Radio Science

Theory, analytical investigation, and performance of the complementary derivatives method for reducing reflection errors from nonuniform grid domains in finite difference methods

Authors


Abstract

[1] The central finite difference is used very often to approximate first-order differential equations, and it results in a second-order truncation error for a uniform grid size. Nonuniform grids are used for simulating structures with large aspect ratios or problems with large field gradients in order to improve computational efficiency. However, changing the grid size increases the truncation error at the interface between domains having different grid sizes. The error at the interface is manifested as a spurious reflection from the grid boundary, thus decreasing the simulation accuracy. The complementary derivatives method (CDM) was originally introduced as a robust discretization technique to eliminate any spurious errors arising from the changing grid sizes. In this paper, we review the theory of the CDM. We investigate the CDM analytically for the one-dimensional case and derive the fundamental modes of propagation in the numerical solution of the differential equation. Then, we calculate the reflection coefficient from the interface of two domains having different grid sizes with and without the CDM. Different representative numerical examples also demonstrate the efficiency of the CDM in reducing the reflection from the grid boundary and improving the simulation results in different applications.

1. Introduction

[2] Nowadays, there is a widespread demand for high accuracy simulations needed to provide results which can parallel the accuracy of recent measurement equipments (e.g., 145 dB for R&S ZVA24 network analyzer from Rohde & Schwarz, and 136 dB for PNA series Microwave Network Analyzers from Agilent), can measure the shielding effectiveness up to 120 dB, or can measure the reflections from the absorbers that goes below 60 dB. When using the finite difference method [Yee, 1966; Taflove and Hagness, 2005; Mitchell and Griffiths, 1980], the simulation domain is discretized and then the differential equations are approximated with difference equations. The central finite difference is used very often to approximate the first-order differential equations and it results in second-order accuracy. Nonuniform grids are also used frequently, well suited for simulating structures with large aspect ratios or problems with large gradients, and yield greater computational efficiency. However, using different grid sizes increases the truncation error at the interface between domains having different grid sizes. The augment of truncation error at the interface is manifested as a spurious reflection from the grid boundary, thus decreasing the simulation accuracy.

[3] The grid size is chosen by considering two criteria: First, the grid size must resolve all structures in the simulation domain and, second, it must also resolve the highest frequency present in the simulation. Using a uniform grid suffers from flexibility when electric or magnetic fields (or both) have large gradients within a limited volume. For example, monolithic microwave integrated circuits (MMIC), in the vicinity of current sources, sharp edges and corners of conductive and dielectric objects, simulation of vias and bond wires in high-frequency electronic packages, and detailed, high dynamic range simulation in biomedical applications, need a very fine mesh to resolve the abrupt changes of fields in a very small volume. Considering the limitation of the computer memory and speed, a very fine mesh for the whole domain renders this method not very attractive for this large class of electromagnetic problems. A more computationally feasible solution would be to use refined mesh in certain subregions. The computational domain size is reduced by using different grid sizes in the simulation domain such that a coarse mesh is used for discretizing the domains with small field variations and a fine mesh is used for discretizing the domains with large field variations.

