working model in 1D with only density

basic_model_only_density/parameters.json

+{
+    "resolution" : 100,
+    "system_size" : 1,
+    "elasticity" : 1.0,
+    "viscosity" : 1.0,
+    "zero_rho_boundary": false,
+    "friction" : 1.0,
+    "lamda": -5.0,
+    "diffusion_rho" : 1.0,
+    "turnover_rho" : 2.0,
+    "average_rho" : 1.0,
+    "saturation_rho" : 1.0,
+    "noise_level": 0.0,
+    "timestep" : 0.01,
+    "savetime": 0.1,
+    "maxtime" : 10.0
+}
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basic_model_only_density/tissue.py

+import numpy as np
+import dolfin as df
+import progressbar
+import os
+import h5py
+import matplotlib.pyplot as plt
+from matplotlib.widgets import Slider
+
+df.set_log_level(df.LogLevel.ERROR)
+df.parameters['form_compiler']['optimize'] = True
+
+class Tissue(object):
+    def __init__(self, parameters):
+        # read in parameters
+        for key in parameters:
+            setattr(self, key, parameters[key])
+        
+        self.mesh = df.IntervalMesh(self.resolution, 
+                                    -self.system_size/2, 
+                                    self.system_size/2)
+
+        scalar_element = df.FiniteElement('P', self.mesh.ufl_cell(), 1)
+        vector_element = df.VectorElement('P', self.mesh.ufl_cell(), 1)
+        # u, v, rho
+        mixed_element = df.MixedElement([vector_element, 
+                                         vector_element, 
+                                         scalar_element])
+
+        # define function space with this mixed element
+        self.function_space = df.FunctionSpace(self.mesh, mixed_element)
+        
+        if self.zero_rho_boundary:
+            self.rho_boundary = 0.0
+        else:
+            self.rho_boundary = self.average_rho
+        self.bc = df.DirichletBC(self.function_space.sub(2), 
+                                 df.Constant(self.rho_boundary), 
+                                 'on_boundary')
+        
+        self.function0 = df.Function(self.function_space)
+        self.function  = df.Function(self.function_space)
+        
+
+    def advection(self, conc, vel, tconc):
+        return df.inner(df.div(vel * conc), tconc)
+
+    def active_stress(self, rho):
+        return (self.lamda * rho/(rho + self.saturation_rho) 
+                * df.Identity(1))
+
+    def epsilon(self, v):
+        return df.sym(df.nabla_grad(v))
+    
+    def passive_stress(self, u, v):
+        eps_u, eps_v = self.epsilon(u), self.epsilon(v)
+        elastic_stress = self.elasticity * eps_u 
+        viscous_stress = self.viscosity * eps_v
+        return (elastic_stress + viscous_stress)
+    
+    def stress(self, u, v, rho):
+        return (self.passive_stress(u, v) + self.active_stress(rho))
+
+    def diffusion_reaction_rho(self, rho, trho):
+        # NOTE: implement a step-function of density in the reaction
+        return (self.diffusion_rho * df.inner(df.nabla_grad(rho), 
+                                              df.nabla_grad(trho)) 
+                + self.turnover_rho * df.inner(rho - self.average_rho, 
+                                               trho)
+                )
+
+    def setup_initial_conditions(self):
+        zero_vector = df.Constant((0.0,))
+        u0 = df.interpolate(zero_vector, 
+                            self.function_space.sub(0).collapse())
+        
+        if self.zero_rho_boundary:
+            base_rho = 0.0
+        else:
+            base_rho = self.average_rho
+        rho0 = df.interpolate(df.Expression(
+                            'base_rho + cos(PI*x[0]/L)', 
+                            L=self.system_size,
+                            base_rho=base_rho,
+                            degree=1, PI=np.pi), 
+                            self.function_space.sub(2).collapse())
+                    
+        # add noise
+        noise_rho = (self.noise_level 
+                     * (2 * np.random.random(rho0.vector().size()) - 1))
+        rho0.vector()[:] = rho0.vector()[:] + noise_rho
+        
+        VFS = self.function_space.sub(1).collapse()
+        v0  = df.Function(VFS)
+        tv = df.TestFunction(VFS)
+        vform = (self.friction * df.inner(v0, tv) 
+                    + df.inner(self.stress(u0, v0, rho0), 
+                               self.epsilon(tv))
+                ) * df.dx
+        df.solve(vform == 0, v0)
+        df.assign(self.function0, [u0, v0, rho0])
+
+    def setup_weak_forms(self):
+        u0, v0, rho0  = df.split(self.function0)
+        u,  v, rho    = df.split(self.function)
+        tu, tv, trho  = df.TestFunctions(self.function_space)
+                  
+        uform = (df.inner((u - u0)/self.timestep, tu)
+                - df.inner(v0, tu))
+        
+        vform = (self.friction * df.inner(v, tv) 
+                + df.inner(self.stress(u, v, rho), 
+                           self.epsilon(tv)))
+
+        rhoform = (df.inner((rho - rho0)/self.timestep, trho)
+                + self.advection(rho0, v0, trho)
+                + self.diffusion_reaction_rho(rho, trho))
+        
+        self.form = (uform + vform + rhoform) * df.dx
+            
+
+    def solve(self, DIR=''):
+        self.uFile   = df.XDMFFile(os.path.join(DIR, 
+                            '%s_displacement.xdmf' % self.timestamp))
+        self.vFile   = df.XDMFFile(os.path.join(DIR, 
