Accurate and Stable Matrix-Free Time-Domain Method Independent of Element Shape for General Electromagnetic Analysis Jin Yan
Dan Jiao
Abstract — In this paper, we present a new timedomain method that is naturally matrix free, i.e., requiring no matrix solution, regardless of whether the discretization is a structured grid or an unstructured mesh. Its matrix-free property is independent of the element shape used for discretization, and its implementation is straightforward. No interpolations, projections, and mass lumping are required. The accuracy and stability of the proposed method are theoretically analyzed and shown to be guaranteed. In addition, no dual mesh is needed and the tangential continuity of the fields is satisfied across the element interface. The flexible framework of the proposed method also allows for a straightforward extension to higher-order accuracy in both electric and magnetic fields. Numerical experiments have validated the accuracy and generality of the proposed matrix-free method.
1
INTRODUCTION
In time-domain methods for electromagnetic analysis, the finite-difference time-domain (FDTD) method [1, 2] has been a popular choice due to its simplicity and merit of being matrix-free (free of a system matrix solution). The traditional FDTD method requires a structured grid. Its generalization to an unstructured mesh has been extensively studied. Many of the non-orthogonal FDTD methods require a dual mesh that satisfies a certain relationship with the primary mesh. Such a dual mesh may not exist in a general unstructured mesh. For cases where the dual mesh exists, the accuracy of the resulting scheme can be low since between E and H, one of them cannot be centered by the integration loop of the other, and be perpendicular to the loop area. Although local interpolation and projection techniques have been developed to more accurately find the dual field from the primary field, and vice versa [3], the accuracy of such techniques is still limited in an arbitrary unstructured mesh. In addition to accuracy, the stability of the nonorthogonal FDTD methods has been studied. It is shown in [4] that as long as the discrete curl-curl operator supports complex-valued eigenvalues, an explicit scheme is unconditionally unstable. Since any real-valued but unsymmetrical matrix can have ∗ School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA, email:
[email protected].
∗
complex-valued eigenvalues that come in conjugate pairs, once the discretized curl-curl operator is unsymmetrical, the resultant explicit time marching is unstable. An unsymmetrical matrix can support negative eigenvalues, which also make a traditional explicit marching absolutely unstable. This point will be made clear in this paper. The unsymmetrical curl-curl operator is common in existing nonorthogonal FDTD methods. As a consequence, it remains a research problem how to ensure both accuracy and stability while preserving the matrixfree property of an FDTD-like method in an arbitrary unstructured mesh. The finite-element method in time domain (TDFEM) is capable of handling unstructured meshes [5]. However, either in its first-order mixed E-B form or second-order vector wave equation based form [5], the TDFEM requires the solution of a mass matrix, thus not being matrix-free in nature. The mass-lumping techniques are error prone for irregularly shaped elements and the orthogonal vector basis functions [6] make use of approximate integrations. In addition, there exists a class of Discontinuous Galerkin time-domain methods, which only involves the solution of local matrices of a small size. However, this is achieved by not enforcing the tangential continuity of the fields across the element interface at each time instant. In contrast, both FDTD and TDFEM impose the tangential continuity of the fields across the interfaces of discretization cells. In this paper, we present a new matrix-free timedomain method that is different from prior attempts. The method has a mass matrix that is naturally diagonal, regardless of whether the discretization is structured or irregular. Its matrixfree property is independent of the element shape, and its implementation is straightforward. The tangential continuity of the fields is also enforced across the element interface at each time instant. In addition, no dual mesh is needed in the proposed method. The flexible framework of the proposed method also allows for a straightforward extension to higher-order accuracy in both E and H. Equally important, we have developed a new time-marching scheme to overcome the absolute instability caused
