# Solving electrostatics problems with scuff-static

scuff-static is a tool within the scuff-em code suite for solving a broad class of electrostatics problems.

The calculations that scuff-static can perform include the following:

• Compute the capacitance matrix (i.e. the self- and mutual- capacitances) of a collection of conductors.

• Compute the DC polarizability of a conducting or dielectric body.

• Compute the electrostatic potential and field at arbitrary user-specified points in the vicinity of conducting or dielectric bodies, with the conductors maintained at arbitrary user-specified potentials and (optionally) an arbitrary user-specified external forcing field.

• Compute the C-matrix, a sort of electrostatic version of the "T-matrix" used to characterize the scattering properties of bodies at nonzero frequencies. The C-matrix was shown in this paper to be related to quantum-mechanical entanglement entropy.

As a technical detail, we note that the implementation of scuff-static actually differs in some significant ways from the other codes in the scuff-em suite; in particular, as compared to the scuff-em core library, scuff-static uses different basis functions and a different formulation of the boundary-element method, as appropriate for zero-frequency problems. (More specifically, scuff-static expands surface electric charge densities on PEC and dielectric surfaces using pulse'' basis functions, which are constant on individual triangles and vanishing everywhere else.) However, from the implementation standpoint, it turns out that the calculations needed to implement the electrostatics calculations in scuff-static are a proper subset of the calculations already implemented in scuff-em. Moreover, from the user's standpoint, the work needed to set up a scuff-static problem (create surface meshes, write geometry files, etc.) is similar to the setup needed for the nonzero-frequency codes in the scuff-em suite. This is why it makes sense to package these codes together.

Here is a brief technical memo discussing the implementation of scuff-static, including both the underlying BEM electrostatics formulation and the execution of the various types of calculation (capacitance, polarizability, etc.) that the code can do.

# 1. scuff-static command-line options

### Common options

scuff-static recognizes the following subset of the list of commonly accepted options to scuff-em command-line codes.

--geometry --TransFile --EPFile

### Options requesting capacitance-matrix output

--CapFile MyCapacitanceMatrix.dat --TransFile MyTransFile.trans

If you specify a file name using --CapFile, scuff-static will compute the full capacitance matrix for your geometry and write the data to the specified file. (The file will be overwritten if it already exists.)

The optional --TransFile option may be used to specify a list of geometrical transformations to be applied to your geometry. In this case, the full capacitance matrix will be computed for each specified transformation.

### Options requesting polarizability output

--PolFile MyPolFile.dat

If you specify a file name using --PolFile, scuff-static will compute the DC polarizability of each object in your geometry and write the data to the specified file. (The file will be overwritten if it already exists.)

### Options requesting computation of electrostatic fields

--EPFile MyEPFile --PlotFile MyPlotFile.pp --PotFile MyPotFile --ConstField [X|Y|Z] --PhiExt PhiFile

If you specify a list of field evaluation points using --EPFile MyEPFile, then scuff-static will compute the electrostatic potential and field at each evaluation point and write the results to a file named MyEPFile.out. (The file will be overwritten if it already exists.) The file will contain a header explaining how to interpret its contents.

The --PlotFile option may be used to request creation of a gmsh visualization file plotting the induced charge density on all conducting and dielectric surfaces in the geometry. If you say --PlotFile MyPlotFile.pp, then you will get a file named MyPlotFile.pp which may be opened in gmsh for visualization purposes.

By default, all conductor surfaces will be held at zero potential. If you wish to set one or more conductor surfaces to non-zero potential, you may do this by saying --PotFile MyPotFile (here "pot" is short for "potential). MyPotFile should be simply a list of (surface label, potential value) pairs, like this:

UpperSurface 1.2 LowerSurface -3.4

where UpperSurface and LowerSurface are the labels you assigned to the surfaces in question in the .scuffgeo file. (The label is the string following the OBJECT or SURFACE keyword in the .scuffgeo file.) This would set the conductor surface UpperSurface to a potential of 1.2 volts and the conductor surface LowerSurface to a potential of -3.4 volts.

By default, the calculation will be performed with no external electrostatic field. You can say e.g. --ConstField X to request that the calculation be performed in the presence of a constant unit-strength electrostatic field pointing in the positive $x$ direction.

Alternatively, you may use the --PhiExt option to specify the name of a file describing a more complicated (non-constant) external field.