# Effective “thickness” of finite-difference material elements

I have a 3D finite-difference formulation (for time-harmonic Maxwell using a standard Yee cell grid.) A "perfect electrical conductor" condition can be implemented by enforcing the desired electric field components to 0. (This can also be approximated by making the effective $\epsilon_r$ have a really large imaginary part but for a PEC it's easiest and most stable to just enforce the 0-field magnitude condition.)

If I define a single field component (say Ex of a single Yee cell) to be PEC, I am wondering what the effective dimensions\volume of the defined material would be for a grid-spacing of $dx$. Clearly it's not just a infinitesimally thin line of PEC, but at the same time it does not seem like it would be a full grid cell thick (i.e. be a block of PEC with volume $dx^3$) since the field components around it are not PEC.

Please let me know if the question is unclear at all and I'll try to clarify further. Thanks!

• Are you going to be enforcing the condition on just one, or all four of the Ex edges of the Yee cell? Also, in practice, are you going to do this for a single cell, a line of cells, a plane of cells? – jeffdk Feb 28 '13 at 3:55
• The Yee cell only has a single Ex component. I would be interested in hearing the answer for all of those situations if possible actually! – Costis Feb 28 '13 at 5:29

## 1 Answer

I think a good way to visualize or extend the behavior of the Yee cell is to recognize that it's a special kind of mixed finite element method, where (i) space has been discretized with axis aligned hexahedra (ii) curl conforming functions ("edge elements") have been used to model the electric field E (iii) divergence conforming elements ("facet elements") have been been used to model the magnetic flux B and (iv) all the volume integrals arising from the FEM testing process have been approximated by lumping the integration (each cube has 8 equally weighted quadrature points, one at each node).

PEC boundary conditions get applied to the edges, allowing you to coarsely trace out wires and such by following the edges of the hexahedra (so IMO, your intuition that shorting a single Ex makes a thin PEC line is correct - FDTD won't model this especially accurately because it does not incorporate the field singularities of a thin wire without some modifications, but to first/zeroth order, a wire filament is the effect you'll get). To enforce a sheet of pec, you'd short out both transverse components on whatever collection of facets best approximates your surface (note this will automatically enforce the normal component of B to be zero on that sheet, too)

The paper "A finite element method based on whitney forms to solve Maxwell equations in the time domain" by Man-Fai Wong, Odile Picon and Victor Fouad Hanna[1] has some nice discussion about the similarities between mixed finite elements and Yee's FDTD. You might also find it instructive to read about "finite integration theory"/FIT methods[2] for modeling Maxwell's equations, which are a similar generalization of the Yee method (I think the commercial software "CST microwave studio" is an FIT code.. at least the transient solver part)

Reference:

• Thanks for the great answer! As a quick followup: Is there a rule of thumb for estimating the "approximate" thickness of a sheet of PEC in a FD Yee cell method or the "thickness" of the thin wire represented by a line of PEC. Namely, if I am implementing a dipole antenna structure for instance, what sort of actual 3D structure (say implemented in a commercial FEM solver for instance or constructed in actuality) would present the same input impedance / scattering parameters at the driving port? Also, what would the effective cross-section of the wire be? (circular? square?) – Costis Feb 28 '13 at 22:29
• I am not sure what e.g. the input impedance would be, you could maybe find it out via (numerical) experiment. For instance, put a 50ohm lumped port model in your FDTD, then record/ifft s11 to find zin, and consult your antenna handbook to determine the equivalent wire parameters (diameter/coating/whatever). I'd expect the impedance to be a complex function of your grid parameters - not only lattice spacing/cell anisotropy, but also distance to your PML/ABC and all its parameters (layers, grading, conductivity, etc). All of those things get lumped together into that impulse response. – rchilton1980 Feb 28 '13 at 22:59
• I have a somewhat low opinion of FDTD for modeling cross section, though it depends on the nature of the structure under consideration. All the wiggles of stairstepping the geometry adds spurious scattering, plus the intrinsic dispersion (and dispersion anisotropy) of the grid, inexactness of the PML/ABC termination, and leakage of the TF/SF interface stack the deck a bit against FDTD if you need accuracy. I'd stick to MoM for that. On the other hand, FDTD is great for modeling like ground penetrating radar, which is already pretty dispersive begin with (and MoM can't touch it anyway). – rchilton1980 Feb 28 '13 at 23:07