Accurate Matrix-Free Time-Domain Method in. Three-Dimensional Unstructured Meshes. Jin Yan and Dan Jiao. School of Electrical and Computer Engineering, ...
Accurate Matrix-Free Time-Domain Method in Three-Dimensional Unstructured Meshes Jin Yan and Dan Jiao School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA
Abstract—We develop an accurate 3-D matrix-free timedomain method independent of the element shape used for discretization. The accuracy and stability of the proposed method are shown to be theoretically guaranteed. No dual mesh is needed. The tangential continuity of the fields is satisfied across the element interface. Numerical experiments on unstructured 3D meshes with arbitrarily shaped tetrahedron and triangular prism elements have validated the accuracy and generality of the proposed method.
I. I NTRODUCTION
Fig. 1: H points and directions determined based on e.
The finite-difference time-domain (FDTD) method has been a popular choice for time-domain analysis due to its simplicity and matrix-free (free of matrix solution) property. But it requires a structured grid. The generalization of the FDTD to arbitrary unstructured meshes remains to be a research problem. The finite-element method in time domain (TDFEM) [1] has no difficulty in handling irregular meshes, but it requires the solution of a mass matrix. There also exists a class of Discontinuous Galerkin Time-Domain methods, which only involves the solution of local matrices of small sizes. However, this is achieved without enforcing the tangential continuity of the fields across the element interface at each time instant, which has its implication in either accuracy or computational efficiency. In this work, we develop a new time domain method that is matrix free in nature independent of element shape for analyzing general 3-D problems. Both accuracy and stability of this new matrix-free method are theoretically guaranteed in unstructured meshes.
interface. The diag({µ}) is a diagonal matrix whose i-th entry is the permeability at rhi point. To discretize Ampere’s law, we let it be satisfied at rei (i = 1, 2, ..., Ne ) points, and then take the dot product of the resultant with unit vector eˆi at each point, we obtain the following discretization of Ampere’s law
II. P ROPOSED G ENERAL F RAMEWORK Consider a general 3-D electromagnetic problem discretized into arbitrarily shaped elements. First, we discretize Faraday’s law byPexpanding the electric field E in each element as m E = j=1 ej Nj , where m is the basis number, ej is the unknown coefficient of the j-th vector basis Nj . Substituting the expansion of E into Faraday’s law, evaluating H at point ˆ i (i = 1, 2, ..., Nh ) direction, we obtain rhi along h ∂{h} , (1) ∂t where {e} stands for the global vector of unknown coefficients ei (i = 1, 2, ..., Ne ), {h} denotes the vector of discrete H ˆ i (i = 1, 2, ..., Nh ), whose i-th entry is hi = H(rhi ) · h and Se is a sparse matrix whose ij-th entry is Se,ij = ˆ i ·{∇×Nj }(rhi ). The number of nonzero entries in each row h of Se is m. During the assembling of Se across all elements, the tangential continuity of E is enforced across the element Se {e} = −diag({µ})
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∂{e} + diag({σ}){e} + {j}, (2) ∂t in which Sh is of size Ne × Nh representing the discretized eˆi ·{∇×H}(rei ), the i-th entry of vector {e} is ei = E(rei )·ˆ ei , while ji = eˆi · J(rei ), and the diag({}) and diag({σ}) are the diagonal matrices whose i-th entry is the permittivity, and conductivity respectively at point rei . The (1) and (2) can be solved in a leapfrog way which is matrix-solution free. We can also combine the two to solve as n σ o ∂ {e} ∂ 2 {e} 1 ∂{j} +diag +S {e} = −diag , 2 ∂t ∂t ∂t (3) where S = diag({ 1 })Sh diag({ µ1 })Se . Sh {h} = diag({})
III. P ROPOSED 3-D F ORMULATIONS In this section, we present detailed 3-D formulations of (1) and (2). The accuracy of (1) is guaranteed with commonly used curl-conforming vector basis functions [1], [2] for generating H at any point along any direction. We hence can use this freedom to choose the H points and directions in a way to ensure the accuracy of (2). Our approach is 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, 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 ˆ i . As a result, point as rhi , and the tangential direction as h the eˆi · ∇ × H at ei ’s point can be accurately discretized as
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eˆi · {∇ × H}(rei ) = (hm1 + hm2 )/lim + (hn1 + hn2 )/lin , (4)
