add one patch example

LDDsimulation/UseCaseBase.py

+"""base class for LDD simulation use cases.
+
+This module defines a base class for the LDDsimulation usecases.
+"""
+
+import dolfin as df
+import mshr
+import numpy as np
+import sympy as sym
+import typing as tp
+import functools as ft
+import domainPatch as dp
+import LDDsimulation as ldd
+import helpers as hlp
+
+class UseCaseBase(object):
+    """Base class for LDD simulation use cases.
+    """
+    

TP-one-patch/TP-one-patch-alterantive.py

+#!/usr/bin/python3
+import dolfin as df
+import mshr
+import numpy as np
+import sympy as sym
+import typing as tp
+import domainPatch as dp
+import LDDsimulation as ldd
+import functools as ft
+import helpers as hlp
+#import ufl as ufl
+
+# init sympy session
+sym.init_printing()
+
+solver_tol = 1E-7
+
+############ GRID #######################ü
+mesh_resolution = 51
+timestep_size = 0.005
+number_of_timesteps = 300
+# decide how many timesteps you want analysed. Analysed means, that we write out
+# subsequent errors of the L-iteration within the timestep.
+number_of_timesteps_to_analyse = 0
+starttime = 0
+
+Lw = 1000 #/timestep_size
+Lnw=Lw
+
+l_param_w = 80
+l_param_nw = 80
+
+include_gravity = True
+
+
+##### Domain and Interface ####
+# global simulation domain domain
+sub_domain0_vertices = [df.Point(-1.0,-1.0), #
+                        df.Point(1.0,-1.0),#
+                        df.Point(1.0,1.0),#
+                        df.Point(-1.0,1.0)]
+# interface between subdomain1 and subdomain2
+interface12_vertices = [df.Point(-1.0, 0.0),
+                        df.Point(1.0, 0.0) ]
+# subdomain1.
+sub_domain1_vertices = [interface12_vertices[0],
+                        interface12_vertices[1],
+                        sub_domain0_vertices[2],
+                        sub_domain0_vertices[3] ]
+
+# vertex coordinates of the outer boundaries. If it can not be specified as a
+# polygon, use an entry per boundary polygon. This information is used for defining
+# the Dirichlet boundary conditions. If a domain is completely internal, the
+# dictionary entry should be 0: None
+subdomain1_outer_boundary_verts = {
+    0: [interface12_vertices[1],
+        sub_domain0_vertices[2],
+        sub_domain0_vertices[3], #
+        interface12_vertices[0]]
+}
+# subdomain2
+sub_domain2_vertices = [sub_domain0_vertices[0],
+                        sub_domain0_vertices[1],
+                        interface12_vertices[1],
+                        interface12_vertices[0] ]
+
+subdomain2_outer_boundary_verts = {
+    0: [interface12_vertices[0], #
+        sub_domain0_vertices[0],
+        sub_domain0_vertices[1],
+        interface12_vertices[1]]
+}
+# subdomain2_outer_boundary_verts = {
+#     0: [interface12_vertices[0], df.Point(0.0,0.0)],#
+#     1: [df.Point(0.0,0.0), df.Point(1.0,0.0)], #
+#     2: [df.Point(1.0,0.0), interface12_vertices[1]]
+# }
+# subdomain2_outer_boundary_verts = {
+#     0: None
+# }
+
+# list of subdomains given by the boundary polygon vertices.
+# Subdomains are given as a list of dolfin points forming
+# a closed polygon, such that mshr.Polygon(subdomain_def_points[i]) can be used
+# to create the subdomain. subdomain_def_points[0] contains the
+# vertices of the global simulation domain and subdomain_def_points[i] contains the
+# vertices of the subdomain i.
+subdomain_def_points = [sub_domain0_vertices,#
+                      sub_domain1_vertices,#
+                      sub_domain2_vertices]
+# in the below list, index 0 corresponds to the 12 interface which has index 1
+interface_def_points = [interface12_vertices]
+
+# if a subdomain has no outer boundary write None instead, i.e.
+# i: None
+# if i is the index of the inner subdomain.
+outer_boundary_def_points = {
+    # subdomain number
+    1 : subdomain1_outer_boundary_verts,
+    2 : subdomain2_outer_boundary_verts
+}
+
+# adjacent_subdomains[i] contains the indices of the subdomains sharing the
+# interface i (i.e. given by interface_def_points[i]).
+adjacent_subdomains = [[1,2]]
+isRichards = {
+    1: False, #
+    2: False
+    }
+
+
+viscosity = {#
+# subdom_num : viscosity
+    1 : {'wetting' :1,
+         'nonwetting': 1}, #
+    2 : {'wetting' :1,
+         'nonwetting': 1}
+}
+
+porosity = {#
+# subdom_num : porosity
+    1 : 1,#
+    2 : 1
+}
+
+# Dict of the form: { subdom_num : density }
+densities = {
+    1: {'wetting': 1,  #997,
+        'nonwetting': 1}, #1225},
+    2: {'wetting': 1,  #997,
+        'nonwetting': 1}, #1225},
+}
+
+gravity_acceleration = 9.81
+
+L = {#
+# subdom_num : subdomain L for L-scheme
+    1 : {'wetting' :Lw,
+         'nonwetting': Lnw},#
+    2 : {'wetting' :Lw,
+         'nonwetting': Lnw}
+}
+
+lambda_param = {#
+# subdom_num : lambda parameter for the L-scheme
+    1 : {'wetting' :l_param_w,
+         'nonwetting': l_param_nw},#
+    2 : {'wetting' :l_param_w,
+         'nonwetting': l_param_nw}
+}
+
+## relative permeabilty functions on subdomain 1
+def rel_perm1w(s):
+    # relative permeabilty wetting on subdomain1
+    return s**2
+
+def rel_perm1nw(s):
+    # relative permeabilty nonwetting on subdomain1
+    return (1-s)**2
+
+_rel_perm1w = ft.partial(rel_perm1w)
+_rel_perm1nw = ft.partial(rel_perm1nw)
+
+subdomain1_rel_perm = {
+    'wetting': _rel_perm1w,#
+    'nonwetting': _rel_perm1nw
+}
+## relative permeabilty functions on subdomain 2
+def rel_perm2w(s):
+    # relative permeabilty wetting on subdomain2
+    return s**3
+def rel_perm2nw(s):
+    # relative permeabilty nonwetting on subdosym.cos(0.8*t - (0.8*x + 1/7*y))main2
+    return (1-s)**3
+
+_rel_perm2w = ft.partial(rel_perm2w)
+_rel_perm2nw = ft.partial(rel_perm2nw)
+
+subdomain2_rel_perm = {
+    'wetting': _rel_perm2w,#
+    'nonwetting': _rel_perm2nw
+}
+
+## dictionary of relative permeabilties on all domains.
+relative_permeability = {#
+    1: subdomain1_rel_perm,
+    2: subdomain2_rel_perm
+}
+
+
+# definition of the derivatives of the relative permeabilities
+# relative permeabilty functions on subdomain 1
+def rel_perm1w_prime(s):
+    # relative permeabilty on subdomain1
+    return 2*s
+
+def rel_perm1nw_prime(s):
+    # relative permeabilty on subdomain1
+    return -2*(1-s)
+
+# # definition of the derivatives of the relative permeabilities
+# # relative permeabilty functions on subdomain 1
+def rel_perm2w_prime(s):
+    # relative permeabilty on subdomain1
+    return 3*s**2
+
+def rel_perm2nw_prime(s):
+    # relative permeabilty on subdomain1
+    return -3*(1-s)**2
+
+_rel_perm1w_prime = ft.partial(rel_perm1w_prime)
+_rel_perm1nw_prime = ft.partial(rel_perm1nw_prime)
+_rel_perm2w_prime = ft.partial(rel_perm2w_prime)
+_rel_perm2nw_prime = ft.partial(rel_perm2nw_prime)
+
+subdomain1_rel_perm_prime = {
+    'wetting': _rel_perm1w_prime,
+    'nonwetting': _rel_perm1nw_prime
+}
+
+
+subdomain2_rel_perm_prime = {
+    'wetting': _rel_perm2w_prime,
+    'nonwetting': _rel_perm2nw_prime
+}
+
+# dictionary of relative permeabilties on all domains.
