finitestrain.pyΒΆ
In this script we solve the nonlinear Saint Venant-Kichhoff problem on a unit square domain (optionally with a circular cutout), clamped at both the left and right boundary in such a way that an arc is formed over a specified angle. The configuration is constructed such that a symmetric solution is expected.
9from nutils import mesh, function, solver, export, cli, testing
10import numpy
11
12def main(nelems:int, etype:str, btype:str, degree:int, poisson:float, angle:float, restol:float, trim:bool):
13 '''
14 Deformed hyperelastic plate.
15
16 .. arguments::
17
18 nelems [10]
19 Number of elements along edge.
20 etype [square]
21 Type of elements (square/triangle/mixed).
22 btype [std]
23 Type of basis function (std/spline).
24 degree [1]
25 Polynomial degree.
26 poisson [.25]
27 Poisson's ratio, nonnegative and stricly smaller than 1/2.
28 angle [20]
29 Rotation angle for right clamp (degrees).
30 restol [1e-10]
31 Newton tolerance.
32 trim [no]
33 Create circular-shaped hole.
34 '''
35
36 domain, geom = mesh.unitsquare(nelems, etype)
37 if trim:
38 domain = domain.trim(function.norm2(geom-.5)-.2, maxrefine=2)
39 bezier = domain.sample('bezier', 5)
40
41 ns = function.Namespace()
42 ns.x = geom
43 ns.angle = angle * numpy.pi / 180
44 ns.lmbda = 2 * poisson
45 ns.mu = 1 - 2 * poisson
46 ns.ubasis = domain.basis(btype, degree=degree).vector(2)
47 ns.u_i = 'ubasis_ki ?lhs_k'
48 ns.X_i = 'x_i + u_i'
49 ns.strain_ij = '.5 (u_i,j + u_j,i)'
50 ns.energy = 'lmbda strain_ii strain_jj + 2 mu strain_ij strain_ij'
51
52 sqr = domain.boundary['left'].integral('u_k u_k d:x' @ ns, degree=degree*2)
53 sqr += domain.boundary['right'].integral('((u_0 - x_1 sin(2 angle) - cos(angle) + 1)^2 + (u_1 - x_1 (cos(2 angle) - 1) + sin(angle))^2) d:x' @ ns, degree=degree*2)
54 cons = solver.optimize('lhs', sqr, droptol=1e-15)
55
56 energy = domain.integral('energy d:x' @ ns, degree=degree*2)
57 lhs0 = solver.optimize('lhs', energy, constrain=cons)
58 X, energy = bezier.eval(['X_i', 'energy'] @ ns, lhs=lhs0)
59 export.triplot('linear.png', X, energy, tri=bezier.tri, hull=bezier.hull)
60
61 ns.strain_ij = '.5 (u_i,j + u_j,i + u_k,i u_k,j)'
62 ns.energy = 'lmbda strain_ii strain_jj + 2 mu strain_ij strain_ij'
63
64 energy = domain.integral('energy d:x' @ ns, degree=degree*2)
65 lhs1 = solver.minimize('lhs', energy, lhs0=lhs0, constrain=cons).solve(restol)
66 X, energy = bezier.eval(['X_i', 'energy'] @ ns, lhs=lhs1)
67 export.triplot('nonlinear.png', X, energy, tri=bezier.tri, hull=bezier.hull)
68
69 return lhs0, lhs1
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
finitestrain.py
(view log). To select quadratic splines and a cutout add python3
finitestrain.py btype=spline degree=2 trim=yes
(view log).
77if __name__ == '__main__':
78 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.
86class test(testing.TestCase):
87
88 @testing.requires('matplotlib')
89 def test_default(self):
90 lhs0, lhs1 = main(nelems=4, etype='square', btype='std', degree=1, poisson=.25, angle=10, restol=1e-10, trim=False)
91 with self.subTest('linear'): self.assertAlmostEqual64(lhs0, '''
92 eNpjYMAE5ZeSL/HqJ146YeB4cbvhl/PzjPrOcVy8da7b4Og5W6Osc/rGt88+MvY+u+yC7NlcQ+GzEsYP
93 z/w3nn1mvon7mdsXJM8oG304vdH45Oluk2WnlU1bTgMAv04qwA==''')
94 with self.subTest('non-linear'): self.assertAlmostEqual64(lhs1, '''
95 eNpjYMAEZdrKl2/p37soY1h84aKh2/lmI4Zz7loq5y0MD55rNtI652Rcefa48aUzzZcjzj4ylDjrYnz6
96 jIBJ8Zl2E9Yzty9InlE2+nB6o/HJ090my04rm7acBgAKcSdV''')
97
98 @testing.requires('matplotlib')
99 def test_mixed(self):
100 lhs0, lhs1 = main(nelems=4, etype='mixed', btype='std', degree=1, poisson=.25, angle=10, restol=1e-10, trim=False)
101 with self.subTest('linear'): self.assertAlmostEqual64(lhs0, '''
102 eNpjYICAqxfbL+Xov7kIYi80OA+mtxleOA+iVxjNPBdncOdc6sXT51yNgs8ZGX89e8/Y66zqBaOz/Ya8
103 Z4WMX575ZTz5zAqTgDPKRh9O374geWaj8cnT3SbLTiubtpwGAJ6hLHk=''')
104 with self.subTest('non-linear'): self.assertAlmostEqual64(lhs1, '''
105 eNpjYIAA7fv2l6UMEi6C2H8N7l0A0VcMzc+D6H4jznPyhpfOdelwnm80EjznYTz57CnjG2eWX0o/+9VQ
106 +KyT8cUzzCbZZ2abiJ9RNvpw+vYFyTMbjU+e7jZZdlrZtOU0AJN4KHY=''')
107
108 @testing.requires('matplotlib')
109 def test_spline(self):
110 lhs0, lhs1 = main(nelems=4, etype='square', btype='spline', degree=2, poisson=.25, angle=10, restol=1e-10, trim=False)
111 with self.subTest('linear'): self.assertAlmostEqual64(lhs0, '''
112 eNpjYMAOrl3J0vmixaY7QS9N545+w9VaA5eLXYZp51MvVl/I1F164YeBxAVlI//zzMZB52KN35+dd+H9
113 2Vd6b85yGx0/a22cd/aXMetZH5PTZ7ZfaDmzTL/nzFGj3DPPje3OLDBhPvPC5N7p2xckz/gZsJwRML5z
114 Wstk++m7JlNPK5u2nAYATqg9sA==''')
115 with self.subTest('non-linear'): self.assertAlmostEqual64(lhs1, '''
116 eNpjYMAOnLUP6ejq9ukI67vflTVQvdRt0H8h3fDBOT7trReK9adeyDFcez7YaN+5X0Z7z7oYB5/9rKx9
117 ztdA6Fyq0dqzScbGZ78bLzmja5J8RvzSrjN9BgvOfDFKP/PTWOfMSpO3p8+YbDx9+4LkGT8DljMCxndO
118 a5lsP33XZOppZdOW0wApLzra''')