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:
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:
We start by importing the necessary modules.
23from nutils import mesh, function, solver, export, cli, testing
24import treelog
25
26def main(nelems:int, etype:str, btype:str, degree:int):
27 '''
28 Laplace problem on a unit square.
29
30 .. arguments::
31
32 nelems [10]
33 Number of elements along edge.
34 etype [square]
35 Type of elements (square/triangle/mixed).
36 btype [std]
37 Type of basis function (std/spline), availability depending on the
38 selected element type.
39 degree [1]
40 Polynomial degree.
41 '''
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
.
49 domain, geom = 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
.
56 ns = function.Namespace()
57 ns.x = geom
58 ns.basis = domain.basis(btype, degree=degree)
59 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:
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.
70 res = domain.integral('basis_n,i u_,i d:x' @ ns, degree=degree*2)
71 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:
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.
83 sqr = domain.boundary['left'].integral('u^2 d:x' @ ns, degree=degree*2)
84 sqr += domain.boundary['top'].integral('(u - cosh(1) sin(x_0))^2 d:x' @ ns, degree=degree*2)
85 cons = 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.
93 lhs = 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.
102 bezier = domain.sample('bezier', 9)
103 x, u = bezier.eval(['x_i', 'u'] @ ns, lhs=lhs)
104 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.
109 err = domain.integral('(u - sin(x_0) cosh(x_1))^2 d:x' @ ns, degree=degree*2).eval(lhs=lhs)**.5
110 treelog.user('L2 error: {:.2e}'.format(err))
111
112 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).
120if __name__ == '__main__':
121 cli.run(main)
Once a simulation is developed and tested, it is good practice to save a few
strategic return values for regression testing. The nutils.testing
module, which builds on the standard unittest
framework, facilitates
this by providing nutils.testing.TestCase.assertAlmostEqual64()
for the
embedding of desired results as compressed base64 data.
129class test(testing.TestCase):
130
131 @testing.requires('matplotlib')
132 def test_default(self):
133 cons, lhs, err = main(nelems=4, etype='square', btype='std', degree=1)
134 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
135 eNrbKPv1QZ3ip9sL1BgaILDYFMbaZwZj5ZnDWNfNAeWPESU=''')
136 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
137 eNoBMgDN/7Ed9eB+IfLboCaXNKc01DQaNXM14jXyNR82ZTa+NpI2oTbPNhU3bjf7Ngo3ODd+N9c3SNEU
138 1g==''')
139 with self.subTest('L2-error'):
140 self.assertAlmostEqual(err, 1.63e-3, places=5)
141
142 @testing.requires('matplotlib')
143 def test_spline(self):
144 cons, lhs, err = main(nelems=4, etype='square', btype='spline', degree=2)
145 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
146 eNqrkmN+sEfhzF0xleRbDA0wKGeCYFuaIdjK5gj2aiT2VXMAJB0VAQ==''')
147 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
148 eNqrkmN+sEfhzF0xleRbrsauxsnGc43fGMuZJJgmmNaZ7jBlN7M08wLCDLNFZh/NlM0vmV0y+2CmZV5p
149 vtr8j9kfMynzEPPF5lfNAcuhGvs=''')
150 with self.subTest('L2-error'):
151 self.assertAlmostEqual(err, 8.04e-5, places=7)
152
153 @testing.requires('matplotlib')
154 def test_mixed(self):
155 cons, lhs, err = main(nelems=4, etype='mixed', btype='std', degree=2)
156 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
157 eNorfLZF2ucJQwMC3pR7+QDG9lCquAtj71Rlu8XQIGfC0FBoiqweE1qaMTTsNsOvRtmcoSHbHL+a1UD5
158 q+YAxhcu1g==''')
159 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
160 eNorfLZF2ueJq7GrcYjxDJPpJstNbsq9fOBr3Gh8xWS7iYdSxd19xseMP5hImu5UZbv1xljOxM600DTW
161 NN/0k2mC6SPTx6Z1pnNMGc3kzdaaPjRNMbMyEzWzNOsy223mBYRRZpPNJpktMks1azM7Z7bRbIXZabNX
162 ZiLmH82UzS3Ns80vmj004za/ZPYHCD+Y8ZlLmVuYq5kHm9eahwDxavPF5lfNAWFyPdk=''')
163 with self.subTest('L2-error'):
164 self.assertAlmostEqual(err, 1.25e-4, places=6)