+ka_prime = {
+    1: subdomain1_rel_perm_prime,
+    2: subdomain2_rel_perm_prime,
+}
+
+
+
+def saturation(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return df.conditional(pc > 0, 1/((1 + pc)**(1/(index + 1))), 1)
+
+
+def saturation_sym(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return 1/((1 + pc)**(1/(index + 1)))
+
+
+# derivative of S-pc relationship with respect to pc. This is needed for the
+# construction of a analytic solution.
+def saturation_sym_prime(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return -1/((index+1)*(1 + pc)**((index+2)/(index+1)))
+
+
+# note that the conditional definition of S-pc in the nonsymbolic part will be
+# incorporated in the construction of the exact solution below.
+S_pc_sym = {
+    1: ft.partial(saturation_sym, index=1),
+    2: ft.partial(saturation_sym, index=2),
+    # 3: ft.partial(saturation_sym, index=2),
+    # 4: ft.partial(saturation_sym, index=1)
+}
+
+S_pc_sym_prime = {
+    1: ft.partial(saturation_sym_prime, index=1),
+    2: ft.partial(saturation_sym_prime, index=2),
+    # 3: ft.partial(saturation_sym_prime, index=2),
+    # 4: ft.partial(saturation_sym_prime, index=1)
+}
+
+sat_pressure_relationship = {
+    1: ft.partial(saturation, index=1),
+    2: ft.partial(saturation, index=2),
+    # 3: ft.partial(saturation, index=2),
+    # 4: ft.partial(saturation, index=1)
+}
+
+#
+# def saturation(pc, n_index, alpha):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pc > 0, 1/((1 + (alpha*pc)**n_index)**((n_index - 1)/n_index)), 1)
+#
+# # S-pc-relation ship. We use the van Genuchten approach, i.e. pc = 1/alpha*(S^{-1/m} -1)^1/n, where
+# # we set alpha = 0, assume m = 1-1/n (see Helmig) and assume that residual saturation is Sw
+# def saturation_sym(pc, n_index, alpha):
+#     # inverse capillary pressure-saturation-relationship
+#     #df.conditional(pc > 0,
+#     return 1/((1 + (alpha*pc)**n_index)**((n_index - 1)/n_index))
+#
+#
+# # derivative of S-pc relationship with respect to pc. This is needed for the
+# # construction of a analytic solution.
+# def saturation_sym_prime(pc, n_index, alpha):
+#     # inverse capillary pressure-saturation-relationship
+#     return -(alpha*(n_index - 1)*(alpha*pc)**(n_index - 1)) / ( (1 + (alpha*pc)**n_index)**((2*n_index - 1)/n_index) )
+#
+# # note that the conditional definition of S-pc in the nonsymbolic part will be
+# # incorporated in the construction of the exact solution below.
+# S_pc_sym = {
+#     1: ft.partial(saturation_sym, n_index=3, alpha=0.001),
+#     2: ft.partial(saturation_sym, n_index=6, alpha=0.001),
+#     # 3: ft.partial(saturation_sym, n_index=3, alpha=0.001),
+#     # 4: ft.partial(saturation_sym, n_index=3, alpha=0.001),
+#     # 5: ft.partial(saturation_sym, n_index=3, alpha=0.001),
+#     # 6: ft.partial(saturation_sym, n_index=3, alpha=0.001)
+# }
+#
+# S_pc_sym_prime = {
+#     1: ft.partial(saturation_sym_prime, n_index=3, alpha=0.001),
+#     2: ft.partial(saturation_sym_prime, n_index=6, alpha=0.001),
+#     # 3: ft.partial(saturation_sym_prime, n_index=3, alpha=0.001),
+#     # 4: ft.partial(saturation_sym_prime, n_index=3, alpha=0.001),
+#     # 5: ft.partial(saturation_sym_prime, n_index=3, alpha=0.001),
+#     # 6: ft.partial(saturation_sym_prime, n_index=3, alpha=0.001)
+# }
+#
+# sat_pressure_relationship = {
+#     1: ft.partial(saturation, n_index=3, alpha=0.001),
+#     2: ft.partial(saturation, n_index=6, alpha=0.001),
+#     # 3: ft.partial(saturation, n_index=3, alpha=0.001),
+#     # 4: ft.partial(saturation, n_index=3, alpha=0.001),
+#     # 5: ft.partial(saturation, n_index=3, alpha=0.001),
+#     # 6: ft.partial(saturation, n_index=3, alpha=0.001)
+# }
+#
+
+
+#############################################
+# Manufacture source expressions with sympy #
+#############################################
+x, y = sym.symbols('x[0], x[1]')  # needed by UFL
+t = sym.symbols('t', positive=True)
+
+# construction of differentiable characteristic function.
+def dist_to_epsilon_thickening_of_subdomain0_complement(x, y, epsilon):
+    """ calculates the (euklidian distance)^2 of a point x,y to the epsilon
+        thickening of the complement of the domain.
+    """
+    is_inside = ((1-sym.Abs(x) > epsilon) & (1-sym.Abs(y) > epsilon))
+    sym.Piecewise()
+
+p_e_sym = {
+    1: {'wetting': -5 - (1+t*t)*(1 + x*x + y*y),
+        'nonwetting': -2 -t*(1-y + x**2)**2},
+    2: {'wetting': -5.0 - (1.0 + t*t)*(1.0 + x*x),
+        'nonwetting': -2 -t*(1 + x**2)**2 - sym.sqrt(2+t**2)*(1+y)**2*x**2*y**2},
+}
+
+pc_e_sym = dict()
+for subdomain, isR in isRichards.items():
+    if isR:
+        pc_e_sym.update({subdomain: -p_e_sym[subdomain]['wetting'].copy()})
+    else:
+        pc_e_sym.update({subdomain: p_e_sym[subdomain]['nonwetting'].copy()
+                                        - p_e_sym[subdomain]['wetting'].copy()})
+
+
+symbols = {"x": x,
+           "y": y,
+           "t": t}
+# turn above symbolic code into exact solution for dolphin and
+# construct the rhs that matches the above exact solution.
+exact_solution_example = hlp.generate_exact_solution_expressions(
+                        symbols=symbols,
+                        isRichards=isRichards,
+                        symbolic_pressure=p_e_sym,
+                        symbolic_capillary_pressure=pc_e_sym,
+                        saturation_pressure_relationship=S_pc_sym,
+                        saturation_pressure_relationship_prime=S_pc_sym_prime,
+                        viscosity=viscosity,
+                        porosity=porosity,
+                        relative_permeability=relative_permeability,
+                        relative_permeability_prime=ka_prime,
+                        densities=densities,
+                        gravity_acceleration=gravity_acceleration,
+                        include_gravity=include_gravity,
+                        )
+source_expression = exact_solution_example['source']
+exact_solution = exact_solution_example['exact_solution']
+initial_condition = exact_solution_example['initial_condition']
+
+# Dictionary of dirichlet boundary conditions.
+dirichletBC = dict()
+# similarly to the outer boundary dictionary, if a patch has no outer boundary
+# None should be written instead of an expression.
+# This is a bit of a brainfuck:
+# dirichletBC[ind] gives a dictionary of the outer boundaries of subdomain ind.
+# Since a domain patch can have several disjoint outer boundary parts, the
+# expressions need to get an enumaration index which starts at 0.
+# So dirichletBC[ind][j] is the dictionary of outer dirichlet conditions of
+# subdomain ind and boundary part j.
+# Finally, dirichletBC[ind][j]['wetting'] and dirichletBC[ind][j]['nonwetting']
+# return the actual expression needed for the dirichlet condition for both
+# phases if present.
+
+# subdomain index: {outer boudary part index: {phase: expression}}
+for subdomain in isRichards.keys():
+    # if subdomain has no outer boundary, outer_boundary_def_points[subdomain] is None
+    if outer_boundary_def_points[subdomain] is None:
+        dirichletBC.update({subdomain: None})
+    else:
+        dirichletBC.update({subdomain: dict()})
+        # set the dirichlet conditions to be the same code as exact solution on
+        # the subdomain.
