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Method of Moments and Finite Element Methods are two of the most used methods in computational electromagnetics to solve electromagnetic equations. As it is known, in FEM sparse matrixes are used while MoM uses the full-matrixes.

My question is, why MoM uses full-matrix while a sparse matrix is obtained in FEM? Is there any advantage or disadvantage between these types of solutions?

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    $\begingroup$ I know very little about MoM, but if it is generating a fully coupled matrix system, then the basis functions used in the MoM formulation must be very non-local, making all degrees of freedom coupled to each other. In contrast, FEM is designed with assumptions weak enough to allow for basis functions that are non zero only in small regions clustered around each of the nodal points in the mesh. $\endgroup$
    – Paul
    Apr 30 '20 at 1:13
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Method of Moments is the name given to the Boundary Element Method (BEM) in the Electromagnetism community. Since you are using Green functions for BEM you get fully populated and asymmetric matrices.

Some advantages that are commonly mentioned for BEM are:

  • It reduces the dimensionality of the mesh. This might look to lead to smaller matrices but, in general, they are more expensive compared to sparse matrices for FEM. This might be an advantage when dealing with remeshing such as fracture mechanics and shape optimization.

  • You can deal naturally with unbounded domains, such as exterior acoustics, antenna radiation, or seismic wave propagation.

You can add Fast Multipole Method to improve BEM performance. And, I would say that is the way to go instead of just using plain BEM nowadays.

References

Liu, Y. (2009). Fast multipole boundary element method: theory and applications in engineering. Cambridge university press.

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    $\begingroup$ +1 for the community specific reference. To be fair, fast integration like FMM/PME is almost always used for large integral equations like this. However, this is less about the method and more about numerics. $\endgroup$ May 2 '20 at 4:05
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    $\begingroup$ The issue you will run into with the FMM-based iterative solution approach is that of conditioning. BEM matrices for anything more complex than a conducting sphere are often very poorly conditioned, causing iterative solvers to stagnate regardless of how fast you can apply your BEM matrix to a vector. Designing effective preconditioners to ameliorate this problem is extremely difficult for general problems; one may instead opt to redesign the BEM equations themselves to improve conditioning. $\endgroup$ Aug 24 at 10:24
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Advantages to MoM/BEM:

  1. BEM naturally incorporates boundaries at infinity, while FEM needs to truncate the domain at some point. This can either be a feature of a bug. The truncation means that FEM naturally gets to model periodic structures very easily, but not open boundaries; vice versa for BEM.
  2. Computational effort for BEM usually depends on the complexity of your physical geometry of interest (i.e., the boundaries). In contrast, FEM grows in proportion to the physical volume of interest.
  3. BEM requires a FULL system matrix while FEM is usually SPARSE. One way to think of this is that every source point in BEM depends on what every other source is doing. In contrast, a single node in FEM depends only on what its neighbors are doing.
  4. BEM is hard to get working reliably. Every little feature you add to BEM tends to require a lot of careful, mathematical analysis. In contrast, the most difficult part of FEM is usually the meshing. Once you get that part figured out, the rest of the simulation kind of takes care of itself. Thus, the barrier to entry is a lot easier with FEM if you want to code it yourself.
  5. Stability is arguably a little better with FEM than BEM. For example, sharp corners are particularly nasty because the charge density wants to shoot off to infinity. This can scramble your convergence with BEM. In contrast, FEM usually doesn't react so badly to that sort of thing.
  6. Floating conductors are inherently difficult to model with FEM, but BEM can do it easily.

There's probably more pros/cons, but those are the big ones I've encountered when writing up my own simulation codes.

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