- Slob, E.C.
- Impulse Time-Domain Electromagnetics of Continuous Media
- Impulse Time-Domain Electromagnetics of Continuous Media - Alex Shvartsburg - Google книги
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In this section, a series of CPML absorption experiments is carried out in a vacuum with relatively high-frequency pulses. Because the dispersion is confirmed when using conventional CPML, it is indispensable to improve CPML parameter settings for qualitative understanding. To verify the existence of dispersion and the feasibility of parameter setting in a CPML, numerical simulation experiments in a 3D vacuum were implemented.

As shown in Fig. The maximum frequency of the source is close to the frequency that satisfies numerical dispersion conditions in a uniform grid. This scheme highlights the impact of dispersion from stretching grids. The time of a1 — c1 is 0. Corresponding to Fig. Figure 3 displays snapshot aggregations of Hz the Z component in the magnetic field combining 3 groups of parameter settings pa, pb, pc and 4 moments. The XY plane of the snapshots is perpendicular to the Z coordinate in a vacuum and at the same depth as the transmitting coil.

The color bar indicates the values of Hz. Figure 3 a1—c1 displays the moment that the Gauss pulse has been transmitted and electromagnetic waves have just entered the inner CPML, where there is no obvious reflection from the absorption boundary. The wave field characteristics for these three different parameter settings are almost the same, and the wave fronts of Hz are circular, which indicates that the dispersion is negligible in the internal forward area with a uniform grid.

## Slob, E.C.

For the next moment in time see Fig. The lines p1, p2 and p3 indicate that the coordinates of the head wave and p1 and p2 are closer to the PEC than p3, which indicates that the propagation velocity of the electromagnetic wave is slower in Fig. This effect is demonstrated in Fig. The time when the head wave arrives shows that the propagation velocity in conditions pa and pb are the same, although faster than in pc.

Then, at the moment in time of Fig. This phenomenon is the typical dispersion that the electromagnetic field is continuously inducing after the wave front has gone through the grids. If the grids are in a vacuum, the components of the electromagnetic field will increase exponentially. In fact, parameter settings of pa, pb and pc share the same conductivity, which will produce equal damping effects on an electromagnetic wave and dynamic equilibriums on absorption and dispersion. Relative to Fig.

Regardless, the inference of Eq. The wave field is similar to Fig. When we compare the dispersions between Fig. Next, we employ the energy of the entire wave field to estimate the dispersion standard and absorption quantity in the CPML see Fig. Low energy implies the successful absorption of the CPML after the source is turned off the moment of T1.

Within the time that the head waves propagate from the inner CPML to the outer CPML the period from T1 to T2 , the decline rates of energy in pa-pf gradually accelerate and differ little from each other because of the gradually enhanced absorption efficiency in the CPML, whereas the dispersion is not at all prominent. Shortly afterwards, the head waves reflect to the inner CPML the period from T2 to T3 , the energy gradients reach maximum and the curves of pa-pf begin to appear significantly different when electromagnetic waves are mainly absorbed by the outer CPML and different dispersions appear.

The energy gradients of pa-pf continuously decrease after the moment of T3; moreover, the curves are significantly distinguishable and comprehensively display the absorption effects from various parameters. To answer these questions, the next section will provide the interpretation employing a low-frequency pulse.

A severe challenge to low-frequency electromagnetic modeling is the large skin depth. Although CPML can obtain wonderful absorption characteristics for low-frequency electromagnetic waves, it is beyond their abilities to ideally absorb electromagnetic waves through only 8 layers. Therefore, experiments were conducted to evaluate the absorption effects of the CPML with a low-frequency pulse in a vacuum. CPMLs are generally set as 8 layers due to the restrictions of computer memory. The modeling areas are similar to the high-frequency experiments. The Hz component in Fig.

It is meaningful to highlight the approximation between modeling results and analytical solutions; accordingly, the relative error 13 is introduced:. According to Error t in Fig. Based on Fig. Finally, the general optimal parameters for the Gauss pulse are as follows:. Therefore, the CPML guarantees that the main energy of a low-frequency electromagnetic pulse can be absorbed, taking into account the high-frequency component.

