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EM solver, FDTD vs FEM

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metebalci:
I think Sonnet uses MoM and Ansys uses FEM and/or MoM for electronics (but FDTD for photonics in Lumerical). Recent Ansys blog post (https://www.ansys.com/blog/what-is-fdtd) says also FEM and MoM are "the predominant gold standard" in electronics. Why is FDTD not used in EM simulations in electronics ?

selcuk:
Those are commercial products and companies usually advertise that their method is the best one.

FDTD is being used in EM simulations. There is an open source tool for solving electromagnetic fields using the FDTD method.

https://www.openems.de

I'm using this one as well. It is not as user friendly or advanced as the commercial tools but does most of the job.

Nominal Animal:
The computational electromagnetics article at Wikipedia provides a concise summary:

"An eigenvalue problem formulation of computational electromagnetics (CEM) allows us to calculate steady state normal modes in a structure. Transient response and impulse field effects are more accurately modeled by CEM in time domain by FDTD. Curved geometrical objects are treated more accurately as finite elements FEM, or non-orthogonal grids. Beam propagation method (BPM) can solve for the power flow in waveguides. CEM is application specific, even if different techniques converge to the same field and power distributions in the modeled domain."

If we created a graph of all the possible methods and approaches and transforms, you'd see they are all connected.  The differences are exactly in their relative weaknesses and strengths, especially when computational complexity (wrt. memory and number of calculations needed) and accuracy in specific cases (like surfaces, shock waves, discontinuities in general) is considered.

Combining different methods in a single simulation is also very, very hard to do efficiently/effectively.  It is much easier to use a specific approach in the entire simulation, because then all the data structures and algorithms can be implemented and optimized for that specific approach.  It is definitely not just changing a function call or two per time step; everything tends to change with the approach, if you care about computational efficiency.

niconiconi:

--- Quote ---Recent Ansys blog post says also FEM and MoM are "the predominant gold standard" in electronics. [...] Why is FDTD not used in EM simulations in electronics?

--- End quote ---

It's better to ask "what" before asking "why". If the question itself is flawed, the answer is meaningless. As a matter of fact, FDTD has many applications in EM simulations for electronics. The theoretical foundation of FDTD itself is also no less valid than FEM/MoM, since it literally simulates Maxwell's equations directly. Given the appropriate inputs, the robustness of the method has already been demonstrated by thousands of peer-reviewed papers published in the last 30 years. Commercial products based on FDTD also exist for electronics design, such as EMPIRE.

So the real question is: Why is FDTD not commercially used as widely as FEM/MoM in electronics design? What are the tradeoffs? Here are some reasons that I can think of.

First, FDTD is a time-domain, general-purpose method. Given the correct inputs, it often can compute your solution from Maxwell's equations directly by brute-force, very little math analysis is needed. The simulation results also show the propagation of E&M field step by step, making it very straightforward. Meanwhile, to use FEM/MoM successfully, often, you need to do some math analysis about your problem first (often involves Green's function) before you can program a field solver. As a result, a basic FDTD solver is both more general-purpose and much more easier to program than a basic FEM/MoM solver. For large-scale problems, FDTD is also the method of choice since its computational complexity is O(n), where n is the number of cells in the simulation box. Meanwhile, FEM/MoM needs linear algebra and matrix multiplication inversion, which has a complexity around O(n^3) (for the basic textbook algorithm). However, on the other hand, it also means that if you have a very particular problem and the problem is not large, you can save a lot of work by doing some math first, then you can design a very specialized and efficient FEM/MoM algorithm for a particular problem, which is more efficient than FDTD. It's also why FEM/MoM was historically used first because practical FDTD was not feasible until computers have enough memory. For example, NEC can solve wire antenna problems within minutes because it's a very narrow problem - FDTD will be much slower. Meanwhile, if you want to simulate the specific absorption ratio of radio waves of a human head from a mobile phone, or to find the radar cross-section of an aircraft using a full-scale model, FDTD is the right choice.

