A Novel Dual-Field Time-Domain Finite-Element ... - IEEE Xplore

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A Novel Dual-Field Time-Domain Finite-Element Domain-Decomposition Method for Computational Electromagnetics Zheng Lou, Student Member, IEEE, and Jian-Ming Jin, Fellow, IEEE

Abstract—A novel dual-field time-domain finite-element domain-decomposition method is presented for numerical simulation of electromagnetic phenomena. The method divides the computation domain into several non-overlapping subdomains and computes both the electric and magnetic fields in each subdomain by solving second-order vector wave equations. A leapfrog-like scheme is employed in the time marching to update the alternating electric and magnetic fields. Adjacent subdomains are related to each other by the equivalent surface currents on the subdomain interfaces, which are also updated in a similar fashion. The method requires minimum communications between the subdomains, thus, is highly suitable for parallel computations. The proposed domain decomposition method is particularly attractive for the numerical simulation of finite arrays and large-scale electromagnetic problems. Numerical stability and dispersion error analyses are conducted and several radiation examples are presented to demonstrate the accuracy and efficiency of the method. Index Terms—Domain decomposition (DD), finite-element method (FEM), radiation analysis, time-domain simulation.

I. INTRODUCTION

R

ECENT developments in the time-domain finite-element method (TDFEM) have made it a powerful and versatile numerical technique for simulating a variety of complicated electromagnetic problems [1]-[11]. Traditionally, the TDFEM can be formulated as either an explicit [12], [13] or an implicit [2] method. Most of the explicit TDFEM, together with the closely related finite-volume time-domain (FVTD) method [14], [15], are based on the dual formulation of the first-order Maxwell’s equations. Both electric field and magnetic field (or equivalently electric flux density and magnetic flux density ) are updated in the time marching. Usually, and are discretized on staggered grids in both space and time due to the discretization of the first-order derivatives in Maxwell’s equations. On the other hand, an implicit TDFEM is based on solving the second-order vector wave equations using Galerkin’s method. A major advantage of an implicit TDFEM is that it can be made unconditionally stable if the Newmark-beta method is used for temporal discretization [16]. Moreover, the numerical dispersion error associated with an implicit TDFEM can be systematically reduced by utilizing Manuscript received November 9, 2005; revised February 10, 2006. This work was supported by a grant from the Air Force Office of Scientific Research via the MURI Program under Contract FA9550-04-1-0326. The authors are with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2991 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2006.875922

higher-order vector basis functions [2]. Despite these advantages, the capability of an implicit TDFEM is limited by its relatively high computational complexity, especially in the simulation of large problems. Since a linear system of equations needs to be solved at each time step, the computational cost associated with an implicit TDFEM is usually considerably higher than that with the finite-difference time-domain (FDTD) or an explicit TDFEM for the same discretization density. Since the system matrix to be inverted is the same at each time step, one would ideally use a direct solver to factorize the system matrix before time marching and then reuse the factorization repeatedly at each time step to minimize the total computation time. For large problems, however, the use of a direct solver becomes impractical because the memory required to factorize a sparse matrix resulting from a FEM assembling scales as where stands for the dimension of the computation domain. Therefore, an implicit TDFEM often has to resort to iterative solvers for large problems to avoid excessive memory usage. An alternative approach to solving this problem is to divide the original computation domain into several smaller subdomains. With a reduced size, each smaller subdomain problem can then be factorized and solved using a sparse direct solver and the overall computational complexity can be reduced as compared to the original single-domain problem. To further reduce the computation time, the subdomain problems can be distributed on a massively parallel computing system and solved in parallel. Therefore, a robust and efficient domain decomposition method is of great interest for the TDFEM simulation of large-scale electromagnetic problems. The domain decomposition method itself is a broad research subject and has received much attention in the advent of parallel computation. In mathematics, domain decomposition is a general term referring to a numerical method intended for solving a partial differential equation (PDE) problem by dividing the original problem into smaller subproblems. Numerous domain decomposition methods based on different decomposition strategies have been designed and implemented [17]–[29]. One class of domain decomposition techniques are known as the Schwartz methods. Existing Schwartz methods can be further categorized into overlapping, additive, and multiplicative Schwartz methods [18]. The classical alternating Schwartz method is an iterative method which alternatingly solves several overlapping subdomain problems with inhomogenous Direchlet boundary conditions in each iteration [17]. An extension to the basic Schwartz algorithm is to employ a non-overlapping partitioning and replace the Direchlet boundary condition with a transmission boundary condition. Such a domain decomposition method has been applied first to optical control

