-mesh : outer boundary conditions, spacings

gd1 needs to know

• what the boundaries of the computational volume shall be,
• what the default spacing between mesh-planes shall be,
• what the boundary conditions at the outer planes of the computational volume shall be.

You specify these values in this section.

 ##############################################################################
 # Flags: nomenu, noprompt, nomessage,                                        #
 ##############################################################################
 # section -mesh                                                              #
 ##############################################################################
 ## Bounding box:                                                             #
 #   pxlow = undefined      , pylow = undefined      , pzlow = undefined      #
 #   pxhigh= undefined      , pyhigh= undefined      , pzhigh= undefined      #
 #  volume= (undefined, undefined, undefined, undefined, undefined, undefined)#
 #      __zlowref = undefined                                                 #
 #      __zhighref= undefined                                                 #
 ## Boundary conditions:                                                      #
 #   cxlow = electric, cylow = electric, czlow = electric                     #
 #   cxhigh= electric, cyhigh= electric, czhigh= electric                     #
 ## periodic bc's for eigenvalues:                                            #
 #   xperiodic= no , xphase= undefined                                        #
 #   yperiodic= no , yphase= undefined                                        #
 #   zperiodic= no , zphase= undefined                                        #
 # # # # #                                                                    #
 # xspacing= undefined, xminspacing= undefined                                #
 # yspacing= undefined, yminspacing= undefined                                #
 # zspacing= undefined, zminspacing= undefined                                #
 # graded  = no                                                               #
 #   xgraded= no , ygraded= no , zgraded= no                                  #
 # xqfgraded= 1.20, xdmaxgraded= undefined                                    #
 # yqfgraded= 1.20, ydmaxgraded= undefined                                    #
 # zqfgraded= 1.20, zdmaxgraded= undefined                                    #
 # perfectmesh= no                                                            #
 # geoscale= 1.0                                                              #
 ## Commands: # # # # # # # # # #                                             #
 # xfixed(N, X0, X1)  -- N fixed meshplanes between X0 and X1                 #
 # yfixed(N, Y0, Y1)  -- N fixed meshplanes between Y0 and Y1                 #
 # zfixed(N, Z0, Z1)  -- N fixed meshplanes between Z0 and Z1                 #
 ##############################################################################
 # listplanes, return, help                                                   #
 ##############################################################################

• pxlow, pxhigh, pylow, pyhigh, pzlow, pzhigh, volume
  (Plane at XLOW ...) These values are the coordinates of the planes that bound the computational volume.
  These parameters are mandatory.
  Alternatively to specifying via pxlow.., you can specify the volume of your computational volume as volume= (XL, XH, YL, YH, ZL, ZH).
  gd1 ignores all parts of items that are specified outside of the box with these boundary planes.
• __zlowref, __zhighref
  Special parameters. If these parameters are given, the mesh is generated such that the mesh below __zlowref and above __zhighref is translational invariant.
• cxlow, cxhigh, cylow, cyhigh, czlow, czhigh
  (Condition at XLOW ...) These are the boundary conditions at the boundary planes of the computational volume. Possible values are electric, magnetic.
• xperiodic, yperiodic, zperiodic
  Possible values are yes, no. Toggles the application of periodic boundary conditions between the lower and upper boundary planes in x-, y- or z-direction. You can compute with more than one ?periodic= yes. This is only implemented for eigenvalue computations. For time-domain computation, these parameters should be set to no.
• xphase, yphase, zphase
  Specification of the phase (in degrees) to enforce between the lower and upper boundary planes, when ?periodic= yes.
• spacing
  The default spacing, that gd1 should use for discretising the computational volume. In between fixed meshplanes, the gridspacing will be about spacing.
  This parameter is mandatory.
  Of course, specifying a small spacing will lead to a good discretisation. Be aware however, that the memory requirement is proportional to $1/(spacing)^3$ and the CPU-time approximately to $1/(spacing)^{4}$.
• minspacing
  The smallest mesh-spacing allowed.
  gd1 tries to place meshplanes approximately homogeneously over the volume. It is guaranteed, that there are meshplanes at the boundary planes of bricks, and where you enforce them via xfixed, yfixed, zfixed, If the distance between such fixed meshplanes is more than minspacing. Otherwise, one of these fixed meshplanes is deleted, it is the meshplane with the higher coordinate value.

  If this parameter minspacing is not given, gd1 uses the value of spacing/10 instead.

• graded
  Possible values are yes, no.
  If graded= yes, the space between fixed meshplanes will be filled with a graded mesh. The ratio of adjacent mesh-spacing will be approx. qfgraded, The largest mesh-spacing will be less or equal dmaxgraded.
• xgraded, ygraded, zgraded
  Same as graded , except only grid planes in x-, y- or z-constant are considered for grading.
• qfgraded, dmaxgraded
  These are the parameters that are mentioned above in the discussion of graded.
• perfectmesh
  If specified as yes, the material boundaries are approximated as smooth patches. If specified as no, the material distribution in a cell is approximated for each cell as one out of 73 possible materialdistributions. The meshing and computation with perfectmesh= no is somewhat faster and is more thoroughly tested. In both cases, the error in the computed resonant frequencies is reduced by about a factor of ten as compared to the common known approximation with diagonal fillings. Unfortunately, in some configurations unphysical fields are computed when perfectmesh= yes, so use with care. The default is perfectmesh= no.
• geoscale
  Each coordinate that you specify (each parameter which has the dimension of a length) is multiplied by this factor. Each parameter which has the dimension of 1/length, is multiplied by 1/geoscale.
• xfixed, yfixed, zfixed
  The meshgenerator of gd1 fills the computational volume in principle homogeneous (if graded= no). It is only guaranteed, that the borders of bricks lie on meshplanes. Therefore, if you have a geometry, where geometric items other than bricks are in, you can improve the mesh by enforcing meshplanes with these commands.
  Syntax:
  xfixed(NUMBER, LOW, HIGH)
yfixed(NUMBER, LOW, HIGH)
zfixed(NUMBER, LOW, HIGH)
  • NUMBER:
    The number of meshplanes to place in between LOW, HIGH. The total number of meshplanes enforced by such a command is exactly NUMBER. In the case of xfixed, the meshplanes are positioned at positions $x_i$:
    $$\begin{eqnarray*} x_i &=& LOW+(i-1) \frac{HIGH-LOW}{\max(1,NUMBER-1)} ;\; \; i= 1, (1), NUMBER \end{eqnarray*}$$
    
    

  • LOW, HIGH:
    The first and last coordinates of the meshplanes to be enforced.
  It is possible to specify NUMBER $\le$ 0, in this case nothing happens.
  It is possible to specify NUMBER=1, in this case a single meshplane at LOW is enforced.
  It is not necessary that LOW is really lower than HIGH.
• listplanes
  Lists the position of gridplanes that will be used by gd1.

Example

 -mesh
     spacing= 1e-3
     pxlow= 1e-2, pxhigh= 0
     pylow= 2e-2, pyhigh= 0
     pzlow= 3e-2, pzhigh= 0
     
     cxlow= ele, cxhigh= mag
     cylow= ele, cyhigh= mag
     zperiodic= yes, zphase= 120