POLARIS Example - Sphere and Bicone

The geometry consists of a sphere with radius \(r=0.1\,m\) and a symmetrical biconical body with the same base radius \(r\) and height \(h=r\). The distance between the sphere and the tip of the conical body \(d=r/4=0.025\,m\). The conducting sphere has a fixed potential of \(V=100\,kV\), the potential of the conducting bicone is unknown.

In order to see how to generate the sphereBicone.yaml input file with the preprocessor of POLARIS, check the Preprocessor Tutorial.

Main run

With the provided input files, POLARIS computes the electrostatic interaction between the two bodies.

polaris --input=sphereBicone.mesh --settings=sphereBicone.yaml --output=sphereBicone.vtkhdf

polaris -i sphereBicone.mesh -s sphereBicone.yaml -o sphereBicone.vtkhdf

#===================================================================
# INPUT
#-------------------------------------------------------------------
# Reading grid and settings ... done
# Grid with 1792 elements
#-------------------------------------------------------------------
# Mesh info
# Element type occurrences:
#     Tria6   :  1792
# Bounding box: [{-1.00000e-01,-1.00000e-01,-1.00000e-01} ---
#                { 1.00000e-01, 1.00000e-01, 3.25000e-01}]
# Mesh metrics:
#             Name Unit              Min     ID          Max     ID
#          Measure [L*L]          0.0001    512       0.0002   1546
#     Corner angle [degree]      43.8180   1280      86.0257   1280
# Condition number [-]            1.0042    575       1.7290   1344
#     Mutual angle [degree]       0.0004   1328      90.3316   1285
#-------------------------------------------------------------------
# Execution info
# Using 1 threads per rank
# Using 1 MPI ranks
# Input:                            0.048s
#===================================================================
# CALCULATION OF SURFACE DENSITY
#-------------------------------------------------------------------
# Creation of calculator ... done
# BEM preparation:                  4.513s
#-------------------------------------------------------------------
# Computation of RHS ... done, |b| = 3577.7
# RHS computation:                  0.003s
#-------------------------------------------------------------------
# Solving main system ...
#-------------------------------------------------------------------
#    Iter           res
Sys     0 3.53343187e-01
Sys     1 2.55590337e-01
Sys     2 2.38317295e-01
Sys     3 1.68033547e-01
Sys     4 4.85218423e-02
Sys     5 1.85808848e-02
Sys     6 8.28104461e-03
Sys     7 3.25397664e-03
Sys     8 1.23607373e-03
Sys     9 5.91776832e-04
Sys    10 2.90999506e-04
Sys    11 1.78527080e-04
Sys    12 1.23419575e-04
Sys    13 9.25279023e-05
# System solver stopped with reason 'relative tolerance'
#           after 14 iterations, |x| = 3.9909
# Solving:                          0.202s
#===================================================================
# CALCULATION OF SURFACE POTENTIAL AND E-FIELD
#-------------------------------------------------------------------
# Solving system to compute normal E-field ...
#-------------------------------------------------------------------
#   Iter           res
EN     0 1.00000000e+00
EN     1 4.47713685e-13
# Solver for normal E-field stopped with reason 'relative tolerance'
#           after 1 iterations
# Normal E-field:                   0.012s
#-------------------------------------------------------------------
# Computation of surface potential ...
# Solver for potential stopped with reason 'direct solve'
#           after 1 iterations
# Potential:                        0.002s
#-------------------------------------------------------------------
# Computation tangential E-field
# Tangential E-field:               0.003s
#===================================================================
# POST-PROCESSING ...
#-------------------------------------------------------------------
# Electrodes
#-------------------------------------------------------------------
# Name              Appl. voltage     Total charge
  Sphere            +1.00000000E+02   +1.28699990E+02
#-------------------------------------------------------------------
# Floating potentials
#-------------------------------------------------------------------
# Name              Appl. charge      Voltage
  Bicone            +0.00000000E+00   +4.56740593E+01
#-------------------------------------------------------------------
# Dielectrics
#-------------------------------------------------------------------
# Name              Avg. voltage      Total charge
  Air               +7.75023111E+01   +1.28770006E+02
#-------------------------------------------------------------------
# Integral forces
#-------------------------------------------------------------------
# Name             Fx               Fy               Fz
  Sphere           -9.26932525E-12  -1.35112987E-11  +3.15752308E-08
  Bicone           -1.60255212E-13  -2.07069570E-13  -2.84796114E-08
  Air              +9.42958046E-12  +1.37183683E-11  -3.09561941E-09
  SUM              +0.00000000E+00  +0.00000000E+00  +0.00000000E+00
#-------------------------------------------------------------------
# Field extrema
#-------------------------------------------------------------------
  Range of potential: [ +4.56740593E+01 : +1.00000000E+02 ]
  Max of Eout field:     2.75176162E+03
  Max of Ein  field:     0.00000000E+00
#-------------------------------------------------------------------
# Post processing:                  0.090s
#===================================================================
# NORMAL TERMINATION
#-------------------------------------------------------------------
# Number of elements = 1792
# Total runtime:                    4.873s
# Total memory consumption:
#       -- Peak memory per rank:  57.844MiB
#       -- Total peak memory:     57.844MiB
# Solution data:
#       -- sphereBicone.vtkhdf
#-------------------------------------------------------------------
# License valid until: <date> (<days left> days remaining)

