IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
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On the Static Loop Modes in the Marching-on-in-Time Solution of the Time-Domain Electric Field Integral Equation Yifei Shi, Member, IEEE, Hakan Bağcı, Member, IEEE, and Mingyu Lu, Senior Member, IEEE
Abstract—When marching-on-in-time (MOT) method is applied to solve the time-domain electric field integral equation, spurious internal resonant and static loop modes are always observed in the solution. The internal resonant modes have recently been studied by the authors; this letter investigates the static loop modes. Like internal resonant modes, static loop modes, in theory, should not be observed in the MOT solution since they do not satisfy the zero initial conditions; their appearance is attributed to numerical errors. It is discussed in this letter that the dependence of spurious static loop modes on numerical errors is substantially different from that of spurious internal resonant modes. More specifically, when Rao–Wilton–Glisson functions and Lagrange interpolation functions are used as spatial and temporal basis functions, respectively, errors due to space-time discretization have no discernible impact on spurious static loop modes. Numerical experiments indeed support this discussion and demonstrate that the numerical errors due to the approximate solution of the MOT matrix system have dominant impact on spurious static loop modes in the MOT solution. Index Terms—Electric field integral equation, marching-on-intime, numerical analysis, static loop modes, time domain.
I. INTRODUCTION
N
UMERICAL solutions of time-domain integral equation (TDIE) are often obtained using the marching-on-intime (MOT) scheme [2]. When MOT is applied in solving the time-domain electric field integral equation (TD-EFIE), spurious internal resonant and static loop modes are almost always observed in the solution [3]. Internal resonant modes can be eliminated from the solution space by linearly combining the EFIE with the magnetic field integral equation [3]. Also, techniques making use of Helmholtz decomposition [4] and/or Calderon identities [5] have been developed to eliminate static loop modes from the solution space of TD-EFIE. Manuscript received November 12, 2013; revised February 04, 2014; accepted February 08, 2014. Date of publication February 11, 2014; date of current version February 26, 2014. This work was supported in part by the National Science Foundation under Grant ECCS 1303142 and the Center for Uncertainty Quantification in Computational Science and Engineering at KAUST. Y. Shi and H. Bağcı are with the Division of Computer, Electrical, and Mathematical Sciences and Engineering (CEMSE), King Abdullah University of Science and Engineering (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail:
[email protected];
[email protected]). M. Lu is with the Department of Electrical and Computer Engineering, West Virginia University Institute of Technology, Montgomery, WV 25136 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LAWP.2014.2305716
One can argue that the aforementioned methods are useful but not necessary because neither internal resonant nor static loop modes satisfy the zero initial conditions enforced by the MOT scheme and, thus, are allowed in the solution. It has been conjectured in [3] that appearance of spurious internal resonant and static loop modes in the MOT solution is attributed to numerical errors. In [1], this conjecture has been verified for internal resonant modes via numerical experiments. In this letter, theoretical analysis and numerical experiments are carried out to verify this conjecture for static loop modes. In the MOT solution of TD-EFIE, the dependence of spurious static loop modes on numerical errors is substantially different from that of spurious internal resonant modes. The theoretical analysis presented in this letter shows that errors due to space-time discretizations and numerical integrations have no discernible impact on spurious static loop modes, when the TD-EFIE is discretized classically, i.e., when Rao–Wilton–Glisson (RWG) [6] functions and Lagrange interpolation [7] functions are used respectively as spatial and temporal basis functions to represent the electric current density induced on the surface of a perfect electrically conducting (PEC) scatterer. Thus, if the conjecture in [3] holds true, spurious static loop modes are expected to be very weak in the MOT solution if there are no other error sources. To verify this theoretical prediction, numerical experiments are designed in which numerical errors only stem from the space-time discretizations and numerical integrations. As expected, in these numerical experiments, the spurious static loop modes observed in the MOT solution are very weak; their magnitudes are only slightly above the level of machine precision. In another set of numerical experiments, an additional error source (i.e., other than those produced by the space-time discretizations and numerical integrations) is introduced: The accuracy of the iterative solution of the MOT matrix system at each time-step during time marching is adjusted. Indeed, these experiments clearly show that the magnitudes of spurious static loop modes are dominantly dictated by the accuracy of the MOT matrix system solution. In summary, the contributions of this work are twofold. 1) It clearly demonstrates that the spurious static loop modes are generated in the MOT solution of TD-EFIE because of numerical errors. 2) Meanwhile, it also shows that the dependence of spurious static loop modes on the numerical errors is substantially different from that of spurious internal resonant modes as reported in [1].