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LOU AND JIN: NOVEL DFDD TDFE DOMAIN-DECOMPOSITION METHOD

problems [19] and then electromagnetic problems [20]. In the generalized Schwartz theory, domain decomposition can be interpreted as a Krylov space preconditioner [21], [22]. Many domain decomposition methods can be formulated and analyzed in this framework [20]. Another class of domain decomposition methods are the Schur complement methods (also known as the iterative substructuring methods) [23], [24]. In the Schur complement method, the internal elimination is carried out independently in each subdomain. This results in a reduced system relating the interface unknowns, which is usually solved by an iterative solver. Another method known as the finite-element tearing and interconnecting (FETI) method [25]–[29] is closely related to the Schur complement method. It uses Lagrange multipliers to enforce continuity conditions on the subdomain interface and to formulate the reduced global problem. The FETI method was originally proposed to solve large computational mechanics and acoustics problems [25]. It was recently extended to solve Maxwell’s equations in the frequency domain [27], [28] and time domain [29]. One disadvantage associated with the aformentioned domain decomposition methods is that all of them rely on some sort of iterative mechanism to obtain an approximate solution to the true solution. Therefore, the efficiency of such methods depends heavily on the convergence property of the iterations, which in turn depends on the shape and size of the subdomains and the total number of subdivisions [24], [26]. Moreover, solving a global interface problem, as required by some of the domain decomposition methods, may be undesirable from the viewpoint of parallelism. Solving such problems demands using a parallel iterative solver, which requires global communications between processors. This may impose a limitation on the overall scalability of the method despite that each subdomain problem can be solved independently. For time-domain (TD) simulations, these problems become even more severe when a linear system needs to be solved at each time step. Although the local matrices for subdomain problems can be factorized only once at the beginning of the computation, the global problem needs to be solved repeatedly at each time step. As a result, the iterative solver can easily become the bottleneck of the entire computation. Finally, the necessity of implementing a parallel iterative solver also increases the complexity of the computer program. In this paper, we propose a novel domain decomposition scheme, which is different from the two traditional classes of methods described above. Our formulation is based on the dual-field second-order vector wave equations for each subdomain. Both the electric and magnetic fields are solved on the same spatial grid but they are staggered in the time domain. That is, the electric field is solved at integer time indices and the magnetic field is solved at half integer time indices, as is typically done in the FDTD method. Such an arrangement results in a leapfrog time-marching scheme to update the equivalent surface currents on the subdomain interfaces. With the equivalent surface currents known on the interfaces, the electric and magnetic fields in each subdomain can be updated independently. The method is most efficient if the number of subdomains is chosen such that each subdoamin problem is small enough to be solved by a sparse direct solver. By doing so, the system matrix for each subdomain is pre-factorized

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Fig. 1. Domain decomposition with two subdomains.

and stored in the memory before time marching. At each time step, the subdomain problems are solved efficiently using the local pre-factorized matrices. The communication between two adjacent subdomains is realized by exchanging the equivalent surface currents on the interface. The CPU time for such an operation is usually insignificant compared to the solving time for the subdomain problems. Therefore, the proposed method is highly scalable for parallel computations. In the following sections, we first formally present the formulation of the dual-field domain decomposition (DFDD) TDFEM. Next, the stability condition for the method is derived in the stability analysis. It is followed by the numerical dispersion analysis and the scalability tests. Finally, several radiation examples are presented as applications of the proposed method. II. FORMULATION The proposed TD finite-element domain decomposition method is described in this section. A computation domain consisting of two subdomains will be considered first for the illustrative purpose. Arbitrary partitioning of the computation domain can be handled in the same fashion. The formulation is followed by the stability and numerical dispersion analyses. Finally, the performance of the method on serial and parallel computers is studied. A. Domain Decomposition Fig. 1 shows a general computation domain denoted by and bounded by a metallic surface and an impedance sur. An arbitrary artificial boundary breaks into two face and . Let us denote the electric and magsubdomains netic fields in subdomain as and . The can be written in second-order wave equation in subdomain the following dual format:

(1) (2) denotes the interior excitation in subdomain where is boundary condition for the metallic surface

. The

(3) (4)

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and the boundary condition for the impedance surface

is

(5) (6)

Testing (1) and (2) with vector basis function weak-form wave equations

yields the

(12)

where (13) (14) (7)

. After discretization are the equivalent surface currents on in space, (11) and (12) can be written in the matrix form as

(15)

(8)

On , we enforce the impedance boundary condition (5) and (6). Setting and , (5) and (6) become the first-order absorbing boundary conditions (ABC). , we can make use of Maxwell’s equations On the interface

(9) (10)

(16) where the matrix entries are given by (17) (18)

and rewrite (7) and (8) as

(19) (20) (21) (22) (23) (11)

(24)

LOU AND JIN: NOVEL DFDD TDFE DOMAIN-DECOMPOSITION METHOD

where vectors are given by

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and the right-hand side

(25) (26) (27) (28) (29) Here we should point out that we have assigned degrees of freedom to both sides of the interface . As a result, the equivaare not uniquely determined on lent surface currents and . To avoid this ambiguity, we evaluate the surface integrals in , i.e., the exterior side of the interface. Such a (27)–(29) on choice allows each subdomain to evaluate its surface equivalent currents using the electromagnetic fields from its neighbors. Next we discretize (15) and (16) in the time domain using the Newmark-beta method. To construct a valid time-marching scheme, we discretize the electric field on integer time indices and the magnetic field on half integer time . This results in an indices updating scheme in a half-step leapfrog fashion

is upsecond substep, the magnetic field at time index dated using the magnetic field at previous time indices and and the magnetic current on the interface at previous and ). The above time-marching scheme time indices ( can be implemented as the following. at time index : For each subdomain Step 1) Set Step 2) Step 3) Step 4) Step 5)