[4] Using different grid sizes improves the computational efficiency of the simulation, however, at the expense of increasing the truncation error at the interface between domains having different grid sizes. The truncation error at the interface is manifested as a spurious reflection from the grid boundary. Several papers have introduced different methods to reduce this spurious reflection or grid-truncation error [Sheen, 1991; Lee, 1990; Li et al., 1993; Xiao and Vahldieck, 1993; Jiang and Arai, 1998; Namiki and Ito, 2001; Okoniewski et al., 1997; Heinrich et al., 1996]. In the work of Sheen [1991], the grid size in the second domain is reduced by one-third of the grid size in the first domain and spatial derivatives of the fields at the interface are expressed by central difference approximations to achieve second-order accuracy. However, in this method, the reduced grid size is limited to specific numbers which limits its applicability to specific geometries that conform to specialized grid. In the works of Lee [1990] and Li et al. [1993], the derivative of magnetic field at the electric field position is approximated by fitting a second-degree polynomial to the magnetic fields at three points. The coefficient in the error term of this approximation, however, is large, which limits the grid size reduction factor. In the work of Xiao and Vahldieck [1993], two methods were introduced to maintain the second-order accuracy. One method uses a specific mesh ratio between two regions to obtain the central finite differences, while the second uses a universal grading scheme with continuously variable lattice size, however, a demonstration of the performance of these method were not reported. In the work of Jiang and Arai [1998], the computational accuracy was improved by interpolating the magnetic field components between the fine mesh and coarse mesh, which cannot guarantee the second-order accuracy. In the work of Namiki and Ito [2001], a high-order implicit scheme was enforced at the boundary to reduce the truncation error. In the work of Okoniewski et al. [1997], a numerically derived three-dimensional subgridding scheme was introduced but without theoretical limits on its potential. In the work of Heinrich et al. [1996], the characteristic impedances of two waveguides with infinitely thin and square center conductors are calculated using conformal mapping and mode matching, respectively. Then, the values of minimum grid size and amplification factor were optimized to obtain good approximations of the characteristic impedances using the FDTD method. But there is no explanation whether these values can also be used for other experiments or whether each experiment needs different optimized values, thus limiting the generality and practicality of this method. More recently, two new methods were presented by Bérenger [2006] and Pierantoni and Rozzi [2007] that were based on Huygens' equivalent currents and Hilbert Space formulations, respectively. While these methods do not suffer from the constraints of earlier methods, the error bounds arising from the nonuniform grid domains have yet to be investigated.

[5] In this paper, first we review the theory of the complementary derivatives method (CDM) [Kermani and Ramahi, 2006a, 2006b, 2005], we then present a full analytical investigation of the method for the one-dimensional case. We derive the fundamental modes of propagation in the numerical solution of the differential equation and calculate the reflection coefficient from the interface of two domains having different grid sizes when CDM is applied. The performance of the CDM is demonstrated by showing a comparison between the reflection coefficients for the numerical solution of the wave equation, calculated with and without the CDM.

2. Theory of Complementary Derivatives Method

[6] In this section, and without loss of generality, we demonstrate the application of the CDM on the classical FDTD method based on the Yee scheme [Yee, 1966] for solving Maxwell's equations. Maxwell's curl equations for an isotropic, source-free medium with permittivity of ɛ and permeability of μ are:

equation image
equation image

First we consider the one-dimensional case. The arrangement of electric and magnetic fields along, say the x axis, is shown in Figure 1. The updating equations are based on discretizing the derivative operators in time and space. The first-order time derivative is discretized using the central difference scheme achieving second-order accuracy. The same procedure is used for the space derivatives and second-order accuracy is similarly achieved if the grid size remains unchanged.

Figure 1.

The arrangement of E and H fields along the x axis in the FDTD domain based on the Yee scheme. The grid size changes from Δ to ΔR at grid boundary.

[7] To this end, let us construct a computational domain composed of two regions with boundary x0 as the interface between the regions (shown by dashed line in Figure 1). We assume that the cell size changes from Δ to ΔR = Δ′ at the interface. Using Taylor series expansion, we express the magnetic fields on both sides of the boundary as

equation image
equation image

By subtracting (3) from (4), and after some algebraic manipulations we can write the derivative of H′(x0) as

equation image

[8] Next, we construct a second domain in which the grid size changes from Δ to ΔR = Δ″. Using identical procedure in finding the derivative of the magnetic field, H′(x0), for the new cell size, we have

equation image

As we see in (5) and (6), the truncation error is first order. The arithmetic average of (5) and (6) can be obtained as follows:

equation image

As a sufficient condition for canceling the first-order truncation error, the third term on the right-hand side of (7) should be zero, resulting in the following identity:

equation image

[9] We define complementary derivatives as the two derivatives defined at the interface of two different computational domains of the same structure. Both domains have identical grid size, henceforth referred to as the common grid size, on one side of the interface (to the left when considering Figure 1). On the other side of the interface, the grid sizes are different subject to the condition that their arithmetic mean is equal to the common grid size. Averaging the two complementary derivatives at the interface achieves second-order accuracy.

[10] The above procedure requires two separate simulations which is computationally unattractive. A single simulation implementation of the CDM is possible as explained next.

