diffusion/laplace_iga_interactive.py

Description

Laplace equation with Dirichlet boundary conditions solved in a single patch NURBS domain using the isogeometric analysis (IGA) approach, using commands for interactive use.

This script allows the creation of a customisable NURBS surface using igakit built-in CAD routines, which is then saved in custom HDF5-based files with .iga extension.

Notes

The create_patch function creates a NURBS-patch of the area between two coplanar nested circles using igakit CAD built-in routines. The created patch is not connected in the orthoradial direction. This is a problem when the disconnected boundary is not perpendicular to the line connecting the two centres of the circles, as the solution then exhibits a discontinuity along this line. A workaround for this issue is to enforce perpendicularity by changing the start angle in function igakit.cad.circle (see the code down below for the actual trick). The discontinuity disappears.

Usage Examples

Default options, storing results in this file’s parent directory:

$ python3 sfepy/examples/diffusion/laplace_iga_interactive.py

Command line options for tweaking the geometry of the NURBS-patch & more:

$ python3 sfepy/examples/diffusion/laplace_iga_interactive.py --R1=0.7 --C2=0.1,0.1 --viewpatch

View the results using:

$ python3 resview.py concentric_circles.vtk

source code

#!/usr/bin/env python
r"""
Laplace equation with Dirichlet boundary conditions solved in a single patch
NURBS domain using the isogeometric analysis (IGA) approach, using commands
for interactive use.

This script allows the creation of a customisable NURBS surface using igakit
built-in CAD routines, which is then saved in custom HDF5-based files with
.iga extension.

Notes
-----

The ``create_patch`` function creates a NURBS-patch of the area between two
coplanar nested circles using igakit CAD built-in routines. The created patch
is not connected in the orthoradial direction. This is a problem when the
disconnected boundary is not perpendicular to the line connecting the two
centres of the circles, as the solution then exhibits a discontinuity along
this line. A workaround for this issue is to enforce perpendicularity by
changing the start angle in function ``igakit.cad.circle`` (see the code down
below for the actual trick). The discontinuity disappears.

Usage Examples
--------------

Default options, storing results in this file's parent directory::

  $ python3 sfepy/examples/diffusion/laplace_iga_interactive.py

Command line options for tweaking the geometry of the NURBS-patch & more::

  $ python3 sfepy/examples/diffusion/laplace_iga_interactive.py --R1=0.7 --C2=0.1,0.1 --viewpatch

View the results using::

  $ python3 resview.py concentric_circles.vtk
"""

from argparse import RawDescriptionHelpFormatter, ArgumentParser

import os
import sys
sys.path.append('.')

import numpy as nm
from sfepy import data_dir
from sfepy.base.ioutils import ensure_path
from sfepy.base.base import IndexedStruct
from sfepy.discrete import (FieldVariable, Integral, Equation,Equations,
                            Problem)
from sfepy.discrete.iga.domain import IGDomain
from sfepy.discrete.common.fields import Field
from sfepy.terms import Term
from sfepy.discrete.conditions import Conditions, EssentialBC
from sfepy.solvers.ls import ScipyDirect
from sfepy.solvers.nls import Newton

def create_patch(R1, R2, C1, C2, order=2, viewpatch=False):
    """
    Create a single 2d NURBS-patch of the area between two coplanar nested
    circles using igakit.

    Parameters
    ----------
    R1 : float
        Radius of the inner circle.
    R2 : float
        Radius of the outer circle.
    C1 : list of two floats
        Coordinates of the center of the inner circle given as [x1, y1].
    C2 : list of two floats
        Coordinates of the center of the outer circle given as [x2, y2].
    order : int, optional
        Degree of the NURBS basis functions. The default is 2.
    viewpatch : bool, optional
        When set to True, display the NURBS patch. The default is False.

    Returns
    -------
    None.

    """

    from sfepy.discrete.iga.domain_generators import create_from_igakit
    import sfepy.discrete.iga.io as io
    from igakit.cad import circle, ruled
    from igakit.plot import plt as iplt
    from numpy import pi

    # Assert the inner circle is inside the outer one
    inter_centers = nm.sqrt((C2[0]-C1[0])**2 + (C2[1]-C1[1])**2)
    assert R2>R1, "Outer circle should have a larger radius than the inner one"
    assert inter_centers<R2-R1, "Circles are not nested"

    # Geometry Creation
    centers_direction = [C2[0]-C1[0], C2[1]-C1[1]]
    if centers_direction[0]==0 and centers_direction[1]==0:
        start_angle = 0.0
    else:
        start_angle = nm.arctan2(centers_direction[1], centers_direction[0])
    c1 = circle(radius=R1, center=C1, angle=(start_angle, start_angle + 2*pi))
    c2 = circle(radius=R2, center=C2, angle=(start_angle, start_angle + 2*pi))
    srf = ruled(c1,c2).transpose() # make the radial direction first

    # Refinement
    insert_U = insert_uniformly(srf.knots[0], 6)
    insert_V = insert_uniformly(srf.knots[1], 6)
    srf.refine(0, insert_U).refine(1, insert_V)

    # Setting the NURBS-surface degree
    srf.elevate(0, order-srf.degree[0] if order-srf.degree[0] > 0 else 0)
    srf.elevate(1, order-srf.degree[1] if order-srf.degree[1] > 0 else 0)

    # Sfepy .iga file creation
    nurbs, bmesh, regions = create_from_igakit(srf, verbose=True)

    # Save .iga file in sfepy/meshes/iga
    filename_domain = data_dir + '/meshes/iga/concentric_circles.iga'
    io.write_iga_data(filename_domain, None, nurbs.knots, nurbs.degrees,
                      nurbs.cps, nurbs.weights, nurbs.cs, nurbs.conn,
                      bmesh.cps, bmesh.weights, bmesh.conn, regions)

    if viewpatch:
        iplt.use('matplotlib')
        iplt.figure()
        iplt.plot(srf)
        iplt.show()

def insert_uniformly(U, n):
    """
    Find knots to uniformly add to U.
    [Code from igakit/demo/venturi.py file]

    Given a knot vector U and the number of uniform spans desired,
    find the knots which need to be inserted.

