Add advection-diffusion as another usage example (#1138)

Co-authored-by: Tom Gustafsson <[email protected]>

docs/examples/ex50.py

@@ -0,0 +1,113 @@
+# Copyright (C) 2024 Radost Waszkiewicz and Jan Turczynowicz
+# This software is published under BSD-3-clause license
+
+# Folowing code solves advection-diffusion problem for
+# temperature distribution around a cold sphere in warm
+# liquid. Liquid flow is modelled using Stokes flow field.
+# Thermal difusivity to advection ratio is controlled by
+# Peclet number.
+
+import numpy as np
+
+from pathlib import Path
+from skfem import MeshTri, Basis, ElementTriP1, BilinearForm
+from skfem import asm, solve, condense
+from skfem.helpers import grad, dot
+
+# Define the Peclet number
+peclet = 30
+
+#
+# Code for creating mesh which we load
+#
+
+# floor_depth = 5.0
+# floor_width = 5.0
+# ball_size = 1.0
+# ball_segments = 100
+# mesh_size = 0.01
+# far_mesh = 0.5
+
+# box_points = [
+#         ([0, -floor_depth], far_mesh),
+#         ([floor_width, -floor_depth], far_mesh),
+#         ([floor_width, floor_depth], far_mesh),
+#         ([0, floor_depth], mesh_size),
+#     ]
+
+# phi_values = np.linspace(0, np.pi, ball_segments)
+# ball_points = ball_size * np.column_stack((np.sin(phi_values), np.cos(phi_values)))
+# mesh_boundary = np.vstack((
+#     np.array([p for p,s  in box_points])
+#     , ball_points))
+
+# # Create the geometry and mesh using pygmsh
+# with pygmsh.geo.Geometry() as geom:
+#     poly = geom.add_polygon(
+#         mesh_boundary,
+#         mesh_size=([s for p,s in box_points]) + ([mesh_size] * len(ball_points)),
+#     )
+
+#     raw_mesh = geom.generate_mesh()
+
+# # Convert the mesh to a skfem MeshTri object and define boundaries
+# mesh = MeshTri(
+#     raw_mesh.points[:, :2].T, raw_mesh.cells_dict["triangle"].T
+# ).with_boundaries(
+#     {
+#         "left": lambda x: np.isclose(x[0], 0),  # Left boundary condition
+#         "bottom": lambda x: np.isclose(x[1], -floor_depth),  # Bottom boundary condition
+#         "ball": lambda x: x[0] ** 2 + x[1] ** 2 < 1.1 * ball_size**2,
+#     }
+# )
+
+mesh = MeshTri.load(Path(__file__).parent / 'meshes' / 'cylinder_stokes.msh')
+
+# Define the basis for the finite element method
+basis = Basis(mesh, ElementTriP1())
+
+
+@BilinearForm
+def advection(k, l, m):
+    """Advection bilinear form."""
+
+    # Coordinate fields
+    r, z = m.x
+
+    u = 1  # velocity scale
+    a = 1  # ball size
+
+    # Stokes flow around a sphere of size `a`
+    w = r ** 2 + z ** 2
+    v_r = ((3 * a * r * z * u) / (4 * w**0.5)) * ((a / w) ** 2 - (1 / w))
+    v_z = u + ((3 * a * u) / (4 * w**0.5)) * (
+        (2 * a**2 + 3 * r**2) / (3 * w) - ((a * r) / w) ** 2 - 2
+    )
+
+    return (l * v_r * grad(k)[0] + l * v_z * grad(k)[1]) * 2 * np.pi * r
+
+
+@BilinearForm
+def claplace(u, v, w):
+    """Laplace operator in cylindrical coordinates."""
+    r = abs(w.x[1])
+    return dot(grad(u), grad(v)) * 2 * np.pi * r
+
+
+# Identify the interior degrees of freedom
+interior = basis.complement_dofs(basis.get_dofs({"bottom", "ball"}))
+
+# Assemble the system matrix
+A = asm(claplace, basis) + peclet * asm(advection, basis)
+
+# Boundary condition
+u = basis.zeros()
+u[basis.get_dofs("bottom")] = 1.0
+u[basis.get_dofs("ball")] = 0.0
+
+u = solve(*condense(A, x=u, I=interior))
+
+if __name__ == "__main__":
+
+    mesh.draw(boundaries=True).show()
+    basis.plot(u, shading='gouraud', cmap='viridis').show()

docs/listofexamples.rst

@@ -454,6 +454,24 @@ This example solves the advection equation on a periodic square mesh.
 See the source code of :exlink:`42`
 for more information.
 
+Example 50: Advection-diffusion in non-uniform flow
+---------------------------------------------------
+
+This example solves the advection-diffusion problem
+for the temperature distribution around a cold sphere
+in a warm liquid. The liquid flow is modeled using
+the Stokes flow field. The thermal diffusivity to
+advection ratio is controlled by the Peclet number.
+The problem is solved in a cylindrical coordinate
+system (hence a different definition of the Laplacian).
+
+.. figure:: https://github.com/turczyneq/fem_advection_diffusion/assets/51670923/271b5b86-71f1-49fb-b32e-3c9d012a29ee
+
+   Temperature distribution of Example 50. Liquid behind the sphere is colder than the incoming one.
+
+See the source code of :exlink:`50` for more information.
+
+
 Heat transfer
 =============
 

tests/test_examples.py

@@ -408,3 +408,11 @@ class TestEx49(TestCase):
     def runTest(self):
         import docs.examples.ex49 as ex49
         self.assertLess(np.abs(ex49.l2), 0.0012)
+
+
+class TestEx50(TestCase):
+
+    def runTest(self):
+        import docs.examples.ex50 as ex50
+        self.assertAlmostEqual(ex50.u.max(),
+                               1.0026836382652844)