+        for outer_boundary_ind in outer_boundary_def_points[subdomain].keys():
+            dirichletBC[subdomain].update(
+                {outer_boundary_ind: exact_solution[subdomain]}
+                )
+
+
+# def saturation(pressure, subdomain_index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pressure < 0, 1/((1 - pressure)**(1/(subdomain_index + 1))), 1)
+#
+# sa
+
+write_to_file = {
+    'meshes_and_markers': True,
+    'L_iterations': True
+}
+
+output_string = "./output/alternative_example_number_of_timesteps{}_".format(number_of_timesteps)
+
+# initialise LDD simulation class
+simulation = ldd.LDDsimulation(tol = 1E-14, LDDsolver_tol=solver_tol, debug=False)
+simulation.set_parameters(output_dir = output_string,#
+    subdomain_def_points = subdomain_def_points,#
+    isRichards = isRichards,#
+    interface_def_points = interface_def_points,#
+    outer_boundary_def_points = outer_boundary_def_points,#
+    adjacent_subdomains = adjacent_subdomains,#
+    mesh_resolution = mesh_resolution,#
+    viscosity = viscosity,#
+    porosity = porosity,#
+    L = L,#
+    lambda_param = lambda_param,#
+    relative_permeability = relative_permeability,#
+    saturation = sat_pressure_relationship,#
+    starttime = starttime,#
+    number_of_timesteps = number_of_timesteps,
+    number_of_timesteps_to_analyse = number_of_timesteps_to_analyse,
+    timestep_size = timestep_size,#
+    sources = source_expression,#
+    initial_conditions = initial_condition,#
+    dirichletBC_expression_strings = dirichletBC,#
+    exact_solution = exact_solution,#
+    densities=densities,
+    include_gravity=include_gravity,
+    write2file = write_to_file,#
+    )
+
+simulation.initialise()
+# simulation.write_exact_solution_to_xdmf()
+simulation.run()

TP-one-patch/TP-one-patch-linear-koefficients.py

+#!/usr/bin/python3
+import dolfin as df
+import mshr
+import numpy as np
+import sympy as sym
+import typing as tp
+import domainPatch as dp
+import LDDsimulation as ldd
+import functools as ft
+import helpers as hlp
+#import ufl as ufl
+
+# init sympy session
+sym.init_printing()
+
+
+solver_tol = 1E-12
+
+############ GRID #######################ü
+mesh_resolution = 60
+timestep_size = 0.01
+number_of_timesteps = 140
+# decide how many timesteps you want analysed. Analysed means, that we write out
+# subsequent errors of the L-iteration within the timestep.
+number_of_timesteps_to_analyse = 10
+starttime = 0
+
+Lw = 1  #/timestep_size
+Lnw=Lw
+
+l_param_w = 40
+l_param_nw = 40
+
+include_gravity = True
+debugflag = False
+analyse_condition = True
+
+output_string = "./output/linear_coefficients_number_of_timesteps{}_".format(number_of_timesteps)
+
+##### Domain and Interface ####
+# global simulation domain domain
+sub_domain0_vertices = [df.Point(-1.0,-1.0), #
+                        df.Point(1.0,-1.0),#
+                        df.Point(1.0,1.0),#
+                        df.Point(-1.0,1.0)]
+
+subdomain0_outer_boundary_verts = {
+    0: [sub_domain0_vertices[0],
+        sub_domain0_vertices[1],
+        sub_domain0_vertices[2],
+        sub_domain0_vertices[3],
+        sub_domain0_vertices[0]]
+}
+
+# list of subdomains given by the boundary polygon vertices.
+# Subdomains are given as a list of dolfin points forming
+# a closed polygon, such that mshr.Polygon(subdomain_def_points[i]) can be used
+# to create the subdomain. subdomain_def_points[0] contains the
+# vertices of the global simulation domain and subdomain_def_points[i] contains the
+# vertices of the subdomain i.
+subdomain_def_points = [sub_domain0_vertices]
+# in the below list, index 0 corresponds to the 12 interface which has index 1
+interface_def_points = None
+
+# if a subdomain has no outer boundary write None instead, i.e.
+# i: None
+# if i is the index of the inner subdomain.
+outer_boundary_def_points = {
+    # subdomain number
+    0 : subdomain0_outer_boundary_verts
+}
+
+# adjacent_subdomains[i] contains the indices of the subdomains sharing the
+# interface i (i.e. given by interface_def_points[i]).
+adjacent_subdomains = None
+isRichards = {
+    0: False, #
+    }
+
+viscosity = {#
+# subdom_num : viscosity
+    0 : {'wetting' :1,
+         'nonwetting': 1}, #
+}
+
+porosity = {#
+# subdom_num : porosity
+    0: 1,#
+}
+
+# Dict of the form: { subdom_num : density }
+densities = {
+    0: {'wetting': 1,  #997,
+        'nonwetting': 1}, #1225}
+}
+
+gravity_acceleration = 9.81
+
+L = {#
+# subdom_num : subdomain L for L-scheme
+    0: {'wetting' :Lw,
+         'nonwetting': Lnw},#
+}
+
+lambda_param = {#
+# subdom_num : lambda parameter for the L-scheme
+    0: {'wetting' :l_param_w,
+         'nonwetting': l_param_nw},#
+}
+
+## relative permeabilty functions on subdomain 1
+def rel_perm1w(s):
+    # relative permeabilty wetting on subdomain1
+    return s
+
+def rel_perm1nw(s):
+    # relative permeabilty nonwetting on subdomain1
+    return (1-s)
+
+_rel_perm1w = ft.partial(rel_perm1w)
+_rel_perm1nw = ft.partial(rel_perm1nw)
+
+subdomain1_rel_perm = {
+    'wetting': _rel_perm1w,#
+    'nonwetting': _rel_perm1nw
+}
+
+## dictionary of relative permeabilties on all domains.
+relative_permeability = {#
+    0: subdomain1_rel_perm,
+}
+
+# definition of the derivatives of the relative permeabilities
+# relative permeabilty functions on subdomain 1
+def rel_perm1w_prime(s):
+    # relative permeabilty on subdomain1
+    return 1
+
+def rel_perm1nw_prime(s):
+    # relative permeabilty on subdomain1
+    return -1
+
+_rel_perm1w_prime = ft.partial(rel_perm1w_prime)
+_rel_perm1nw_prime = ft.partial(rel_perm1nw_prime)
+
+subdomain1_rel_perm_prime = {
+    'wetting': _rel_perm1w_prime,
+    'nonwetting': _rel_perm1nw_prime
+}
+
+# dictionary of relative permeabilties on all domains.
+ka_prime = {
+    0: subdomain1_rel_perm_prime,
+}
+
+
+
+def saturation(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return df.conditional(pc > 0, df.conditional(pc<=1, 1-pc, 0), 1)
+
+
+def saturation_sym(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return 1-pc
+
+
+# derivative of S-pc relationship with respect to pc. This is needed for the
+# construction of a analytic solution.
+def saturation_sym_prime(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return -1
+
+
+# def saturation(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pc > 0, -index*pc, 1)
+#
+#
+# def saturation_sym(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return -index*pc
+#
+#
+# # derivative of S-pc relationship with respect to pc. This is needed for the
+# # construction of a analytic solution.
+# def saturation_sym_prime(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return -index
+
+
+# note that the conditional definition of S-pc in the nonsymbolic part will be
+# incorporated in the construction of the exact solution below.
