The Time-Domain Method of Moments

Most moment method codes solve integral equations in the frequency domain, but it is also possible to use the method of moments to solve time-domain integral equations [1-30]. Consider a perfect electric conductor in free space excited by an incident field . This incident field induces a current on the surface S of the conductor that in turn radiates a scattered field. Enforcing the boundary condition on the total magnetic field or electric field on S gives rise to a time domain magnetic field integral equation (TDMFIE) or a time domain electric field integral equation (TDEFIE), respectively.

(1)

(2)

Note that in the principal value, we essentially exclude the part where the source and observation points are the same (i.e. R=0). Since , and , it is always true that . The main difficulty in extending the approach used to solve the frequency domain integral equations comes from the retarded time variable. However, the TDIEs can be solved numerically by means of a marching-on-in-time (MOT) procedure.

Like the method of moments in the frequency domain, the MoM-TD method discretizes the scatterers or targets into segments or patches. The time axis is then divided into equal increments or time steps. The triangular patches and vector basis functions proposed by Rao-Wilton-Glisson (RWG) [31] are commonly used to discretize the current in space and time by expanding the current  as a finite linear combination of products of spatial basis functions  and temporal basis functions

(3)

where the temporal basis functions are generally versions of the same function shifted by a certain number of time steps,  with , .

To determine the expansion coefficients , Galerkin testing functions are applied in space and point matching is applied at times , , leading to a set of matrix equations that can be written as,

(4)

The vector [V] contains the known incident field quantities and the terms of the Z-matrix are functions of the geometry. The unknown coefficients of the induced current are the terms of the [I] vector. These values are obtained by solving the system of equations iteratively. For example, Andriulli [28] proposed an explicit iterative scheme,

(5)

where  is a  vector,  is a  matrix relating the currents on the body at time . The current coefficient vector  can be obtained once the current coefficient vectors ,  are known.

The TDIEs have applications and limitations similar to their frequency domain counterparts. The EFIE is suitable for closed and open bodies, while the MFIE is only suitable for smooth, closed bodies. MoM-TD techniques are not very effective when applied to arbitrary configurations with complex geometries or inhomogeneous dielectrics. They also are not well-suited for analyzing the interior of conductive enclosures or thin plates with wire attachments on both sides. However, the time domain MoM is especially well suited for dealing with fast transient electromagnetic fields incident on or radiated from structures in free space.

References

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