CHAPTER FIVE Sampling Biorthogonal Time Domain Method (SBTD) The finite difference time domain (FDTD) method was proposed by K. Yee [1] in 1966. The simplicity of the FDTD method in mathematics has proved to be its great advantage. The method does not involve any integral equations, Green’s functions, singularities, nor matrix equations. Neither does it involve functional or variational principles. In addition the FDTD proves to be versatile when used in complicated geometries. The computational issues associated with the FDTD are the radiation boundary conditions or absorption boundary conditions for open structures, numer- ical dispersion, and stability conditions. Its major drawbacks include its massive memory consumption and huge computational time. In these regard wavelets offer significant improvements to the FDTD. It will be shown that the Yee-based FDTD is identical to the Galerkin method using Haar wavelets. Since the Haar bases are discontinuous, the slow decay of the frequency components and the Gibbs phenomena of the Haar basis prevent the use of a coarse mesh in the FDTD. In contrast, the Daubechies-based sampling functions are contin- uous basis functions with fast decay in both the spatial and spectral domains. Thus a more efficient time domain method can be derived: the sampling biorthogonal time domain (SBTD) algorithm. 5.1 BASIS FDTD FORMULATION For a lossy medium with a conductivity σ , we begin with Maxwell’s two curl equa- tions µ ∂H ∂t =−∇×E, (5.1.1)  ∂E ∂t + σ E =∇×H. (5.1.2) 189 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X 190 SAMPLING BIORTHOGONAL TIME DOMAIN METHOD (SBTD) We obtain by the leapfrog method [2] a set of finite difference equations k+(1/2) H x  , m + 1 2 , n + 1 2  = k−(1/2) H x  , m + 1 2 , n + 1 2  + t µz  k E y  , m + 1 2 ,(n + 1)  − k E y  , m + 1 2 , n  − t µy  k E z  , (m + 1), n + 1 2  − k E z  , m, n + 1 2  , k+(1/2) H y   + 1 2 , m, n + 1 2  = k−(1/2) H y   + 1 2 , m, n + 1 2  + t µx  k E z   + 1, m, n + 1 2  − k E z  , m, n + 1 2  − t µz  k E x   + 1 2 , m, n + 1  − k E x   + 1 2 , m, n  , (5.1.3) k+(1/2) H z   + 1 2 , m + 1 2 , n  = k−(1/2) H z   + 1 2 , m + 1 2 , n  + t µy  k E x   + 1 2 , m + 1, n  − k E x   + 1 2 , m, n  − t µx  k E y   + 1, m + 1 2 , n  − k E y  , m + 1 2 , n  , k+1 E x   + 1 2 , m, n  =  1 − (σ t /2) 1 + (σ t /2)  k E x   + 1 2 , m, n  +  1 1 + (σ t/2)  t y  k+(1/2) H z   + 1 2 , m + 1 2 , n  − k+(1/2) H z   + 1 2 , m − 1 2 , n  BASIS FDTD FORMULATION 191 − t z  k+(1/2) H y   + 1 2 , m, n + 1 2  − k+(1/2) H y   + 1 2 , m, n − 1 2  , k+1 E y  , m + 1 2 , n  =  1 − σ t 2 1 + σ t 2  k E y  , m + 1 2 , n  +  1 1 + σ t 2   t z  k+(1/2) H x  , m + 1 2 , n + 1 2  − k+(1/2) H x  , m + 1 2 , n − 1 2  − t x  k+(1/2) H z   + 1 2 , m + 1 2 , n  − k+(1/2) H z   − 1 2 , m + 1 2 , n  , (5.1.4) k+1 E z  , m, n + 1 2  =  1 − σ t 2 1 + σ t 2  k E z  , m, n + 1 2  +  1 1 + σ t 2   t x  k+(1/2) H y   + 1 2 , m, n + 1 2  − k+(1/2) H y   − 1 2 , m, n + 1 2  − t y  k+(1/2) H x  , m + 1 2 , n + 1 2  − k+(1/2) H x  , m − 1 2 , n + 1 2  . In the equations above, indexes l, m,andn are the node numbers in the x, y,andz directions, respectively, and the leftscript k denotes the time step. Note that a central difference scheme has been used in all of the finite difference equations. Figure 5.1 illustrates a unit cell in the FDTD lattice where the electric and magnetic fields are spaced apart by a half-grid in each dimension. At the interface of two media (e.g., at the boundary between a conductor and a dielectric), the average values of  and 192 SAMPLING BIORTHOGONAL TIME DOMAIN METHOD (SBTD) H 2 3 6 8 7 65 4 8 2 4 1 5 Magnetic cell Electric cell 3 E FIGURE 5.1 Standard Yee-FDTD lattice. σ are used:  = (  1 +  2 ) /2andσ = ( σ 1 + σ 2 ) /2. To ensure the stability of the time-stepping algorithm of (5.1.1) and (5.1.2), a time increment is chosen to satisfy