finitestrain.py¶

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.

 9 import nutils, numpy 

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, Poisson’s ratio, wedge angle, Newton tolerance, and a boolean flag for a circular cutout.

 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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 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, angle: 'bend angle (degrees)' = 20, restol: 'residual tolerance' = 1e-10, trim: 'create circular-shaped hole' = False): domain, geom = nutils.mesh.unitsquare(nelems, etype) if trim: domain = domain.trim(nutils.function.norm2(geom-.5)-.2, maxrefine=2) bezier = domain.sample('bezier', 5) ns = nutils.function.Namespace() ns.x = geom ns.angle = angle * numpy.pi / 180 ns.lmbda = 2 * poisson ns.mu = 1 - 2 * poisson ns.ubasis = domain.basis(btype, degree=degree).vector(2) ns.u_i = 'ubasis_ki ?lhs_k' ns.X_i = 'x_i + u_i' ns.strain_ij = '.5 (u_i,j + u_j,i)' ns.energy = 'lmbda strain_ii strain_jj + 2 mu strain_ij strain_ij' sqr = domain.boundary['left'].integral('u_k u_k d:x' @ ns, degree=degree*2) 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) cons = nutils.solver.optimize('lhs', sqr, droptol=1e-15) energy = domain.integral('energy d:x' @ ns, degree=degree*2) lhs0 = nutils.solver.optimize('lhs', energy, constrain=cons) X, energy = bezier.eval(['X_i', 'energy'] @ ns, lhs=lhs0) nutils.export.triplot('linear.png', X, energy, tri=bezier.tri, hull=bezier.hull) ns.strain_ij = '.5 (u_i,j + u_j,i + u_k,i u_k,j)' ns.energy = 'lmbda strain_ii strain_jj + 2 mu strain_ij strain_ij' energy = domain.integral('energy d:x' @ ns, degree=degree*2) lhs1 = nutils.solver.minimize('lhs', energy, lhs0=lhs0, constrain=cons).solve(restol) X, energy = bezier.eval(['X_i', 'energy'] @ ns, lhs=lhs1) nutils.export.triplot('nonlinear.png', X, energy, tri=bezier.tri, hull=bezier.hull) 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 (view log).

 68 69 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.

  77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 class test(nutils.testing.TestCase): @nutils.testing.requires('matplotlib') def test_default(self): lhs0, lhs1 = main(nelems=4, angle=10) with self.subTest('linear'): self.assertAlmostEqual64(lhs0, ''' eNpjYICB8ku8+icMthvOM+K42G1ga6Rv/Mh42YVcQwnj/8bzTW5fUDbaaNxtomwK18CQfCnxkuPFL+f7 zt06d/Rc1rnbZ73Pyp4VPvvwzOwz7mckz3w4ffL0stMtpwGSOirA''') with self.subTest('non-linear'): self.assertAlmostEqual64(lhs1, ''' eNpjYICBMu1b+jKGFw2bjdy1LICkk/Fx4+bLjwxdjAVM2k1uX1A22mjcbaJsCtfAoHz53sXiC27nGc6p nD94Tutc5dlLZyLOSpw9fab4DOsZyTMfTp88vex0y2kA6e4nVQ==''') @nutils.testing.requires('matplotlib') def test_mixed(self): lhs0, lhs1 = main(nelems=4, angle=10, etype='mixed') with self.subTest('linear'): self.assertAlmostEqual64(lhs0, ''' eNoBZACb/wAAAADV0WwvAAChMAAAtjEAAKgyXjBl0UUyMjPeMyXQjzESM/ozqDQjMtvQsTOLNCM1AAAA AIfS7NEAAM/RAADQzwAAmc7czsvOU871zUrNMs0NzenMk8xQzPDLGczJy6bLhMsZ2Sx5''') with self.subTest('non-linear'): self.assertAlmostEqual64(lhs1, ''' eNoBZACb/wAAAAAr3xowAAD9MAAA1DEAAI8yHzGKLIEySDPKM6fS9TFCMwM0mzQjMtvQsTOLNCM1AAAA AD/TYNEAAN7QAAA3zwAACc7SzgnPEc6TzdjMZ80TzdHMa8wXzPDLGczJy6bLhMthnih2''') @nutils.testing.requires('matplotlib') def test_spline(self): lhs0, lhs1 = main(nelems=4, angle=10, degree=2, btype='spline') with self.subTest('linear'): self.assertAlmostEqual64(lhs0, ''' eNpjYECAa1e+aE3Qu6Nfa9BlmHoxU/eHgbIRs3Gs8bwLr/S4jayNfxn7mGy/sEz/qNFz4wUmL0xuX/Az EDDWMrlromyKZAxDlg6bbppOw1WXi2nnqy8svSBxwf980Ln3Z9+ffXP2+Nm8s6xnT59pOdNzJveM3Rnm M/dOS55hOXPn9PbTU0+3nAYAZeQ9sA==''') with self.subTest('non-linear'): self.assertAlmostEqual64(lhs1, ''' eNpjYEAAZ21dXWF9WYNug3RDPu1i/RzDYKNfRi7Gn5V9DVKNkoy/G+uaiF/qM/hi9NN4pckZk9sX/AwE jLVM7poomyIZw3BIp0/H/a7qpf4LD85tvTD1wtrz+87tPRt8Vvuc0Lm1Z43PLjmTfGbXmQVn0s/onHl7 euNpyTMsZ+6c3n566umW0wB4sDra''')