# cahnhilliard.pyΒΆ

In this script we solve the Cahn-Hiilliard equation, which models the unmixing of two phases under the effect of surface tension.

  6 7 8 9 10 11 12 13 14 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 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 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 from nutils import mesh, function, solver, sample, export, cli, testing import numpy, treelog, itertools, enum, typing class stab(enum.Enum): none = '0' # for educational purposes only linear = '.5 dc^2 (6 - 6 c^2 + 8 c dc - 3 dc^2) / epsilon^2' optimal = '.5 dc^2 (1 - dc^2 / 12) / epsilon^2' def main(nelems:int, etype:str, btype:str, degree:int, epsilon:typing.Optional[float], contactangle:float, timestep:float, mtol:float, seed:int, circle:bool, stab:stab): ''' Cahn-Hilliard equation on a unit square/circle. .. arguments:: nelems [20] Number of elements along domain edge. etype [square] Type of elements (square/triangle/mixed). btype [std] Type of basis function (std/spline), with availability depending on the configured element type. degree [2] Polynomial degree. epsilon [] Interface thickness; defaults to an automatic value based on the configured mesh density if left unspecified. contactangle [90] Wall contact angle in degrees. timestep [.01] Time step. mtol [.01] Threshold value for chemical potential peak to peak difference, used as a stop criterion. seed [0] Random seed for the initial condition. circle [no] Select circular domain as opposed to a unit square. stab [linear] Stabilization method (linear/optimal/none). ''' mineps = 1./nelems if epsilon is None: treelog.info('setting epsilon={}'.format(mineps)) epsilon = mineps elif epsilon < mineps: treelog.warning('epsilon under crititical threshold: {} < {}'.format(epsilon, mineps)) domain, geom = mesh.unitsquare(nelems, etype) bezier = domain.sample('bezier', 5) # sample for plotting ns = function.Namespace() if not circle: ns.x = geom else: angle = (geom-.5) * (numpy.pi/2) ns.x = function.sin(angle) * function.cos(angle)[[1,0]] / numpy.sqrt(2) ns.epsilon = epsilon ns.ewall = .5 * numpy.cos(contactangle * numpy.pi / 180) ns.cbasis, ns.mbasis = function.chain([domain.basis('std', degree=degree)] * 2) ns.c = 'cbasis_n ?lhs_n' ns.dc = 'cbasis_n (?lhs_n - ?lhs0_n)' ns.m = 'mbasis_n ?lhs_n' ns.F = '.5 (c^2 - 1)^2 / epsilon^2' ns.dF = stab.value ns.dt = timestep nrg_mix = domain.integral('F d:x' @ ns, degree=7) nrg_iface = domain.integral('.5 c_,k c_,k d:x' @ ns, degree=7) nrg_wall = domain.boundary.integral('(abs(ewall) + c ewall) d:x' @ ns, degree=7) nrg = nrg_mix + nrg_iface + nrg_wall + domain.integral('(dF - m dc - .5 dt epsilon^2 m_,k m_,k) d:x' @ ns, degree=7) numpy.random.seed(seed) lhs = numpy.random.normal(0, .5, ns.cbasis.shape) # initial condition with treelog.iter.plain('timestep', itertools.count()) as steps: for istep in steps: E = sample.eval_integrals(nrg_mix, nrg_iface, nrg_wall, lhs=lhs) treelog.user('energy: {0:.3f} ({1[0]:.0f}% mixture, {1[1]:.0f}% interface, {1[2]:.0f}% wall)'.format(sum(E), 100*numpy.array(E)/sum(E))) x, c, m = bezier.eval(['x_i', 'c', 'm'] @ ns, lhs=lhs) export.triplot('phase.png', x, c, tri=bezier.tri, clim=(-1,1)) export.triplot('chempot.png', x, m, tri=bezier.tri) if numpy.ptp(m) < mtol: break lhs = solver.optimize('lhs', nrg, arguments=dict(lhs0=lhs), lhs0=lhs, tol=1e-10) return lhs 

If the script is executed (as opposed to imported), nutils.cli.run() calls the main function with arguments provided from the command line.