the inequality ct ≤ 1  1/x 2 + 1/y 2 + 1/z 2 , (5.1.5) where c is the velocity of light in the computational space. Equation (5.1.5) will be derived in the next section. A Gaussian pulse E z = e −(t−t 0 ) 2 /T 2 is chosen as the excitation pulse and is imposed upon the rectangular region under the port to be excited. The finite difference mesh must be truncated because of the finite ability of com- puters to solve across very large or even infinite 3D volumes. The field components tangential to the truncation planes cannot be evaluated from the FDTD equations above since they would require for their evaluation the values of field components outside the mesh. The tangential electric field components on the truncation planes must be specified in such a way that outgoing waves are not reflected; this is known as an absorbing boundary condition (ABC). There are many ABCs, including the Mur absorbing boundary conditions [3] and the perfectly matched layer absorbing BASIS FDTD FORMULATION 193 boundary conditions (PML) of Berenger [4], among others. Here we have specified the boundary values of the fields according to the Engquist-Majda unconditionally stable, absorbing boundary condition [5] φ k+1 0 = φ k 1 + ct −x ct +x (φ k+1 1 − φ k 0 ), where φ 0 and φ 1 are the tangential electric field components at the mesh wall and at the first node within the wall, respectively. Figure 5.2 depicts a system consisting of three coupled microstrip lines. In the direction of signal propagation, the y direction, we have chosen the parameters t , x, y,andz such that the wave travels one spatial step in approximately five temporal steps; this choice in turn requires a priori calculation in order to obtain the approximate wave velocity in the direction of propagation. At the top and side boundaries, the local velocity of light at the calculated node is used as the approx- imate wave velocity. Without a loss of generality, the time domain solution for this six-port system is obtained by means of the following procedures: (1) Initialize (at t = kt = 0) all fields to 0. (2) Impose Gaussian excitation on port 1: • H k+(1/2) is calculated from the FDTD equations. • E k+1 is calculated from the FDTD equations. • The tangential E field is set to 0 on the ground plane and the absorbing boundary condition is used on the truncation planes. • Store port voltages V (1) i (kt) at the reference plane of port i (i = 1, 2, 3, 4, 5, 6), where a port voltage V i has been obtained by numerically integrating the vertical electric field beneath the center of port i. ref. plane L 1 ref. plane x z y 1 2 3 4 5 6 =3.5 d W W W S S t FIGURE 5.2 Coupled three-line system. 194 SAMPLING BIORTHOGONAL TIME DOMAIN METHOD (SBTD) • Store port currents I (1) i (kt) at the reference plane of port i (i = 1, 2, 3), where a port current I i has been obtained by numerical integration of the magnetic field around the strip surface of port i in the reference plane. • k → k +1, repeat the previous steps 1 through 5 until the pulse and induced waves pass through the reference plane of ports (4, 5, 6) completely. (3) Impose a Gaussian excitation on port 2 and repeat the above six procedures. Store all of the port voltages V (2) i (i = 1, 2, 3, 4, 5, 6) and port currents I (2) i (i = 1, 2, 3). In the previous items the superscripts (1) and (2) represented port 1 excitation and port 2 excitation, respectively. The Yee algorithm has been modified and extended into many versions and deriva- tives, including the nonuniform mesh FDTD, and the finite volume time domain (FVTD) method [6], nonorthogonal mesh [7], and the like. The transmission line matrix (TLM) method was proposed by Peter Johns [8] in 1971 independently of Yee’s work. Nonetheless, it was proven that the TLM is equivalent to FDTD method. In handling lossy structures, the TLM needs to use artificial “stubs”; this