-energy: compute energy in E or H fields

In this section you may compute the stored energy for an electric or magnetic field. The result of the computation is stored in symbolic variables that may be used eg. for computation of user defined figures of merit.

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 # Flags: nomenu, noprompt, nomessage,                                        #
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 # section: -energy                                                           #
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 # symbol   = h_1                                                             #
 # quantity = h                                                               #
 # solution = 1                                                               #
 # bbxlow =    -1.0000e+30, bbylow =    -1.0000e+30, bbzlow =    -1.0000e+30  #
 # bbxhigh=     1.0000e+30, bbyhigh=     1.0000e+30, bbzhigh=     1.0000e+30  #
 #                                                                            #
 #                                                                            #
 # @henergy : undefined                 (symbol: undefined, m: 1)             #
 # @eenergy : undefined                 (symbol: undefined, m: 1)             #
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 # doit, ?, return, end, help, ls                                             #
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• symbol= QUAN_ISOL:
  This is the full name of the symbol to be processed.
• quantity= QUAN:
  This is the first part of the "symbol".
• solution= ISOL:
  This is the last part of the "symbol", the index of the "symbol".

• bbxlow=,bbxhigh=,bbylow=bbyhigh=,bbzlow=,bbzhigh=:
  Specifies the coordinates of a bounding box. Only the fields that lie within the box are used for integrating the energy.
• doit: Performs the computation. The specified field is loaded from file and its energy density is integrated over the computational volume.

  If the integrated field was an electric field, the result of the computation is stored in the symbolic variable @eenergy. If the integrated field was a magnetic field, the result of the compuation is stored in the symbolic variable @henergy.

The energy for a magnetic field is computed as:

$$@henergy = \frac{1}{2m} \int \mu H^2 \; dV$$ (2.1)

The energy for an electric field is computed as:

$$@eenergy = \frac{1}{2m} \int \varepsilon E^2 \; dV$$ (2.2)

The time-averaging factor m is 2 for resonant fields, and 1 for nonresonant fields.

When you are computing resonant fields without periodic boundary conditions, the fields that gd1.pp processes are the electric fields at a time t=0 and the magnetic fields at a time t=T/4, where T=1/f. This means, you 'see' both the electric and magnetic fields at a time where they are at a maximum. To get the total stored energy for resonant fields, one has to integrate $(1/2) \mu H^2 + (1/2) \varepsilon E^2$ over the volume at one instant time. But E and H are not at the same time. They are offset by T/4. The factor of 1/m, with m=2 for resonant fields, accounts for the effect of integrating both the electric and magnetic fields at their (time) peak values.

When you are computing time dependent fields, the fields that gd1.pp (normally) processes are electric and magnetic fields at (almost) the same time (almost, because there is a time-offset of half of the used timestep, but the used timestep is very small). For time dependent fields, the electric and magnetic fields are not at their time-peak values, but more important, the electric and magnetic fields are at the same time. The stored energy can therefore be expressed directly by integrating $(1/2) \mu H^2 + (1/2) \varepsilon E^2$ over the volume. No factor 'm' is needed.

You cannot and do not need to specify that 'm'. gd1.pp does this for you.

Example To compute the stored energy in the first electric field found in the database, we say:

 -energy, symbol= e_1, doit