+S_pc_sym_handle = {
+    0: ft.partial(saturation_sym, index=1),
+}
+
+S_pc_sym_prime_handle = {
+    0: ft.partial(saturation_sym_prime, index=1),
+}
+
+sat_pressure_relationship = {
+    0: ft.partial(saturation, index=1),
+}
+
+
+#############################################
+# Manufacture source expressions with sympy #
+#############################################
+x, y = sym.symbols('x[0], x[1]')  # needed by UFL
+t = sym.symbols('t', positive=True)
+
+epsilon_x_inner = 0.7
+epsilon_x_outer = 0.99
+epsilon_y_inner = epsilon_x_inner
+epsilon_y_outer = epsilon_x_outer
+
+def mollifier(x, epsilon):
+    """ one d mollifier """
+    out_expr = sym.exp(-1/(1-(x/epsilon)**2) + 1)
+    return out_expr
+
+mollifier_handle = ft.partial(mollifier, epsilon=epsilon_x_inner)
+
+pw_sym_x = sym.Piecewise(
+    (mollifier_handle(x), x**2 < epsilon_x_outer**2),
+    (0, True)
+)
+
+pw_sym_y = sym.Piecewise(
+    (mollifier_handle(y), y**2 < epsilon_y_outer**2),
+    (0, True)
+)
+
+def mollifier2d(x, y, epsilon):
+    """ one d mollifier """
+    out_expr = sym.exp(-1/(1-(x**2 + y**2)/epsilon**2) + 1)
+    return out_expr
+
+mollifier2d_handle = ft.partial(mollifier2d, epsilon=epsilon_x_outer)
+
+pw_sym2d_x = sym.Piecewise(
+    (mollifier2d_handle(x, y), x**2 + y**2 < epsilon_x_outer**2),
+    (0, True)
+)
+
+zero_on_epsilon_shrinking_of_subdomain = sym.Piecewise(
+    (mollifier_handle(sym.sqrt(x**2 + y**2)+2*epsilon_x_inner), ((-2*epsilon_x_inner<sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<=epsilon_x_inner))),
+    (mollifier_handle(sym.sqrt(x**2 + y**2)-2*epsilon_x_inner), ((epsilon_x_inner<sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_epsilon_shrinking_of_subdomain_x = sym.Piecewise(
+    (mollifier_handle(x+2*epsilon_x_inner), ((-2*epsilon_x_inner<x) & (x<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=x) & (x<=epsilon_x_inner))),
+    (mollifier_handle(x-2*epsilon_x_inner), ((epsilon_x_inner<x) & (x<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_epsilon_shrinking_of_subdomain_y = sym.Piecewise(
+    (1, y<=-2*epsilon_x_inner),
+    (mollifier_handle(y+2*epsilon_x_inner), ((-2*epsilon_x_inner<y) & (y<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=y) & (y<=epsilon_x_inner))),
+    (mollifier_handle(y-2*epsilon_x_inner), ((epsilon_x_inner<y) & (y<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_shrinking = zero_on_epsilon_shrinking_of_subdomain #zero_on_epsilon_shrinking_of_subdomain_x + zero_on_epsilon_shrinking_of_subdomain_y
+gaussian = pw_sym2d_x# pw_sym_y*pw_sym_x
+cutoff = gaussian/(gaussian + zero_on_shrinking)
+
+# # construction of differentiable characteristic function.
+# def smooth_characteristic_func_on_epsilon_shrinking_of_subdomain0(x, y, epsilon_x_inner, epsilon_y_inner, epsilon_x_outer, epsilon_y_outer):
+#     dist_to_complement_x = ft.partial(mollifier, epsilon=epsilon_x_inner)
+#     dist_to_complement_y = ft.partial(mollifier, epsilon=epsilon_y_inner)
+#     dist_to_complement = dist_to_complement_y(y)*dist_to_complement_x(x)
+#     dist_to_eps_shrinking_x = ft.partial(zero_outside_epsilon_thickening_of_subdomain, epsilon=epsilon_x_outer)
+#     dist_to_eps_shrinking_y = ft.partial(zero_outside_epsilon_thickening_of_subdomain, epsilon=epsilon_y_outer)
+#     dist_to_eps_shrinking = dist_to_eps_shrinking_y(y)*dist_to_eps_shrinking_x(x)
+#     return dist_to_complement/(dist_to_eps_shrinking + dist_to_complement)
+#
+
+# def dist_to_epsilon_thickening_of_subdomain0_complement(x, y, epsilon):
+#     """ calculates the (euklidian distance)^2 of a point x,y to the epsilon
+#         thickening of the complement of the domain.
+#     """
+#     is_inside = ((1-sym.Abs(x) > epsilon) & (1-sym.Abs(y) > epsilon))
+#     sym.Piecewise((0, is_inside))
+
+# p_e_sym = {
+#     0: {'wetting': (-3 - (1+t*t)*(1 + x*x + y*y))*cutoff,
+#         'nonwetting': (-1 -t*(1+y + x**2)**2)*cutoff},
+# }
+
+p_e_sym = {
+    0: {'wetting': -3 -(sym.cos(2*t-x - 2*y)*sym.sin(3*(1+y)/2*sym.pi)*sym.sin(5*(1+x)/2*sym.pi))**2,
+        'nonwetting': -1 -(sym.cos(t-x -y)*sym.sin(3*(1+y)/2*sym.pi)*sym.sin(5*(1+x)/2*sym.pi))**2},
+}
+
+
+print(f"\n\n\nsymbolic type is {type(p_e_sym[0]['wetting'])}\n\n\n")
+# # pw_sym_x*pw_sym_y
+# p_e_sym = {
+#     0: {'wetting': -3*pw_sym2d_x + 0*t,
+#         'nonwetting': -1*pw_sym_x*pw_sym_y+ 0*t},
+# }
+
+# p_e_sym = {
+#     0: {'wetting': -3*cutoff + 0*t,
+#         'nonwetting': -1*zero_on_shrinking+ 0*t},
+# }
+
+
+pc_e_sym = dict()
+for subdomain, isR in isRichards.items():
+    if isR:
+        pc_e_sym.update({subdomain: -p_e_sym[subdomain]['wetting'].copy()})
+    else:
+        pc_e_sym.update({subdomain: p_e_sym[subdomain]['nonwetting'].copy()
+                                        - p_e_sym[subdomain]['wetting'].copy()})
+
+
+
+S_pc_sym = {
+    0: sym.Piecewise(
+        (1, pc_e_sym[0]<= 0),
+        (S_pc_sym_handle[0](pc_e_sym[0]), ((0<pc_e_sym[0])& (pc_e_sym[0] < 1))),
+        (0, True)
+        )
+}
+
+S_pc_sym_prime = {
+    0: sym.Piecewise(
+        (S_pc_sym_prime_handle[0](pc_e_sym[0]), ((pc_e_sym[0] > 0)& (pc_e_sym[0] < 1))),
+        (0, True)
+        )
+}
+
+symbols = {"x": x,
+           "y": y,
+           "t": t}
+
+
+# turn above symbolic code into exact solution for dolphin and
+# construct the rhs that matches the above exact solution.
+exact_solution_example = hlp.generate_exact_solution_expressions(
+                        symbols=symbols,
+                        isRichards=isRichards,
+                        symbolic_pressure=p_e_sym,
+                        symbolic_capillary_pressure=pc_e_sym,
+                        symbolic_S_pc_relationship=S_pc_sym,
+                        symbolic_S_pc_relationship_prime=S_pc_sym_prime,
+                        viscosity=viscosity,
+                        porosity=porosity,
+                        relative_permeability=relative_permeability,
+                        relative_permeability_prime=ka_prime,
+                        densities=densities,
+                        gravity_acceleration=gravity_acceleration,
+                        include_gravity=include_gravity,
+                        )
+source_expression = exact_solution_example['source']
+exact_solution = exact_solution_example['exact_solution']
+initial_condition = exact_solution_example['initial_condition']
+
+# Dictionary of dirichlet boundary conditions.
+dirichletBC = dict()
+# similarly to the outer boundary dictionary, if a patch has no outer boundary
+# None should be written instead of an expression.
+# This is a bit of a brainfuck:
+# dirichletBC[ind] gives a dictionary of the outer boundaries of subdomain ind.
+# Since a domain patch can have several disjoint outer boundary parts, the
+# expressions need to get an enumaration index which starts at 0.
+# So dirichletBC[ind][j] is the dictionary of outer dirichlet conditions of
+# subdomain ind and boundary part j.
+# Finally, dirichletBC[ind][j]['wetting'] and dirichletBC[ind][j]['nonwetting']
+# return the actual expression needed for the dirichlet condition for both
+# phases if present.
+
+# subdomain index: {outer boudary part index: {phase: expression}}
+for subdomain in isRichards.keys():
+    # if subdomain has no outer boundary, outer_boundary_def_points[subdomain] is None
+    if outer_boundary_def_points[subdomain] is None:
+        dirichletBC.update({subdomain: None})
+    else:
+        dirichletBC.update({subdomain: dict()})
+        # set the dirichlet conditions to be the same code as exact solution on
+        # the subdomain.
