platewithhole.py

In this script we solve the linear plane strain elasticity problem for an infinite plate with a circular hole under tension. We do this by placing the circle in the origin of a unit square, imposing symmetry conditions on the left and bottom, and Dirichlet conditions constraining the displacements to the analytical solution to the right and top. The traction-free circle is removed by means of the Finite Cell Method (FCM).

10from nutils import mesh, function, solver, export, cli, testing
11from nutils.expression_v2 import Namespace
12import numpy
13import treelog
14
15
16def main(nelems: int, etype: str, btype: str, degree: int, traction: float, maxrefine: int, radius: float, poisson: float):
17    '''
18    Horizontally loaded linear elastic plate with FCM hole.
19
20    .. arguments::
21
22       nelems [9]
23         Number of elements along edge.
24       etype [square]
25         Type of elements (square/triangle/mixed).
26       btype [std]
27         Type of basis function (std/spline), with availability depending on the
28         selected element type.
29       degree [2]
30         Polynomial degree.
31       traction [.1]
32         Far field traction (relative to Young's modulus).
33       maxrefine [2]
34         Number or refinement levels used for the finite cell method.
35       radius [.5]
36         Cut-out radius.
37       poisson [.3]
38         Poisson's ratio, nonnegative and strictly smaller than 1/2.
39    '''
40
41    domain0, geom = mesh.unitsquare(nelems, etype)
42    domain = domain0.trim(function.norm2(geom) - radius, maxrefine=maxrefine)
43
44    ns = Namespace()
45    ns.δ = function.eye(domain.ndims)
46    ns.x = geom
47    ns.define_for('x', gradient='∇', normal='n', jacobians=('dV', 'dS'))
48    ns.lmbda = 2 * poisson
49    ns.mu = 1 - poisson
50    ns.ubasis = domain.basis(btype, degree=degree).vector(2)
51    ns.u = function.dotarg('lhs', ns.ubasis)
52    ns.X_i = 'x_i + u_i'
53    ns.strain_ij = '(∇_j(u_i) + ∇_i(u_j)) / 2'
54    ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
55    ns.r2 = 'x_k x_k'
56    ns.R2 = radius**2 / ns.r2
57    ns.k = (3-poisson) / (1+poisson)  # plane stress parameter
58    ns.scale = traction * (1+poisson) / 2
59    ns.uexact_i = 'scale (x_i ((k + 1) (0.5 + R2) + (1 - R2) R2 (x_0^2 - 3 x_1^2) / r2) - 2 δ_i1 x_1 (1 + (k - 1 + R2) R2))'
60    ns.du_i = 'u_i - uexact_i'
61
62    sqr = domain.boundary['left,bottom'].integral('(u_i n_i)^2 dS' @ ns, degree=degree*2)
63    cons = solver.optimize('lhs', sqr, droptol=1e-15)
64    sqr = domain.boundary['top,right'].integral('du_k du_k dS' @ ns, degree=20)
65    cons = solver.optimize('lhs', sqr, droptol=1e-15, constrain=cons)
66
67    res = domain.integral('∇_j(ubasis_ni) stress_ij dV' @ ns, degree=degree*2)
68    lhs = solver.solve_linear('lhs', res, constrain=cons)
69
70    bezier = domain.sample('bezier', 5)
71    X, stressxx = bezier.eval(['X_i', 'stress_00'] @ ns, lhs=lhs)
72    export.triplot('stressxx.png', X, stressxx, tri=bezier.tri, hull=bezier.hull)
73
74    err = domain.integral(function.stack(['du_k du_k dV', '∇_j(du_i) ∇_j(du_i) dV'] @ ns), degree=max(degree, 3)*2).eval(lhs=lhs)**.5
75    treelog.user('errors: L2={:.2e}, H1={:.2e}'.format(*err))
76
77    return err, cons, lhs

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 platewithhole.py (view log). To select mixed elements and quadratic basis functions add python3 platewithhole.py etype=mixed degree=2 (view log).

86if __name__ == '__main__':
87    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.

 96class test(testing.TestCase):
 97
 98    def test_spline(self):
 99        err, cons, lhs = main(nelems=4, etype='square', btype='spline', degree=2, traction=.1, maxrefine=2, radius=.5, poisson=.3)
100        with self.subTest('l2-error'):
101            self.assertAlmostEqual(err[0], .00033, places=5)
102        with self.subTest('h1-error'):
103            self.assertAlmostEqual(err[1], .00672, places=5)
104        with self.subTest('constraints'):
105            self.assertAlmostEqual64(cons, '''
106                eNpjaGBoYGBAxvrnGBow4X89g3NQFSjQwLAGq7i10Wus4k+NfM8fNWZgOGL89upc47WX0ozvXjAzPn1e
107                1TjnPACrACoJ''')
108        with self.subTest('left-hand side'):
109            self.assertAlmostEqual64(lhs, '''
110                eNpbZHbajIHhxzkGBhMgtgdi/XPypyRPvjFxO/PccPq5Vn2vcxr6luf+6xmcm2LMwLDQePf5c0bTzx8x
111                5D7vaTjnnIFhzbmlQPH5xhV39Y3vXlxtJHoh2EjvvLXR63MbgOIbjRdfrTXeecnUeO+Fn0Yrzj818j1/
112                FCh+xPjt1bnGay+lGd+9YGZ8+ryqcc55AK+AP/0=''')
113
114    def test_mixed(self):
115        err, cons, lhs = main(nelems=4, etype='mixed', btype='std', degree=2, traction=.1, maxrefine=2, radius=.5, poisson=.3)
116        with self.subTest('l2-error'):
117            self.assertAlmostEqual(err[0], .00024, places=5)
118        with self.subTest('h1-error'):
119            self.assertAlmostEqual(err[1], .00739, places=5)
120        with self.subTest('constraints'):
121            self.assertAlmostEqual64(cons, '''
122                eNpjaGDADhlwiOEU1z8HZusbgukkg5BzRJqKFRoa1oD1HzfceA5NH9FmgKC10SuwOdONpM7DxDYa77gM
123                MueoMQPDEePzV2Hic42XXmoynnQRxvc3dryQbnz3Aoj91Mj3vJnx6fOqxjnnAQzkV94=''')
124        with self.subTest('left-hand side'):
125            self.assertAlmostEqual64(lhs, '''
126                eNoNzE8og3EcBvC3uUo5rNUOnBSK9/19n0Ic0Eo5oJBmRxcaB04kUnPgoETmT2w7LVrtMBy4auMw+35/
127                7/vaykFSFEopKTnIe/jU01PPU6FNWcQIn+Or5CBfSqCGD1uDYhi7/KbW+dma5aK65gX6Y8Po8HSzZQ7y
128                vBniHyvFV9aq17V7TK42O9kwFS9YUzxhjXIcZxLCnIzjTsfxah/BMFJotjUlZYz6xYeoPqEPKaigbKhb
129                9lOj9NGa9KgtVmqJH9UT36gcp71dEr6HaVS5GS8f46AcQ9itx739SQXdBL8dRqeTo1odox35poh2yJVh
130                apEueucsRWWPgpJFoLKPNzeHC/fU+yl48pDyMi6dCFbsBNJODNu2iawOoE4PoVdP4kH/UkZeaEDaUJQG
131                zMg/DouRUg==''')