 102 103 if __name__ == '__main__': 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.

 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 class test(testing.TestCase): @testing.requires('matplotlib') def test_initial(self): lhs = main(nelems=3, etype='square', btype='std', degree=2, epsilon=None, contactangle=90, timestep=1, mtol=float('inf'), seed=0, circle=False, stab=stab.linear) self.assertAlmostEqual64(lhs, ''' eNoBxAA7/xM3LjTtNYs3MDcUyt41uc14zjo0LzKzNm812jFhNNMzwDYgzbMzV8o0yCM1rzWeypE3Tcnx L07NzTa4NlMyETREyrPIGMxYMl82VDbjy1/M8clZyf3IRjday6XLmMl6NRnJDs1Ayh00WMu1yQHRUDSs MKIz7MoEzM/KCMxwyvjIlzLQyxTJdjQ5yjEwWjX3MTk2n8kwNMbKTsoay1DMWDC8ycM1eTQyyb42NzdK NmLN5skSNs/LXDbnMuw19DNKNREtGTfui1ut''') @testing.requires('matplotlib') def test_square(self): lhs = main(nelems=3, etype='square', btype='std', degree=2, epsilon=None, contactangle=90, timestep=1, mtol=.1, seed=0, circle=False, stab=stab.linear) self.assertAlmostEqual64(lhs, ''' eNqbZTbHzMHsiGmpCd9V1gszzWaZ2ZjtMQ01eXV+xbk0szSgzAaTDxdNTkue1jbTMpM15TJqP/335PeT 100vmyqYaJ3tPNV1svNknmmKqYJR+On3J01Pmp9MMY0y/WIYCOSZn7Q82XCi8UTXiSkn5pxYBISovJYT rSd6T0wD8xae6ATCCSemn5gLlusFwiknZp9YcGIpEE4Ewhkn5p1YfGIFEKLyAN6wcSE=''') @testing.requires('matplotlib') def test_contactangle(self): lhs = main(nelems=3, etype='square', btype='std', degree=2, epsilon=None, contactangle=45, timestep=1, mtol=.1, seed=0, circle=False, stab=stab.linear) self.assertAlmostEqual64(lhs, ''' eNqzNsszkzZbbfrdOOus6Jlss5lmPmbPTQtNtp6be8bZrNTss6mW6SMDv9OnTokDZRpMbxl7nNE89fTk ItNHpl0mT8+fOzX3ZP7J3yb+ph1G206zn7I+KXWyyOSeibK+1ulzJyVP/joRZhJp0m6yyeSyyXsgDAfy 2kw2mlw0eWvyxiTLJNtkgslmk3Mmz4CwzqTeZLbJNpOzJo+AcIrJVJO1JkdMbpi8BsLlJitM9gHNeGLy 2eQLkLfSZL/JFZOnJl+BEAAJrlyi''') @testing.requires('matplotlib') def test_mixedcircle(self): lhs = main(nelems=3, etype='mixed', btype='std', degree=2, epsilon=None, contactangle=90, timestep=1, mtol=.1, seed=0, circle=True, stab=stab.linear) self.assertAlmostEqual64(lhs, ''' eNrTM31uImDqY1puGmwia1prssNY37TERNM01eSOkYuJlck6Q1ED9TP9px+fOmq82FjtfKFJiM6CK70m BsZixmUXgk9XnMo7VX6661zL+cZz58+ln0s6e/PM7DOvjDTOvTz97tS8c6xn9pzYemLHiQMn9p9YDyS3 nth4YteJbUCRHUByO5DcfGLDieUnlpyYA2RtP7HpxJ4T64Aih8Bwz4k1QPF5QJ3rgap3ntgCVAHRe+bE biBr5YmDQBMBKJ13Eg==''')