# 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.png', 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 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.

  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 92 93 94 95 96 97 98 99 100 class test(nutils.testing.TestCase): @nutils.testing.requires('matplotlib') def test_default(self): cons, lhs = main(nelems=4) with self.subTest('constraints'): self.assertAlmostEqual64(cons, ''' eNpjYICDBnzwhykMMhCpAwEBQ08XYg==''') with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, ''' eNpjYICBFGMxYyEgTjFebDLBpB2IF5tkmKaYJgJxhukPOIRrYBA1CjJgYFh3/vXZMiMVQwaGO+e6zvYY 2QBZR86VnO2FsorPAgAXLB7S''') @nutils.testing.requires('matplotlib') def test_mixed(self): cons, lhs = main(nelems=4, etype='mixed') with self.subTest('constraints'): self.assertAlmostEqual64(cons, ''' eNpjYACCBiBkQMJY4A9TGGQgUgcCAgBVTxdi''') with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, ''' eNpjYGBgSDKWNwZSQKwExAnGfSbLTdpNek2WmWSYppgmAHGG6Q84BKpk4DASN2Bg2K/JwHDrPAPDj7Mq hnlGRddenpt+ts/I0nChyrlzJWcdDbuNYjUOnSs/CwB0uyJb''') @nutils.testing.requires('matplotlib') def test_quadratic(self): cons, lhs = main(nelems=4, degree=2) with self.subTest('constraints'): self.assertAlmostEqual64(cons, ''' eNpjYMAADQMJf5iiQ4ZB5kJMCAAkxE4W''') with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, ''' eNpjYEAHlUauhssMuw2nAvEyQ1fDSqMsY1NjJWNxYzEgVgKys4xlTThNfhu/NX4HxL+NOU1kTRabzDaZ bNJj0g3Ek4HsxSa8ptym7KZMYMgOZPOaZpimm6aYJoFhCpCdYboFCDfDIYj3AwNiOJDhviGPQbf+RV0G Bv1LpRe+nFc8x22UY5hv8F6PgUHw4sTzU859PZtldNGQ3XCCPgNDwYWf5/TPTTtbYvTKUNpwP1DE8cLT c2Lnes62Gf01NDW8BxRRunD6HPO5KqjIA6CIAlSkw+ifobnhI6CI3IWT55jOVQBF/hqaGT4EishfOAVU U3EWAA5lcd0=''') @nutils.testing.requires('matplotlib') def test_poisson(self): cons, lhs = main(nelems=4, poisson=.4) with self.subTest('constraints'): self.assertAlmostEqual64(cons, ''' eNpjYICDBnzwhykMMhCpAwEBQ08XYg==''') with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, ''' eNpjYIABC+M1RkuN1hhZGE8xyTKJAOIpJomm4aaBQJxo+gMO4RoYJhu/MWRgEDmXe+a18QKj//8Tzoqe YTLZCmR5n/13msVkG5DldfbPaQC28iVf''')