+                            '%s_velocity.xdmf' % self.timestamp))
+        self.rhoFile = df.XDMFFile(os.path.join(DIR, 
+                            '%s_density.xdmf' % self.timestamp))
+        
+        self.setup_initial_conditions()
+        self.setup_weak_forms()
+        
+        # time-variables
+        self.time = 0.0
+        savesteps = int(self.savetime/self.timestep)
+        maxsteps = int(self.maxtime/self.timestep)
+        u, v, rho = self.function0.split(deepcopy=True)
+        self.uFile.write_checkpoint(u, 'displacement', self.time)
+        self.vFile.write_checkpoint(v, 'velocity', self.time)
+        self.rhoFile.write_checkpoint(rho, 'density', self.time)
+
+        for steps in progressbar.progressbar(range(1, maxsteps + 1)):
+            df.solve(self.form == 0, self.function, self.bc)
+            self.function0.assign(self.function)
+            self.time += self.timestep            
+            u, v, rho = self.function0.split(deepcopy=True)
+            if steps % savesteps == 0:
+                self.uFile.write_checkpoint(u, 'displacement', 
+                                            self.time, append=True)
+                self.vFile.write_checkpoint(v, 'velocity', 
+                                            self.time, append=True)
+                self.rhoFile.write_checkpoint(rho, 'density', 
+                                              self.time, append=True)
+            # move mesh
+            dr = df.project(v*self.timestep, self.function_space.sub(0).collapse())
+            df.ALE.move(self.mesh, dr)
+
+        self.uFile.close()
+        self.vFile.close()
+        self.rhoFile.close()        
+
+
+def viz_tissue(params, DIR=''):
+    savesteps = int(params['maxtime']/params['savetime'])
+    times = np.arange(savesteps+1) * params['savetime']    
+    # Read mesh geometry from h5 file
+    var = 'density'
+    h5 = h5py.File(os.path.join(DIR, 
+                                '%s_%s.h5' % (params['timestamp'], var)), "r")
+    # should be in the loop if remeshing
+    topology = np.array(h5['%s/%s_0/mesh/topology'%(var,var)])
+    geometry = []
+    for i in range(len(times)):
+        geometry.append(np.array(h5['%s/%s_%d/mesh/geometry'%(var,var,i)]))
+    h5.close()
+    geometry = np.array(geometry)
+    geometry, zeros= np.dsplit(geometry, 2)
+    
+    mesh = df.IntervalMesh(params['resolution'], 
+                                -params['system_size']/2, 
+                                params['system_size']/2)
+    
+    # Read data
+    u = np.zeros((len(times), mesh.num_vertices(), 1))
+    v = np.zeros_like(u)
+    rho = np.zeros((len(times), mesh.num_vertices()))
+
+    uFile   = df.XDMFFile(os.path.join(DIR, 
+                                       '%s_displacement.xdmf' % params['timestamp']))
+    vFile   = df.XDMFFile(os.path.join(DIR, 
+                                       '%s_velocity.xdmf' % params['timestamp']))
+    rhoFile = df.XDMFFile(os.path.join(DIR, 
+                                       '%s_density.xdmf' % params['timestamp']))
+
+    # Reading data
+    print('Reading data...')
+    for steps in progressbar.progressbar(range(savesteps+1)):
+        mesh.coordinates()[:] = geometry[steps]
+        SFS = df.FunctionSpace(mesh, 'P', 1)
+        VFS = df.VectorFunctionSpace(mesh, 'P', 1)
+        ui, vi, rhoi = df.Function(VFS), df.Function(VFS), df.Function(SFS)
+            
+        uFile.read_checkpoint(ui, 'displacement', steps)
+        vFile.read_checkpoint(vi, 'velocity', steps)
+        rhoFile.read_checkpoint(rhoi, 'density', steps)
+        u_vec = ui.compute_vertex_values(mesh)
+        u[steps] = u_vec.reshape(1, int(u_vec.shape[0])).T
+        v_vec = vi.compute_vertex_values(mesh)
+        v[steps] = v_vec.reshape(1, int(v_vec.shape[0])).T
+        rho[steps] = rhoi.compute_vertex_values(mesh)
+    uFile.close()
+    vFile.close()
+    rhoFile.close()
+    
+    
+    fig, axes = plt.subplots(2, 1, sharex=True, figsize=(8,8))
+    axes[-1].set_xlabel(r'$x$')
+    axes[0].set_ylabel(r'$\rho$')
+    axes[1].set_ylabel(r'$v$')
+    
+    axes[0].set_xlim(np.min(geometry), np.max(geometry))
+    axes[0].set_ylim(np.min(rho), np.max(rho))
+    axes[1].set_ylim(np.min(v), np.max(v))
+    
+    rhoplot, = axes[0].plot(geometry[0], rho[0], 'g-', ms=3)
+    velplot, = axes[1].plot(geometry[0], v[0],   'r-', ms=3)
+
+    def update(value):
+        ti = np.abs(times-value).argmin()
+        rhoplot.set_ydata(rho[ti])
+        rhoplot.set_xdata(geometry[ti])
+        velplot.set_ydata(v[ti])
+        velplot.set_xdata(geometry[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)
+    plt.show()        
+        
+    
+if __name__ == '__main__':
+    import json, datetime
+    assert os.path.isfile('parameters.json'), \
+        'parameters.json file not found'
+    
+    # load the parameters
+    with open('parameters.json') as jsonFile:                    
+        params = json.load(jsonFile)
+    
+    timestamp = '123456' #datetime.datetime.now().strftime("%d%m%y-%H%M%S")
+    params['timestamp'] = timestamp
+        
+    tissue = Tissue(params)
+    tissue.solve()
+
+    with open(params['timestamp'] + '_parameters.json', "w") as fp:
+        json.dump(params, fp, indent=4)   
+
+    viz_tissue(params)
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