The time vs. frequency domain between FDTD and FEM/MoM is also a consideration. FDTD is great for ultra-wideband simulations since it works in the time domain and simulates a large number of frequencies simultaneously, making it very efficient under the right conditions. But it turned out to be a double-edged sword. The biggest limitation of basic FDTD is that the timestep size must be both small enough to resolve the smallest frequency, and to resolve the smallest physical object. It works great for most microwave components, but can be very problematic if you have an electrically-small object. This would require an impractical number of timesteps. Want to simulate an SMD capacitor at 1 MHz instead of 600 MHz? Want to directly simulate the 30 micrometer solder mask on a 1.6 mm circuit board without using any simplification? Good luck... There are some modifications to FDTD that would better handle these situations, but so far they haven't seen widespread use to my best knowledge. Another problem is that material and device characteristics are often frequency-dependent and are given in the frequency domain, such as permittivity or S-parameters, rather than in time domain. If you use frequency-domain methods, they can be accepted directly. But for time-domain methods modeling it would be a headache because you would need to develop an equivalent wideband time-domain model, or find a way to do frequency-domain operations in time-domain such as by using convolution. For example, for modeling permittivity, frequency-domain methods can simulate the material separately at several frequencies, but FDTD needs to find a numerical fit using the Drude/Lorentz material model.

Yet another aspect is meshing. FDTD is generally done on a structured mesh, in particular the Cartesian mesh, because the formulation of FDTD itself involves the use of an staggered grid of E and H field. There are some variants that use other grid shapes with limited use. Meanwhile, FEM/MoM uses unstructured grid so it's more flexible on modeling irregular objects without suffering from the staircase effect.

So in summary.

FDTD:
+General-purpose, simulates Maxwell's equations from first principles.
+Simple to program, little math analysis.
+Wideband solution in a single simulation run.
+Show the evolution of E&M field in time.
+No linear algebra, scale directly to very large problems.
-Often slow and overkill for small-scale problems, or problems with specialized solutions.
-Extremely slow or impractical at low frequency, for electrically-small object, or both. Improvements exist but not widely implemented.
-Modeling frequency-dependent material or circuit properties is not straightforward.
-Use structured mesh and often Cartesian mesh, which is less flexible than unstructured ones. To work around the problem of irregular shapes, "conformal FDTD" exists, but to my best knowledge are used for one-off research projects, and not widely implemented.

FEM/MoM:
+Already Invented and used before FDTD (from an era when computers had little memory)
+Fast and efficient for problems with specialized solutions.
+Straightforward modeling of frequency-dependent material or circuit properties.
+Use unstructured mesh, which is more flexible for irregular shapes.
-Difficult to program, need math analysis.
-Solvers are often designed for special-purpose applications, new math analysis may be needed for each new application.
-Need linear algebra explicitly, scale to large-scale problems can be difficult.
-Wideband solution needs multiple simulation runs.

For some reasons, the EDA industry has chosen FEM/MoM as the preferred solution.

Disclaimer: I don't really understand any of the underlying E&M involved, take these claims with a grain of salt. However, this is based on my literature reading during my attempt to improve the speed of openEMS's FDTD engine (this work is still ongoing, stay tuned for doubled and even tripled simulation speed), so I believe my description is generally accurate.

metebalci:

--- Quote from: Nominal Animal on February 25, 2024, 12:09:24 pm ---If we created a graph of all the possible methods and approaches and transforms, you'd see they are all connected.  The differences are exactly in their relative weaknesses and strengths, especially when computational complexity (wrt. memory and number of calculations needed) and accuracy in specific cases (like surfaces, shock waves, discontinuities in general) is considered.

--- End quote ---

Is there an application or area specific or common reason to mainly select one of these methods or does it all depend on the problem (and the type of results one looking for) ? For example, in antenna research or radar etc. is one of these methods mainly used ?

The reason I wonder this is, I understand if a company or research group most often develops, uses and promotes only one method, but as far as I understand Ansys has all types of solvers but still either offers or recommends (I only know Ansys a bit from the free version, I dont know if it is possible to use any method for electronics in a commercially licensed version) FDTD for photonics and FEM/MoM for high-freq electronics.

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