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where lim and lin are the two side lengths of the loop shown in Fig. 1. Hence, the accuracy of (2) is ensured. From (4), each row of Sh has only four nonzero elements, which are Sh,ij = 1/lij where j denotes the global index of the Hpoint associated with ei . No dual mesh for H is needed. The H points and directions chosen do not form a mesh either. For the construction of (1), if we employ the commonly used zeroth-order vector bases, since the resultant H is a constant in each element, we cannot obtain accurate H at the desired points, and along the desired directions shown in Fig. 1. This problem can be readily alleviated by using higher-order vector bases [2]. The first-order vector bases are sufficient for use. However, unlike the zeroth-order vector bases, they do not completely satisfy ei = E(rei ) · eˆi . This property is required because to connect (2) to (1), the {e} should refer to the same vector. Since ei = E(rei ) · eˆi in (2), so is the ei in (1). For this to be true, the vector bases should satisfy eˆi · Nj (rei ) = δij . The edge degrees of freedom among the first-order vector bases naturally satisfy this property. However, other degrees of freedoms in general do not. Since their definitions are not unique either, we can modify them to suit the need of this work as follows. Among the 20 first-order vector bases in a tetrahedron [2], there are two degrees of freedom located at the center of each face. We keep one basis as before, but modify the other basis. Take the face formed by vertices 1, 2, and 3 as an example. The two face bases we construct are Nf1 = 4.5ξ1 Ω1 with projection direction eˆ = tˆ1 , and Nf2 = cξ2 ξ3 ∇ξ1 with eˆ = (ˆ nf × Ω1 )/||ˆ nf × Ω1 ||. Here, ξi is the volume coordinate at vertex i, Ωi is the normalized zerothorder basis along edge i opposite to node i, tˆi denotes the unit vector tangential to edge i, n ˆ f denotes the unit vector normal to the face, and c is the normalization coefficient that makes Nf2 · eˆ = 1 at the face center. In a triangular prism, among the 36 first-order vector bases, the two degrees of freedom respectively located at the top triangular patch center, the prism center, and the bottom patch center do not satisfy the desired property. Similarly, for the three sets, we keep the first basis in each set the same as before, but modify the second basis. Take the top face as an example, the two bases we construct are Nf1 = 4.5ξ1 ζ1 (2ζ1 − 1)W1 with eˆ = tˆ1 , and Nf2 = cξ2 ξ3 ζ1 (2ζ1 − 1)∇ξ1 with eˆ = (ˆ nf × W1 )/||ˆ nf × W1 ||. Here, ζ1 = 1 on the top triangle and 0 on the lower one, Wi is the normalized zeroth-order basis. The basic idea of this change is to make eˆi · Nj (rei ) = δij satisfied by choosing appropriate basis direction and projection direction of the second basis. Since Sh 6= STe , which is true in general to ensure the accuracy in a unstructured mesh, this will make a traditional explicit marching unstable. This problem is solved in this work by performing a backward-difference based time marching but using a central-difference time step. The former is stable for arbitrarily large time step, while the latter avoids solving the system equation. To be specific, the backward difference yields a system equation of (I + M){e}n+1 = f , where M = ∆t2 D−1 S with diagonal matrix D = I + ∆tdiag({ σ }). The ptime step of a central-difference scheme satisfies ∆t < 1/ kSk, and hence making ||M|| less than 1, allowing for
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Fig. 3: (a) Triangular prism (top view). (b) Simulated fields. the system solution to be obtained by a small number of sparse matrix-vector multiplications as {e}n+1 = (I − M + M2 − · · · + (−M)k ){f }. Notice that every vector can be computed from previous vector by multiplying M. As a result, no matrix solution is needed. IV. N UMERICAL VALIDATION We first simulate a free-space wave propagation problem in a 3-D domain (1 m × 0.5 m × 0.75 m) discretized into 350 tetrahedra as illustrated in Fig. 2(a). The incident E = yˆf (t − x/c) where f (t) = 2(t − t0 )exp(−(t − t0 )2 /τ 2 ) with t0 = 4τ and τ = 1.6 × 10−11 s. An analytical absorbing boundary condition which is the known tangential field is applied on the outermost boundary. The time step used is ∆t = 2.0 × 10−11 s, and the number of expansion terms k is 9. In Fig. 2(b), we plot the entire solution error measured by k{e} − {e}anal k/k{e}anal k at each time instant, where {e}anal denotes the analytical solution, and both vectors contain electric fields at all the Ne points. The error is shown to be less than 3% across the entire time window, validating the accuracy of the proposed method. The second example is discretized into 5, 022 triangular prisms, the mesh of which is shown in Fig. 3(a) from a top view. A square conductor of 5 × 107 S/m conductivity is at the center of the second layer filled by a material of dielectric constant 4. The rest of the two layers are air regions. The top and bottom boundaries are PEC while PMC is imposed on the other four sides. A current source with the same f (t) is injected but with τ = 2.0 × 10−12 s. The ∆t = 5.0 × 10−16 s since the structure has a µm-dimension, and k = 9. Fig. 3(b) compares the simulated eletric fields in comparison with TDFEM. Excellent agreement is observed. R EFERENCES [1] J. M. Jin, The finite element method in electromagnetics. Wiley, 2002. [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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