    Parameters
    ----------
    U : numpy.ndarray
        Original knot vector for a C^p-1 space.
    n : int
        Target number of uniformly-spaced knot spans.

    Returns
    -------
    Knots to be inserted into U
    """
    U0 = U
    dU=(U.max()-U.min())/float(n) # target dU in knot vector
    idone=0
    while idone == 0:
        # Add knots in middle of spans which are too large
        Uadd=[]
        for i in range(len(U)-1):
            if U[i+1]-U[i] > dU:
                Uadd.append(0.5*(U[i+1]+U[i]))
        # Now we add these knots (once only, assumes C^(p-1))
        if len(Uadd) > 0:
            U = nm.sort(nm.concatenate([U,nm.asarray(Uadd)]))
        else:
            idone=1
        # And now a little Laplacian smoothing
        for num_iterations in range(5):
            for i in range(len(U)-2):
                if abs(U0[U0.searchsorted(U[i+1])]-U[i+1]) > 1.0e-14:
                    U[i+1] = 0.5*(U[i]+U[i+2])
    return nm.setdiff1d(U,U0)

helps = {
    'output_dir' :
    'output directory',
    'R1' :
    'Inner circle radius [default: %(default)s]',
    'R2' :
    'Outer circle radius [default: %(default)s]',
    'C1' :
    'centre of the inner circle [default: %(default)s]',
    'C2' :
    'centre of the outer circle [default: %(default)s]',
    'order' :
    'field approximation order [default: %(default)s]',
    'viewpatch' :
    'generate a plot of the NURBS-patch',
}

def main():
    parser = ArgumentParser(description=__doc__.rstrip(),
                            formatter_class=RawDescriptionHelpFormatter)
    parser.add_argument('-o', '--output-dir', default='.',
                        help=helps['output_dir'])
    parser.add_argument('--R1', metavar='R1',
                        action='store', dest='R1',
                        default='0.5', help=helps['R1'])
    parser.add_argument('--R2', metavar='R2',
                        action='store', dest='R2',
                        default='1.0', help=helps['R2'])
    parser.add_argument('--C1', metavar='C1',
                        action='store', dest='C1',
                        default='0.0,0.0', help=helps['C1'])
    parser.add_argument('--C2', metavar='C2',
                        action='store', dest='C2',
                        default='0.0,0.0', help=helps['C2'])
    parser.add_argument('--order', metavar='int', type=int,
                        action='store', dest='order',
                        default=2, help=helps['order'])
    parser.add_argument('-v', '--viewpatch',
                        action='store_true', dest='viewpatch',
                        default=False, help=helps['viewpatch'])
    options = parser.parse_args()

    # Creation of the NURBS-patch with igakit
    R1 = eval(options.R1)
    R2 = eval(options.R2)
    C1 = list(eval(options.C1))
    C2 = list(eval(options.C2))
    order = options.order
    viewpatch = options.viewpatch
    create_patch(R1, R2, C1, C2, order=order, viewpatch=viewpatch)

    # Setting a Domain instance
    filename_domain = data_dir + '/meshes/iga/concentric_circles.iga'
    domain = IGDomain.from_file(filename_domain)

    # Sub-domains
    omega = domain.create_region('Omega', 'all')
    Gamma_out = domain.create_region('Gamma_out', 'vertices of set xi01',
                                     kind='facet')
    Gamma_in = domain.create_region('Gamma_in', 'vertices of set xi00',
                                    kind='facet')

    # Field (featuring order elevation)
    order_increase = order - domain.nurbs.degrees[0]
    order_increase *= int(order_increase>0)
    field = Field.from_args('fu', nm.float64, 'scalar', omega,
                            approx_order='iga', space='H1',
                            poly_space_base='iga')

    # Variables
    u = FieldVariable('u', 'unknown', field) # unknown function
    v = FieldVariable('v', 'test', field, primary_var_name='u') # test function

    # Integral
    integral = Integral('i', order=2*field.approx_order)

    # Term
    t = Term.new('dw_laplace( v, u )', integral, omega, v=v, u=u)

    # Equation
    eq = Equation('laplace', t)
    eqs = Equations([eq])

    # Boundary Conditions
    u_in  = EssentialBC('u_in', Gamma_in, {'u.all' : 7.0})
    u_out = EssentialBC('u_out', Gamma_out, {'u.all' : 3.0})

    # solvers
    ls = ScipyDirect({})
    nls_status = IndexedStruct()
    nls = Newton({}, lin_solver=ls, status=nls_status)

    # problem instance
    pb = Problem('potential', equations=eqs, active_only=True)

    # Set boundary conditions
    pb.set_bcs(ebcs=Conditions([u_in, u_out]))

    # solving
    pb.set_solver(nls)
    status = IndexedStruct()
    state = pb.solve(status=status, save_results=True, verbose=True)

    # Saving the results to a classic VTK file
    filename = os.path.join(options.output_dir, 'concentric_circles.vtk')
    ensure_path(filename)
    pb.save_state(filename, state)

if __name__ == '__main__':
    main()