necessity is inconvenient. Because of its simplicity and popularity, only the standard FDTD will be discussed in the text. 5.2 STABILITY ANALYSIS FOR THE FDTD An unstable solution may occur owing to an improper choice of the time step t for the space intervals x, y,andz. The instability is not due to an accumulation of errors, but to causality. The analysis is conducted on plane waves, and is quite general, since any wave may be expressed as a superposition of plane waves. Let us write FDTD in terms of time–space eigenvalue problems. Space eigenvalues must be located in stable regions. The two curl equations in a lossless medium are written in their component forms ∇×H =  ∂E ∂t , ∂ E z ∂t = 1   ∂ H y ∂x − ∂ H x ∂y  , ∂ E y ∂t = 1   ∂ H x ∂z − ∂ H z ∂x  , ∂ E x ∂t = 1   ∂ H z ∂y − ∂ H y ∂z  , and ∇×E =−µ ∂H ∂t , STABILITY ANALYSIS FOR THE FDTD 195 ∂ H x ∂t = 1 µ  ∂ E y ∂z − ∂ E z ∂y  , ∂ H y ∂t = 1 µ  ∂ E z ∂x − ∂ E x ∂z  , ∂ H z ∂t = 1 µ  ∂ E x ∂y − ∂ E y ∂x  . In the rest of this section, we will only attack 2D problems. In doing so, we will be able to capture the essence of the algorithms without spending too much time and effort on tedious details. In 2D problems we will deal with only three rather than six equations. The extension of the 2D formulation into 3D problems is straightforward, but time-consuming. Consider a 2D TM (z) wave, namely ∂ ∂z = 0, H z = 0. The remaining three equations are ∂ E z ∂t = 1   ∂ H y ∂x − ∂ H x ∂y  , ∂ H x ∂t =− 1 µ ∂ E z ∂y , ∂ H y ∂t = 1 µ ∂ E z ∂x . Using the center difference Yee scheme and simplified notations, we obtain n+1 E z i, j − n E z i, j t = 1   n+(1/2) H y i+(1/2), j − n+(1/2) H y i−(1/2), j x − n+(1/2) H x i, j +(1/2) − n+(1/2) H x i, j −(1/2) y  , n+(1/2) H x i, j +(1/2) − n−(1/2) H x i, j +(1/2) t =− 1 µ n E z i, j +1 − n E z i, j y , n+(1/2) H y i+(1/2), j − n−(1/2) H y i+(1/2), j t = 1 µ n E z i+1, j − n E z i, j x . (5.2.1) C ASE 1. TIME EIGENVALUE PROBLEM Separating the time derivatives in the pre- ceding equations, we arrive at n+1 E z i, j − n E z i, j t = λ n+(1/2) E z i, j , (5.2.2) 196 SAMPLING BIORTHOGONAL TIME DOMAIN METHOD (SBTD) n+(1/2) H x i, j +(1/2) − n−(1/2) H x i, j +(1/2) t = λ n H x i, j +(1/2) , (5.2.3) n+(1/2) H y i+(1/2), j − n−(1/2) H y i+(1/2), j t = λ n H y i+(1/2), j . (5.2.4) The general form of (5.2.2) through (5.2.4) is n+(1/2) V i − n−(1/2) V i t = λ n V i . (5.2.5) Let us define a factor q i = n+(1/2) V i n V i . (5.2.6) In order to have a stable solution of (5.2.2) through (5.2.4), we must meet the condi- tion |q i |≤1. Substituting (5.2.6) into (5.2.5), we have n+(1/2) V i n V i − n−(1/2) V i n V i = λt, or equivalently q i − 1 q i = λt. Thus we obtain q 2 i − λtq i − 1 = 0, q i = λt 2 ±  1 +  λt 2  2 . (5.2.7) In order to have |q i |≤1, we need  Re{λ}=0 − 2 t ≤ Im{λ}≤ 2 t . Letting λ = µ + jν, (5.2.7) gives q i = j νt 2 ±  1 − (νt) 2 4 . C ASE 2. SPACE EIGENVALUE PROBLEM The right-hand side of (5.2.1) provides the following eigenvalue equations STABILITY ANALYSIS FOR THE FDTD 197 H y i+(1/2), j − H y i−(1/2), j x − H x i, j +(1/2) − H x i, j −(1/2) y = λ E z i, j , (5.2.8) E z i, j +1 − E z i, j y =−λµH x i, j +(1/2) , (5.2.9) E z i+1, j − E z i, j x = λµH y i+(1/2), j . (5.2.10) Again, a nonplane wave can be expanded into a superposition of plane waves. Thus we may work with the following plane waves: E z I,J = E z e j (k x I  x +k y J y) , H x I,J = H x e j (k x I  x +k y J y) , H y I,J = H y e j (k x I  x +k y J y) . (5.2.11) Substitution of (5.2.11) into (5.2.8–5.2.10) leads to E z = j 2 λ  H y x sin k x x 2 − H x y sin k y y 2  , H x =−j 2E z λµ y sin k y y 2 , H y = j 2E z λµ x sin k x x 2 , and λ 2 =− 4 µ   sin k x x/2 x  2 +  sin k y y/2 y  2  . Note that |sin(·)|≤1. Hence for ∀ k x , k y , Re{λ}=0 Im{λ}≤2v   1 x  2 +  1 y  2  1/2 . C ASE 3. NUMERICAL STABILITY Relating the time eigenvalue problem to the space eigenvalue problem, we have the 2D stability condition 2v   1 x  2 +  1 y  2  1/2 ≤ 2 t , 198 SAMPLING BIORTHOGONAL TIME DOMAIN METHOD (SBTD) namely t ≤ 1 v  ( 1/ x ) 2 + ( 1/y ) 2 . For 3D cases, the stability condition is t ≤ 1 v  ( 1/ x ) 2 + ( 1/y ) 2 + ( 1/z ) 2 =  v √ 3 , if x = y = z = l. For 1D, this condition reduces to t ≤ x v . 