Calculate from using (27) Update using (30) Set Calculate and from using (28) and (29) using (31). Step 6) Update stands for the neighbors of subdomain In Steps 1 and 4, . In the case of two subdomains, and . These two steps amount to the exchanging of surface currents on the interface between two adjacent subdomains. Note that the continuity of tangential electromagnetic fields across the interface is implied, but not explicitly enforced, in Steps 1 and 4. This is a distinct feature of the proposed domain decomposition scheme. To expedite the execution time at each time step, the system matrices for each subdomain are pre-factorized using a sparse direct solver and stored in memory. Note that the system matrices for the electric and magnetic fields are generally different. Therefore, they need to be factorized and stored separately. The above formulation can be easily generalized to multiple subdomains. Suppose that the computation domain is divided non-overlapping subdomains and each subdomain into has neighbors and thus interfaces. Then for each subdomain the above formulation still remains valid except that (27), (28), and (29) now need to be modified as

(32)

(33)

(30)

(34) Note that importing the electric current at the subdomain interface to update the electric field in the subdomain and vice versa has been used in the parallel FDTD [30], although the specific formulation here is very different because of the use of non-staggered grids. B. Stability Analysis (31) As represented in (30) and (31), each time step in the time marching consists of two substeps. In the first substep, the elecis updated using the electric field at tric field at time index ) and the electric current on previous time indices ( and the interface at previous time indices and . In the

It is well-known that a TDFEM system employing the Newmark-beta method with is unconditionally stable [2], meaning the time step can be chosen independent of the spatial in (30) and (31) discretization. Accordingly, we set to be in our stability analysis. Further, we ignore the lossy and excitation terms in (30) and (31), since these terms will not affect the stability criteria. We again consider the simplest case where the

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computation domain contains only two subdomains and . By examing (30) and (31), it is easily seen that the electric field and the magnetic field in are coupled together through in the interface . The coupling equations for and can be written as

Defining a combined field vector and combined matrices

(47) (35)

we can write (45) and (46) in a compact format as

(48)

(36) or where matrices

and

are given by (49) (37) (38)

and where and as eigenvalues of further reduce (49) to

and

. By denoting the , respectively, we can

Let us consider a time-harmonic plane wave solution with (50) (39) (40) The discretized fields can be written as (41) (42)

It is easily seen from (37) and (38) that and satisfy . Therefore, is an anti-symmetric matrix. Given that is a positive definite and symmetric matrix and is a are purely real and symmetric matrix, it can be proven that are purely imaginary. In order for to be real for arbitrary and , (50) requires that (51)

Using (41) and (42), (35) and (36) can be reduced to

(52) Therefore the stability condition is determined by

(43)

(44) or

(45)

(46)

(53) stands for the spectral radius of . The above analwhere ysis indicates that the updating system represented by (35) and (36) is conditionally stable and the stability condition is deter. It mined by the spectral property of the matrix is worth noting that both and are highly sparse matrices whose entries are zero except for those associated with the de. Thus, matrices and grees of freedom on the interface are also highly sparse matrices. By using the properties of the FEM matrices, it can be rigorously proven that matrix depends only on the spatial discretization immediately next to the interface. For example, if a three-dimensional (3-D) tetrahedral mesh is employed, the stability condition depends only on the tetrahedrons immediately connected to the interface. Note

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TABLE I COMPARISON OF THE STABILITY CONDITIONS FOR A UNIFORM AND NONUNIFORM 1-D MESH

Fig. 2. Nodal configuration for the two elements sharing an interface.

that such a stability condition is less stringent than that of the FDTD method or the explicit TDFEM where the time step is determined by the smallest geometry/discretization in the entire computation domain. In the antenna simulation, for example, if the small feeding structures dictate using a highly-dense mesh around the feed regions, then the computation domain can be partitioned such that the feed regions are enclosed in each individual subdomains and are separated from all the interfaces. In this way, the stability condition will be determined by the sparse portion of the mesh instead of the smallest mesh size in the feed region. To validate the stability condition that we have derived, we simulate a 1-D wave propagation problem using the proposed domain decomposition method. A 0.32 m-long computation domain is divided into two subdomains. In the first case, we emm throughout ploy a uniform mesh with mesh size the entire computation domain. In the second case, we employ where decreases a non-uniform mesh with linearly from 1.0 at the interface to 0.06 at the two ends of the computation domain. In both cases, linear basis functions are and the calused on each cell. The spectral radius of matrix culated stability threshold are shown in Table I. The stability condition for the uniform and non-uniform discretizations are found to be almost identical to each other since both of them have the same mesh size at the interface. It is also observed that both predictions are consistent with numerical simulation results. depends on For 3-D problems, the spectral property of both the shape and dimensions of the element and the properties of the basis functions employed. For an equilateral tetrahedron with a side length of , the calculated stability condition is when the lowest-order basis functions are used. Through proper scaling, the stability condition for an arbitrary interface mesh can be estimated. A similar estimation can be made when higher-order basis functions are used. C. Numerical Dispersion Analysis In the proposed domain decomposition method, the decomposition of the original computation domain into several individual subdomains is made possible by enforcing appropriate inhomogenous Neumann boundary conditions (9) and (10) on the subdomain interfaces. In the continuous domain, such a decomposition is exact, i.e., (11) and (12) are equivalent to (1) and (2). In the discretized domain, however, the subdomain problems will no longer match the original problem exactly due to discretization errors. This is different from many other domain decomposition methods such as the FETI, which attempts to solve the same discretized system as the original single-domain

