA greatly and does not sacrifice the accuracy when the error threshold err1, err2 are chosen appropriately. Some electrically large problems have been solved successfully by this method. Further research on the choice of err1, err2 in LMLFMA is our future work. Although the strict formula about L c is not available for a given object, the LMLFMA is still a promising method for large scale problems. Acknowledgement This work was supported by NSFC (No ), Research Funding (No DZ0212).

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