Fields

A number of fields can be implemented in your simulation, including different types of electromagnetic and electrostatic fields. Thus, there also exist a variety of ways to incorporate fields in VSim. In this section, we will cover general concepts surrounding fields, including:

• Boundary conditions

  • Symmetry boundaries

  • Fully periodic systems

  • Modeling a time-dependent or space-dependent value on a boundary

• Signal creation

  • Creating a circularly polarized pulse

  • Launching a wave from the x-upper boundary of a simulation

We will then look at an example of field blocks in action. We will first look at an electrostatic field, which is created through an EmField block. We will then look at an electromagnetic field that uses the MultiField block. This block in particular allows the user to control more parameters for their fields, make changes pertaining to processing efficiency, or even develop and apply their own algorithms. This object incorporates three general sub-objects:

• Field

  Specification of data.

• FieldUpdater

  Update operations to be performed on the the data.

• UpdateStep

  Algorithmic sequencing of the update operations.

Boundary Conditions

You can implement Dirichlet, Neumann, MAL, and other boundary conditions for your simulation, just like in visual setup.

We will provide a few examples of common boundary conditions you may want to incorporate in your simulation, including:

• Symmetry boundaries

• Fully periodic systems

• Time-dependent or space-dependent values on a boundary

• Launching a wave from the x-upper boundary

Symmetry Boundaries

A Neumann boundary can act as a symmetry boundary. For instance, placing Neumann boundaries appropriately on three faces of a cube will turn the simulation into a symmetric simulation of an 1/8th of a larger cube.

Fully Periodic Systems

You can model a system that is periodic in all directions–that is, one that has no boundaries. In a fully periodic system, however, the overall charge must be zero, as demonstrated in this equation:

(2)\[\int^{x+L}_x \frac{\partial E}{\partial x} dx = E(x+L) - E(x) = \frac{1}{\epsilon_0} \int^{x+L}_x \rho dx = \frac{L}{\epsilon_0} \left< \rho \right>\]

where \(\left<\rho\right>\) is the average charge density over the integration interval. Zero net charge can be enforced by specifying the electrostatic solver parameter enforceZeroNetCharge = true.

For more information, see [BL04].

Modeling a Time-Dependent or Space-Dependent Value on a Boundary

An electromagnetic boundary condition may use time signals or spatial profiles. For example, the emPlaneWave.pre file’s xLowerWaveLauncher boundary condition input block uses a spatial profile, and so kind = variable.

The function parameter in a boundary condition input block also may be an expression, and the expression may be any standard function of space and time.

The emPlaneWave.pre file uses the kind = expression technique.

<BoundaryCondition xLowerWaveLauncher>
  # Value given by function
  kind = variable
  lowerBound = [0 0 0] # Lower limits
  upperBound = [1 $NY+1$ $NZ+1$] # Upper limits
  components = [1] # Ey polarized
  <STFunc component1>
    kind = expression
    expression = 1.e6*sin(OMEGA*t)
  </STFunc>
</BoundaryCondition>

For the complete content of the emPlaneWave.pre file, see the example Electromagnetic Plane Wave in VSimBase.

Signal Creation Examples

Creating a Circularly Polarized Pulse

The steps to create a circularly polarized pulse are:

1. In the electromagnetic boundary condition input block, change the components parameter vector to launch both a y and z component: components = [1 2].

2. Give the amplitudes parameter vector a value for each component: amplitudes = [AMPLITUDE AMPLITUDE].

3. Give the phases parameter a value for each component. To make a circularly polarized pulse, the phases for the two components should be \(\pi/2\) radians (90 degrees) apart: phases = [0. 1.57].

The widths parameter controls the spatial extent of the wave.

Launching a Wave from X-Upper Boundary of Simulation

The steps to change a wave launcher from the x-lower to the x-upper boundary follow:

1. Change the lower and upper bounds of the wave launcher boundary condition to the x-upper end of the grid.

   Note

   The lower bounds of a boundary condition at the x-upper end of the grid are in the physical domain.