problem. To study the accuracy of the proposed domain decomposition method, we investigate the numerical dispersion error associated with the subdomain interfaces and compare it with the numerical dispersion error of the TDFEM itself. The numerical dispersion error associated with a FEM system in general depends on the shape and topology of the specific mesh employed during the discretization procedure. Instead of providing a general 2-D and 3-D numerical dispersion analysis, we will focus on the numerical dispersion analysis in 1-D. This will not only simplify the analysis significantly but also provide more insight into the dispersion properties of the methods. Let us consider the interface between two adjacent subdoamins and the two elements immediately neighbor to the interface, as shown in Fig. 2. For the two local elements (assuming linear basis functions), (35) and (36) are reduced to

(54)

(55) where

(56) Next we follow the same approach as we used in the stability analysis by assuming a plane wave solution with (57) (58) In the discretized domain (59) (60)

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where and are temporal and spatial cell sizes, respectively. Substituting (59) and (60) into (54) and (55) gives

In the case where temporal discretization error dominates the and the dispersion error dispersion error, we have for the subdomain interface is

(68) (61)

and the dispersion error for the standard TDFEM is

(69)

(62) Combining (61) and (62) together yields

The above analysis indicates that the numerical dispersion errors for both the subdomain interfaces and standard FEM are second order with respect to and . Furthermore, in the case where the spatial discretization error dominates the dispersion error, the level of the dispersion error on the interface is the same as that for the standard FEM grid. To confirm the statement, we compute the numerical solution of a 1-D wave propagation problem and plot its RMS error against the analytical solution in Fig. 3. The RMS error is defined as

(63)

(70)

Solving (63) for numerical wavenumber , we have (64), shown at the bottom of the page. Following the same procedure, we can derive the dispersion relationship for the standard TDFEM employing the same discretization and basis functions, which is

where is the analytical solution. In the simulation, we . The number of subdomains varies from choose 1 to 16 with corresponds to the standard FEM. All the error curves in Fig. 3 exhibit the same slop with respect to and the levels for all the curves are comparable to one another. Note that increasing the number of domains does not significantly increase the numerical dispersion error given a fixed spatial discretization. The magnitude difference in the field solutions on two sides case) is shown in Fig. 4. As we of the interface (for have pointed out, the subdomain solutions may disagree on the interface since the continuity condition across the interface is not explicitly enforced. Fig. 4 shows that the discontinuity on the interface is about 40 dB below the true solution when a typdiscretization is employed. It also indicates that ical the discontinuity level can be reduced in a second-order manner with respect to the element size . The same behavior is obtained for cases where late-time are non-trivial. The conclusions we drew from the 1-D analysis can be extended to the 2-D and 3-D cases. Here we perform a convergence test for a 3-D wave propagation problem in an empty coaxial waveguide. The length of the coaxial waveguide is , the wavelength at the center frequency. An artificial interface placed at the middle of the waveguide breaks the computation

(65) We are interested in the dispersion error defined as . To further simplify (64) and (65), . This corresponds we first consider the case where to the case where the spatial discretization error dominates the dispersion error. Under this condition, the dispersion error for the subdomain interface is

(66) and the dispersion error for the standard TDFEM is

(67)

(64)

LOU AND JIN: NOVEL DFDD TDFE DOMAIN-DECOMPOSITION METHOD

Fig. 3. RMS error for the 1-D wave propagation problem.

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Fig. 5. RMS error for the 3-D wave propagation in an empty coaxial waveguide. Waveguide length:  , outer and inner radii: 0:2 and 0:087 . Solid lines: standard TDFEM. Dashed lines: domain decomposition method with M = 2.

other for a fixed discretization and increasing the number of subdomains does not significantly increase the total dispersion error. From these observations, we conclude that the proposed dual-field domain-decomposition (DFDD) method in general preserves the accuracy of the TDFEM although it may produce slightly different numerical results from the single-domain analysis. D. Serial and Parallel Performance

Fig. 4. Magnitude difference in the field solutions on two sides of the interface.

domain into two subdomains. Both subdomains are discretized with tetrahedral elements and the hierarchical vector basis functions [31] of varying orders are used in the convergence test. The RMS errors for the two-subdomain calculation is plotted as dashed lines in Fig. 5. For comparison, the RMS error associated with the standard TDFEM calculation is also plotted as solid lines in the same figure. Again, we found that both methods exhibit similar convergence behavior with respect to mesh size and element order. The level of dispersion errors for the two calculations are also comparable with each other. It should be noted that due to the presence of discretization error the numerical results obtained by using the domain docomposition method will be in general different from that by using the standard TDFEM. The difference may be interpreted as the difference resulting from two different discretizations of the same continuous-domain problem. As we have demonstrated in the numerical dispersion analysis, this difference will eventually vanish as both methods employing a denser discretization or higher element orders. Furthermore, we have found that the dispersion error for the two methods are comparable to each

