Accurate Matrix-Free Time-Domain Method with. Traditional Vector Bases in Unstructured Meshes. Jin Yan and Dan Jiao. School of Electrical and Computer ...
Accurate Matrix-Free Time-Domain Method with Traditional Vector Bases in Unstructured Meshes Jin Yan and Dan Jiao School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Abstract—In this paper, we develop a new time-domain method that is naturally matrix free and independent of element shape used for discretization. No extra modification on traditional vector bases is required. Its implementation is straightforward. The accuracy and stability of the proposed method are shown to be guaranteed. Numerical experiments on a variety of unstructured tetrahedral meshes have validated the accuracy and generality of the proposed method.
I. I NTRODUCTION In time-domain methods for electromagnetic analysis, the finite-difference time-domain (FDTD) method [1] has been a popular choice due to its simplicity and merit of being matrix-free. However, its extension to an unstructured mesh has not been made generic. The finite-element method in time domain (TDFEM) is capable of handling unstructured meshes, but it requires the solution of a mass matrix. The discontinuous galerkin time-domain methods only involve the solution of local matrices of small sizes. However, they do not enforce the tangential continuity of the fields at each time instant. In this paper, we develop a new time-domain method that does not require a matrix solution in an arbitrary unstructured mesh. Furthermore, this capability is achieved without changing existing vector basis functions for expanding fields. No dual mesh is needed, and the tangential continuity of the fields is satisfied across the element interface. The accuracy and stability of the proposed method are also guaranteed. II. P ROPOSED M ATRIX -F REE M ETHOD To discretize Faraday’s law, we expand P the electric field E m in each element by vector bases as E = j=1 uj Nj , where uj is the unknown coefficient of the j-th vector basis Nj , and m is the basis number in each element. Substituting the expansion of E into Faraday’s law, evaluating H at point rhi , and then taking the dot product of the resultant with unit vector ˆ i (i = 1, 2, ..., Nh ), we obtain h Se {u} = −diag({µ})
∂{h} , ∂t
(1)
ˆi · where Se is a sparse matrix whose entries are Se,ij = h {∇ × Nj }(rhi ) in which i denotes the global index of the H-point, while j is the global index of the E’s vector basis function. The {h} is a global vector of length Nh whose i-th ˆ i , and diag({µ}) is a diagonal matrix of entry is H(rhi ) · h the permeability. To discretize Ampere’s law, we apply it at rei (i = 1, 2, ..., Ne ) points, and then take the dot product of the
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resultant with unit vector eˆi at each point, obtaining eˆi · {∇ × H}(rei ) = (rei )
∂ei + σ(rei )ei + eˆi · J(rei ), (2) ∂t
where ei = E(rei ) · eˆi ,
(3)
which is the E at point rei along the eˆi direction. To accurately obtain an arbitrary ei via (2), we define a rectangular H-loop normal to eˆi and centering the ei ’s location as shown in Fig. 1. Regardless of the shape of the element, such a rectangular loop can always be defined for ei . Then the eˆi · ∇ × H in (2) can be accurately discretized as the Fig. 1: H points and directions for generating ei . following eˆi · {∇ × H}(rei ) = (hm1 + hm2 )/lim + (hn1 + hn2 )/lin , (4) where lim is the distance between hm1 and hm2 , while lin is the distance between hn1 and hn2 as illustrated in Fig. 1. With (4), (2) can be written as ∂{e} + diag({σ}){e} + {j}, (5) ∂t in which Sh is a sparse matrix of size Ne × Nh . Each row of Sh has only four nonzero elements. Based on (4), they are Sh,ij = 1/lij , where j denotes the global index of the H-point associated with the ei , and lij is simply two times the distance between the H-point (rhj ) and the E-point (rei ). In (5), the i-th entry of {j} is eˆi · J(rei ), and the diag({}) and diag({σ}) are the diagonal matrices of permittivity, and conductivity respectively. To connect (5) with (1), we need to build a relationship between {e} and {u}. If zeroth-order vector bases are used, the {u} is nothing but E(rei )· eˆi , and hence {u} = {e}. However, if we use the zeroth-order vector bases, since the resultant H is a constant in each element, we cannot generate the H accurately at the desired points along the desired directions as shown in Fig. 1. We hence propose to use higher-order vector bases. First-order bases suffice. However, they do not satisfy the property of ui = E(rei ) · eˆi since for this to be true, the vector bases should satisfy Nj (rei ) · eˆi = δji . Although the edge degrees of freedom satisfy this property, other degrees of freedom in the first-order vector bases do not satisfy. This
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Sh {h} = diag({})
AP-S 2015
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Electric field (V/m)
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Fig. 2: Simulation of a 3D box with a tetrahedron mesh. (a) Electric fields. (b) Entire solution error v.s. time. −9
1.5
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where P has entries Pij = Nj (rei ) · eˆi . To obtain {u} from {e}, we need to solve (6). The P can be easily solved because it is a block diagonal matrix with each block size being either 1 or 2. To explain, consider the first-order vector basis functions in a tetrahedral element. In each element, there are 12 edge degrees of freedom and 2 face degrees of freedom on each facet. If the i-th basis is an edge basis, Pij = δij , and hence the corresponding row (column) has only one nonzero element at the diagonal entry. If the i-th basis is a face basis, the Pij is nonzero only for j = i, and j equal to the global index of the other face basis on the same facet. Hence, P can be easily inverted to obtain {u} = P−1 {e} by inverting the small 2 by 2 matrices for each pair of face bases, and using the reciprocal of the diagonal entries corresponding to the edge bases. To emphasize the simplicity of building P−1 , we denote it by a sparse matrix Q. As a result, (1) can be rewritten as:
x 10
−1 0
(6)
Electric field (V/m)
{e} = P{u},
1
||{e}−{e}anal||/||{e}anal||
problem can be solved by modifying the original first-order vector bases. Here, we solve the problem without changing the original vector bases asP follows. From ei = E(rei ) · eˆi = uj (Nj (rei ) · eˆi ), we obtain the following relationship between {e} and {u}
0.5 0 −0.5
Point 1 (Proposed) Point 2 (Proposed) Point 1 (Analytical) Point 2 (Analytical)
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Fig. 3: Simulation of a 3D sphere with a tetrahedron mesh. (a) Electric fields. (b) Entire solution error v.s. time.