Posterior run

Additionally, with polaris.post the potential and the electric field can be evaluated at all points provided in sphereBicone.xyz by calling

polaris.post --input=sphereBicone.vtkhdf --settings=sphereBicone.yaml --points=sphereBicone.xyz

polaris.post -i sphereBicone.vtkhdf -s sphereBicone.yaml -p sphereBicone.xyz

#===================================================================
# INPUT
#-------------------------------------------------------------------
# Reading solution file, point cloud and settings ...
# Input:                            0.002s
#-------------------------------------------------------------------
# Execution info
# Using 1 threads per rank
# Using 1 MPI ranks
#===================================================================
# FMM-BASED FIELD EVALUATION
#-------------------------------------------------------------------
# Preparation:                      0.014s
# FMM setup:                        0.070s
# FMM apply:                        0.293s
#-------------------------------------------------------------------
# Field extrema
#-------------------------------------------------------------------
  Range of potential:   [ +3.33994562E+01 : +8.11559983E+01 ]
  Range of E-field(X):  [ -2.61322138E+02 : +2.61474660E+02 ]
  Range of E-field(Y):  [ -2.61331192E+02 : +2.61517425E+02 ]
  Range of E-field(Z):  [ -1.51245814E+03 : -3.71822882E+01 ]
  Max eField magnitude:    1.51245816E+03
#===================================================================
# NORMAL TERMINATION
#-------------------------------------------------------------------
# Number of evaluation points: 1089
# Total runtime:                    0.382s
# Total memory consumption:
#       -- Peak memory per rank:  48.641MiB
#       -- Total peak memory:     48.641MiB
# Solution data:
#       -- "../share/examples/sphereBicone/sphereBicone.csv"
#-------------------------------------------------------------------
# License valid until: <date> (<days left> days remaining)

Results

Both result files, sphereBicone.vtkhdf and sphereBicone.csv can be post-processed with ParaView.

#-------------------------------------------------------------------
# Floating potentials
#-------------------------------------------------------------------
# Name              Appl. charge      Voltage        
  Bicone            +0.00000000E+00   +4.56739990E+01

We see that calculated floating potential on the bicone is approximately \(45.674\,kV\) (note that everything is in \(kV\)) and thus very close to the results reported in [1]. The full set of results can be found in the result file sphereBicone.vtkhdf.

Conducting Sphere and Bicone

Main run

Posterior run

Results