In the serial implementation of the domain decomposition method, each subdomain problem is solved sequentially on a single processor. In the pre-processing stage, the system matrices for each subdomain are assembled and pre-factorized one by one and stored in the memory. In the time-marching stage, the electromagnetic fields in each domain are also updated in a sequential manner during each time step. As a result, the total CPU time is the linear summation of the CPU time spent on each subdomain and similarly the peak memory usage is the linear summation of the memory allocated for each subdomain problem. Since for most of the sparse direct solvers the complexity for matrix factorization increases with much , the total factorization time will be reduced faster than by breaking a large single-domain problem into several smaller subdomain problems. Similarly, the peak memory usage will also be reduced by means of domain decomposition. We test the performance of the serial domain decomposition code on an Intel-Itanium II 1.5 GHz processor. For matrix factorization, we use the sparse direct solver in the SGI’s scientific computing software library (SCSL), which turns out to be a highly efficient sparse direct solver. The computation domain box bounded by perfectly under test is a electric conducting (PEC) surfaces. We divide the computation domains where , and domain evenly into record the corresponding memory usage and CPU time. The matrix factorization time, which only needs to be done once, and the CPU time spent in each time step, which needs to be repeated

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Fig. 6. Peak memory usage for the serial implementation of the domain decomposition method.

during the time marching, are recorded separately. Fig. 6 shows that the peak memory usage drops quickly with an increasing number of subdomains until the size of the subdomain problems falls into the region where the scaling performance of the direct solver is nearly linear. The existence of such a linear region can be easily observed by varying the size of the original , single-domain problem. For a larger problem while for the memory usage starts to level off at a larger , the memory usage starts a smaller problem to level off at a smaller , leaving the size of the subdomain problems relatively constant at the turning point. A similar pattern can be observed in Fig. 7 where the total CPU time for mafor various system sizes. trix factorization is plotted versus Fig. 8 shows the CPU time spent per each time step during the time marching, which is dominated by the matrix solving time. As expected, the summation of the matrix solving time for individual subdomains remains relatively constant for a given system size. As we mentioned earlier, one of the major advantages of the proposed domain decomposition method is that it can be easily implemented on a massively parallel computing systems. In the parallel implementation of the method, the subdomain problems are distributed among multiple processors. Each processor handles its own local subdomain problems individually. No global problem needs to be solved. The only communication required by the method is the exchange of surface equivalent currents on the interface between adjacent subdomains. Compared to the matrix solving time, the overhead for communication is usually negligible. This results in a highly efficient parallelized computer code that can be used to solve large electromagnetic problems. The scalability tests are conducted on two computer systems with different architectures: a SGI Altix 350 system (global shared-memory architecture) and an Apple Xserve system (cluster distributed-memory architecture). In both tests, each processor is assigned to handle only one subdomain problem. The size of the computation domain is the same as in the serial

Fig. 7. CPU time during the matrix factorization for the serial implementation of the domain decomposition method.

Fig. 8. CPU time per step during the time marching for the serial implementation of the domain decomposition method.

test. The spatial discretization results in 284 777 unknowns in the single-domain calculation. The CPU speedup (compared to the single-domain) versus the number of processors is shown in Fig. 9. On both computer systems, the factorization speedup is greater than the number of processors (efficiency greater than 100%). This is due to the non-linear scaling property of the sparse direct solver as we have observed in Fig. 7. The speedup for the time marching increases linearly with the number of processors, which indicates the overhead for exchanging the surface currents on the interfaces is indeed small compared to the matrix solving time in the subdomains. When tested on 64 processors [Fig. 9(b)], the time-marching speedup starts to deviate from the linear curve (dashed-line) as a result of increased overhead on inter-processor communications. III. NUMERICAL EXAMPLES In this section, we consider several radiation problems to demonstrate the application of the DFDD TDFEM. The wave-

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TABLE II MEMORY REQUIREMENT AND CPU TIME FOR THE MONOPOLE AND MICROSTRIP PATCH ANTENNAS

examples in this section are calculated on a SGI Altix 350 system that uses Intel Itanium II 1.5 GHz processors. A. Monopole Antenna First, we consider a monopole antenna radiating on a finite ground plane. The 15 mm-long monopole antenna is formed by extending the central conductor of a coaxial cable, which provides the TEM excitation. The monopole antenna is placed at the center of a 80 mm 80 mm square ground plane. The computation domain is a bounded by a 100 mm 100 mm 66 mm ABC box. A standard FEM discretization results in 170,809 unknowns. Using the proposed DFDD method, the computation domain is evenly divided into 9 subdomains, with the central subdmain contains the feed and the monopole antenna. As shown in Table II, with the number of unknowns reduced to about 22 000 for each subdomain, the peak memory usage is reduced from 1.4 GB to 9 130 MB. The factorization time and time-marching time per time step are also reduced from 280.2 s and 1.6 s to 9 16.0 s and 9 0.15 s, respectively. The calculated reflection coefficient at the coaxial port over the frequency range 2.5–7.5 GHz is plotted in Fig. 10(b). The reflection coefficient calculated by using the standard TDFEM is also plotted for comparison. The two curves almost overlay on each other, showing good agreement between the two methods. B. Microstrip Patch Array

