ellipt2d 2.1.0
A package to solve second order elliptic problems in 2D
License: MIT license
Requires Python: >=3.5
Keywords: ellipt2d

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ellipt2d
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A package for solving second order elliptic problems in 2D


* Free software: MIT license
* Documentation: https://ellipt2d.readthedocs.io.


Features
--------

* Supports triangulated domain generated by the _pytriangle: https://github.com/pletzer/pytriangle package
* Stiffness is a tensor
* Can output solution to VTK file
* Simple, easy to use

How to solve an elliptic problem with Ellipt2d
----------------------------------------------

Define the domain. We recommend to use the pytriangle package to triangule a domain. This involves specifying 
boundary points and segments. If there are holes in the domain, then you'll need to specify those as well. As an example, we'll 
triangulate an annulus::

    import triangle
    import numpy
    
    # number of outer and inner poloidal points
    nto, nti = 16, 8
    
    dto, dti = 2*numpy.pi / nto, 2*numpy.pi / nti
    to = numpy.linspace(0., 2*numpy.pi - dto, nto)
    ti = numpy.linspace(0., 2*numpy.pi - dti, nti)
    
    # outer boundary, points and segments go counterclockwise
    bound_pts = [(numpy.cos(t), numpy.sin(t)) for t in to]
    bound_seg = [(i, i + 1) for i in range(nto - 1)] + [(nto - 1, 0)] # close the contour
    
    # add the inner boundary, points and segments go clockwise
    bound_pts += [(0.3*numpy.cos(t), 0.3*numpy.sin(t)) for t in ti]
    bound_seg += [(i, i+1) for i in range(nto, nto + nti - 1)] + [(nto + nti - 1, nto)]
    
    # now create the triangulation
    grid = triangle.Triangle()
    grid.set_points(bound_pts)
    grid.set_segments(bound_seg)
    grid.set_holes([(0., 0.)])
    grid.triangulate()

Create an Ellipt2d instance, here a Laplace operator::

    import ellipt2d
    
    # - div F . grad u + g = s
    # F = [[fxx, fxy], [fxy, fyy]] and g are defined on cells and s on nodes
    # here fxx = fyy = 1 and fxy = g = 0
    nnodes = grid.get_num_points()
    ncells = grid.get_num_triangles()
    fxx = fyy = numpy.ones(ncells)
    fxy = g = numpy.zeros(ncells)
    s = numpy.zeros(nnodes)
    
    equ = ellipt2d.Ellipt2d(grid=grid, fxx=fxx, fxy=fxy, fyy=fyy, g=g, s=s)
  
Set the boundary conditions. You only need to specify boundary conditions if they are different from the zero flux boundary conditions. Here we specify the 
Dirichlet boundary conditions on the outer contour (first nto - 1 points)::

    # gather all the points that are on the external boundary where the radius is larger than 0.31
    nodes = grid.get_points() # [(x, y, 0 or 1), ...]
    ext_pts = [(i, nodes[i][0][0], nodes[i][0][1]) for i in range(nnodes) if nodes[i][1] == 1 and nodes[i][0][0]**2 + nodes[i][0][1]**2 > 0.31**2]
    # Dirichlet BC is cos of angle
    db = {bp[0]: numpy.cos(numpy.arctan2(bp[2], bp[1])) for bp in ext_pts}
    equ.setDirichletBoundaryConditions(db)

Solve the linear system::

    u = equ.solve()
    
Save the solution in a VTK file for plotting with Paraview or VisIt::

    equ.saveVTK(filename='sol.vtk', solution=u, sol_name='u')
    
.. image:: images/sol.png
   :width: 600    

Credits
-------

This package was created with Cookiecutter_ and the `audreyr/cookiecutter-pypackage`_ project template.

.. _Cookiecutter: https://github.com/audreyr/cookiecutter
.. _`audreyr/cookiecutter-pypackage`: https://github.com/audreyr/cookiecutter-pypackage


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History
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2.0.0 (2020-12-15)
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* First release on PyPI.