[11] Let us consider a computational domain with two different grid sizes separated by an interface as shown in Figure 2. A second-order accurate E field interpolation at the interface (x = x0) is obtained using the symmetric H fields at x = x0 − Δ/2 and x = x0 + Δ/2 as

equation image

Here, we assumed that there is an H field node at x = x0 + Δ/2. In the case where an H field node does not coincide with the location x = x0 + Δ/2, we make use of the H fields at two nodes that exist, at x = x0 + (Δ/2−δL) and x = x0 + (Δ/2 + δR). To see how this is accomplished, we express the derivative at x0 using two different differencing schemes. The first expression for the derivative uses the points x = x0 − Δ/2 and x = x0 + (Δ/2−δL), resulting in

equation image

The second expression for the H field derivatives uses the points x = x0 − Δ/2 and x = x0 + (Δ/2 + δR), resulting in

equation image

The arithmetic mean of (10) and (11) gives:

equation image

To cancel the first-order truncation error, the third term of the right-hand side of (12) should be zero. To achieve this, we require that

equation image

Therefore, the magnetic fields which are used in calculating the complementary derivatives should be in the same distance from the point x = x0 + Δ/2. Now, we try to find a straightforward technique to translate the equation (13) as a numerical recipe. First, we suppose that the grid size changes from Δ to Δ′ (Figure 2). The position of H field in the kth cell (as measured from the interface) of the refined grid section can be written as

equation image

If we assume that the H fields of the k1th and k2th (k1 < k2) cells are used in calculating the complementary derivatives, δL and δR, as defined in Figure 2, can be written as

equation image
equation image

Enforcing (13), we have

equation image

If we set Δ′ = αΔ, where α is defined as the cell size reduction factor, and after some manipulation, we have:

equation image

which gives the possible choices for cell size reduction factors. Note that k1 and k2 do not necessarily represent adjacent cells. For example a reduction factor of α = 1/4 results in k1 + k2 = 3, which we can have two different sets of complementary methods of (1) k1 = 0, and k2 = 3 or (2) k1 = 1, and k2 = 2 (Figure 3).We also observe that the grid size in Figure 2 is changing from Δ′ to Δ ( Δ′ < Δ ). Similarly, the CDM can also be used when the grid size increases at grid boundary (Δ′ > Δ ). The cell size increase factor can be β = k1 + k2 + 1.

Figure 2.

FDTD E and H field nodes used for the implementation of CDM in one simulation.

Figure 3.

Two different sets of complementary derivatives for a grid size reduction factor of α = 1/4.

[12] To implement the CDM in the two- and three-dimensional FDTD formulations when the grid sizes only change in two directions, we use a similar procedure to the one-dimensional case. Consider the following partial differential equation

equation image

Suppose that the grid size in the x direction changes from Δx to ΔxR = Δx′ at x0 and the grid size in the y direction changes from Δy to ΔyR = Δy′ at y0 (Figure 4). Discretizing (18) at (x0, y0) using the central difference scheme results in

equation image

Similar to the one-dimensional case, we assume there is another domain in which the grid size in the x direction changes from Δx to ΔxR = Δx″ at x0 and the grid size in the y direction changes from Δy to ΔyR = Δy″ at y0 (Figure 4). Discretization of (18) at (x0, y0) gives

equation image

Arithmetic average of (19) and (20) gives

equation image

The truncation error can be improved to the second-order if the relation between grid sizes in the x direction is

equation image

and the relation between grid sizes in the y direction is

equation image
Figure 4.

Two-dimensional discretized structure. The grid size in the x direction changes from Δx to ΔxR at x0 and in the y direction changes from Δy to ΔyR at y0.

[13] We can use the same procedure to implement the CDM even if the grid sizes are changing in three directions. But in the Maxwell's equations, each updating equation of the electric field (magnetic field) components uses the magnetic field (electric fields) components in only two different directions. In other words, a two-dimensional grid is needed to update each electric, or magnetic field components. Therefore, implementing CDM is required for at most two directions in the Maxwell's equations. It is clear if in the Yee lattice, we select the magnetic fields on the grid boundaries and the electric fields inside the cells, the same procedure can be used to develop the CDM and it will be applied on the electric fields.