Fig. 9. CPU speedup for the parallel implementation of the domain decomposition method tested on (a) SGI Altix 350 system and (b) Apple Xserve cluster. The dashed-line in both figures stands for a linear speedup.

guide feed to the radiation systems, such as coaxial cables, are modeled accurately by enforcing the waveguide port boundary conditions on the waveguide ports [10]. The open free space is terminated using the first-order ABC for efficiency. For better absorbing performance, perfectly matched layers (PML) can be used. It can be shown that the DFDD method can incorporate the PML regions without modifying its formulation. All the

One important application of the proposed domain decomposition method is the analysis of finite antenna array structures. Each unit cell in an array structure is naturally a subdomain, which is repeated in one or two dimensions to form the entire array. Small feeding structures are usually confined inside the unit cell, making it possible to place the interfaces away from the feeding structures. Here we use the domain decomposition method to simulate a 2 2 microstip patch array fed by coaxial cables. The geometry of the patch array is depicted in Fig. 11. The array is excited from port I (lower-left) and matched at port II–IV. In the domain decomposition method, the original computation domain is conveniently partitioned into four subpadomains, each containing one unit cell. The calculated rameter is plotted in Fig. 12. Again, we obtain good agreement between the domain decomposition method and the standard TDFEM. The performance of the DFDD is shown in Table II. C. Vlasov Antenna In the last example, we calculate the return loss from a coaxial-fed circular waveguide antenna, which is also referred to as the Vlasov antenna. The configuration of this antenna

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Fig. 12. S

parameter for the 2 by 2 microstrip patch array.

Fig. 10. Monopole antenna over a finite ground plane. (a) Front view. (b) Calculated reflection coefficient at the coaxial port. Fig. 13. Return loss at the coaxial port for the Vlasov antenna.

2

Fig. 11. 2 2 microstrip patch array fed by coaxial cables. (a) Top view. (b) = 1:5 mm,  = Side view. Coaxial feed dimensions: r = 0:48 mm, r 1:86.

is described in [11] and [32]. The antenna consists of a 50feed whose inner and outer radii are 0.21 and 0.49 cm, respectively. This coaxial feed is connected to a segment of a coaxial waveguide whose cross section is gradually increased

and transformed into a circular waveguide, which excites the mode in the waveguide. The radiating aperture is formed by cutting the circular waveguide at an angle of . The Vlasov antenna represents an example where the geometry of the antenna imposes a difficulty in the numerical simulation. The radius of the circular waveguide is about 30 and 50 times larger than the outer and inner radii of the coaxial feed, respectively. The spatial discretization results in an extremely nonuniform mesh, which in turn yields an ill-conditioned linear system to solve. The situation becomes more severe when a Newmark-beta method is used in an implicit TDFEM scheme since the system matrix becomes the combination of the mass and stiffness matrices. This means a larger iteration number for convergence is required if an iterative solver is used to solve the linear system. On the other hand, if an explicit TDFEM is used for the simulation, the maximum time step would be undesirably limited by the smallest elements at the tip of the coaxial feed. In the domain decomposition method, however, we are able to partition the computation domain into several smaller subdomains, each with a relatively more uniform mesh. This enables us to use a direct solver instead of an iterative solver to solve the subdomain problems efficiently. Moreover,

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TABLE III MEMORY REQUIREMENT AND CPU TIME FOR THE VLASOV ANTENNA

the stability condition becomes more relaxed compared to the explicit TDFEM since the time step is now determined by the element size at the interface. In our simulation, the original single-domain problem has approximately 1 million unknowns. We divide the computation domain into 2, 4, 8, and 16 subdomains. In the last case, the size of the subdomain problem is reduced to about 70 000 unknowns. The calculated return loss of obtained from the eight-subdomain calculation is shown in Fig. 13. As a reference, the same parameter calculated by VOLMAX, a hybrid FETD/FDTD code [33], is also plotted in Fig. 13. The two results agree with each other very well except for at the higher end of the frequency range. The peak memory usage, factorization and time-marching time, together with the are given in number of subdomain unknowns for varying Table III. IV. CONCLUSION A novel DFDD TDFEM is presented for the numerical simulation of electromagnetic problems. The method is based on the dual formulation of the second-order vector wave equations. Both the electric and magnetic fields are solved in each subdomain. The electric field is solved at integer time indices and magnetic field is solved at half integer time indices. Such an arrangement enables the development of a leapfrog time-marching scheme to update the equivalent surface currents on the subdomain interfaces. With the equivalent surface currents known on the interfaces, the electric and magnetic fields in each subdomain can be updated independently. The proposed DFDD TDFEM has several advantages. First, the method is a highly efficient time-domain method intended for solving large electromagnetic problem. Since the subdomain problems are usually much smaller than the original single-domain problem, we are able to use a sparse direct solver to solve the subdomain problems. Instead of solving the linear system repeatedly at each time step, we are now able to reuse the matrix factorizations and significantly reduce the computation time during the time-marching procedure. Moreover, the proposed method is highly parallizable. Unlike other time-domain decomposition methods, the proposed method does not require solving a global problem at each time step. Instead, the method only requires solving local subdomain problems and exchanging the surface currents on the interfaces between adjacent subdomains, both of which are local operations. This enables one to develop a highly efficient parallelized computer program without requiring a parallel linear solver. Next, the method is conditionally stable with the stability condition depending on the spatial discretization immediately next to the interface. Compared to

