# Which type of meshing is more suited for simulation of electromagnetic metamaterial unit cells?

Electromagnetic metamaterials are resonant, periodic structures (repetition of a unit cell) involving both dielectric and conducting elements, and are used and simulated in frequency ranges from microwave (a few hundred MHz) up to optical and more (up to PHz) frequencies. These structures are simulated in frequency domain using FEM.

My question is, considering the above properties of these structures, what properties (qualitatively) should the meshing of the unit cells of metamaterials have generally? More specifically, what are the benefits and drawbacks of using tetrahedral meshing versus hexahedral? (comparison)

• Related: What makes a good finite element? by Jonathan Shewchuk
– Paul
Jul 12, 2014 at 0:47
• A comment to address the generality of the statement, electromagnetic metamaterials (as well as other materials) can be simulated using different methods, not limited to FEM-only. Such methods include FDTD (finite difference time domain) and Method of Moments (integral equation formulations). Those methods have their own advantages and disadvantages. For example, MoM allows for meshing of only the object itself without meshing of the background. Apr 5, 2017 at 21:26

## 2 Answers

As for any other domain, your mesh needs to be fine enough to resolve the features you have. This means that the mesh has to be finer than the geometric details of your unit cell, and it needs to be finer than the wavelengths of the waves you consider.

Beyond this, the question of tets vs hexes is a minor issue. Hexes are generally more accurate, but if the geometry is difficult, hex meshes are more difficult to generate. Of course, in your case, the unit cell is likely a pretty simple geometry, and in that case I'd go with hex meshes.

The more interesting question is: are you going to use any of the existing libraries for simulating your physical model, or are you going to do it all yourself? If the former, you should just go with the kind of cells the library supports, if it is restricted to one kind.

The main advantage of the tetrahedral vs hexahedral is the mesh generation, there are automatic mesh generators for tetrahedral meshes that give "good" elements. This process is not that easy for the hexahedral case and you will need to do a bigger effort to get a nice mesh. Nevertheless, is not a good idea to use linear tetrahedrals since they have linear interpolators (and constant derivatives). In the case of quadratic shape functions I am not sure about which one is better. Although, complete Lagrangian elements are in general better (in terms of accuracy) than their serendipity counterparts.

For high-order elements, like in the Spectral Element Method, you will need less nodes per wavelength. These elements are less sensitive to distorsion, but they need special positions for the nodes and the advantages are mainly for quadrilateral/hexahedral.