+        for outer_boundary_ind in outer_boundary_def_points[subdomain].keys():
+            dirichletBC[subdomain].update(
+                {outer_boundary_ind: exact_solution[subdomain]}
+                )
+
+
+# def saturation(pressure, subdomain_index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pressure < 0, 1/((1 - pressure)**(1/(subdomain_index + 1))), 1)
+#
+# sa
+
+write_to_file = {
+    'meshes_and_markers': True,
+    'L_iterations': True
+}
+
+
+
+# initialise LDD simulation class
+simulation = ldd.LDDsimulation(tol = 1E-14, LDDsolver_tol=solver_tol, debug=debugflag)
+simulation.set_parameters(output_dir = output_string,#
+    subdomain_def_points = subdomain_def_points,#
+    isRichards = isRichards,#
+    interface_def_points = interface_def_points,#
+    outer_boundary_def_points = outer_boundary_def_points,#
+    adjacent_subdomains = adjacent_subdomains,#
+    mesh_resolution = mesh_resolution,#
+    viscosity = viscosity,#
+    porosity = porosity,#
+    L = L,#
+    lambda_param = lambda_param,#
+    relative_permeability = relative_permeability,#
+    saturation = sat_pressure_relationship,#
+    starttime = starttime,#
+    number_of_timesteps = number_of_timesteps,
+    number_of_timesteps_to_analyse = number_of_timesteps_to_analyse,
+    timestep_size = timestep_size,#
+    sources = source_expression,#
+    initial_conditions = initial_condition,#
+    dirichletBC_expression_strings = dirichletBC,#
+    exact_solution = exact_solution,#
+    densities=densities,
+    include_gravity=include_gravity,
+    write2file = write_to_file,#
+    )
+
+simulation.initialise()
+# simulation.write_exact_solution_to_xdmf()
+simulation.run(analyse_condition=analyse_condition)

TP-one-patch/TP-one-patch-purely-postive-pc.py

+#!/usr/bin/python3
+import dolfin as df
+import mshr
+import numpy as np
+import sympy as sym
+import typing as tp
+import domainPatch as dp
+import LDDsimulation as ldd
+import functools as ft
+import helpers as hlp
+#import ufl as ufl
+
+# init sympy session
+sym.init_printing()
+
+
+solver_tol = 1E-6
+
+############ GRID #######################
+
+mesh_resolution = 40
+timestep_size = 0.001
+number_of_timesteps = 10
+# decide how many timesteps you want analysed. Analysed means, that we write out
+# subsequent errors of the L-iteration within the timestep.
+number_of_timesteps_to_analyse = 1
+starttime = 0
+
+Lw = 0.25  #/timestep_size
+Lnw=Lw
+
+l_param_w = 40
+l_param_nw = 40
+
+include_gravity = False
+debugflag = True
+analyse_condition = True
+
+use_case="TP-one-patch-purely-positive-pc"
+output_string = "./output/pure_positive_pc_number_of_timesteps{}_".format(number_of_timesteps)
+
+##### Domain and Interface ####
+# global simulation domain domain
+sub_domain0_vertices = [df.Point(-1.0, -1.0),  #
+                        df.Point(1.0, -1.0),  #
+                        df.Point(1.0, 1.0),  #
+                        df.Point(-1.0, 1.0)]
+
+subdomain0_outer_boundary_verts = {
+    0: [sub_domain0_vertices[0],
+        sub_domain0_vertices[1],
+        sub_domain0_vertices[2],
+        sub_domain0_vertices[3],
+        sub_domain0_vertices[0]]
+}
+
+# list of subdomains given by the boundary polygon vertices.
+# Subdomains are given as a list of dolfin points forming
+# a closed polygon, such that mshr.Polygon(subdomain_def_points[i]) can be used
+# to create the subdomain. subdomain_def_points[0] contains the
+# vertices of the global simulation domain and subdomain_def_points[i] contains the
+# vertices of the subdomain i.
+subdomain_def_points = [sub_domain0_vertices]
+# in the below list, index 0 corresponds to the 12 interface which has index 1
+interface_def_points = None
+
+# if a subdomain has no outer boundary write None instead, i.e.
+# i: None
+# if i is the index of the inner subdomain.
+outer_boundary_def_points = {
+    # subdomain number
+    0 : subdomain0_outer_boundary_verts
+}
+
+# adjacent_subdomains[i] contains the indices of the subdomains sharing the
+# interface i (i.e. given by interface_def_points[i]).
+adjacent_subdomains = None
+isRichards = {
+    0: False, #
+    }
+
+viscosity = {#
+# subdom_num : viscosity
+    0 : {'wetting' :1,
+         'nonwetting': 1}, #
+}
+
+porosity = {#
+# subdom_num : porosity
+    0: 1,#
+}
+
+# Dict of the form: { subdom_num : density }
+densities = {
+    0: {'wetting': 1,  #997,
+        'nonwetting': 1}, #1225}
+}
+
+gravity_acceleration = 9.81
+
+L = {#
+# subdom_num : subdomain L for L-scheme
+    0: {'wetting' :Lw,
+         'nonwetting': Lnw},#
+}
+
+lambda_param = {#
+# subdom_num : lambda parameter for the L-scheme
+    0: {'wetting' :l_param_w,
+         'nonwetting': l_param_nw},#
+}
+
+## relative permeabilty functions on subdomain 1
+def rel_perm1w(s):
+    # relative permeabilty wetting on subdomain1
+    return s**2
+
+def rel_perm1nw(s):
+    # relative permeabilty nonwetting on subdomain1
+    return (1-s)**2
+
+_rel_perm1w = ft.partial(rel_perm1w)
+_rel_perm1nw = ft.partial(rel_perm1nw)
+
+subdomain1_rel_perm = {
+    'wetting': _rel_perm1w,#
+    'nonwetting': _rel_perm1nw
+}
+
+## dictionary of relative permeabilties on all domains.
+relative_permeability = {#
+    0: subdomain1_rel_perm,
+}
+
+# definition of the derivatives of the relative permeabilities
+# relative permeabilty functions on subdomain 1
+def rel_perm1w_prime(s):
+    # relative permeabilty on subdomain1
+    return 2*s
+
+def rel_perm1nw_prime(s):
+    # relative permeabilty on subdomain1
+    return -2*(1-s)
+
+_rel_perm1w_prime = ft.partial(rel_perm1w_prime)
+_rel_perm1nw_prime = ft.partial(rel_perm1nw_prime)
+
+subdomain1_rel_perm_prime = {
+    'wetting': _rel_perm1w_prime,
+    'nonwetting': _rel_perm1nw_prime
+}
+
+# dictionary of relative permeabilties on all domains.
+ka_prime = {
+    0: subdomain1_rel_perm_prime,
+}
+
+
+
+def saturation(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return df.conditional(pc > 0, 1/((1 + pc)**(1/(index + 1))), 1)
+
+def saturation_sym(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return 1/((1 + pc)**(1/(index + 1)))
+
+
+# derivative of S-pc relationship with respect to pc. This is needed for the
+# construction of a analytic solution.
+def saturation_sym_prime(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return -1/((index+1)*(1 + pc)**((index+2)/(index+1)))
+
+
+# def saturation(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pc > 0, -index*pc, 1)
+#
+#
+# def saturation_sym(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return -index*pc
+#
+#
+# # derivative of S-pc relationship with respect to pc. This is needed for the
+# # construction of a analytic solution.
+# def saturation_sym_prime(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return -index
+
+
+# note that the conditional definition of S-pc in the nonsymbolic part will be
+# incorporated in the construction of the exact solution below.
