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''')