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II. MOT SOLUTION OF TD-EFIE AND STATIC LOOP MODES Let represent a closed PEC surface residing in the free space with permittivity , permeability , and speed of light . Surface is illuminated by an incident electric field . As a result of the illumination, electric current density is induced on , and it generates scattered field . Total electric field satisfies the boundary condition on , yielding the TD-EFIE
Fig. 1. Illustration of a static loop mode
composed of 11 RWG functions.
and represent the lengths of the edges where RWG functions are defined. The static loop mode belongs to the null space of (3) since the left-hand-side of (3) becomes zero when is used in the place of
(1) is the distance between the observation In (1), and source position vectors and on , is the outward normal direction of at , and and are del operators acting on and . TD-EFIE (1) is numerically solved using the “classical” MOT scheme. Classical MOT scheme expands the unknown current density in terms of RWG basis functions [6] in space and (shifted) Lagrange interpolation functions [7] in time (2) Inserting (2) into (1) and testing the resulting equation in space with , , at time yields
(3) Equation (3) represents a “discretized” TD-EFIE. In what follows, it is shown that static loop modes belong to the null space of this discretized TD-EFIE. Consider a loop composed of 11 RWG basis functions located on a part of (shown by arrows in Fig. 1). Note that each RWG function is defined at an edge and covers the two triangular patches that share that edge [6]. The static loop mode defined over these 11 RWG functions in Fig. 1 can be mathematically formulated as
(6) The first term on the left-hand-side of (6) is zero because Lagrange interpolation is exact when there is no temporal variation; in other words, (Condition 1). The (Consecond term is zero because dition 2), by noting that the divergence of RWG functions is a constant over any triangular patch and by making use of (5). This analysis clearly shows that static loop modes analytically belong to the null space of the classically discretized TD-EFIE. It should be emphasized here that internal resonant modes belong to the null space of the same discretized TD-EFIE only approximately. As a result, internal resonant and static loop modes have different behaviors in the MOT solution of TD-EFIE. To arrive at an MOT scheme, zero initial conditions are assumed, that is, and are assumed zero for and . With the assumption of zero initial conditions, the testing time takes values , with as the time-step size. Then, the discretized TD-EFIE in (3) is reorganized to the MOT format (7) Here, mension dimensions
and with , and , with
are vectors of di, , are matrices of
(4) Here,
,
, are constants that satisfy (5)
(8)
SHI et al.: STATIC LOOP MODES IN MOT SOLUTION OF TD-EFIE
At time-step , the MOT matrix system (7) is solved for the unknown vector . In theory, static loop modes should not be observed in the MOT solution of TD-EFIE, because these modes do not satisfy the zero initial conditions enforced by the MOT scheme for . It is conjectured that numerical errors built up during time marching may establish the necessary initial conditions and, in turn, introduce the spurious static loop modes in the MOT solution. It should also be noted here that an MOT solver projects numerical errors onto the solution through the incident field. In other words, numerical errors would stay zero if the incident field is zero. In the following, numerical errors of the MOT scheme described above are classified into two groups and investigated separately. Group A: These are all the possible numerical errors in (3). Group A includes errors due to the space-time discretization in (2) and due to numerical evaluation of the surface integrations in (3), and possible aliasing errors due to sampling the incident field in space and time. It is analytically shown from (6) that the first two error types have zero projection onto the static loop modes. Note that Conditions 1 and 2 [see (6)] can only be enforced at the machine precision level because of the numerical operations (regardless of the type of integration rule used). As shown by our numerical results, this has very insignificant effect on the amplitude of the static modes observed in the MOT solution. It should also be clear from (2) and (3) that all the errors in Group A are linear with respect to the incident field. Group B: These are all the possible numerical errors induced after (3) is reorganized to (7). Group B includes numerical errors due to solving the MOT matrix system (7) for . If a direct method such as Gaussian elimination or LU decomposition is used to solve the MOT matrix system (7), Group-B errors are linear with respect to the incident field. If an iterative solver such as GMRES or TFQMR is used, Group-B errors are nonlinear. The nonlinearities due to Euclidean norm (used to judge the convergence of the solution) and the thresholding operation (used to terminate the iterations) are weak as far as the solution is concerned. However, the error in the solution is strongly nonlinear, as verified by numerical experiments in Section III. From the discussion above, it can be concluded that the magnitude of Group-B errors dictates the amplitude of the spurious static loop modes. Consequently, if Group-B errors are suppressed using a direct method or an iterative method with a high accuracy threshold, the amplitudes of static loop modes are expected to be small. These conclusions are verified by the numerical experiments in Section III. III. NUMERICAL RESULTS In this section, the MOT solver described in [8] is used. The scatterer is a PEC sphere with radius 1 m and centered at the spatial origin. The surface current density on the sphere is discretized using RWG basis functions; the temporal basis function is the linear Lagrange interpolation function, and the time-step is 0.51 ns. The -component of the current density at , , is recorded. The excitation is a plane wave propagating toward -direction. Here, is the amplitude, and represents a modulated Gaussian pulse expressed as , where is the duration,
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Fig. 2. MOT results for three incident fields with Group-B errors disabled.