t0 )exp(−(t − t0 )2 /τ 2 ) with t0 = 4τ and τ = 1.6 × 10−11 s. The outermost boundary is truncated by the analytically known tangential field. The time step used is ∆t = 2.0 × 10−11 s, and the number of expansion terms is 9. The electric field at two randomly selected observations points (rp1 = ∂{h} . (7) (0.197, 0.056, 0.066) m and rp2 = (0.364, 0.353, 0.201)) m Se Q{e} = −diag({µ}) ∂t along the directions of eˆp1 = (−0.033, 0.643, 0.766) and The (5) and (7) can then be marched on in time in a leapeˆp2 = (0.397, −0.877, −0.272) are plotted in Fig. 2(a) in comfrog fashion. It can also be combined to solve in a second-order parison with analytical data. Excellent agreement is observed. equation, which yields To further verify the accuracy of the proposed method, we n σ o ∂ {e} 1 ∂{j} calculate k{e} − {e}ref k/k{e}ref k which is the relative error ∂ 2 {e} +diag +S {e} = −diag , of the entire solution vector, where {e} is the entire unknown ∂t2 ∂t ∂t (8) vector solved from this method, while {e}ref denotes the where S = diag({ 1 })Sh diag({ µ1 })Se Q. Since Se is not STh , reference analytical solution. Fig. 2(b) plots the entire solution it can be proved that a conventional explicit marching of error with respect to time. Less than 3% error is observed, (1) and (5) as well as (8) is absolutely unstable. However, verifying the accuracy of the proposed method. if we choose Sh = STe , the accuracy cannot be guaranteed A wave propagation problem in a 3-D sphere of 0.12 in a general unstructured mesh. This dilemma is solved m radius is then simulated. The sphere is discretized into as follows without sacrificing the matrix-free merit of the 1, 987 tetrahedra with 3, 183 edges. The incident E has the proposed method. We start with a backward-difference based same waveform as that of the previous example but with discretization of (8), but using a central-difference based time τ = 2.0 × 10−9 s. The boundary is again terminated by step. The former permits the use of any large time step analytical tangential E. The time step used is ∆t = 2.0×10−12 without making the simulation unstable. However, it requires s and the number of expansion terms is 9. The locations of the the solution of (I + M){e}n+1 = f , where M = ∆t2 D−1 S two observation points are rp1 = (0, −0.098, −0.057) m with with diagonal matrix D = I + ∆tdiag({ σ }). By using the eˆp1 = (0, 0.866, 0.499), and rp2 = (−0.039, 0.005, 0.102) time step of a traditional central-difference scheme, we can m with eˆp2 = (−0.971, −0.185, −0.154). Fig. 3(a) shows achieve the same matrix-free feature of p the central-difference that the electric fields solved by the proposed method are in based explicit marching since ∆t < 1/ kSk, making ||M|| excellent agreement with analytical results. Fig. 3(b) plots the less than 1. Hence, the time marching can be performed by entire solution error with time. Less than 2% error is observed a small number of sparse matrix-vector multiplications as in the entire time window. {e}n+1 = (I − M + M2 − · · · + (−M)k ){f }, without the R EFERENCES need of inverting or factorizing a matrix. III. N UMERICAL VALIDATION We first simulate a free-space wave propagation problem in a 3-D box (1 m by 0.5 m by 0.75 m) discretized into 350 tetrahedra. The incident E = yˆf (t − x/c) where f (t) = 2(t −
[1] K. S. Yee, “Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media”, TAP, May 1966. [2] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher Order Interpolatory Vector Bases for Computational Electromagnetics”, TAP, 1997.
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