Introduction:

This implements Vectorized algorithms for regular and conforming tessellations of d-orthotopes and their faces with high-order orthotopes or simplicial elements, Cuvelier F., 2019

More documentation is available on fc_hypermesh Python package dedicated web page.

Installation:

The fc_hypermesh Python package is available from the Python Package Index, so to install/upgrade simply do

pip install fc_hypermesh -U

Thereafter, it’s possible to run one of the demo functions

import fc_hypermesh
fc_hypermesh.demos.demo01()

Examples usage:

• Meshing the rectangle [-1,1]x[0,1] by simplices with 12+1 points in x-axis and 5+1 points in y-axis:

  from fc_hypermesh import OrthMesh
  oTh=OrthMesh(2,[12,5],type='simplex',box=[[-1,1],[0,1]])
  print(oTh)

  The output of the print(oTh) command is:

  OrthMesh object
          d : 2
      order : 1
        box : [[-1.0, 1.0], [0.0, 1.0]]
    mapping : None
  Mesh (order,type,nq,nme) : (1,simplex,78,120)
  Number of 1-faces : 4
    [ 0] (order,type,nq,nme) : (1,simplex,6,5)
    [ 1] (order,type,nq,nme) : (1,simplex,6,5)
    [ 2] (order,type,nq,nme) : (1,simplex,13,12)
    [ 3] (order,type,nq,nme) : (1,simplex,13,12)
  Number of 0-faces : 4
    [ 0] (order,type,nq,nme) : (1,simplex,1,1)
    [ 1] (order,type,nq,nme) : (1,simplex,1,1)
    [ 2] (order,type,nq,nme) : (1,simplex,1,1)
    [ 3] (order,type,nq,nme) : (1,simplex,1,1)

  If matplotlib package is installed one can represent the mesh

  from fc_tools.graphics import set_axes_equal
  import matplotlib.pyplot as plt
  plt.ion()
  plt.figure(1)
  oTh.plotmesh(legend=True)
  plt.figure(2)
  oTh.plotmesh(m=1,legend=True,linewidth=3)
  plt.axis('off')
  set_axes_equal()

• Meshing the rectangular cuboid [-1,1]x[0,1]x[0,2] by simplices with 11+1 points in x-axis, 5+1 points in y-axis and 10+1 points in z-axis:

  from fc_hypermesh import OrthMesh
  oTh=OrthMesh(3,[10,5,10],box=[[-1,1],[0,1],[0,2]])

  If matplotlib package is installed one can represent the mesh

  from fc_tools.graphics import set_axes_equal
  import matplotlib.pyplot as plt
  plt.ion()
  plt.figure(1)
  oTh.plotmesh(legend=True,linewidth=0.5)
  set_axes_equal()
  plt.figure(2)
  oTh.plotmesh(m=2,legend=True,edgecolor=[0,0,0])
  plt.axis('off')
  set_axes_equal()

• Meshing the rectangle [-1,1]x[0,1] by orthotopes with 12+1 points in x-axis, 5+1 points in y-axis and 10+1 points in z-axis:

  from fc_hypermesh import OrthMesh
  oTh=OrthMesh(2,[12,5],type='orthotope',box=[[-1,1],[0,1]])

  If matplotlib package is installed one can represent the mesh

  from fc_tools.graphics import set_axes_equal
  import matplotlib.pyplot as plt
  plt.ion()
  plt.figure(1)
  oTh.plotmesh(legend=True)
  set_axes_equal()
  plt.figure(2)
  oTh.plotmesh(m=1,legend=True,linewidth=3)
  plt.axis('off')
  set_axes_equal()

• Meshing the rectangular cuboid [-1,1]x[0,1]x[0,2] by orthotopes with 11+1 points in x-axis, 5+1 points in y-axis and 10+1 points in z-axis:

  from fc_hypermesh import OrthMesh
  oTh=OrthMesh(3,[10,5,10],type='orthotope', box=[[-1,1],[0,1],[0,2]])

  If matplotlib package is installed one can represent the mesh

  from fc_tools.graphics import set_axes_equal
  import matplotlib.pyplot as plt
  plt.ion()
  plt.figure(1)
  oTh.plotmesh(legend=True,linewidth=0.5)
  set_axes_equal()
  plt.figure(2)
  oTh.plotmesh(m=2,legend=True,edgecolor=[0,0,0])
  plt.axis('off')
  set_axes_equal()

Testing :

There are eight demos functions in the fc_hypermesh.demos module named demo01 to demo08. The source code is in module demos.py. For example, run the following code under Python:

import fc_hypermesh
fc_hypermesh.demos.demo01()