+S_pc_sym = {
+    0: ft.partial(saturation_sym, index=1),
+}
+
+S_pc_sym_prime = {
+    0: ft.partial(saturation_sym_prime, index=1),
+}
+
+sat_pressure_relationship = {
+    0: ft.partial(saturation, index=1),
+}
+
+
+#############################################
+# Manufacture source expressions with sympy #
+#############################################
+x, y = sym.symbols('x[0], x[1]')  # needed by UFL
+t = sym.symbols('t', positive=True)
+
+epsilon_x_inner = 0.7
+epsilon_x_outer = 0.99
+epsilon_y_inner = epsilon_x_inner
+epsilon_y_outer = epsilon_x_outer
+
+def mollifier(x, epsilon):
+    """ one d mollifier """
+    out_expr = sym.exp(-1/(1-(x/epsilon)**2) + 1)
+    return out_expr
+
+mollifier_handle = ft.partial(mollifier, epsilon=epsilon_x_inner)
+
+pw_sym_x = sym.Piecewise(
+    (mollifier_handle(x), x**2 < epsilon_x_outer**2),
+    (0, True)
+)
+pw_sym_y = sym.Piecewise(
+    (mollifier_handle(y), y**2 < epsilon_y_outer**2),
+    (0, True)
+)
+
+def mollifier2d(x, y, epsilon):
+    """ one d mollifier """
+    out_expr = sym.exp(-1/(1-(x**2 + y**2)/epsilon**2) + 1)
+    return out_expr
+
+mollifier2d_handle = ft.partial(mollifier2d, epsilon=epsilon_x_outer)
+
+pw_sym2d_x = sym.Piecewise(
+    (mollifier2d_handle(x, y), x**2 + y**2 < epsilon_x_outer**2),
+    (0, True)
+)
+
+zero_on_epsilon_shrinking_of_subdomain = sym.Piecewise(
+    (mollifier_handle(sym.sqrt(x**2 + y**2)+2*epsilon_x_inner), ((-2*epsilon_x_inner<sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<=epsilon_x_inner))),
+    (mollifier_handle(sym.sqrt(x**2 + y**2)-2*epsilon_x_inner), ((epsilon_x_inner<sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_epsilon_shrinking_of_subdomain_x = sym.Piecewise(
+    (mollifier_handle(x+2*epsilon_x_inner), ((-2*epsilon_x_inner<x) & (x<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=x) & (x<=epsilon_x_inner))),
+    (mollifier_handle(x-2*epsilon_x_inner), ((epsilon_x_inner<x) & (x<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_epsilon_shrinking_of_subdomain_y = sym.Piecewise(
+    (1, y<=-2*epsilon_x_inner),
+    (mollifier_handle(y+2*epsilon_x_inner), ((-2*epsilon_x_inner<y) & (y<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=y) & (y<=epsilon_x_inner))),
+    (mollifier_handle(y-2*epsilon_x_inner), ((epsilon_x_inner<y) & (y<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_shrinking = zero_on_epsilon_shrinking_of_subdomain #zero_on_epsilon_shrinking_of_subdomain_x + zero_on_epsilon_shrinking_of_subdomain_y
+gaussian = pw_sym2d_x# pw_sym_y*pw_sym_x
+cutoff = gaussian/(gaussian + zero_on_shrinking)
+
+# # construction of differentiable characteristic function.
+# def smooth_characteristic_func_on_epsilon_shrinking_of_subdomain0(x, y, epsilon_x_inner, epsilon_y_inner, epsilon_x_outer, epsilon_y_outer):
+#     dist_to_complement_x = ft.partial(mollifier, epsilon=epsilon_x_inner)
+#     dist_to_complement_y = ft.partial(mollifier, epsilon=epsilon_y_inner)
+#     dist_to_complement = dist_to_complement_y(y)*dist_to_complement_x(x)
+#     dist_to_eps_shrinking_x = ft.partial(zero_outside_epsilon_thickening_of_subdomain, epsilon=epsilon_x_outer)
+#     dist_to_eps_shrinking_y = ft.partial(zero_outside_epsilon_thickening_of_subdomain, epsilon=epsilon_y_outer)
+#     dist_to_eps_shrinking = dist_to_eps_shrinking_y(y)*dist_to_eps_shrinking_x(x)
+#     return dist_to_complement/(dist_to_eps_shrinking + dist_to_complement)
+#
+
+# def dist_to_epsilon_thickening_of_subdomain0_complement(x, y, epsilon):
+#     """ calculates the (euklidian distance)^2 of a point x,y to the epsilon
+#         thickening of the complement of the domain.
+#     """
+#     is_inside = ((1-sym.Abs(x) > epsilon) & (1-sym.Abs(y) > epsilon))
+#     sym.Piecewise((0, is_inside))
+
+p_e_sym = {
+    0: {'wetting': (-3 - (1+t*t)*(1 + x*x + y*y)),  #*cutoff,
+        'nonwetting': (-1 -t*(1.1+y + x**2))},  #*cutoff},
+}
+
+# p_e_sym = {
+#     0: {'wetting': -(sym.cos(2*t-x - 2*y)*sym.sin(3*(1+y)/2*sym.pi)*sym.sin(5*(1+x)/2*sym.pi))**2,
+#         'nonwetting': -6*(sym.cos(t-x -y)*sym.sin(3*(1+y)/2*sym.pi)*sym.sin(5*(1+x)/2*sym.pi))**2},
+# }
+
+
+print(f"\n\n\nsymbolic type is {type(p_e_sym[0]['wetting'])}\n\n\n")
+# # pw_sym_x*pw_sym_y
+# p_e_sym = {
+#     0: {'wetting': -3*pw_sym2d_x + 0*t,
+#         'nonwetting': -1*pw_sym_x*pw_sym_y+ 0*t},
+# }
+
+# p_e_sym = {
+#     0: {'wetting': -3*cutoff + 0*t,
+#         'nonwetting': -1*zero_on_shrinking+ 0*t},
+# }
+
+
+pc_e_sym = dict()
+for subdomain, isR in isRichards.items():
+    if isR:
+        pc_e_sym.update({subdomain: -p_e_sym[subdomain]['wetting']})
+    else:
+        pc_e_sym.update({subdomain: p_e_sym[subdomain]['nonwetting']
+                                        - p_e_sym[subdomain]['wetting']})
+
+symbols = {"x": x,
+           "y": y,
+           "t": t}
+# turn above symbolic code into exact solution for dolphin and
+# construct the rhs that matches the above exact solution.
+exact_solution_example = hlp.generate_exact_solution_expressions(
+                        symbols=symbols,
+                        isRichards=isRichards,
+                        symbolic_pressure=p_e_sym,
+                        symbolic_capillary_pressure=pc_e_sym,
+                        saturation_pressure_relationship=S_pc_sym,
+                        saturation_pressure_relationship_prime=S_pc_sym_prime,
+                        viscosity=viscosity,
+                        porosity=porosity,
+                        relative_permeability=relative_permeability,
+                        relative_permeability_prime=ka_prime,
+                        densities=densities,
+                        gravity_acceleration=gravity_acceleration,
+                        include_gravity=include_gravity,
+                        )
+source_expression = exact_solution_example['source']
+exact_solution = exact_solution_example['exact_solution']
+initial_condition = exact_solution_example['initial_condition']
+
+# Dictionary of dirichlet boundary conditions.
+dirichletBC = dict()
+# similarly to the outer boundary dictionary, if a patch has no outer boundary
+# None should be written instead of an expression.
+# This is a bit of a brainfuck:
+# dirichletBC[ind] gives a dictionary of the outer boundaries of subdomain ind.
+# Since a domain patch can have several disjoint outer boundary parts, the
+# expressions need to get an enumaration index which starts at 0.
+# So dirichletBC[ind][j] is the dictionary of outer dirichlet conditions of
+# subdomain ind and boundary part j.
+# Finally, dirichletBC[ind][j]['wetting'] and dirichletBC[ind][j]['nonwetting']
+# return the actual expression needed for the dirichlet condition for both
+# phases if present.
+
+# subdomain index: {outer boudary part index: {phase: expression}}
+for subdomain in isRichards.keys():
+    # if subdomain has no outer boundary, outer_boundary_def_points[subdomain] is None
+    if outer_boundary_def_points[subdomain] is None:
+        dirichletBC.update({subdomain: None})
+    else:
+        dirichletBC.update({subdomain: dict()})
+        # set the dirichlet conditions to be the same code as exact solution on
+        # the subdomain.
+        for outer_boundary_ind in outer_boundary_def_points[subdomain].keys():
+            dirichletBC[subdomain].update(
+                {outer_boundary_ind: exact_solution[subdomain]}
+                )
+
+
+# def saturation(pressure, subdomain_index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pressure < 0, 1/((1 - pressure)**(1/(subdomain_index + 1))), 1)
+#
+# sa
+
+write_to_file = {
+    'meshes_and_markers': True,
+    'L_iterations': True
+}
+
+# initialise LDD simulation class
+simulation = ldd.LDDsimulation(tol = 1E-14, LDDsolver_tol=solver_tol, debug=debugflag)
+simulation.set_parameters(use_case=use_case,
+    output_dir = output_string,#
+    subdomain_def_points = subdomain_def_points,#
+    isRichards = isRichards,#
+    interface_def_points = interface_def_points,#
+    outer_boundary_def_points = outer_boundary_def_points,#
+    adjacent_subdomains = adjacent_subdomains,#
+    mesh_resolution = mesh_resolution,#
+    viscosity = viscosity,#
+    porosity = porosity,#
+    L = L,#
+    lambda_param = lambda_param,#
+    relative_permeability = relative_permeability,#
+    saturation = sat_pressure_relationship,#
+    starttime = starttime,#
+    number_of_timesteps = number_of_timesteps,
+    number_of_timesteps_to_analyse = number_of_timesteps_to_analyse,
+    timestep_size = timestep_size,#
+    sources = source_expression,#
+    initial_conditions = initial_condition,#
+    dirichletBC_expression_strings = dirichletBC,#
+    exact_solution = exact_solution,#
+    densities=densities,
+    include_gravity=include_gravity,
+    write2file = write_to_file,#
+    )
+
+simulation.initialise()
+# simulation.write_exact_solution_to_xdmf()
+simulation.run(analyse_condition=analyse_condition)

TP-one-patch/TP-one-patch.py

+#!/usr/bin/python3
+import dolfin as df
+import mshr
+import numpy as np
+import sympy as sym
+import typing as tp
+import domainPatch as dp
+import LDDsimulation as ldd
+import functools as ft
+import helpers as hlp
+#import ufl as ufl
+
+# init sympy session
+sym.init_printing()
+
+
+solver_tol = 5E-6
+
+############ GRID #######################ü
+mesh_resolution = 30
+timestep_size = 0.0005
+number_of_timesteps = 1000
+# decide how many timesteps you want analysed. Analysed means, that we write out
+# subsequent errors of the L-iteration within the timestep.