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permittivity, and conductivity respectively at point r𝑒𝑖 . The (2) and (3) can be solved in a leapfrog way, which is free of matrix solution. The two can also be combined to solve as ({ 𝜎 }) ∂ {𝑒} ∂ 2 {𝑒} + S {𝑒} = + 𝑑𝑖𝑎𝑔 ∂𝑡2 ({ }) 𝜖 ∂𝑡 ∂{𝑗} 1 , − 𝑑𝑖𝑎𝑔 𝜖 ∂𝑡
Figure 1: H points and directions determined based on 𝑒. by an unsymmetrical curl-curl operator, without sacrificing the matrix-free property and the accuracy of the proposed method. Numerical simulations on various highly unstructured meshes have demonstrated the validity, accuracy, and stability of the proposed new method. 2
PROPOSED METHOD
To discretize Faraday’s law, we expand ∑𝑚 the electric field E in each element as E = 𝑗=1 𝑒𝑗 N𝑗 , where 𝑚 is the number of vector bases in one element, 𝑒𝑗 is the unknown coefficient of the 𝑗-th vector basis N𝑗 . Substituting the expansion of E into Faraday’s law, evaluating magnetic field H at point rℎ𝑖 along ˆ 𝑖 (𝑖 = 1, 2, ..., 𝑁ℎ ) direction, we obtain ℎ ˆ𝑖 ⋅ ℎ
∑
ˆ 𝑖 ⋅𝜇(rℎ𝑖 ) ∂H(rℎ𝑖 ) , (1) {∇×N𝑗 }(rℎ𝑖 )𝑒𝑗 = −ℎ ∂𝑡
which can be compactly written as S𝑒 {𝑒} = −𝑑𝑖𝑎𝑔({𝜇})
∂{ℎ} , ∂𝑡
(4)
where S = 𝑑𝑖𝑎𝑔({ 1𝜖 })Sℎ 𝑑𝑖𝑎𝑔({ 𝜇1 })S𝑒 . It is evident that the mass matrix in (4) is naturally diagonal. 3
DETAILED FORMULATIONS
What is proposed in the previous section is a general framework for developing a matrix-free time domain method. In this section, we present detailed 3-D formulations. To obtain accurate {𝑒} from {ℎ} in (3), we propose to determine H points and directions based on E’s degrees of freedom. Basically, for each 𝑒𝑖 , we define a rectangular loop perpendicular to 𝑒ˆ𝑖 and centering the 𝑒𝑖 ’s location r𝑒𝑖 , as illustrated in Fig. 1. This can always be done regardless of the original element shape. Along each side of the loop, we choose the middle point as rℎ𝑖 , and the tangential ˆ 𝑖 . With such a choice of H’s points direction as ℎ and directions, the 𝑒ˆ𝑖 ⋅ ∇ × H at 𝑒𝑖 ’s point can be accurately evaluated as
(2)
𝑒ˆ𝑖 ⋅{∇×H}(r𝑒𝑖 ) = (ℎ𝑚1 +ℎ𝑚2 )/𝑙𝑖𝑚 +(ℎ𝑛1 +ℎ𝑛2 )/𝑙𝑖𝑛 , (5) where {𝑒} stands for the global vector of unknown where 𝑙𝑖𝑚 and 𝑙𝑖𝑛 are the two side lengths of the coefficients 𝑒𝑖 (𝑖 = 1, 2, ..., 𝑁𝑒 ), {ℎ} denotes the vecloop shown in Fig. 1. The accuracy of (3) is hence ˆ𝑖 tor of discrete H whose 𝑖-th entry is ℎ𝑖 = H(rℎ𝑖 )⋅ ℎ ensured. From (5), each row of Sℎ has only four (𝑖 = 1, 2, ..., 𝑁ℎ ), and S𝑒 is a sparse matrix whose nonzero elements, which are S ℎ,𝑖𝑗 = 1/𝑙𝑖𝑗 where 𝑗 ˆ 𝑖 ⋅ {∇ × N𝑗 }(rℎ𝑖 ). The num𝑖𝑗-th entry is S𝑒,𝑖𝑗 = ℎ denotes the global index of the H-point associated ber of nonzero entries in each row of S𝑒 is 𝑚. The with 𝑒𝑖 . Notice that no dual mesh for H is required. tangential continuity of E is enforced across the ele- The H-points and directions defined here do not ment interface. The 𝑑𝑖𝑎𝑔({𝜇}) is a diagonal matrix make a mesh either. We only need to sample H whose 𝑖-th entry is the permeability at rℎ𝑖 point. at the points along the directions shown in Fig. 1 To discretize Ampere’s law, we apply it at r𝑒𝑖 based on E’s points and directions. (𝑖 = 1, 2, ..., 𝑁𝑒 ) points, and then take the dot prodFor the construction of (2), since the curl of the uct of the resultant with unit vector 𝑒ˆ𝑖 at each commonly used zeroth-order vector bases is conpoint, we obtain the following discretization of Am- stant in each element, the resultant H is also conpere’s law stant in each element, we cannot use zeroth-order bases to obtain accurate H at the desired points ∂{𝑒} +𝑑𝑖𝑎𝑔({𝜎}){𝑒}+{𝑗}, (3) along the desired directions shown in Fig. 1. We Sℎ {ℎ} = 𝑑𝑖𝑎𝑔({𝜖}) ∂𝑡 hence propose to use higher-order vector bases. The in which Sℎ is of size 𝑁𝑒 ×𝑁ℎ and Sℎ {ℎ} represents first-order vector bases are sufficient for use. Howthe discretized 𝑒ˆ𝑖 ⋅ {∇ × H}(r𝑒𝑖 ). The 𝑖-th entry of ever, unlike the zeroth-order vector bases, they do vector {𝑒} is 𝑒𝑖 = E(r𝑒𝑖 ) ⋅ 𝑒ˆ𝑖 . The 𝑖-th entry of {𝑗} not completely satisfy 𝑒𝑖 = E(r𝑒𝑖 ) ⋅ 𝑒ˆ𝑖 . This propis 𝑗𝑖 = 𝑒ˆ𝑖 ⋅ J(r𝑒𝑖 ), and the 𝑑𝑖𝑎𝑔({𝜖}) and 𝑑𝑖𝑎𝑔({𝜎}) erty is required because to connect (3) to (2) withare the diagonal matrices whose 𝑖-th entry is the out any transformation, the {𝑒} should refer to the
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same vector. Since 𝑒𝑖 = E(r𝑒𝑖 )⋅ 𝑒ˆ𝑖 in (3), so is the 𝑒𝑖 in (2). For this to be true, the vector bases should satisfy (6) 𝑒ˆ𝑖 ⋅ N𝑗 (r𝑒𝑖 ) = 𝛿𝑖𝑗 .