2. Adjust the x-upper conducting boundary to zero out only the y component.

3. Adjust the x-lower conducting boundary to zero out both y and z components. Be sure to change both the amplitudes and components vector.

The simulation occurs in the lab frame. This means the pulse will strike the x-lower boundary and reflect back after roughly fifty steps for the given time step and simulation size. To avoid reflections, you need to include an absorbing boundary at the x-lower end.

Defining the EM Field and Boundary Conditions in VSim

Now that we’ve discussed boundary conditions, pre-conditioners, and solvers, we are ready to try adding fields to simulations.

You can implement multiple electric and magnetic fields in your simulation. Here, we will show examples of both the EmField and MultiField objects being used.

Electrostatic Field Example

We will begin with EmField, where we will define an electrostatic field covering the entire grid.

<EmField yseField>
  kind = yeeStaticEmField

  # Setting potential V = 0 on left wall of simulation
  <BoundaryCondition left>
    kind = dirichlet
    minDim = 1
    lowerBounds = [0 0 0]
    upperBounds = [1 NY1 NZ1]
    # NY1 = NY + 1, NZ1 = NZ + 1 (these are where the guard cells are)
    <STFunc voltFunc>
      kind = constantFunc
      amplitude = 0.
    </STFunc>
  </BoundaryCondition>

  # Declaring our solver and preconditioner
  <Solver yseSolver>
    kind = gmres
    precond = multigrid
    smoother = GaussSiedel
    nLevels = 12
    threshold = 0.08
    tolerance = 1e-8
  </Solver>

</EmField>

Electromagnetic Field Example

The MultiField block, in comparison, does need a number of nested blocks that describe the fields involved, as well as their boundary and initial conditions, field updaters, and update steps. Below, we have written up an example with an electric and magnetic field.

<Multifield myFields>
 # The default kind for MultiField is Null, which is what we will use here.

   <Field E>
     kind = regular
     numComponents = 3
     labels = [E_x E_y E_z]

     # Set E_y = 1.0 on y-lower boundary initial condition
     <InitialCondition eInitial>
       kind = constant
       lowerBounds = [-1 0 -1]
       upperBounds = [NX1 1 NZ1] # NY1 = NY + 1, NZ1 = NZ + 1 (guard cells)
       indices = [1]
       amplitudes = [1.0]
     </InitialCondition>

     # Set E_x = 0.0 (conductor) on x-lower boundary
     <BoundaryCondition eBound>
       kind = constant
       lowerBounds = [0 -1 -1]
       upperBounds = [1 NY1 NZ1]
       indices = [0]
       amplitudes = [0.]
     </BoundaryCondition>

   </Field>

   # Define simple magnetic field
   <Field B>
     kind = regular
     numComponents = 3
     labels = [B_x B_y B_z]
   </Field>

   <FieldUpdater yeeAmpere>
     kind = yeeAmpereUpdater
     readFields = [B]
     writeFields = [E]
   </FieldUpdater>

   <FieldUpdater yeeFaraday>
     kind = yeeFaradayUpdater
     readFields = [E]
     writeFields = [B]
   </FieldUpdater>

   <UpdateStep firstStep>
     toDtFrac = 1.0
     updaters = [yeeAmpere]
   </UpdateStep>

   <UpdateStep secondStep>
     toDtFrac = 1.0
     updaters = [yeeFaraday]
   </UpdateStep>

   updateStepOder = [firstStep secondStep]

</MultiField>

The MultiField we defined ended up having the same updaters as an electromagnetic field created using EmField. However, this need not always be the case, as a large number of FieldUpdater, UpdateStep, and other related objects exist for describing non-conventional systems.

If using the MultiField block in an electrostatic simulation, you must incorporate the linearSolveUpdater block. For more details, visit the linearSolveUpdater sections in VSim Reference.

EM Field in Cylindrical Coordinates

When working in cylindrical coordinates, one must use a special FieldUpdater that is designed specifically for the CoordProd grid. In the list of available updaters (found in FieldUpdater), the updaters which work with the CoordProd grid will be named ending in CoordProd.