[14] To demonstrate the performance of CDM, several experiments are presented. In the first experiment we consider a one-dimensional domain of length 750 mm and apply the second-order Higdon's absorbing boundary condition [Higdon, 1986] at both ends of the computational domain to isolate any terminal reflections. The source is positioned 250 mm from the left domain boundary; the monitor point is selected 375 mm from the left domain boundary, 125 mm from the source (Figure 5). The excitation is a differentiated Gaussian pulse, given by

equation image

where td = 100 ps and τ = 20 ps. The grid size in the left side of grid boundary is Δ = 250 μm and on the right size of grid boundary is refined to Δ′ = 125 μm (Figure 5). The simulation result, showing the reflection from the boundary of domains having different grid sizes, is given in Figure 6.

Figure 5.

Simulated structure with different grid sizes. The grid size on the left side of the grid boundary is Δ = 250 μm and on the right side of the grid boundary is Δ′ = 125 μm.

Figure 6.

Reflected Ez from boundary of two domains with different grid sizes of Δ = 250 μm and Δ′ = 125 μm without applying CDM.

[15] Next, we apply the CDM at the boundary of domains. Since the grid size reduction factor is 1/2, the H fields of the first and second cells are used as complementary derivatives (refer to equation (17)). This simulation result is shown in Figure 6. As we see in Figure 6, the application of CDM reduces the reflection from grid boundary by approximately two orders of magnitudes, amounting to an improvement in the simulation accuracy by approximately 40 dB.

[16] In the second numerical experiment, we examine the case in which different sets of complementary derivatives can be utilized. Here, the one-dimensional FDTD domain is 340 mm long and the second-order Higdon's absorbing boundary condition [Higdon, 1986] is applied at both ends of the computational domain to isolate any terminal reflections. The source is positioned at 120 mm from the left domain boundary and a monitor point is selected 50 mm away from the source (Figure 7). The same excitation as the previous experiment is used. The grid size on the left side of the grid boundary is Δ = 200 μm and on the right side of the grid boundary is Δ′ = 50 μm (Figure 7).

Figure 7.

Simulated structure with different grid sizes. The grid size on the left side of the grid boundary is Δ = 200 μm and on the right side of the grid boundary is Δ′ = 50 μm.

[17] In this experiment, the grid size reduction factor is α = 1/4. Figures 3a and 3b show the two different sets of points that can be used as complementary derivatives for a grid size reduction factor of α = 1/4. Figure 8 shows the reflected electric field from the boundary of domains when CDM was not applied and for cases when CDM was applied (different discretization densities) at the grid boundary. As we see in Figure 8, using different methods of CDM have reduced the reflected signal from the boundary between the domains by almost two orders of magnitude, amounting to an improvement in accuracy by approximately 40 dB.

Figure 8.

Reflected Ez from boundary of two domains with different grid sizes of Δ = 200 μm and Δ′ = 50 μm without applying CDM.

[18] In the next numerical experiment, the CDM is applied to the problem of a partially filled parallel plate waveguide of dimensions 420 mm × 30 mm. First, we consider a uniform grid size in the entire computational domain of the guide with Δx = Δy = 1 mm (Figure 9). The numerical results obtained from this case will be considered as the reference solution (Eref). The z-polarized source is positioned at (180 mm, 15 mm) and the monitor point is selected at (200 mm, 15 mm). The second-order Higdon's absorbing boundary condition [Higdon, 1986] is applied on boundaries at x = 0 and x = 420 mm. The parallel plate waveguide is partially filled by a material of ɛr = 10 and the width of 50 mm (Figure 9). The rest of waveguide is empty (ɛr = 1). The excitation is a modulated Gaussian pulse

equation image

with td = 480 ps, τ = 160 ps, and fm = 10 GHz.

Figure 9.

Simulated structure as reference. The grid size is uniform (Δx = Δy = 1 mm) throughout the computational domain. Higdon's absorbing boundary condition (ABC) is applied at the right and left terminals of the domain.

[19] Next, we solve the same problem, but decrease the cell size to the right of an interface positioned at x = 220 mm (Figure 10) to Δx′ = 0.5, 0.25 and 0.125 mm, corresponding to reduction ratios of 1:2, 1:4 and 1:8, respectively. The cell size in the y direction, locations of the source and monitor point, and excitation pulse are all unchanged.

