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.
7 | import nutils
|
The main function defines the parameter space for the script. Configurable parameters are the mesh density (in number of elements along an edge), element type (square, triangle, or mixed), type of basis function (std or spline, with availability depending on element type), polynomial degree, and Poisson’s ratio.
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 | def main(nelems: 'number of elements along edge' = 10,
etype: 'type of elements (square/triangle/mixed)' = 'square',
btype: 'type of basis function (std/spline)' = 'std',
degree: 'polynomial degree' = 1,
poisson: 'poisson ratio < 0.5' = .25):
domain, geom = nutils.mesh.unitsquare(nelems, etype)
ns = nutils.function.Namespace()
ns.x = geom
ns.basis = domain.basis(btype, degree=degree).vector(2)
ns.u_i = 'basis_ni ?lhs_n'
ns.X_i = 'x_i + u_i'
ns.lmbda = 2 * poisson
ns.mu = 1 - 2 * poisson
ns.strain_ij = '(u_i,j + u_j,i) / 2'
ns.stress_ij = 'lmbda strain_kk δ_ij + 2 mu strain_ij'
sqr = domain.boundary['left'].integral('u_k u_k d:x' @ ns, degree=degree*2)
sqr += domain.boundary['right'].integral('(u_0 - .5)^2 d:x' @ ns, degree=degree*2)
cons = nutils.solver.optimize('lhs', sqr, droptol=1e-15)
res = domain.integral('basis_ni,j stress_ij d:x' @ ns, degree=degree*2)
lhs = nutils.solver.solve_linear('lhs', res, constrain=cons)
bezier = domain.sample('bezier', 5)
X, sxy = bezier.eval(['X_i', 'stress_01'] @ ns, lhs=lhs)
nutils.export.triplot('shear.jpg', X, sxy, tri=bezier.tri, hull=bezier.hull)
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).
52 53 | if __name__ == '__main__':
nutils.cli.run(main)
|
Once a simulation is developed and tested, it is good practice to save a few
strategicly chosen return values for routine regression testing. Here we use
the standard unittest
framework, with
nutils.numeric.assert_allclose64()
facilitating the embedding of
desired results as compressed base64 data.
61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 | import unittest
class test(unittest.TestCase):
def test_default(self):
cons, lhs = main(nelems=4)
nutils.numeric.assert_allclose64(cons, 'eNpjYICDBnzwhykMMhCpAwEBQ08XYg==')
nutils.numeric.assert_allclose64(lhs, 'eNpjYICBFGMxYyEgTjFebDLBpB2IF5tkmKaYJg'
'JxhukPOIRrYBA1CjJgYFh3/vXZMiMVQwaGO+e6zvYY2QBZR86VnO2FsorPAgAXLB7S')
def test_mixed(self):
cons, lhs = main(nelems=4, etype='mixed')
nutils.numeric.assert_allclose64(cons, 'eNpjYACCBiBkQMJY4A9TGGQgUgcCAgBVTxdi')
nutils.numeric.assert_allclose64(lhs, 'eNpjYGBgSDKWNwZSQKwExAnGfSbLTdpNek2WmW'
'SYppgmAHGG6Q84BKpk4DASN2Bg2K/JwHDrPAPDj7MqhnlGRddenpt+ts/I0nChyrlzJWcdDbuN'
'YjUOnSs/CwB0uyJb')
def test_quadratic(self):
cons, lhs = main(nelems=4, degree=2)
nutils.numeric.assert_allclose64(cons, 'eNpjYMAADQMJf5iiQ4ZB5kJMCAAkxE4W')
nutils.numeric.assert_allclose64(lhs, 'eNpjYEAHlUauhssMuw2nAvEyQ1fDSqMsY1NjJW'
'NxYzEgVgKys4xlTThNfhu/NX4HxL+NOU1kTRabzDaZbNJj0g3Ek4HsxSa8ptym7KZMYMgOZPOa'
'Zpimm6aYJoFhCpCdYboFCDfDIYj3AwNiOJDhviGPQbf+RV0GBv1LpRe+nFc8x22UY5hv8F6PgU'
'Hw4sTzU859PZtldNGQ3XCCPgNDwYWf5/TPTTtbYvTKUNpwP1DE8cLTc2Lnes62Gf01NDW8BxRR'
'unD6HPO5KqjIA6CIAlSkw+ifobnhI6CI3IWT55jOVQBF/hqaGT4EishfOAVUU3EWAA5lcd0=')
def test_poisson(self):
cons, lhs = main(nelems=4, poisson=.4)
nutils.numeric.assert_allclose64(cons, 'eNpjYICDBnzwhykMMhCpAwEBQ08XYg==')
nutils.numeric.assert_allclose64(lhs, 'eNpjYIABC+M1RkuN1hhZGE8xyTKJAOIpJomm4a'
'aBQJxo+gMO4RoYJhu/MWRgEDmXe+a18QKj//8TzoqeYTLZCmR5n/13msVkG5DldfbPaQC28iVf')
|