How then do CPML impacts the absorption effect of relatively low-frequency electromagnetic waves on earth? Although mathematical derivation may be the most rigorous choice, the reality is that hundreds of reflections occur during a single pulse period and in the target area of the crosswell EM.

Moreover, a large number of parameters with complex combinations are highly variable in the low-frequency experiments and derivation from formulas for the evaluation of absorption effects encounters discrepancies when compared with realistic modeling. The grid number n from the transmitting coil to the innermost CPML was preset because of the elaborate degree of grids and the restricted size of the modeling area.

Therefore, n was treated as constant. As is shown in Fig. Additionally, several significant variables were extracted from Fig. These variables established corresponding standards 1—4 in Fig. Standards 1—3 are based on relative errors whose logarithm operation emphasizes the minimums of modeling error.

## Impulse Time-Domain Electromagnetics of Continuous Media

Figure 7 a1—c1 and Fig. Logarithmic curves C1—C6 connect the minimum or cool color areas in each figure. The area of cool color increases and the whole relative error decreases when n increases. However, because the amplitude of Hz is inversely proportional to the third power of distance, the increase in n causes a decrease of Hz that is reflected from the PEC; therefore, the minimum relative error in Fig.

As a result, it was effective to reinforce the absorption of the primary field by expanding the modeling area. Moreover, Fig. Figure 7 a3—c3 and Fig. A difference from the primary field is that the areas of minimum relative error are mostly distributed as isosceles triangles whose vertex angles in each figure can be joined as a straight line L1—L6.

Therefore, the dispersion remains tiny in low-frequency conditions referred to Eq. In contrast with Fig. Additionally, the overall relative error in Fig. The outlines of Fig. Unfortunately, the conclusions of Fig. As a solution, the results of standards 1—4 were weighted averaged see Fig. The interpretation of this discovery can be seen in Fig. This conclusion also verifies the validity of Eqs 25 and The discussions above organize the strategy of optimal parameters in a CPML. Essentially, the electromagnetic energy is converted into heat energy.

Therefore, the CPML warps the space compared with a uniform grid in a conventional FDTD and the dispersion is an inherent characteristic of the tectonically warped space-time.

## Impulse Time-Domain Electromagnetics of Continuous Media - Alex Shvartsburg - Google книги

Because of the restriction on grid numbers resulting from the memory limits of computers, space-time in CPMLs warps stepwise with slight interfacial reflection. Moreover, the forward results in CPMLs show that the span of a pulse diminishes and a blue shift appears. Nonetheless, we believe that an ingenious scheme will be developed to achieve ideal absorption by simply warping space-time. Considering the relatively low transmitting frequency and small grid interval of the crosswell EM, we used the CPML as an absorbing boundary condition of an FDTD and derived the optimal parameters of the CPML to strengthen the advantages of the absorption of low-frequency electromagnetic waves.

High-frequency experiments proved that dispersion in the CPML exists, especially in the outer layer. Further experiments involving low frequencies established the evaluation standard for absorbing boundary conditions. How to cite this article : Fang, S. Crosswell electromagnetic modeling from impulsive source: Optimization strategy for dispersion suppression in convolutional perfectly matched layer.

We thank anonymous reviewers for their helpful comments.

Author Contributions S. All authors reviewed the manuscript.

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Published online Sep 2. Author information Article notes Copyright and License information Disclaimer. Received Apr 20; Accepted Aug This work is licensed under a Creative Commons Attribution 4. Abstract This study applied the finite-difference time-domain FDTD method to forward modeling of the low-frequency crosswell electromagnetic EM method. Methods First, this section introduces the role and evaluation of pivotal CPML parameters that enable the absorption results to satisfy crosswell EM requirements. Open in a separate window. Figure 1. Figure 2. Results Because the dispersion is severe in conventional CPML, we should take actions to suppress it without sacrificing too much absorption efficiency.

The analytical solution of a magnetic dipole pulse To analyze the absorption mechanisms of CPMLs, it is necessary to exclude medium influences during forward modeling; therefore, it is more dialectical to deduce the analytical solution in a vacuum. The dispersion law from relatively high-frequency modeling experiments In this section, a series of CPML absorption experiments is carried out in a vacuum with relatively high-frequency pulses.

Figure 3. Snapshots of absorption effects in CPMLs at the same depth as the emission source.