Figure 10.

Simulated structure with different grid sizes. The grid size in the y direction is uniform (Δy = 1 mm) on the entire domain. The grid size in the x direction changes from Δx = 1 mm to Δx′ = 0.5, 0.25, and 0.125 mm. Higdon's absorbing boundary condition (ABC) is applied at the right and left terminals of the domain.

[20] We define the normalized error as

equation image

In Figure 11 we show the normalized error as a function of time for the cases with and without the application of CDM. A significant reduction of error in the E field is observed when CDM is applied. Furthermore, it is observed that the error resulting from the application of CDM decreases as the ratio of the grid sizes across the interface approaches unity. While this can be intuitively expected, it can be predicted by the analysis presented in section 3 below. However, it is important to note that the variance in the error as a function of the grid sizes ratio is low. Further observation of equation (12) shows that the error terms O(δ2) decreases as δ decreases, which explains why the errors remains bounded as shown in Figure 11.

Figure 11.

Normalized error in the E field as obtained using the standard FDTD interpolation scheme with and without CDM.

[21] In the next experiment we demonstrate the performance of CDM in a three-dimensional simulation domain. The simulation domain is a box with dimensions of 140 mm × 390 mm × 140 mm (Figure 12). We set the second-order Higdon's absorbing boundary condition [Higdon, 1986] on the outer surfaces of our domain. The source is located at (70 mm, 50 mm, 70 mm); the monitor point is at (70 mm, 55 mm, 70 mm). The excitation is a Gaussian pulse

equation image

with td = 80 ps and τ = 20 ps. The grid size is Δ = 0.5 mm and the simulation result for this discretization will be considered the reference solution. In the next step, we increase the grid size on the left side of grid boundary (in the y direction) to 1 mm. The grid size on the right side of the grid boundary and all measurements are unchanged. In Figure 13, we show the normalized error (26) as a function of time for cases with and without the application of CDM. A significant reduction of reflection is observed.

Figure 12.

Structure of the three-dimensional simulation domain.

Figure 13.

Normalized error in the E field as obtained using the standard FDTD interpolation scheme with and without CDM.

3. Analytical Investigation of the CDM

[22] Our strategy in investigating the CDM analytically is to start with the fundamental modes of propagation in the numerical solution of governing equation. In the work of Vichnevetsky [1981a], the fundamental modes of propagation in the numerical solution of advection equation are calculated. We use the same procedure to calculate the fundamental modes of propagation in the numerical solution of wave equation. Next, the calculated fundamental modes of propagation are used to calculate the reflection coefficients from the grid boundary of two domains with different grid sizes, using the same method as Vichnevetsky [1981b].

3.1. Fundamental Modes of Propagation in the Numerical Solution of the Wave Equation Using the Leapfrog Scheme

[23] The one-dimensional wave equation can be written as:

equation image

where c is the wave speed. The wave equation, which is second order in space and time, can be written as two coupled first-order equations:

equation image
equation image

The leapfrog scheme can be used to solve these equations numerically using the Yee algorithm [Yee, 1966]. To investigate them analytically, we apply the finite difference semidiscretization scheme where only the spatial derivative is approximated with a difference equation resulting in (see Figure 14):

equation image
equation image

where

equation image
equation image

and {uj(t)} ≅ {U(xj,t)}, and {vj+1/2(t)} ≅ {V(xj+1/2,t)}.

Figure 14.

Leapfrog scheme to solve the wave equation.

[24] Next, we define the Fourier transforms equation imagej(Ω) and equation imagej+1/2(Ω) of the semidiscrete numerical solutions {uj(t)} and {vj+1/2(t)}. We assume that {uj(t)} and {vj+1/2(t)} are in the L2 space or square integrable throughout this work, which means the L2 norms

equation image

and

equation image

are finite. Therefore, the Fourier transforms equation imagej(Ω) and equation imagej + 1/2(Ω) of the semidiscrete numerical solutions {uj(t)} and {vj+1/2(t)} exist and are defined as

equation image
equation image

Substituting the Fourier transforms of (31) and (32) we have:

