Convergence testing is a method used to determine the settings required to meet the desired level
of accuracy for simulation results.
As we saw in a previous unit on the design workflow, convergence testing is recommended
as the final step in the design of a component after narrowing down the range of design parameters.
This is because reducing the simulation error often requires increased memory requirements
and simulation time.
Before any convergence testing is done, it's important to first determine an acceptable
level or error for the particular device that is being simulated.
For example, it's not necessary to resolve the sizes of features down to a precision
of 1 nm if you know that the manufacturing process used to fabricate the device can only
achieve an accuracy of 10 nm.
Since the FDTD method solves Maxwell's equations with no approximations, the main sources of
error from FDTD simulations come from the numerical error due to discretization of space
and time, use of artificial PML absorbing boundaries, and error in the broadband material
The discretization of the simulation volume in space by the simulation mesh can introduce
errors since curved surfaces of the structure will not be represented exactly, and there
can also be some monitor interpolation error that's introduced when field components are interpolated from
the locations where they are calculated on the Yee cell.
There's also some discrepancy between the simulated speed of light in a medium compared
to the ideal case due to the discrete mesh which is referred to as grid dispersion.
All of these errors are reduced when a finer mesh with smaller mesh step size is used.
If PML absorbing boundary conditions are used, there can be errors introduced if the boundary
is placed too close to the structure such that there is coupling of fields between evanescent
fields of the structure and the PML material.
There can also be artificial reflection from the PML material.
This error can be reduced by placing the PML boundary farther away from any scattering
objects and increasing the number of layers of PML material used.
The error in material fitting may not be included as part of the convergence testing process since it can typically
be reduced by modifying the material fitting parameters before running any simulations.
Changing the material fit does not typically cause any increase in simulation time.
Material fitting parameters can be set in the Material Explorer window from the Materials
button in the main toolbar.
You can review the details about material fitting in the Material Properties section
of this course.
If you want to quantify the error due to material fitting, you might want to compare the simulation
results between using the broadband material fit to results that you get from from single frequency simulations where the
material fit is not generated.
To quantify the level of convergence, you can start by choosing a result of interest from your
For example, if you are simulating particle scattering, you could choose the result of
interest to be the value of the scattering cross section (sigma) of the particle.
In the case where you already know the correct value of the result analytically, you can
calculate the absolute error in the simulated result using the equation here.
However, you don't usually have an analytic answer to compare simulation results with,
so instead, you can estimate the absolute error due to a particular source of error
by using the following method: Run the simulation, varying one of the parameters related to a
source of error, such as the mesh step size.
If we sweep the mesh step size, reducing the mesh step size over N points in the sweep,
and collect the result at each point, we can estimate the absolute error in the result
for the ith point in the sweep using the equation here where sigma_i would be the scattering
cross section for the ith point in the sweep, and sigma_N would be the cross section at
the final point in the sweep.
Similarly, the relative error between two points in the sweep is given by the equation
If measuring a result from a broadband simulation, you can use this equation to estimate the
absolute error which includes the integration over the wavelength range.
These plots show the relative and absolute error in a Mie scattering simulation as the
distance between the structure and PML is increased by increasing the simulation span,
and as the mesh is made finer by sweeping the mesh accuracy setting.
From these plots you can determine the necessary simulation span and mesh accuracy needed to
meet the acceptable level of error.
A full list of sources of error, and a detailed example of convergence testing can be found
in the link below this video.