the FDTD or the explicit TDFEM, the stability condition is relaxed here. This is especially beneficial for problems where the use of a non-uniform discretization is necessary to resolve locally-confined small details without significantly increasing the overall complexity of the problem. In the proposed DFDD method, the numerical solution of a multi-domain calculation may be different from the solution of the single-domain calculation. The difference can be considered as the result of the discretization errors in the two calculations and such a difference can be reduced as both calculations employing finer discretizations. From our numerical dispersion error analysis, we arrive at the conclusion that the DFDD method preserves the accuracy of the standard implicit TDFEM method, although its solution may be slightly different from that of the single-domain calculation. Similar to the FDTD method and the explicit TDFEM, the proposed DFDD method solves for both the electric and magnetic fields. Although the total number of unknowns is doubled as compared to a standard implicit TDFEM, this overhead is easily compensated by employing multiple subdomains and solving them in serial or distributing them among multiple processors, as we have demonstrated in the scalability test. In addition, simultaneous calculation of the electric and magnetic fields may be desirable in certain applications such as in the near-to-far field transformation where both the electric and magnetic fields are needed to transform local fields into far fields. The application of the proposed domain decomposition method is most likely to be found in the numerical simulation of finite arrays and large-scale electromagnetic problems. We have presented several radiation examples to demonstrate the accuracy and efficiency of the method. Future research will focus on extending the method to general scattering analysis and applying the method to much larger and more complicated electromagnetic problems. ACKNOWLEDGMENT The authors wish to thank Dr. D. Riley of Northrop Grumman Corporation for providing the VOLMAX results. REFERENCES [1] J. Lee, R. Lee, and A. Cangellaris, “Time-domain finite-element methods,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 430–442, Mar. 1997. [2] S. Gedney and U. Navsariwala, “An unconditionally stable finite element time-domain solution of the vector wave equation,” IEEE Microwave Guided Wave Lett., vol. 5, pp. 332–334, Oct. 1995. [3] H. Tsai, Y. Wang, and T. Itoh, “An unconditionally stable extended finite-element time-domain solution of active nonlinear microwave circuits using perfectly matched layers,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 10, pp. 2226–2232, Oct. 2002.