+number_of_timesteps_to_analyse = 5
+starttime = 0
+
+Lw = 0.25  #/timestep_size
+Lnw=Lw
+
+l_param_w = 40
+l_param_nw = 40
+
+include_gravity = True
+debugflag = False
+analyse_condition = True
+
+output_string = "./output/nondirichlet_number_of_timesteps{}_".format(number_of_timesteps)
+
+##### Domain and Interface ####
+# global simulation domain domain
+sub_domain0_vertices = [df.Point(-1.0,-1.0), #
+                        df.Point(1.0,-1.0),#
+                        df.Point(1.0,1.0),#
+                        df.Point(-1.0,1.0)]
+
+subdomain0_outer_boundary_verts = {
+    0: [sub_domain0_vertices[0],
+        sub_domain0_vertices[1],
+        sub_domain0_vertices[2],
+        sub_domain0_vertices[3],
+        sub_domain0_vertices[0]]
+}
+
+# list of subdomains given by the boundary polygon vertices.
+# Subdomains are given as a list of dolfin points forming
+# a closed polygon, such that mshr.Polygon(subdomain_def_points[i]) can be used
+# to create the subdomain. subdomain_def_points[0] contains the
+# vertices of the global simulation domain and subdomain_def_points[i] contains the
+# vertices of the subdomain i.
+subdomain_def_points = [sub_domain0_vertices]
+# in the below list, index 0 corresponds to the 12 interface which has index 1
+interface_def_points = None
+
+# if a subdomain has no outer boundary write None instead, i.e.
+# i: None
+# if i is the index of the inner subdomain.
+outer_boundary_def_points = {
+    # subdomain number
+    0 : subdomain0_outer_boundary_verts
+}
+
+# adjacent_subdomains[i] contains the indices of the subdomains sharing the
+# interface i (i.e. given by interface_def_points[i]).
+adjacent_subdomains = None
+isRichards = {
+    0: False, #
+    }
+
+viscosity = {#
+# subdom_num : viscosity
+    0 : {'wetting' :1,
+         'nonwetting': 1}, #
+}
+
+porosity = {#
+# subdom_num : porosity
+    0: 1,#
+}
+
+# Dict of the form: { subdom_num : density }
+densities = {
+    0: {'wetting': 1,  #997,
+        'nonwetting': 1}, #1225}
+}
+
+gravity_acceleration = 9.81
+
+L = {#
+# subdom_num : subdomain L for L-scheme
+    0: {'wetting' :Lw,
+         'nonwetting': Lnw},#
+}
+
+lambda_param = {#
+# subdom_num : lambda parameter for the L-scheme
+    0: {'wetting' :l_param_w,
+         'nonwetting': l_param_nw},#
+}
+
+## relative permeabilty functions on subdomain 1
+def rel_perm1w(s):
+    # relative permeabilty wetting on subdomain1
+    return s**2
+
+def rel_perm1nw(s):
+    # relative permeabilty nonwetting on subdomain1
+    return (1-s)**2
+
+_rel_perm1w = ft.partial(rel_perm1w)
+_rel_perm1nw = ft.partial(rel_perm1nw)
+
+subdomain1_rel_perm = {
+    'wetting': _rel_perm1w,#
+    'nonwetting': _rel_perm1nw
+}
+
+## dictionary of relative permeabilties on all domains.
+relative_permeability = {#
+    0: subdomain1_rel_perm,
+}
+
+# definition of the derivatives of the relative permeabilities
+# relative permeabilty functions on subdomain 1
+def rel_perm1w_prime(s):
+    # relative permeabilty on subdomain1
+    return 2*s
+
+def rel_perm1nw_prime(s):
+    # relative permeabilty on subdomain1
+    return -2*(1-s)
+
+_rel_perm1w_prime = ft.partial(rel_perm1w_prime)
+_rel_perm1nw_prime = ft.partial(rel_perm1nw_prime)
+
+subdomain1_rel_perm_prime = {
+    'wetting': _rel_perm1w_prime,
+    'nonwetting': _rel_perm1nw_prime
+}
+
+# dictionary of relative permeabilties on all domains.
+ka_prime = {
+    0: subdomain1_rel_perm_prime,
+}
+
+
+
+def saturation(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return df.conditional(pc > 0, 1/((1 + pc)**(1/(index + 1))), 1)
+
+def saturation_sym(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return 1/((1 + pc)**(1/(index + 1)))
+
+
+# derivative of S-pc relationship with respect to pc. This is needed for the
+# construction of a analytic solution.
+def saturation_sym_prime(pc, index):
+    # inverse capillary pressure-saturation-relationship
+    return -1/((index+1)*(1 + pc)**((index+2)/(index+1)))
+
+
+# def saturation(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pc > 0, -index*pc, 1)
+#
+#
+# def saturation_sym(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return -index*pc
+#
+#
+# # derivative of S-pc relationship with respect to pc. This is needed for the
+# # construction of a analytic solution.
+# def saturation_sym_prime(pc, index):
+#     # inverse capillary pressure-saturation-relationship
+#     return -index
+
+
+# note that the conditional definition of S-pc in the nonsymbolic part will be
+# incorporated in the construction of the exact solution below.
+S_pc_sym = {
+    0: ft.partial(saturation_sym, index=1),
+}
+
+S_pc_sym_prime = {
+    0: ft.partial(saturation_sym_prime, index=1),
+}
+
+sat_pressure_relationship = {
+    0: ft.partial(saturation, index=1),
+}
+
+
+#############################################
+# Manufacture source expressions with sympy #
+#############################################
+x, y = sym.symbols('x[0], x[1]')  # needed by UFL
+t = sym.symbols('t', positive=True)
+
+epsilon_x_inner = 0.7
+epsilon_x_outer = 0.99
+epsilon_y_inner = epsilon_x_inner
+epsilon_y_outer = epsilon_x_outer
+
+def mollifier(x, epsilon):
+    """ one d mollifier """
+    out_expr = sym.exp(-1/(1-(x/epsilon)**2) + 1)
+    return out_expr
+
+mollifier_handle = ft.partial(mollifier, epsilon=epsilon_x_inner)
+
+pw_sym_x = sym.Piecewise(
+    (mollifier_handle(x), x**2 < epsilon_x_outer**2),
+    (0, True)
+)
+pw_sym_y = sym.Piecewise(
+    (mollifier_handle(y), y**2 < epsilon_y_outer**2),
+    (0, True)
+)
+
+def mollifier2d(x, y, epsilon):
+    """ one d mollifier """
+    out_expr = sym.exp(-1/(1-(x**2 + y**2)/epsilon**2) + 1)
+    return out_expr
+
+mollifier2d_handle = ft.partial(mollifier2d, epsilon=epsilon_x_outer)
+
+pw_sym2d_x = sym.Piecewise(
+    (mollifier2d_handle(x, y), x**2 + y**2 < epsilon_x_outer**2),
+    (0, True)
+)
+
+zero_on_epsilon_shrinking_of_subdomain = sym.Piecewise(
+    (mollifier_handle(sym.sqrt(x**2 + y**2)+2*epsilon_x_inner), ((-2*epsilon_x_inner<sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<=epsilon_x_inner))),
+    (mollifier_handle(sym.sqrt(x**2 + y**2)-2*epsilon_x_inner), ((epsilon_x_inner<sym.sqrt(x**2 + y**2)) & (sym.sqrt(x**2 + y**2)<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_epsilon_shrinking_of_subdomain_x = sym.Piecewise(
+    (mollifier_handle(x+2*epsilon_x_inner), ((-2*epsilon_x_inner<x) & (x<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=x) & (x<=epsilon_x_inner))),
+    (mollifier_handle(x-2*epsilon_x_inner), ((epsilon_x_inner<x) & (x<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_epsilon_shrinking_of_subdomain_y = sym.Piecewise(
+    (1, y<=-2*epsilon_x_inner),
+    (mollifier_handle(y+2*epsilon_x_inner), ((-2*epsilon_x_inner<y) & (y<-epsilon_x_inner))),
+    (0, ((-epsilon_x_inner<=y) & (y<=epsilon_x_inner))),
+    (mollifier_handle(y-2*epsilon_x_inner), ((epsilon_x_inner<y) & (y<2*epsilon_x_inner))),
+    (1, True),
+)
+
+zero_on_shrinking = zero_on_epsilon_shrinking_of_subdomain #zero_on_epsilon_shrinking_of_subdomain_x + zero_on_epsilon_shrinking_of_subdomain_y
+gaussian = pw_sym2d_x# pw_sym_y*pw_sym_x
+cutoff = gaussian/(gaussian + zero_on_shrinking)
+
+# # construction of differentiable characteristic function.
