Solving Open Boundary Problems With FDTD Absorbing Boundary Conditions

Key Takeaways

• FDTD uses grid-based differential time-domain modeling, and this helps to cover a wide frequency range in one simulation run. 

• In open boundary problems, absorbing boundary conditions (otherwise called radiation boundary conditions) are introduced. 

• FDTD absorbing boundary conditions are helpful in replacing the infinite space that surrounds the solution region with the finite computational domain. 

The radiation of an antenna in free space is an example of an open boundary problem

The finite-difference time-domain (FDTD) method is the most popular computational numerical method used to solve time-dependent electromagnetic field problems. The FDTD method is easy to understand and its algorithm is far less complex to implement compared to other numerical methods such as the finite element method (FEM) or method of moments (MoM) solvers. 

The FDTD method uses grid-based differential time-domain modeling, and this helps to cover a wide frequency range in one simulation run. Maxwell’s equations and wave equations representing electromagnetic problems are modified to simple simultaneous equations associated with each grid present in the solution region. The equations of each grid connect with the terms from the neighboring grids, and the whole 3D Cartesian grid is modeled like that until it reaches the physical boundary of the solution region.

In the case of electromagnetic field problems that are unbounded in space, boundary conditions are required to bring the infinite space to a finite computational domain. FDTD absorbing boundary conditions are introduced to solve open boundary electromagnetic problems so that a solution obtained can be presented as the accurate approximation of the real solution that considers unlimited space surrounding it.

FDTD Absorbing Boundary Conditions

In the FDTD method, the solution region of electromagnetic field problems is discretized with cells or grids. The electromagnetic solution region is discretized in such a way that the size of the grids present in it are shorter than the electromagnetic wavelength and the geometrical details. For each grid, both electric and magnetic fields are defined using mathematical equations and are solved grid-by-grid to reach the final desired solution.

The FDTD algorithm can be summarized as follows:

1. The partial derivatives in the equations representing electromagnetic problems are replaced by central difference equations or finite difference equations. The electric and magnetic fields of each grid are defined. Space staggering and time staggering are used to discretize space and time so that the electric and magnetic fields are staggered in both space and time.
2. The resulting finite difference equations are solved to obtain update equations. The update equations express the electric and magnetic fields of future (unknown values) in terms of past fields (known values).
3. The magnetic fields are evaluated one time step into the future. The magnetic fields transform from future fields to past fields after evaluation. Similarly, the electric fields are also evaluated and the results become past field values.
4. The evaluation of magnetic and electric fields is repeated for the desired duration, and this will give the desired solution for the electromagnetic problem. 

FDTD Absorbing Boundary Conditions in Open Boundary Electromagnetic Problems

As mentioned above, FDTD simulations are driven by update equations. In finite physical domains, the grids and equations to be solved are finite. There is no need for the termination of FDTD grids in electromagnetic field problems associated with spatially limited structures. However, there is no finite physical domain in electromagnetic field problems where the solution region is unbounded in space. The interaction of incident waves with a scattering structure and the radiation of an antenna in free space are examples of open boundary problems.

In open boundary problems, absorbing boundary conditions (otherwise called radiation boundary conditions) are introduced. FDTD absorbing boundary conditions are helpful in replacing the infinite space that surrounds the solution region with the finite computational domain. The solutions obtained by applying FDTD absorbing boundary conditions are accurate approximations of the real solution. 

Ideal Absorbing Boundary Conditions

There are certain conditions that a good absorbing boundary should meet for the termination of grids in open boundary problems. They are:

1. The absorbing boundary should not give much weight to FDTD.
2. It should be capable of meeting the precision required in most engineering applications.
3. A good absorbing boundary condition should be universal. It should be independent of the geometry of the structure and electromagnetic parameters. 
4. Numerical stability is preferred in good absorbing boundary conditions. 

FDTD absorbing boundary conditions are essential for truncating open boundary problems, and the application of these boundary conditions transforms the limited space computation domain equivalent to the unlimited space physical domain.

There are several FDTD absorbing boundary conditions such as the perfectly matched layer (PML) with loss, the surface integral equation on the boundary, and Mur’s absorbing boundary condition. While choosing absorbing boundary conditions, one must think of the accuracy of approximation required in the computational solution compared to the actual solution. Cadence offers 3D FDTD electromagnetic simulation tools for solving spatially limited as well as open boundary electromagnetic problems. 

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