semi-definite. As a result, its eigenvalues 𝜆 are real valued and no less than√zero. In this case, a time step satisfying Δ𝑡 ≤ 2/ 𝜆𝑚𝑎𝑥 can make 𝑧 in (10) bounded by 1, where 𝜆𝑚𝑎𝑥 is the maximum eigenThe edge degrees of freedom among the first-order value, which is also S’s spectral radius. Hence, the vector bases naturally satisfy this property. How- explicit simulation of (4) is stable. However, if S is ever, other degrees of freedoms defined on the face not symmetric such as the one obtained from many or inside the volume, in general, do not. Since non-orthogonal FDTD methods as well as the protheir definitions are not unique either, we can mod- posed new matrix-free method, its eigenvalues eiify them to suit the need of this work as follows. ther are real (can be negative) or come in complexAmong the 20 first-order vector bases in a tetrahe- conjugate pairs. For complex-valued or negative dron [7], there are two degrees of freedom located at eigenvalues 𝜆, the two roots 𝑧1 and 𝑧2 shown in the center of each face which do not satisfy the de- (10) satisfy 𝑧1 𝑧2 = 1 and neither of them has modsired property. Therefore, we propose to keep one ulus equal to 1. As a result, the modulus of one of basis as before, but modify the other one. Take them must be greater than 1, and hence the explicit the face formed by vertices 1, 2, and 3 as an exam- time-domain simulation of (4) must be unstable. The stability problem is solved in this work as ple. The two face bases along with their projection follows without sacrificing the matrix-free merit of directions are built as the following: the proposed method. We perform a backward(7) difference based time marching but using a central𝑒ˆ𝑓1 = 𝑡ˆ1 , N𝑓1 = 4.5𝜉1 Ω1 ; difference time step. The former is stable for large 𝑒ˆ𝑓2 = (ˆ 𝑛𝑓 × Ω1 )/∣∣ˆ 𝑛𝑓 × Ω1 ∣∣, N𝑓2 = 𝑐𝜉2 𝜉3 ∇𝜉1 . (8) time step, while the latter avoids solving the system matrix. To be specific, the backward differHere, 𝜉𝑖 is the volume coordinate at vertex 𝑖, Ω𝑖 is ence yields a system equation of (I + M){𝑒}𝑛+1 = the normalized zeroth-order basis along edge 𝑖 op- 𝑓 , where M = Δ𝑡2 D−1 S with diagonal matrix posite to node 𝑖, 𝑡ˆ𝑖 denotes the unit vector tangen- D = I + Δ𝑡𝑑𝑖𝑎𝑔({ 𝜎 }). The time step of a central𝜖 tial to edge 𝑖, 𝑛 ˆ 𝑓 denotes the unit vector normal to difference scheme satisfies Δ𝑡 < 1/√∥S∥, and the face, and 𝑐 is the normalization coefficient that hence making ∣∣M∣∣ less than 1, allowing for the makes N𝑓2 ⋅ 𝑒ˆ𝑓2 = 1 at the face center. The same system solution to be obtained by a small number modification is applied to the bases on the other of sparse matrix-vector multiplications as {𝑒}𝑛+1 = three faces of the tetrahedron. With this modifica- (I − M + M2 − ⋅ ⋅ ⋅ + (−M)𝑘 ){𝑓 }. Notice that evtion, the completeness of the first-order vector bases ery vector can be computed from previous vector is preserved, and meanwhile the desired property of by multiplying M. As a result, no matrix solution 𝑒𝑖 = E(r𝑒𝑖 ) ⋅ 𝑒ˆ𝑖 is satisfied. is needed. 4
TIME MARCHING AND STABILITY 5 ANALYSIS
Since Sℎ ∕= S𝑇𝑒 , which is true in general to ensure the accuracy in an unstructured mesh, this will make a traditional explicit marching unstable. To prove, we can perform a stability analysis of the central-difference based time discretization of (4). The 𝑧-transform of the central-difference based time marching of (4) results in the following equation: (9) (𝑧 − 1)2 + Δ𝑡2 𝜆𝑧 = 0,
SIMULATION RESULTS
