elasticity.py¶
In this script we solve the linear elasticity problem on a unit square domain, clamped at the left boundary, and stretched at the right boundary while keeping vertical displacements free.
7from nutils import mesh, function, solver, export, cli, testing
8from nutils.expression_v2 import Namespace
9
10
11def main(nelems: int, etype: str, btype: str, degree: int, poisson: float):
12 '''
13 Horizontally loaded linear elastic plate.
14
15 .. arguments::
16
17 nelems [10]
18 Number of elements along edge.
19 etype [square]
20 Type of elements (square/triangle/mixed).
21 btype [std]
22 Type of basis function (std/spline), with availability depending on the
23 configured element type.
24 degree [1]
25 Polynomial degree.
26 poisson [.25]
27 Poisson's ratio, nonnegative and strictly smaller than 1/2.
28 '''
29
30 domain, geom = mesh.unitsquare(nelems, etype)
31
32 ns = Namespace()
33 ns.δ = function.eye(domain.ndims)
34 ns.x = geom
35 ns.define_for('x', gradient='∇', normal='n', jacobians=('dV', 'dS'))
36 ns.basis = domain.basis(btype, degree=degree).vector(2)
37 ns.u = function.dotarg('lhs', ns.basis)
38 ns.X_i = 'x_i + u_i'
39 ns.lmbda = 2 * poisson
40 ns.mu = 1 - 2 * poisson
41 ns.strain_ij = '(∇_j(u_i) + ∇_i(u_j)) / 2'
42 ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
43
44 sqr = domain.boundary['left'].integral('u_k u_k dS' @ ns, degree=degree*2)
45 sqr += domain.boundary['right'].integral('(u_0 - .5)^2 dS' @ ns, degree=degree*2)
46 cons = solver.optimize('lhs', sqr, droptol=1e-15)
47
48 res = domain.integral('∇_j(basis_ni) stress_ij dV' @ ns, degree=degree*2)
49 lhs = solver.solve_linear('lhs', res, constrain=cons)
50
51 bezier = domain.sample('bezier', 5)
52 X, sxy = bezier.eval(['X_i', 'stress_01'] @ ns, lhs=lhs)
53 export.triplot('shear.png', X, sxy, tri=bezier.tri, hull=bezier.hull)
54
55 return 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
elasticity.py
(view log). To select mixed elements and quadratic basis functions add
python3 elasticity.py etype=mixed degree=2
(view log).
64if __name__ == '__main__':
65 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.
74class test(testing.TestCase):
75
76 def test_default(self):
77 cons, lhs = main(nelems=4, etype='square', btype='std', degree=1, poisson=.25)
78 with self.subTest('constraints'):
79 self.assertAlmostEqual64(cons, '''
80 eNpjYMACGsiHP0wxMQBKlBdi''')
81 with self.subTest('left-hand side'):
82 self.assertAlmostEqual64(lhs, '''
83 eNpjYMAEKcaiRmLGQQZCxgwMYsbrzqcYvz672KTMaIKJimG7CQPDBJM75xabdJ3NMO0xSjG1MUw0Beox
84 PXIuw7Tk7A/TXqMfQLEfQLEfQLEfpsVnAUzzHtI=''')
85
86 def test_mixed(self):
87 cons, lhs = main(nelems=4, etype='mixed', btype='std', degree=1, poisson=.25)
88 with self.subTest('constraints'):
89 self.assertAlmostEqual64(cons, '''
90 eNpjYICCBiiEsdFpIuEPU0wMAG6UF2I=''')
91 with self.subTest('left-hand side'):
92 self.assertAlmostEqual64(lhs, '''
93 eNpjYICAJGMOI3ljcQMwx3i/JohSMr51HkQnGP8422eiYrjcJM+o3aToWq/Jy3PLTKafzTDtM0oxtTRM
94 MF2okmJ67lyGacnZH6aOhj9Mu41+mMZq/DA9dO6HaflZAAMdIls=''')
95
96 def test_quadratic(self):
97 cons, lhs = main(nelems=4, etype='square', btype='std', degree=2, poisson=.25)
98 with self.subTest('constraints'):
99 self.assertAlmostEqual64(cons, '''
100 eNpjYCACNIxc+MOUMAYA/+NOFg==''')
101 with self.subTest('left-hand side'):
102 self.assertAlmostEqual64(lhs, '''
103 eNqFzL9KA0EQx/HlLI5wprBJCol/rtfN7MxobZEXOQIJQdBCwfgAItwVStQmZSAvcOmtVW6z5wP4D2yE
104 aKOwEhTnDRz4VvPhp9T/1zeP0ILF5hhSnUK5cQlKpaDvx3DoWvA57Zt128PIMO5CjHvNOn5s1lCpOi6V
105 MZ5PGS/k/1U0qGcqVMIcQ5jhmX4XM8N9N8dvWyFtG3RVjOjADOkNBrQMGV3rlJTKaMcN6NUOqWZHlBVV
106 PjER/0DIDAE/6ICVCjh2Id/ZiBdslY+LrpiOmLaYhJ90IibhNdcW0xHTFTPhUzPhX8h5W3rRuZicV1zO
107 N3bCgXRUeDFedjxvSc/ai/G86jzfWi87Xswfg5Nx3Q==''')
108
109 def test_poisson(self):
110 cons, lhs = main(nelems=4, etype='square', btype='std', degree=1, poisson=.4)
111 with self.subTest('constraints'):
112 self.assertAlmostEqual64(cons, '''
113 eNpjYMACGsiHP0wxMQBKlBdi''')
114 with self.subTest('left-hand side'):
115 self.assertAlmostEqual64(lhs, '''
116 eNpjYMAEFsaTjdcYvTFcasTAsMZI5JyFce6ZKSavjbNMFhhFmDAwZJkknJ1iInom0ZTJJNx0q1GgKQND
117 uKn32UTTf6d/mLKY/DDdZvQDKPbD1OvsD9M/pwGZyh9l''')