is the delay, and is the modulation frequency of the pulse. Three sets of parameters are shown in Fig. 2 for defined in (1). First, LU decomposition method is applied to solve the MOT matrix system in (7). As discussed in Section II, the use of LU decomposition reduces the level of Group-B errors to approximately , which is the machine precision level associated with our double-precision implementation. In Fig. 2, computed for all three excitations using the MOT solver is plotted after normalization with its peak value in time. Spurious static loop modes appearing in the solution have relative magnitudes , , and , for the three excitations, respectively. The appearance of these modes with very small amplitudes in the solution can be explained by the fact that Conditions 1 and 2 [see (6)] can only be enforced at the machine precision level due to the numerical operations involved. These results also demonstrate that Group-A errors are linear with respect to the incident field. The static component of incident field (iii) is about 10 and 100 times that of the incident fields (ii) and (i), respectively. As expected from the linearity of Group A and as clearly shown in Fig. 2, the magnitude of static loop mode in corresponding to the incident field (iii) is approximately 10 and 100 times that corresponding to incident fields (ii) and (i), respectively. Before demonstrating the effect of Group-B errors on static loop modes, a numerical experiment is conducted to show that Group B is nonlinear when GMRES is used in solving the MOT matrix system. In this experiment, three matrix systems , , and are solved, where , , and . First, these matrix systems are solved using the LU decomposition method; let the solutions be denoted by , , and , respectively. Then, the matrix systems are solved using the GMRES method with as the convergence threshold; let the solutions be denoted by , , and , respectively. Taking LU solutions as the reference, three errors , , and are defined. Error vectors , , and
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IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, VOL. 13, 2014
errors: When the GMRES threshold is changed from to , the magnitude of spurious static loop mode grows by about 10 000 times. IV. CONCLUSION
Fig. 3. Numerical results to show that the error due to GMRES is nonlinear.
Appearance of spurious static loop modes in the MOT solution of TD-EFIE is investigated. It is theoretically shown that errors due to space-time discretization carried out using RWG and Lagrange interpolation functions has zero projection onto static loop modes. Numerical experiments demonstrate that errors due to the approximate solution of MOT matrix system using an iterative solver have dominant impact on the generation of spurious static loop modes in the MOT solution. It should be noted here that this letter focuses on the “integral form” of TD-EFIE as given in (1). Investigation of the “derivative form” of TD-EFIE, which is obtained by taking a time derivative of (1), is out of the scope of this letter. ACKNOWLEDGMENT The athors would like to thank the King Abdullah University of Science and Technology (KAUST) Supercomputing Laboratory (KSL) for providing the necessary computational resources. REFERENCES
Fig. 4. MOT results with respect to three Group-B errors.
all have 963 elements. In Fig. 3, 963 elements of and are plotted, and it is clearly shown that and are drastically different, indicating that the error due to GMRES must be nonlinear. In the last numerical experiment, Group-B errors are enabled, and their impact on spurious static loop modes is demonstrated. Incident field (i) is used in three simulations where the MOT matrix system in (7) is solved using different solvers in Fig. 4. obtained for these three simulations are plotted in Fig. 4. When GMRES solver (with or precision) is used, Group-B errors are nonlinear. As a result, corresponding spurious static loop modes are fairly strong despite the fact that incident field (i) has little static component. Moreover, the magnitude of spurious static loop modes is dictated by Group-B
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