+# def smooth_characteristic_func_on_epsilon_shrinking_of_subdomain0(x, y, epsilon_x_inner, epsilon_y_inner, epsilon_x_outer, epsilon_y_outer):
+#     dist_to_complement_x = ft.partial(mollifier, epsilon=epsilon_x_inner)
+#     dist_to_complement_y = ft.partial(mollifier, epsilon=epsilon_y_inner)
+#     dist_to_complement = dist_to_complement_y(y)*dist_to_complement_x(x)
+#     dist_to_eps_shrinking_x = ft.partial(zero_outside_epsilon_thickening_of_subdomain, epsilon=epsilon_x_outer)
+#     dist_to_eps_shrinking_y = ft.partial(zero_outside_epsilon_thickening_of_subdomain, epsilon=epsilon_y_outer)
+#     dist_to_eps_shrinking = dist_to_eps_shrinking_y(y)*dist_to_eps_shrinking_x(x)
+#     return dist_to_complement/(dist_to_eps_shrinking + dist_to_complement)
+#
+
+# def dist_to_epsilon_thickening_of_subdomain0_complement(x, y, epsilon):
+#     """ calculates the (euklidian distance)^2 of a point x,y to the epsilon
+#         thickening of the complement of the domain.
+#     """
+#     is_inside = ((1-sym.Abs(x) > epsilon) & (1-sym.Abs(y) > epsilon))
+#     sym.Piecewise((0, is_inside))
+
+p_e_sym = {
+    0: {'wetting': (-3 - (1+t*t)*(1 + x*x + y*y)),  #*cutoff,
+        'nonwetting': (-1 -t*(1+y + x**2)**2)},  #*cutoff},
+}
+
+# p_e_sym = {
+#     0: {'wetting': -(sym.cos(2*t-x - 2*y)*sym.sin(3*(1+y)/2*sym.pi)*sym.sin(5*(1+x)/2*sym.pi))**2,
+#         'nonwetting': -6*(sym.cos(t-x -y)*sym.sin(3*(1+y)/2*sym.pi)*sym.sin(5*(1+x)/2*sym.pi))**2},
+# }
+
+
+print(f"\n\n\nsymbolic type is {type(p_e_sym[0]['wetting'])}\n\n\n")
+# # pw_sym_x*pw_sym_y
+# p_e_sym = {
+#     0: {'wetting': -3*pw_sym2d_x + 0*t,
+#         'nonwetting': -1*pw_sym_x*pw_sym_y+ 0*t},
+# }
+
+# p_e_sym = {
+#     0: {'wetting': -3*cutoff + 0*t,
+#         'nonwetting': -1*zero_on_shrinking+ 0*t},
+# }
+
+
+pc_e_sym = dict()
+for subdomain, isR in isRichards.items():
+    if isR:
+        pc_e_sym.update({subdomain: -p_e_sym[subdomain]['wetting']})
+    else:
+        pc_e_sym.update({subdomain: p_e_sym[subdomain]['nonwetting']
+                                        - p_e_sym[subdomain]['wetting']})
+
+symbols = {"x": x,
+           "y": y,
+           "t": t}
+# turn above symbolic code into exact solution for dolphin and
+# construct the rhs that matches the above exact solution.
+exact_solution_example = hlp.generate_exact_solution_expressions(
+                        symbols=symbols,
+                        isRichards=isRichards,
+                        symbolic_pressure=p_e_sym,
+                        symbolic_capillary_pressure=pc_e_sym,
+                        saturation_pressure_relationship=S_pc_sym,
+                        saturation_pressure_relationship_prime=S_pc_sym_prime,
+                        viscosity=viscosity,
+                        porosity=porosity,
+                        relative_permeability=relative_permeability,
+                        relative_permeability_prime=ka_prime,
+                        densities=densities,
+                        gravity_acceleration=gravity_acceleration,
+                        include_gravity=include_gravity,
+                        )
+source_expression = exact_solution_example['source']
+exact_solution = exact_solution_example['exact_solution']
+initial_condition = exact_solution_example['initial_condition']
+
+# Dictionary of dirichlet boundary conditions.
+dirichletBC = dict()
+# similarly to the outer boundary dictionary, if a patch has no outer boundary
+# None should be written instead of an expression.
+# This is a bit of a brainfuck:
+# dirichletBC[ind] gives a dictionary of the outer boundaries of subdomain ind.
+# Since a domain patch can have several disjoint outer boundary parts, the
+# expressions need to get an enumaration index which starts at 0.
+# So dirichletBC[ind][j] is the dictionary of outer dirichlet conditions of
+# subdomain ind and boundary part j.
+# Finally, dirichletBC[ind][j]['wetting'] and dirichletBC[ind][j]['nonwetting']
+# return the actual expression needed for the dirichlet condition for both
+# phases if present.
+
+# subdomain index: {outer boudary part index: {phase: expression}}
+for subdomain in isRichards.keys():
+    # if subdomain has no outer boundary, outer_boundary_def_points[subdomain] is None
+    if outer_boundary_def_points[subdomain] is None:
+        dirichletBC.update({subdomain: None})
+    else:
+        dirichletBC.update({subdomain: dict()})
+        # set the dirichlet conditions to be the same code as exact solution on
+        # the subdomain.
+        for outer_boundary_ind in outer_boundary_def_points[subdomain].keys():
+            dirichletBC[subdomain].update(
+                {outer_boundary_ind: exact_solution[subdomain]}
+                )
+
+
+# def saturation(pressure, subdomain_index):
+#     # inverse capillary pressure-saturation-relationship
+#     return df.conditional(pressure < 0, 1/((1 - pressure)**(1/(subdomain_index + 1))), 1)
+#
+# sa
+
+write_to_file = {
+    'meshes_and_markers': True,
+    'L_iterations': True
+}
+
+# initialise LDD simulation class
+simulation = ldd.LDDsimulation(tol = 1E-14, LDDsolver_tol=solver_tol, debug=debugflag)
+simulation.set_parameters(output_dir = output_string,#
+    subdomain_def_points = subdomain_def_points,#
+    isRichards = isRichards,#
+    interface_def_points = interface_def_points,#
+    outer_boundary_def_points = outer_boundary_def_points,#
+    adjacent_subdomains = adjacent_subdomains,#
+    mesh_resolution = mesh_resolution,#
+    viscosity = viscosity,#
+    porosity = porosity,#
+    L = L,#
+    lambda_param = lambda_param,#
+    relative_permeability = relative_permeability,#
+    saturation = sat_pressure_relationship,#
+    starttime = starttime,#
+    number_of_timesteps = number_of_timesteps,
+    number_of_timesteps_to_analyse = number_of_timesteps_to_analyse,
+    timestep_size = timestep_size,#
+    sources = source_expression,#
+    initial_conditions = initial_condition,#
+    dirichletBC_expression_strings = dirichletBC,#
+    exact_solution = exact_solution,#
+    densities=densities,
+    include_gravity=include_gravity,
+    write2file = write_to_file,#
+    )
+
+simulation.initialise()
+# simulation.write_exact_solution_to_xdmf()
+simulation.run(analyse_condition=analyse_condition)