equation image
equation image

After some algebraic manipulation, we have

equation image

First, we suppose that the grid is uniform, which hj = equation imagej = Δ for all j values (Figure 15). Therefore (39) reduce to

equation image

The solution of this recurrence equation can be achieved by seeking the fundamental solutions. To calculate the fundamental solutions we suppose that

equation image

Substituting (41) into (40) results in

equation image

Therefore, equation image(Ω) must satisfy the characteristic equation

equation image

The roots of (43) or the characteristic ratios given by

equation image
equation image

Therefore, we can conclude that the numerical solution of (28) can be expressed as

equation image

which has two fundamental solutions. These two different fundamental solutions describe different propagation properties. One solution, which has the characteristic ratio of

equation image

describes the rightward propagating wave. The other solution has the characteristic ratio of

equation image

Describing the leftward propagating wave.

Figure 15.

Leapfrog scheme to solve the wave equation. A uniform grid is used.

3.2. Calculation of the Reflection Coefficient From Grid Boundary in the CDM Treatment of Grid Boundary in the Wave Equation

[25] In this section, we calculate the reflection coefficient when the CDM is implemented at the interface. At the interface of two grids with different sizes, the semidiscretization approximation (31) is modified to

equation image

Taking the Fourier transform of (49), we have

equation image

which simplifies to

equation image

The Fourier transforms of the following finite difference semidiscretization approximations

equation image
equation image
equation image

are given respectively by

equation image
equation image
equation image

Substituting (54) and (55) in (56), we have

equation image

which is the Fourier transform of the CDM treatment at the interface.

[26] Using the same argument as in the work of Vichnevetsky [1981a], only three fundamental solutions exist. The same procedure as in the works of Vichnevetsky [1981a, 1981b] is used to calculate the reflection coefficient. First, we let equation image0, equation image0, and equation image0 denote the fundamental solutions at the interface. The continuity at the interface X = X0 gives

equation image

We assume equation image1, and equation image2 are the characteristic ratios corresponding to solutions equation imagej, equation imagej to the left of the origin defined by (47) and (48), respectively. Also, suppose that equation imageΔ′ and equation imageΔ″ are the characteristic ratios of the rightward propagating solutions to the right of the origin with grid sizes of Δ′ and Δ″, respectively, given by:

equation image
equation image

To calculate the reflection coefficient, we express equation image1, equation image1 and equation image−1 in terms of equation image0 and equation image0 as

equation image
equation image
equation image

By substituting (58), (61), (62) and (63) into (57), we have

equation image

which results in

equation image

The reflection coefficient for the case without the CDM can be found using an identical procedure, leading to

equation image

[27] To show the performance of CDM for the wave equation, we have compared the reflection coefficients from the interface of two grids having different sizes for the cases with and without CDM. These results are presented in Figure 17. The chosen frequency of the propagating wave is 1 GHz and the grid size to the left of origin (considered at u0 in Figure 16) is Δ = λ/20 (λ is the wavelength at 1 GHz). The two complementary parts have grid sizes of Δ′ = Δ−γ × Δ (refined grid) and Δ″ = Δ + γ × Δ (coarsened grid) for 0 < γ < 1, which satisfy the complementary condition of Δ = (Δ′ + Δ″)/2. As we see in Figure 17, the CDM has significantly reduced the reflection coefficient.

Figure 16.

Implementing CDM on a nonuniform grid.

Figure 17.

Reflection coefficient of the wave equation at grid boundary plotted as a function of the mesh ratio; refined mesh, coarsened mesh, and the CDM treatment of grid boundary.

4. Conclusions

[28] In this work, we presented the complementary derivatives method (CDM) to reduce the reflection error arising from nonuniform grid. The primary advantage of the CDM in comparison to previously developed methods is that it achieves second-order accuracy irrespective of the mesh reduction (or enlargement) ratio. Several numerical experiments were presented which showed a dramatic decrease in the reflected error with an improvement of two orders of magnitude in comparison to the cases where the CDM was not applied. Finally, we presented analytic investigation, for the one-dimensional problem, of the reflections arising at the interface between two different-size grid domains and derived the fundamental modes of propagation in the numerical solution of the differential equation.

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