5.3 FDTD AS MAXWELL’S EQUATIONS WITH HAAR EXPANSION The finite difference time domain (FDTD) formulas of (5.1.3) to (5.1.4) are derived from the two Maxwell curl equations, using the finite difference to approximate the differential operators. In this section we will see that the FDTD can be derived as a special case of wavelet expansion using the Haar system. To simplify our mathematical notation without losing generality, we consider the one-dimensional case, namely the telegraphers’ equations in the frequency domain. The telegraphers’ equations are        − dI dx = jωCV − dV dx = jωLI. (5.3.1) When the finite difference method is applied, the expected result is        I n+1 − I n x =−j ωCV n+(1/2) V n+(1/2) − V n−(1/2) x =−j ωLI n . (5.3.2) Let us expand the unknown current and voltage in terms of Haar scalets, that is, pulse functions  I =  m I m P m (x) V =  m V m+(1/2) P m+(1/2) (x). (5.3.3) [...]... centralized finite difference expression of (5.3.2) Notice that the derivation is totally new and never makes use of the finite difference concept 5.4 FDTD WITH BATTLE–LEMARIE WAVELETS Battle–Lemarie wavelets possess better regularity than Haar wavelets The Battle– Lemarie based time domain method, referred to as the multiresolution time domain (MRTD), improves numerical dispersion of the FDTD significantly [9]... +(1/2) (x) d x ∂x = P x ∂ −m P x ∂x = P ∂ x −m H x ∂x −H = P x 1 − m + x 2 x −m x = δm,m − δm,m +1 1 δ x x 1 − m + x 2 x 1 − m + x 2 − 1 2 dx + 1 2 dx x −m x −δ x − (m + 1) x dx FDTD WITH BATTLE–LEMARIE WAVELETS 201 Thus far we have sufficient knowledge to derive the finite difference equation (5.3.2) By substituting (5.3.3) into (5.3.1), we obtain − Vm+(1/2) m d Pm+(1/2) (x) = jωL dx Im Pm (x) m Multiplying... = P x −k , x where P(x) =  1   1 2   0 if |x| < 1 2 1 2 1 2 if |x| = if |x| > It can easily be seen that ∞ −∞ Pk (x)Pl (x) d x = ( x) δk,l , (5.3.4) which is analogous to the orthogonality for wavelets ∞ −∞ ϕ j,k (x)ϕ j,l (x) d x = δk,l Show (1) If k = l, Pk (x) and Pl (x) have no overlaps; hence ∞ −∞ Pk (x)Pl (x) d x = 0 (5.3.5) (2) If k = l, ∞ −∞ Pk (x)Pl (x) d x = P2 x − k dx x u = x = x... dωϕ(ω)e−iωm+iωx ˆ ∂ ϕ(ω )e−iω [m +(1/2)] eiω x ˆ ∂x ∞ dω dω ∞ −∞ −∞ dx 1 i x(ω+ω ) iω ˆ e ϕ(ω )ϕ(ω)e−i(ωm)−iω [m +(1/2)] ˆ 2π 2π dωϕ(ω)e−iωm ˆ dω δ(ω + ω )(iω )ϕ(ω )e−iω [m +(1/2)] ˆ FDTD WITH BATTLE–LEMARIE WAVELETS = ∞ 1 2π 1 = π −∞ ∞ 203 dωϕ(ω)e−i(ωm) (−iω)ϕ(−ω)eiω[m +(1/2)] ˆ ˆ ω|ϕ(ω)|2 sin ω m − m + ˆ 0 1 2 dω This integral can be evaluated numerically We can rewrite the expression above as ∞ −∞ ϕm (x)... method in Section 5.4 shows an excellent capacity to approximate a precise solution, even at a rate near the Nyquist sampling limit However, in the MRTD the nonsampling properties of the Battle–Lemarie wavelets make the formulation difficult to compute For instance, the two-term finite difference expression in the FDTD has been extended to 18 terms in the MRTD The field quantity at a given node is the sum . of the finite difference concept. 5.4 FDTD WITH BATTLE–LEMARIE WAVELETS Battle–Lemarie wavelets possess better regularity than Haar wavelets. The Battle– Lemarie based time domain method, referred. with Maxwell’s two curl equa- tions µ ∂H ∂t =−∇×E, (5.1.1)  ∂E ∂t + σ E =∇×H. (5.1.2) 189 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons,. computational time. In these regard wavelets offer significant improvements to the FDTD. It will be shown that the Yee-based FDTD is identical to the Galerkin method using Haar wavelets. Since the Haar bases