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
8
9def main(nelems:int, etype:str, btype:str, degree:int, poisson:float):
10 '''
11 Horizontally loaded linear elastic plate.
12
13 .. arguments::
14
15 nelems [10]
16 Number of elements along edge.
17 etype [square]
18 Type of elements (square/triangle/mixed).
19 btype [std]
20 Type of basis function (std/spline), with availability depending on the
21 configured element type.
22 degree [1]
23 Polynomial degree.
24 poisson [.25]
25 Poisson's ratio, nonnegative and strictly smaller than 1/2.
26 '''
27
28 domain, geom = mesh.unitsquare(nelems, etype)
29
30 ns = function.Namespace()
31 ns.x = geom
32 ns.basis = domain.basis(btype, degree=degree).vector(2)
33 ns.u_i = 'basis_ni ?lhs_n'
34 ns.X_i = 'x_i + u_i'
35 ns.lmbda = 2 * poisson
36 ns.mu = 1 - 2 * poisson
37 ns.strain_ij = '(u_i,j + u_j,i) / 2'
38 ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
39
40 sqr = domain.boundary['left'].integral('u_k u_k d:x' @ ns, degree=degree*2)
41 sqr += domain.boundary['right'].integral('(u_0 - .5)^2 d:x' @ ns, degree=degree*2)
42 cons = solver.optimize('lhs', sqr, droptol=1e-15)
43
44 res = domain.integral('basis_ni,j stress_ij d:x' @ ns, degree=degree*2)
45 lhs = solver.solve_linear('lhs', res, constrain=cons)
46
47 bezier = domain.sample('bezier', 5)
48 X, sxy = bezier.eval(['X_i', 'stress_01'] @ ns, lhs=lhs)
49 export.triplot('shear.png', X, sxy, tri=bezier.tri, hull=bezier.hull)
50
51 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).
59if __name__ == '__main__':
60 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.
68class test(testing.TestCase):
69
70 @testing.requires('matplotlib')
71 def test_default(self):
72 cons, lhs = main(nelems=4, etype='square', btype='std', degree=1, poisson=.25)
73 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
74 eNpjYMACGsiHP0wxMQBKlBdi''')
75 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
76 eNpjYMAEKcaiRmLGQQZCxgwMYsbrzqcYvz672KTMaIKJimG7CQPDBJM75xabdJ3NMO0xSjG1MUw0Beox
77 PXIuw7Tk7A/TXqMfQLEfQLEfQLEfpsVnAUzzHtI=''')
78
79 @testing.requires('matplotlib')
80 def test_mixed(self):
81 cons, lhs = main(nelems=4, etype='mixed', btype='std', degree=1, poisson=.25)
82 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
83 eNpjYICCBiiEsdFpIuEPU0wMAG6UF2I=''')
84 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
85 eNpjYICAJGMOI3ljcQMwx3i/JohSMr51HkQnGP8422eiYrjcJM+o3aToWq/Jy3PLTKafzTDtM0oxtTRM
86 MF2okmJ67lyGacnZH6aOhj9Mu41+mMZq/DA9dO6HaflZAAMdIls=''')
87
88 @testing.requires('matplotlib')
89 def test_quadratic(self):
90 cons, lhs = main(nelems=4, etype='square', btype='std', degree=2, poisson=.25)
91 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
92 eNpjYCACNIxc+MOUMAYA/+NOFg==''')
93 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
94 eNqFzL9KA0EQx/HlLI5wprBJCol/rtfN7MxobZEXOQIJQdBCwfgAItwVStQmZSAvcOmtVW6z5wP4D2yE
95 aKOwEhTnDRz4VvPhp9T/1zeP0ILF5hhSnUK5cQlKpaDvx3DoWvA57Zt128PIMO5CjHvNOn5s1lCpOi6V
96 MZ5PGS/k/1U0qGcqVMIcQ5jhmX4XM8N9N8dvWyFtG3RVjOjADOkNBrQMGV3rlJTKaMcN6NUOqWZHlBVV
97 PjER/0DIDAE/6ICVCjh2Id/ZiBdslY+LrpiOmLaYhJ90IibhNdcW0xHTFTPhUzPhX8h5W3rRuZicV1zO
98 N3bCgXRUeDFedjxvSc/ai/G86jzfWi87Xswfg5Nx3Q==''')
99
100 @testing.requires('matplotlib')
101 def test_poisson(self):
102 cons, lhs = main(nelems=4, etype='square', btype='std', degree=1, poisson=.4)
103 with self.subTest('constraints'): self.assertAlmostEqual64(cons, '''
104 eNpjYMACGsiHP0wxMQBKlBdi''')
105 with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
106 eNpjYMAEFsaTjdcYvTFcasTAsMZI5JyFce6ZKSavjbNMFhhFmPz/n2WScHaKieiZRFMmk3DTrUaBpv//
107 h5t6n000/Xf6hymLyQ/TbUY/gGI/TL3O/jD9cxoASiglXw==''')