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| # Movement of a doubly periodic mesh driven by a Monge-Ampère type equation | ||
| # ========================================================================= | ||
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| # In the `previous demo <./monge_ampere_ring.py.html>`__, we demonstrated mesh movement | ||
| # with the Monge-Ampère method, driven by a ring shaped monitor function. In this demo, | ||
| # we solve the same problem but on a doubly periodic mesh. | ||
| # | ||
| # Begin by importing from the namespaces of Firedrake and Movement. :: | ||
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| from firedrake import * | ||
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| from movement import * | ||
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| # Create a doubly periodic mesh with the same resolution as in the previous demo. :: | ||
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| n = 20 | ||
| mesh = PeriodicUnitSquareMesh(n, n) | ||
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| # Define the same monitor function and an instance of the | ||
| # :class:`~movement.monge_ampere.MongeAmpereMover` class. :: | ||
| # class. :: | ||
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| def ring_monitor(mesh): | ||
| alpha = Constant(20.0) | ||
| beta = Constant(200.0) | ||
| gamma = Constant(0.15) | ||
| x, y = SpatialCoordinate(mesh) | ||
| r = (x - 0.5) ** 2 + (y - 0.5) ** 2 | ||
| return Constant(1.0) + alpha / cosh(beta * (r - gamma)) ** 2 | ||
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| rtol = 1.0e-08 | ||
| mover = MongeAmpereMover(mesh, ring_monitor, method="quasi_newton", rtol=rtol) | ||
| mover.move() | ||
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| # This should give command line output similar to the following: | ||
| # | ||
| # .. code-block:: none | ||
| # | ||
| # 0 Volume ratio 11.49 Variation (σ/μ) 9.71e-01 Residual 9.19e-01 | ||
| # 1 Volume ratio 7.98 Variation (σ/μ) 6.71e-01 Residual 5.12e-01 | ||
| # 2 Volume ratio 5.60 Variation (σ/μ) 5.40e-01 Residual 3.58e-01 | ||
| # 3 Volume ratio 7.09 Variation (σ/μ) 4.89e-01 Residual 2.98e-01 | ||
| # 4 Volume ratio 5.60 Variation (σ/μ) 4.54e-01 Residual 2.58e-01 | ||
| # 5 Volume ratio 7.48 Variation (σ/μ) 4.31e-01 Residual 2.22e-01 | ||
| # 6 Volume ratio 6.91 Variation (σ/μ) 4.16e-01 Residual 2.07e-01 | ||
| # 7 Volume ratio 8.46 Variation (σ/μ) 4.03e-01 Residual 1.82e-01 | ||
| # 8 Volume ratio 7.68 Variation (σ/μ) 3.93e-01 Residual 1.71e-01 | ||
| # 9 Volume ratio 7.65 Variation (σ/μ) 3.94e-01 Residual 1.60e-01 | ||
| # 10 Volume ratio 7.51 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 11 Volume ratio 7.49 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 12 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 13 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 14 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 15 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 16 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 17 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 18 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 19 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 20 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 21 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 22 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 23 Volume ratio 7.48 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 24 Volume ratio 7.47 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 25 Volume ratio 7.43 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 26 Volume ratio 7.43 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 27 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 28 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 29 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 30 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 31 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 32 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 33 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 34 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 35 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 36 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 37 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 38 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 39 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 40 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 41 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 42 Volume ratio 7.42 Variation (σ/μ) 3.93e-01 Residual 1.59e-01 | ||
| # 43 Volume ratio 7.58 Variation (σ/μ) 3.95e-01 Residual 1.60e-01 | ||
| # 44 Volume ratio 7.80 Variation (σ/μ) 3.97e-01 Residual 1.60e-01 | ||
| # 45 Volume ratio 8.32 Variation (σ/μ) 4.09e-01 Residual 1.70e-01 | ||
| # 46 Volume ratio 9.19 Variation (σ/μ) 4.22e-01 Residual 1.84e-01 | ||
| # 47 Volume ratio 9.70 Variation (σ/μ) 4.29e-01 Residual 1.78e-01 | ||
| # 48 Volume ratio 9.40 Variation (σ/μ) 4.01e-01 Residual 1.29e-01 | ||
| # 49 Volume ratio 10.47 Variation (σ/μ) 4.03e-01 Residual 1.04e-01 | ||
