Mesh
A mesh partitions the continuous 3D box into discrete rectangular cuboids. In the Finite-Difference Time-Domain (FDTD) algorithm, these cuboids discretize the 3D space and form the basic unit of electromagnetic calculations. These are known as Yee cells (named after Kane S. Yee’s 1966 algorithm).
openEMS supports the standard Cartesian mesh in conventional FDTD algorithm. A non-uniform mesh is supported as an improved to save computational resources, by allowing the use of fine mesh sizes only around small features in the structure.
As an extension the FDTD algorithm, openEMS also supports meshing in Cylindrical coordinates. The is useful for simulating round structures without the “staircasing” error. The cylindrical mesh can be further “subgridded” into smaller cells, achieving resource-saving benefits similar to the non-uniform Cartesian mesh. As a showcase, an antenna for medical Magnetic Resonances Imaging (MRI) has been successfully simulated this way in a research project.
Create a Mesh
Obtain the Mesh Object
We obtain the mesh object by calling GetGrid()
to manipulate it further, and set its unit of measurement to 1 mm (1e-3 meters).
This will be used as the base unit of all coordinates:
mesh = csx.GetGrid()
unit = 1e-3
mesh.SetDeltaUnit(unit)
Decide the Upper Frequency and Mesh Resolution
Let’s pick an arbitrary lower frequency of 100 MHz and an upper frequency of 10 GHz. Based on the previous analysis, one can calculate the desired mesh resolution via the following code:
import math
from openEMS.physical_constants import C0
f_min = 100e6 # post-processing only, not used in simulation
f_max = 10e9 # determines mesh size and excitation signal bandwidth
epsilon_r = 1
v = C0 / math.sqrt(epsilon_r)
wavelength = v / f_max / unit # convert to millimeters
res = wavelength / 10
General Requirements
In general, the mesh must satisfy four requirements:
Frequency Resolution. Its interval must be small enough to resolve the shortest wavelength (highest frequency component) of the signal, so that electromagnetic field details are not missed. Thus, we need several cells per wavelength.
Spatial Resolution. Its interval must be small enough to resolve the shapes of the simulated structure, so that small details of the structure are not missed. Thus, we need at least a few cells around the important shapes (such as the waveguide plates) of the structure. openEMS uses a rectilinear mesh with variable spacing. To save time, only use a fine mesh interval around details on a structure; use a coarse mesh for the rest.
Smoothness. Its interval should change smoothly, not by a sudden jump. It’s recommended that the mesh spacing vary by no more than a factor of 1.5 between adjacent lines.
1/3-2/3 Rule. The edges of a conductor have singularities with strong electric field. To improve FDTD accuracy, ideally, around the metal edge, the metal should occupy 1/3 of a cell, while the vacuum or insulator occupies 2/3 of a cell. More on that later.
As a rule of thumb, we want a mesh resolution of at least:
Note
This is the same “1/10 wavelength” rule of thumb used for determining whether transmission line effects are significant in a distributed circuit.
The relationship between wavelength and frequency is:
in which \(\lambda\) is the wavelength of the electromagnetic wave in meters, \(v\) is the speed of light in a medium, in meters per second, and \(f_\mathrm{max}\) is the highest frequency component of the signals used in simulation.
The speed of light in a medium is given by:
in which \(c_0\) is the speed of light in vacuum, \(\epsilon_r\) is the relative permittivity of the medium (in engineering, it’s sometimes also denoted as a material’s dielectric constant \(D_k = \epsilon_r\)). The \(\mu_r\) term is the medium’s relative permeability, which is usually omitted in engineering for the purpose of signal transmission. Most dielectrics (like plastics or fiberglass) in cables are non-magnetic - unless one is dealing with inductors or circulators.
In vacuum, \(\epsilon_r = 1\) and \(\mu_r = 1\) exactly.
Courant-Friedrichs-Lewy (CFL) Criterion
In general, the CFL criterion governs the stability of FDTD calculations. It states that the simulation timestep can’t be larger than:
where \(v\) is the wave speed, \(\Delta x\), \(\Delta y\), \(\Delta z\) are the distances between mesh lines.
This creates a peculiar limitation: to resolve the smallest mesh cell, one must use a small timestep regardless of frequency.
Even for simulations at low frequencies, as long as the simulated structure has small features (i.e. mesh distance is short), we must use very small timesteps, advancing the simulation only a few nanoseconds per iteration. Under 100 MHz, at the bottom of the VHF band or in the HF band, the required number of iterations becomes impractically large. This means openEMS (and other textbook FDTD solvers with explicit timestepping) is unsuitable if there’s a large mismatch between the signal wavelength and the physical size of the structure, such as a 10 cm circuit board operating at 1 MHz, where the wavelength is 300 m in vacuum.
Note
In openEMS, you only need to define an appropriate mesh. There’s no need to calculate the required timestep size, By default, openEMS uses a modified timestep criterion named Rennings2 is used, not CFL. This improves timestep selection in non-uniform meshes. But the general limitation still applies.
Rennings2 is derived in the unpublished paper [11]_,
see SetTimeStepMethod() for details.
If you really need small cells (e.g. to resolve some important feature of your structure) you will have to live with long execution times, or perhaps FDTD is not the right method for your problem. As a workaround, one may also try extracting the circuit parameters using a higher signal frequency, then using those equivalent-circuit parameters in a low-frequency simulation with a general-purpose linear circuit simulator.