laplace.py

In this script we solve the Laplace equation \(u_{,kk} = 0\) on a unit square domain \(Ω\) with boundary \(Γ\), subject to boundary conditions:

\[ \begin{align}\begin{aligned}u &= 0 && Γ_{\rm left}\\ ∂_n u &= 0 && Γ_{\rm bottom}\\ ∂_n u &= \cos(1) \cosh(x_1) && Γ_{\rm right}\\ u &= \cosh(1) \sin(x_0) && Γ_{\rm top} \end{aligned}\end{align} \]

This case is constructed to contain all combinations of homogenous and heterogeneous, Dirichlet and Neumann type boundary conditions, as well as to have a known exact solution:

\[u_{\rm exact} = \sin(x_0) \cosh(x_1).\]

We start by importing the necessary modules.

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import nutils, numpy

The main function defines the parameter space for the script. Configurable parameters are the mesh density (in number of elements along an edge), element type (square, triangle, or mixed), type of basis function (std or spline, with availability depending on element type), and polynomial degree.

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def main(nelems: 'number of elements along edge' = 10,
         etype: 'type of elements (square/triangle/mixed)' = 'square',
         btype: 'type of basis function (std/spline)' = 'std',
         degree: 'polynomial degree' = 1):

A unit square domain is created by calling the nutils.mesh.unitsquare() mesh generator, with the number of elements along an edge as the first argument, and the type of elements (“square”, “triangle”, or “mixed”) as the second. The result is a topology object domain and a vectored valued geometry function geom.

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  domain, geom = nutils.mesh.unitsquare(nelems, etype)

To be able to write index based tensor contractions, we need to bundle all relevant functions together in a namespace. Here we add the geometry x, a scalar basis, and the solution u. The latter is formed by contracting the basis with a to-be-determined solution vector ?lhs.

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  ns = nutils.function.Namespace()
  ns.x = geom
  ns.basis = domain.basis(btype, degree=degree)
  ns.u = 'basis_n ?lhs_n'

We are now ready to implement the Laplace equation. In weak form, the solution is a scalar field \(u\) for which:

\[∀ v: ∫_Ω v_{,k} u_{,k} - ∫_{Γ_n} v f = 0.\]

By linearity the test function \(v\) can be replaced by the basis that spans its space. The result is an integral res that evaluates to a vector matching the size of the function space.

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  res = domain.integral('basis_n,i u_,i d:x' @ ns, degree=degree*2)
  res -= domain.boundary['right'].integral('basis_n cos(1) cosh(x_1) d:x' @ ns, degree=degree*2)

The Dirichlet constraints are set by finding the coefficients that minimize the error:

\[\min_u ∫_{\Gamma_d} (u - u_d)^2\]

The resulting cons array holds numerical values for all the entries of ?lhs that contribute (up to droptol) to the minimization problem. All remaining entries are set to NaN, signifying that these degrees of freedom are unconstrained.

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  sqr = domain.boundary['left'].integral('u^2 d:x' @ ns, degree=degree*2)
  sqr += domain.boundary['top'].integral('(u - cosh(1) sin(x_0))^2 d:x' @ ns, degree=degree*2)
  cons = nutils.solver.optimize('lhs', sqr, droptol=1e-15)

The unconstrained entries of ?lhs are to be determined such that the residual vector evaluates to zero in the corresponding entries. This step involves a linearization of res, resulting in a jacobian matrix and right hand side vector that are subsequently assembled and solved. The resulting lhs array matches cons in the constrained entries.

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  lhs = nutils.solver.solve_linear('lhs', res, constrain=cons)

Once all entries of ?lhs are establised, the corresponding solution can be vizualised by sampling values of ns.u along with physical coordinates ns.x, with the solution vector provided via the arguments dictionary. The sample members tri and hull provide additional inter-point information required for drawing the mesh and element outlines.

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  bezier = domain.sample('bezier', 9)
  x, u = bezier.eval(['x_i', 'u'] @ ns, lhs=lhs)
  nutils.export.triplot('solution.png', x, u, tri=bezier.tri, hull=bezier.hull)

To confirm that our computation is correct, we use our knowledge of the analytical solution to evaluate the L2-error of the discrete result.

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  err = domain.integral('(u - sin(x_0) cosh(x_1))^2 d:x' @ ns, degree=degree*2).eval(lhs=lhs)**.5
  nutils.log.user('L2 error: {:.2e}'.format(err))

  return cons, lhs, err

If the script is executed (as opposed to imported), nutils.cli.run() calls the main function with arguments provided from the command line. For example, to keep with the default arguments simply run python3 laplace.py (view log). To select mixed elements and quadratic basis functions add python3 laplace.py etype=mixed degree=2 (view log).

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if __name__ == '__main__':
  nutils.cli.run(main)

Once a simulation is developed and tested, it is good practice to save a few strategicly chosen return values for routine regression testing. Here we use the standard unittest framework, with nutils.numeric.assert_allclose64() facilitating the embedding of desired results as compressed base64 data.

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class test(nutils.testing.TestCase):

  @nutils.testing.requires('matplotlib')
  def test_default(self):
    cons, lhs, err = main(nelems=4, etype='square', btype='std', degree=1)
    nutils.numeric.assert_allclose64(cons, 'eNrbKPv1QZ3ip9sL1BgaILDYFMbaZwZj5ZnDW'
      'NfNAeWPESU=')
    nutils.numeric.assert_allclose64(lhs, 'eNoBMgDN/7Ed9eB+IfLboCaXNKc01DQaNXM14j'
      'XyNR82ZTa+NpI2oTbPNhU3bjf7Ngo3ODd+N9c3SNEU1g==')
    numpy.testing.assert_almost_equal(err, 1.63e-3, decimal=5)

  @nutils.testing.requires('matplotlib')
  def test_spline(self):
    cons, lhs, err = main(nelems=4, etype='square', btype='spline', degree=2)
    nutils.numeric.assert_allclose64(cons, 'eNqrkmN+sEfhzF0xleRbDA0wKGeCYFuaIdjK5'
      'gj2aiT2VXMAJB0VAQ==')
    nutils.numeric.assert_allclose64(lhs, 'eNqrkmN+sEfhzF0xleRbrsauxsnGc43fGMuZJJ'
      'gmmNaZ7jBlN7M08wLCDLNFZh/NlM0vmV0y+2CmZV5pvtr8j9kfMynzEPPF5lfNAcuhGvs=')
    numpy.testing.assert_almost_equal(err, 8.04e-5, decimal=7)

  @nutils.testing.requires('matplotlib')
  def test_mixed(self):
    cons, lhs, err = main(nelems=4, etype='mixed', btype='std', degree=2)
    nutils.numeric.assert_allclose64(cons, 'eNorfLZF2ucJQwMC3pR7+QDG9lCquAtj71Rlu'
      '8XQIGfC0FBoiqweE1qaMTTsNsOvRtmcoSHbHL+a1UD5q+YAxhcu1g==')
    nutils.numeric.assert_allclose64(lhs, 'eNorfLZF2ueJq7GrcYjxDJPpJstNbsq9fOBr3G'
      'h8xWS7iYdSxd19xseMP5hImu5UZbv1xljOxM600DTWNN/0k2mC6SPTx6Z1pnNMGc3kzdaaPjRN'
      'MbMyEzWzNOsy223mBYRRZpPNJpktMks1azM7Z7bRbIXZabNXZiLmH82UzS3Ns80vmj004za/ZP'
      'YHCD+Y8ZlLmVuYq5kHm9eahwDxavPF5lfNAWFyPdk=')
    numpy.testing.assert_almost_equal(err, 1.25e-4, decimal=6)