3D and 4D Space-Time Grids for Electromagnetic Field Analysis
Using Finite Integration Method
Tetsuji Matsuo and Takeshi Mifune

Graduate School of Engineering, Kyoto University, Japan
1Introduction
The finite integration (FI) method [1]-[4] has been studied torealize time-domain electromagnetic field computation using
unstructured spatial grid. However, similarly to the FDTD
method, the FI method uses a uniform time-step, which is
restricted by the CFL condition based on the smallest spatial
grid size. Previous works [5], [6] introduced a space-time FI
method that achieves non-uniform time-steps naturally on 3D
and 4D space-time grids with 2D and 3D space. However, it was
found that a rough construction of space-time grid results incomputational inaccuracy because of unphysical wave-reflection.
Ref. [7] proposed improved 3D and 4D space-time grids for
accurate electromagnetic wave computation. This study
proposes another improved FI method.
2Finite Integration Method on a Space–Time Grid
The coordinate system is denoted by:(ct, x, y, z) = (x0, x1, x2, x3)(1)
where c = 1 / √(ε0μ0), and ε0 and μ0 are the electric and magnetic
constants. The Maxwell equations are given as :dF = 0 , dG = J(2)F = − Σi=13 Eidx0dxi + Σj=13 Bjdxkdxl,
G = Σi=13 Hidx0dxi + Σj=13 Djdxkdxl,(3)
J = cρdx1dx2dx3 − Σj=13Jjdx0dxkdxl
Bwhere ( 1, B2, B3) = cB, (D1, D2, D3) = cD, and ρ is the electriccharge density; (j, k, l) is a cyclic permutation of (1, 2, 3).
The integrated form of (3) is given as:∫ F = 0 , ∫G = ∫ J∂Ωp∂ΩdΩd

Type B grid has dual edges that are not orthogonal tocorresponding primal faces as shown in Fig. 2. This study
examines two types of constitutive relations described by Eqs.
(9) and (10).
(9)
exi = εrε0 dxi / Δl , eyi = εrε0 dyi / Δl (i = 1, 2)
ex1 = εrε0 (dx1 + dxy/2) , ey1 = εrε0 (dy1 − dxy/2) ,ex2 = εrε0 [Δl dx2 − (1−Δl) dy2] / (1−2Δl) ,(10)ey2 = εrε0 [Δl dy2 − (1−Δl) dx2] / (1−2Δl)where d and e are the electric flux and the integration of electricfield given by Eq. (5), respectively. Equation (10) gives moreaccurate approximation of constitutive relation than Eq. (9).
Fig. 3 portrays distributions of discrepancy ΔB3 between B3
obtained in the same way as in [8] by the FDTD method and theFI method with the two space-time grids. The type A grid yields
numerical error whereas the numerical error is suppressed by thetype B grid with Eq. (10) type relation.

primalgrid(Δx0)2/8
dualgridΔl(I)

(II)

(a)

(4)

where Ωp and Ωd are hypersurfaces in space-time; theirboundaries ∂Ωp and ∂Ωd are represented by the faces of primaland dual grids in the FI method. The electromagnetic variables
are defined in the FI method as:
f = ∫SpF , g = ∫SdG,(5)
where Sp and Sd are the faces of primal and dual grids.
The Hodge dual grid [6] is used to express the constitutive
equation simply as:∫Sdcrdx0dxj / ∫Spdxkdxl = −∫Sddxkdxl / ∫Spcrdx0dxj = a (6)cr = 1 / √(εrμr)(7)where a is a constant determined for each pair of Sp and Sd; (j, k,l) is a cyclic permutations of (1, 2, 3); εr and μr are the specificpermittivity and permeability. Thereby,f = Zg / a(8)where Z = √(μrμ0/εrε0) is the impedance.

(III)

Δlx2
irregularity

x1
domain (I)domain (III)domain (II)Δx0

domain (III)domain (I)domain (III)domain (I)

Δx0/2

x0

Δx1 = Δx2 = 1

x1, x2

primalgridΔl(Δx0)2/8

dual
grid
(I)

(III)

(II)

domain (III)domain (II)(b)Δl3

3D Space–Time Grid with 2D Space
Figure 1 illustrates space-time grids where domains (I) and(II) have uniform time-steps Δx0 and Δx0/2, respectively;
domain (III) is the connecting domain. The grid shown in Fig.
1(a) was proposed in [5] that causes unphysical wave-reflection
because of spatial irregularity as reported in [5]. This paper
proposes the grid shown in Fig. 1(b) to suppress the unphysicalreflection. The former and latter grids are called type A and type
B in this article.

x2x1
domain (I)Δx0

Δx0/2

x0
x1, x2Δx1 = Δx2 = 1

Fig. 1. Space-time grids: (a) type A and (b) type B.

ex2 dx2
dy2ey2xex1

Δx1=10.008
0.006
0.004
0.002
0-0.002-0.004-0.006-0.00830
20100-30 -20
2x-10-10 0-20
10120 30 -30
xB30.008
0.006
0.004
0.002
0-0.002-0.004(b)-0.006-0.00830
20100-30 -20
2x-10-10 0-20
10 20
1x30 -30
B30.008
0.006
0.004
0.002
0-0.002-0.004(c)-0.006-0.008x1x60050040030020010000-100-20-20 -100010 20 30 40 50 60-20 -10

3x0.008
0.006
0.004
0.002
0-0.002-0.004-0.006-0.008040030020010000-1000
10 20 30 40 50 60-20 -10
1x0.008
0.006
0.004
0.002
0-0.002-0.004-0.006-0.008Fig. 3. Discrepancy of B3 between space-time FI method and
FDTD method: (a) type A grid, (b) type B grid with Eq. (9)relation, and (c) type B grid with Eq. (10) relation.

4D Space–Time Grid with 3D Space
Figure 4 illustrates 4D space-time grids of type B at thecorner of domain (III). A wave propagation is simulated
similarly to [6], where electromagnetic wave is scattered by a
cubic pore with εr = 1 surrounded by dielectric with εr = 5.
Figures 5(b)(c) and (d) portray distributions of B3 given by thespace-time FI method with type A and B grids. For comparison,Fig. 5(a) depicts the distribution obtained using the FDTD
method with the same uniform spatial grid and time-step as indomain (II), which is restricted by the smallest permittivity. The
FDTD method requires about two times as much computation
time as the space-time FI method. The type B grid with Eq. (10)type relation yields more accurate distribution than type A. The
numerical instability is not observed even after x0 = 105Δx0.

-0.5-1010 20 30 40 50 60x000(b)50-20 -100.5

160-20

11(a)

30
20
100-30 -20
2x-10-10 0-20
10 20
1x30 -30

4x2

30Fig. 4. Space-time grid of type B where solid and dashed lines
represent primal and dual grids: (left) primal grid, (right) primaland dual grids.

Fig. 2. Coner of domain (III).
B3
0.008
0.006
0.004
0.002
0-0.002-0.004(a)-0.006-0.008xx2Δl0x1x2

dx1

xΔx2 Δl=1dey1 xy exyx2

dy1

1

0.5

0-0.5

-1010 20 30 40 50 60
1x(c)
(d)
Fig. 5. Scattering of B3: (a) FDTD method, (b) type A grid, (c)type B grid with Eq. (9) relation, and type B grid with Eq. (10)relation.

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