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[4] F. Edelvik, G. Ledfelt, P. Lotstedt, and D. Riley, “An unconditionally stable subcell model for arbitrarily oriented thin wires in the FETD method,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 1797–1805, Aug. 2003. [5] D. White and M. Stowell, “Full-wave simulation of electromagnetic coupling effects in RF and mixed-signal ICs using a time-domain finiteelement method,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1404–1413, May 2004. [6] D. Jiao, A. Ergin, B. Shanker, E. Michielssen, and J. Jin, “A fast timedomain higher-order finite-element-boundary-integral method for 3-D electromagnetic scattering analysis,” IEEE Trans. Antennas Propag., vol. 50, no. 9, pp. 1192–1202, Sep. 2002. [7] D. Jiao, J. M. Jin, E. Michielssen, and D. Riley, “Time-domain finite-element simulation of three-dimensional scattering and radiation problems using perfectly matched layers,” IEEE Trans. Antennas Propag., vol. 51, no. 2, pp. 296–305, Feb. 2003. [8] D. Riley and J. Jin, “Modeling of magnetic loss in the finite-element time-domain method,” Microwave Opt. Tech. Lett., vol. 46, no. 2, pp. 165–168, Jul. 2005. [9] R. Petersson and J. Jin, “A three-dimensional time-domain finite element formulation for periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 12–19, Jan. 2006. [10] Z. Lou and J. Jin, “An accurate waveguide port boundary condition for the time-domain finite-element method,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 3014–3023, Sep. 2005. [11] ——, “Modeling and simulation of broadband antennas using the timedomain finite element method,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4099–4110, Dec. 2005. [12] A. Cangellaris, C. Lin, and K. Mei, “Point-matched time-domain finite element methods for electromagnetic radiation and scattering,” IEEE Trans. Antennas Propag., vol. 35, no. 10, pp. 1160–1173, Oct. 1987. [13] C. Chan, J. Elson, and H. Sangani, “An explicit fintie-difference time-domain method using Whitney elements,” in Proc. IEEE APS Int. Symp. Dig., Jul. 1994, vol. 3, pp. 1768–1771. [14] K. Yee and J. Chen, “The finite-difference time-domain and the finitevolume time-domain methods in solving Maxwell’s equations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 354–363, Mar. 1997. [15] D. Baumann, C. Fumeaux, P. Leuchtmann, and R. Vahldieck, “Finitevolume time-domain modelling of a broadband double-ridged horn antenna,” Int. J. Numer. Modelling, vol. 17, no. 3, pp. 285–298, May/Jun. 2004. [16] N. Newmark, “A method of computation for structural dynamics,” J. Eng. Mechanics Division, vol. 85, pp. 67–94, Jul. 1959. [17] H. Schwarz, Gesammelte Mathematische Abhandlungen. Berlin, Germany: Springer, 1890, vol. 2, pp. 133–143. [18] D. Stefanica, “Domain decomposition methods for mortar finite elements,” Ph.D. dissertation, Dept. Math., New York Univ., New York, 2000. [19] J. Benamou and B. Despres, “A domain decomposition method for the Helmholtz equation and related optimal control problems,” J. Comput. Phys., vol. 136, no. 1, pp. 68–82, 1997. [20] S. Lee, M. Vouvakis, and J. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys., vol. 203, no. 1, pp. 1–21, 2005. [21] M. Dryja and O. Widlund, “Some domain decomposition algorithms for elliptic problems,” Iterative Methods for Large Linear Systems, pp. 273–291, 1990, Edited by David R. Kincaid and Linda J. Hayes. [22] B. Smith, P. Bjørstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations. Cambridge, U.K.: Cambridge Univ. Press, 1996. [23] B. Wohlmuth, A. Toselli, and O. Widlund, “An iterative substructuring method for Raviart-Thomas vector fields in three dimensions,” SIAM J. Numer. Anal., vol. 37, no. 5, pp. 1657–1676, 2000. [24] L. Margetts, “Parallel finite element analysis,” Ph.D. dissertation, School of Civil Engineering, Univ. Manchester, Manchester, U.K., 2002. [25] C. Farhat, “A method of finite element tearing and interconnecting and its parallel solution algorithm,” Int. J. Numer. Method. Eng., vol. 32, no. 6, pp. 1205–1227, Oct. 1991. [26] C. Farhat, N. Maman, and G. Brown, “Mesh partitioning for implicit computations via iterative domain decomposition: impact and optimization of the subdomain aspect ratio,” Int. J. Numer. Method. Eng., vol. 38, no. 6, pp. 989–1000, Mar. 1995. [27] C. Wolfe, U. Navsariwala, and S. Gedney, “A parallel finite-element tearing and interconnecting algorithm for solution of the vector wave equation with PML absorbing medium,” IEEE Trans. Antennas Propag., vol. 48, no. 2, pp. 278–284, Feb. 2000.

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Zheng Lou (S’05) was born in Wuhan, Hubei, China, in 1978. He received the B.E. degree in electrical engineering from the University of Science and Technology of China, Hefei, in 2001 and the M.S. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 2003, where he is currently working toward the Ph.D. degree in electrical engineering. Since 2002, he has been a Research Assistant with the Center for Computational Electromagnetics at the University of Illinois at Urbana-Champaign. His research interests include numerical simulation of antennas and microwave devices, time-domain finite element methods, and periodic structures. Mr. Lou received the Raj Mittra Outstanding Graduate Research Award from the Department of Electrical and Computer Engineering in 2005. Jian-Ming Jin (S’87–M’89–SM’94–F’01) received the B.S. and M.S. degrees in applied physics from Nanjing University, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from the University of Michigan, Ann Arbor, in 1989. He is a Professor of Electrical and Computer Engineering and Associate Director of the Center for Computational Electromagnetics at the University of Illinois at Urbana-Champaign. He has authored and co-authored over 150 papers in refereed journals and 15 book chapters. He has also authored The Finite Element Method in Electromagnetics (New York: Wiley, 1st edition 1993, 2nd edition 2002) and Electromagnetic Analysis and Design in Magnetic Resonance Imaging (Boca Raton, FL: CRC, 1998), co-authored Computation of Special Functions (New York: Wiley, 1996), and co-edited Fast and Efficient Algorithms in Computational Electromagnetics Norwood, MA: Artech, 2001). His current research interests include computational electromagnetics, scattering and antenna analysis, electromagnetic compatibility, bioelectromagnetics, and magnetic resonance imaging. Dr. Jin is a member of Commision B of USNC/URSI and Tau Beta Pi. He was a recipient of the 1994 National Science Foundation Young Investigator Award and the 1995 Office of Naval Research Young Investigator Award. He also received the 1997 Xerox Junior Research Award and the 2000 Xerox Senior Research Award presented by the College of Engineering, University of Illinois at Urbana-Champaign, and was appointed as the first Henry Magnuski Outstanding Young Scholar in the Department of Electrical and Computer Engineering in 1998. He was a Distinguished Visiting Professor in the Air Force Research Laboratory in 1999. His name is often listed in the University of Illinois at Urbana-Champaign’s List of Excellent Instructors . He was elected by ISI as one of the world’s most cited authors in 2002. He served as an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and Radio Science and is also on the Editorial Board for Electromagnetics and Microwave and Optical Technology Letters. He was the Symposium Co-chairman and Technical Program Chairman of the Annual Review of Progress in Applied Computational Electromagnetics in 1997 and 1998, respectively.