| # 50 Volume ratio 9.84 Variation (σ/μ) 3.72e-01 Residual 8.48e-02 | ||
| # 51 Volume ratio 10.24 Variation (σ/μ) 3.87e-01 Residual 7.56e-02 | ||
| # 52 Volume ratio 9.07 Variation (σ/μ) 3.61e-01 Residual 5.80e-02 | ||
| # 53 Volume ratio 9.81 Variation (σ/μ) 3.73e-01 Residual 4.61e-02 | ||
| # 54 Volume ratio 8.79 Variation (σ/μ) 3.56e-01 Residual 3.42e-02 | ||
| # 55 Volume ratio 9.36 Variation (σ/μ) 3.63e-01 Residual 2.55e-02 | ||
| # 56 Volume ratio 8.79 Variation (σ/μ) 3.52e-01 Residual 1.92e-02 | ||
| # 57 Volume ratio 9.10 Variation (σ/μ) 3.57e-01 Residual 1.44e-02 | ||
| # 58 Volume ratio 8.79 Variation (σ/μ) 3.51e-01 Residual 1.11e-02 | ||
| # 59 Volume ratio 8.96 Variation (σ/μ) 3.53e-01 Residual 8.45e-03 | ||
| # 60 Volume ratio 8.79 Variation (σ/μ) 3.50e-01 Residual 6.54e-03 | ||
| # 61 Volume ratio 8.89 Variation (σ/μ) 3.51e-01 Residual 5.04e-03 | ||
| # 62 Volume ratio 8.79 Variation (σ/μ) 3.49e-01 Residual 3.88e-03 | ||
| # 63 Volume ratio 8.85 Variation (σ/μ) 3.50e-01 Residual 2.99e-03 | ||
| # 64 Volume ratio 8.80 Variation (σ/μ) 3.49e-01 Residual 2.27e-03 | ||
| # 65 Volume ratio 8.83 Variation (σ/μ) 3.49e-01 Residual 1.73e-03 | ||
| # 66 Volume ratio 8.80 Variation (σ/μ) 3.49e-01 Residual 1.29e-03 | ||
| # 67 Volume ratio 8.82 Variation (σ/μ) 3.49e-01 Residual 9.63e-04 | ||
| # 68 Volume ratio 8.80 Variation (σ/μ) 3.49e-01 Residual 6.87e-04 | ||
| # 69 Volume ratio 8.82 Variation (σ/μ) 3.49e-01 Residual 4.97e-04 | ||
| # 70 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 3.34e-04 | ||
| # 71 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 2.29e-04 | ||
| # 72 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 1.41e-04 | ||
| # 73 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 8.91e-05 | ||
| # 74 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 4.84e-05 | ||
| # 75 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 2.72e-05 | ||
| # 76 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 1.43e-05 | ||
| # 77 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 1.07e-05 | ||
| # 78 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 7.66e-06 | ||
| # 79 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 5.59e-06 | ||
| # 80 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 3.89e-06 | ||
| # 81 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 2.76e-06 | ||
| # 82 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 1.86e-06 | ||
| # 83 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 1.28e-06 | ||
| # 84 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 8.23e-07 | ||
| # 85 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 5.42e-07 | ||
| # 86 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 3.29e-07 | ||
| # 87 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 2.04e-07 | ||
| # 88 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 1.13e-07 | ||
| # 89 Volume ratio 8.81 Variation (σ/μ) 3.49e-01 Residual 6.46e-08 | ||
| # Solver converged in 89 iterations. | ||
| # | ||
| # Again, plot the adapted mesh: :: | ||
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| import matplotlib.pyplot as plt | ||
| from firedrake.pyplot import triplot | ||
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| fig, axes = plt.subplots() | ||
| triplot(mover.mesh, axes=axes) | ||
| axes.set_aspect(1) | ||
| plt.savefig("monge_ampere_periodic-adapted_mesh.jpg") | ||
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| # .. figure:: monge_ampere_periodic-adapted_mesh.jpg | ||
| # :figwidth: 60% | ||
| # :align: center | ||
| # | ||
| # Observe that the outer boundary of the mesh is no longer square - each boundary | ||
| # segment has been warped. However, you might be able to convice yourself that the warp | ||
| # of the left boundary matches that of the right and that the warp of the top boundary | ||
| # matches that of the bottom. To be absolutely certain, let's check that the area of the | ||
| # mesh matches expectations. We can do this by simply integrating unity over the domain | ||
| # associated with the adapted mesh. :: | ||
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| import numpy as np | ||
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| expected_area = 1.0 | ||
| assert np.isclose(assemble(Constant(1.0) * dx(domain=mover.mesh)), expected_area) | ||
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| # .. rubric:: Exercise | ||
| # | ||
| # Looking at the solver output above, you might notice that the residual progress stalls | ||
| # for quite a few iterations before descending. Why might this be? Set up this demo and | ||
| # the previous one to record the residual values during the iteration. Re-run them and | ||
| # create a plot to compare the convergence progress on the same axes. | ||
| # | ||
| # In the `next demo <./monge_ampere_3d.py.html>`__, we will demonstrate | ||
| # that the Monge-Ampère method can also be applied in three dimensions. | ||
| # | ||
| # This tutorial can be dowloaded as a `Python script <monge_ampere_periodic.py>`__. | ||
| # | ||
| # | ||
| # .. rubric:: References | ||
| # | ||
| # .. bibliography:: | ||
| # :filter: docname in docnames |
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