The proposed method has been extensively validated on various unstructured meshes. As an example, a sphere discretized into tetrahedral elements is simulated, whose mesh is shown in Fig. 2. The discretization results in 3,183 edges and 1,987 tetrahedrons. We set up a free-space wave propagation problem in the given mesh to validate the accuracy of the proposed method against analytical results. The incident E has an expression of E = 𝑦ˆ𝑓 (𝑡 − 𝑓 (𝑡) = 2(𝑡 − 𝑡0 ) exp(−(𝑡 − 𝑡0 )2 /𝜏 2 ), where 𝜆 is the eigenvalue of S. The two roots of (9) 𝑥/𝑐), where −9 𝜏 = 2.0 × 10 s, 𝑡0 = 4𝜏 , and 𝑐 denotes the speed can be readily found to be of light. The outermost boundary of the mesh is √ 2 2 2 truncated by analytical E fields. The time step 2 − Δ𝑡 𝜆 ± Δ𝑡 𝜆(Δ𝑡 𝜆 − 4) . (10) used is Δ𝑡 = 2.0 × 10−12 s, which is the same as 𝑧1,2 = 2 that used in a traditional TDFEM method. The The S resulting from a traditional FDTD in a struc- number of expansion terms 𝑘 is 9. The two degrees tured grid or a TDFEM is symmetric and real- of freedom of the electric field, whose indices in vecvalued, thus Hermitian. Furthermore, it is positive tor {𝑒} are 1, and 9,762 respectively, are plotted in
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Figure 2: Illustration of the mesh of a sphere structure (tetrahedron).
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Figure 4: Entire solution error v.s. time. Acknowledgments
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This work was supported by a grant from NSF under award No. 1065318, and a grant from DARPA under award N00014-10-1-0482B.
Electric field (V/m)
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References
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[1] K. S. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, no. 3, pp. 302–307, 1966.
Point 1 (Proposed) Point 2 (Proposed) Point 1 (Analytical) Point 2 (Analytical)
0.5
1 Time (s)
1.5 −8
x 10
Figure 3: Electric fields of the sphere simulated from the proposed method in comparison with analytical results.
Fig. 3 in comparison with analytical data. Excellent agreement can be observed. In Fig. 4, we plot the entire solution error versus time. This error is defined as ∣∣{𝑒}𝑡ℎ𝑖𝑠 (𝑡) − {𝑒}𝑟𝑒𝑓 (𝑡)∣∣ , Error𝑒𝑛𝑡𝑖𝑟𝑒 (𝑡) = ∣∣{𝑒}𝑟𝑒𝑓 (𝑡)∣∣
(11)
[2] A. Taflove and S. C. Hagness, “Computational electrodynamics: The finite-difference timedomain method,” Artech House, Boston, MA, 2000. [3] M. Madsen, “Divergence preserving discrete surface integral methods for maxwell’s equations using nonorthogonal grids,” J. Computat.Phys., vol. 119, pp. 34–45, 1995. [4] S. Gedney and J. Roden, “Numerical stability of nonorthogonal fdtd methods,” IEEE Trans. on AP, vol. 48, no. 2, pp. 231–239, 2000. [5] D. Jiao and J. Jin, The finite element method in electromagnetics. John Wiley & Sons, 2002, ch. Finite element analysis in time domain, pp. 529–584.
where {𝑒} denotes the entire unknown vector solved [6] D. Jiao and J. Jin, “Three-dimensional orthogonal vector basis functions for time-domain fifrom the proposed method, while {𝑒}𝑟𝑒𝑓 is the refnite element solution of vector wave equations,” erence analytical solution. Less than 3% error is obIEEE Trans. Antennas and Propag., vol. 51, served in the entire time window. It is evident that no. 1, pp. 59–66, 2003. the proposed method is not just accurate at certain points, but accurate at all points in the computa- [7] R. D. Graglia, D. R. Wilton, and A. F. Peterson, tional domain for all time instants simulated. Note “Higher order interpolatory vector bases for that the center peak error is due to zero passing, computational electromagnetics,” IEEE Trans. thus a comparison with close to zero fields at the Antennas and Propag., vol. 45, no. 3, pp. 329– specific time instant. 342, 1997.
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