The Finite Element Time Domain Method

The finite-element time-domain (FETD or TDFEM) method combines the advantages of a time-domain technique with the versatile spatial discretization options of the finite element method. A variety of FETD methods have been proposed. These schemes generally fall into two categories. Methods in the first category directly discretize the time-dependent Maxwell"s equations, yielding an explicit, conditionally stable, time-marching algorithm that can be viewed as a generalization of the finite-difference time-domain (FDTD) method for unstructured grids [1-10]. Methods in the second category discretize the second-order vector wave (curl-curl) equation, obtained by eliminating one of the field variables from Maxwell's equations. The solution of a linear system of equations is required at each time step, but this implicit method can be formulated to be unconditionally stable [11-23].

The explicit FETD has less computational complexity, however the maximum time-step must be constrained to insure stability and it can be relatively difficult to achieve convergence. In implicit FETD methods, the time step is not constrained by a stability criterion and these methods can be extended to higher-orders in a relatively straightforward manner. However, implicit schemes have greater computational complexity because they require a global linear system of equations to be solved at each time step. This can make the simulation of large-scale electromagnetic problems relatively inefficient.

Lou [24] presented a dual-field FETD formulation that computes both the electric and magnetic fields in a leapfrog fashion. This formulation has the advantages of implicit FETD schemes while reducing the computational complexity significantly when the computational domain is split into non-overlapping smaller subdomains. He extended the domain decomposition to the element level in [25].

For unbounded problems, the truncated boundaries of the FETD computational domain need to be properly treated. This is usually done by using conventional absorbing boundary operators [26, 27], boundary integral (BI) methods [28], or PML boundaries [29-37]. Absorbing boundary operators are easy to implement but can exhibit large reflection errors [27]. Boundary integral methods are theoretically exact but computationally expensive. PML implementations yield very small reflection errors in other time-domain methods, such as the FDTD method. However, the FETD formulation of the PML has not been thoroughly investigated.

References

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