Benchmarking:

• Bench of the OrthMesh constructor for a 3-dimensional orthotope tessellated with (order 1) simplices.

  import fc_hypermesh
  fc_hypermesh.bench(3,range(20,170,20),type='simplex',box=[[-1,1],[-1,1],[-1,1]])

  The output of this code is:

  #---------------------------------------------------------------------------
  #    computer: zbook17
  #      system: Ubuntu 18.04.3 LTS (x86_64)
  #   processor: Intel(R) Core(TM) i7-4800MQ CPU @ 2.70GHz
  #              (1 procs/4 cores by proc/2 threads by core)
  #         RAM: 31.3 Go
  #    software: Python
  #     release: 3.7.6
  #---------------------------------------------------------------------------
  # fc_hypermesh.OrthMesh constructor with
  #   d      =3
  #   type   =simplex
  #   order  =1
  #   box    =[[-1, 1], [-1, 1], [-1, 1]]
  #   mapping=None
  #---------------------------------------------------------------------------
  #date:2019-12-29_14-13-35
  #nbruns:5
  #numpy:      i8        i8        i8            f8
  #format: {:>7d}   {:>10d}   {:>10d}      {:11.3f}
  #labels:      N        nq       nme   OrthMesh(s)
               20      9261     48000         0.196
               40     68921    384000         0.209
               60    226981   1296000         0.302
               80    531441   3072000         0.411
              100   1030301   6000000         0.576
              120   1771561  10368000         0.840
              140   2803221  16464000         1.197
              160   4173281  24576000         1.679

• Bench of the OrthMesh constructor for a 3-dimensional orthotope tessellated with 3-order simplices.

  import fc_hypermesh
  fc_hypermesh.bench(3,range(10,61,10),order=3,type='simplex',box=[[-1,1],[-1,1],[-1,1]])

  The output of this code is:

  #---------------------------------------------------------------------------
  #    computer: zbook17
  #      system: Ubuntu 18.04.3 LTS (x86_64)
  #   processor: Intel(R) Core(TM) i7-4800MQ CPU @ 2.70GHz
  #              (1 procs/4 cores by proc/2 threads by core)
  #         RAM: 31.3 Go
  #    software: Python
  #     release: 3.7.6
  #---------------------------------------------------------------------------
  # fc_hypermesh.OrthMesh constructor with
  #   d      =3
  #   type   =simplex
  #   order  =3
  #   box    =[[-1, 1], [-1, 1], [-1, 1]]
  #   mapping=None
  #---------------------------------------------------------------------------
  #date:2019-12-29_14-26-34
  #nbruns:5
  #numpy:      i8        i8        i8            f8
  #format: {:>7d}   {:>10d}   {:>10d}      {:11.3f}
  #labels:      N        nq       nme   OrthMesh(s)
               10     29791      6000         0.143
               20    226981     48000         0.222
               30    753571    162000         0.269
               40   1771561    384000         0.357
               50   3442951    750000         0.500
               60   5929741   1296000         0.733

• Bench of the OrthMesh constructor for a 5-dimensional orthotope tessellated with (order 1) orthotopes.

  import fc_hypermesh
  fc_hypermesh.bench(5,[5,10,15,20,25,27],type='orthotope',box=[[-1,1],[-1,1],[-1,1],[-1,1],[-1,1]])

  The output of this code is:

  #---------------------------------------------------------------------------
  #    computer: zbook17
  #      system: Ubuntu 18.04.3 LTS (x86_64)
  #   processor: Intel(R) Core(TM) i7-4800MQ CPU @ 2.70GHz
  #              (1 procs/4 cores by proc/2 threads by core)
  #         RAM: 31.3 Go
  #    software: Python
  #     release: 3.7.6
  #---------------------------------------------------------------------------
  # fc_hypermesh.OrthMesh constructor with
  #   d      =5
  #   type   =orthotope
  #   order  =1
  #   box    =[[-1, 1], [-1, 1], [-1, 1], [-1, 1], [-1, 1]]
  #   mapping=None
  #---------------------------------------------------------------------------
  #date:2019-12-29_14-18-30
  #nbruns:5
  #numpy:      i8        i8        i8            f8
  #format: {:>7d}   {:>10d}   {:>10d}      {:11.3f}
  #labels:      N        nq       nme   OrthMesh(s)
                5      7776      3125         0.427
               10    161051    100000         0.529
               15   1048576    759375         0.844
               20   4084101   3200000         1.789
               25  11881376   9765625         4.296
               27  17210368  14348907         5.958