first attempt

moving_domain/linear_growth.py

+import dolfin as df
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.widgets import Slider
+import progressbar
+
+df.set_log_level(df.LogLevel.ERROR)
+df.parameters['form_compiler']['optimize'] = True
+
+# parameters
+tmax, dt, L, dx, b, m, D = 10, 0.01, 1, 0.01, 0.01, 2, 0.01
+Nt = int(tmax/dt)
+Nx = int(L/dx)
+
+# mesh, function space, functions
+mesh = df.IntervalMesh(Nx, 0, L)
+function_space = df.FunctionSpace(mesh, 'P', 1)
+c, tc = df.Function(function_space), df.TestFunction(function_space)
+
+# initial condition
+c0 = df.interpolate(df.Expression('1 + 0.2*cos(m * pi * x[0]/L)', pi = np.pi, L = L, m = m, degree=1), function_space)
+
+# arrays
+times = np.linspace(0, tmax, Nt + 1)
+x_array = np.zeros((Nt+1, Nx+1))
+x_array[0] = mesh.coordinates()[:, 0]
+c_array = np.zeros((Nt+1, Nx+1))
+c_array[0] = c0.compute_vertex_values(mesh)
+
+# velocity 
+v = df.Expression('b/(L + b *t)*x[0]', b=b, L=L, t=0, degree=0)
+vh = df.project(v, function_space)
+
+# form
+cform = (df.inner((c - c0)/dt, tc) 
+        + D * df.inner((c).dx(0), (tc).dx(0))
+        + df.inner((vh*c0).dx(0), tc) 
+        )* df.dx
+
+# time stepping
+for i in progressbar.progressbar(range(1, Nt+1)):
+    v.t = times[i]
+    vh.assign(df.project(v, function_space))
+    df.solve(cform == 0, c)
+    c_array[i] = c.compute_vertex_values(mesh)
+    c0.assign(c)
+    df.ALE.move(mesh, df.project(vh*dt, function_space))
+    x_array[i] = mesh.coordinates()[:, 0]
+
+
+# exact solution
+c_exact = np.zeros((Nt + 1, Nx + 1))
+for j in range(0, Nt+1):
+    l = L + b * times[j]
+    c_exact[j] = (L/l) * (1 + 0.2 * np.cos(m * np.pi * x_array[j]/l) * np.exp(-m**2 * np.pi**2 * D * times[j]/(L * l)))
+
+
+# plotting
+fig, ax = plt.subplots(1,1,figsize=(8,6))
+ax.set_xlabel(r'$x$')
+ax.set_ylabel(r'$c(x,t)$')
+ax.set_xlim(np.min(x_array), np.max(x_array))
+ax.set_ylim(min(np.min(c_exact), np.min(c_array)), max(np.max(c_array), np.max(c_exact)))
+ax.grid(True)
+
+cplot, = ax.plot(x_array[0], c_array[0], '--',label = 'Numerical solution')
+c_exactplot, = ax.plot(x_array[0], c_exact[0],label = 'Exact solution')
+
+def update(value):
+        ti = np.abs(times-value).argmin()
+        cplot.set_xdata(x_array[ti])
+        cplot.set_ydata(c_array[ti])
+        c_exactplot.set_xdata(x_array[ti])
+        c_exactplot.set_ydata(c_exact[ti])
+        plt.draw()
+        
+sax = plt.axes([0.1, 0.92, 0.7, 0.02])
+slider = Slider(sax, r'$t/\tau$', min(times), max(times),
+        valinit=min(times), valfmt='%3.1f',
+        fc='#999999')
+slider.drawon = False
+slider.on_changed(update)
+ax.legend(loc=0)
+plt.show()
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