uiae

TP-TP-2-patch-constant-solution/TP-TP-2-patch-constant-solution.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 ufl as ufl
+
+# init sympy session
+sym.init_printing()
+
+##### 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]]
+}
+
+# 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
+    }
+
+
+############ GRID #######################ü
+mesh_resolution = 41
+timestep_size = 0.01
+number_of_timesteps = 100
+# 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 = 11
+starttime = 0
+
+viscosity = {#
+# subdom_num : viscosity
+    1 : {'wetting' :1,
+         'nonwetting': 1}, #
+    2 : {'wetting' :1,
+         'nonwetting': 1}
+}
+
+densities = {
+    1: {'wetting': 1,
+        'nonwetting': 1},
+    2: {'wetting': 1,
+        'nonwetting': 1},
+    # 3: {'wetting': 1},
+    # 4: {'wetting': 1}
+}
+
+gravity_acceleration = 9.81
+
+porosity = {#
+# subdom_num : porosity
+    1 : 1,#
+    2 : 1
+}
+
+L = {#
+# subdom_num : subdomain L for L-scheme
+    1 : {'wetting' :0.25,
+         'nonwetting': 0.25},#
+    2 : {'wetting' :0.25,
+         'nonwetting': 0.25}
+}
+
+l_param = 40
+lambda_param = {#
+# subdom_num : lambda parameter for the L-scheme
+    1 : {'wetting' :l_param,
+         'nonwetting': l_param},#
+    2 : {'wetting' :l_param,
+         'nonwetting': l_param}
+}
+
+## 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**2
+def rel_perm2nw(s):
+    # relative permeabilty nonwetting on subdosym.cos(0.8*t - (0.8*x + 1/7*y))main2
+    return (1-s)**2
+
+_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 2*(l_param_w1-s)
+
+_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: subdomain1_rel_perm_prime,
+}
+
+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=3, 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=3, 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=3, 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)
+}
+
+
+# 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(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(capillary_pressure > 0,
+#     return 1/((1 + (alpha*pc)**n_index)**((n_index - 1)/n_index))
+#
+# S_pc_rel = {#
+#     1: ft.partial(saturation_sym, n_index = 3, alpha=0.001),# n= 3 stands for non-uniform porous media
+#     2: ft.partial(saturation_sym, n_index = 6, alpha=0.001) # n=6 stands for uniform porous media matrix (siehe Helmig)
+# }
+
+# S_pc_rel_sym = {#
+#     1: ft.partial(saturation_sym, n_index = sym.Symbol('n'), alpha = sym.Symbol('a')),# n= 3 stands for non-uniform porous media
+#     2: ft.partial(saturation_sym, n_index = sym.Symbol('n'), alpha = sym.Symbol('a')) # n=6 stands for uniform porous media matrix (siehe Helmig)
+# }
+
+
+# # this function needs to be monotonically decreasing in the capillary_pressure.
+# # since in the richards case pc=-pw, this becomes as a function of pw a mono
+# # tonically INCREASING function like in our Richards-Richards paper. However
+# # since we unify the treatment in the code for Richards and two-phase, we need
+# # the same requierment
+# # for both cases, two-phase and Richards.
+# 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=2),
+#     # 5: 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=2),
+#     # 5: 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=2),
+#     # 5: ft.partial(saturation, index=1)
+# }
+
+# exact_solution = {
+#     1: {'wetting': '1.0 - (1.0 + t*t)*(1.0 + x[0]*x[0] + x[1]*x[1])'},
+#     2: {'wetting': '1.0 - (1.0 + t*t)*(1.0 + x[0]*x[0])'},
+#     3: {'wetting': '1.0 - (1.0 + t*t)*(1.0 + x[0]*x[0])'},
+#     4: {'wetting': '1.0 - (1.0 + t*t)*(1.0 + x[0]*x[0])'},
+#     5: {'wetting': '1.0 - (1.0 + t*t)*(1.0 + x[0]*x[0] + x[1]*x[1])'}
+# }
+#
+# initial_condition = {
+#     1: {'wetting': '-(x[0]*x[0] + x[1]*x[1])'},
+#     2: {'wetting': '-x[0]*x[0]'},
+#     3: {'wetting': '-x[0]*x[0]'},
+#     4: {'wetting': '-x[0]*x[0]'},
+#     5: {'wetting': '-(x[0]*x[0] + x[1]*x[1])'}
+# }
+
+#############################################
+# Manufacture source expressions with sympy #
+#############################################
+x, y = sym.symbols('x[0], x[1]')  # needed by UFL
+t = sym.symbols('t', positive=True)
+
+p_e_sym = {
+    1: {'wetting': -3 + 0*t,
+        'nonwetting': -1+ 0*t},
+    2: {'wetting': -3+ 0*t,
+        'nonwetting': -1+ 0*t},
+    # 3: {'wetting': 1.0 - (1.0 + t*t)*(1.0 + x*x)},
+    # 4: {'wetting': 1.0 - (1.0 + t*t)*(1.0 + x*x)},
+    # 5: {'wetting': 1.0 - (1.0 + t*t)*(1.0 + x*x + y*y)}
+}
+
+# pc_e_sym = {
+#     1: -1*p_e_sym[1]['wetting'],
+#     2: -1*p_e_sym[2]['wetting'],
+#     # 3: -1*p_e_sym[3]['wetting'],
+#     # 4: -1*p_e_sym[4]['wetting'],
+#     # 5: -1*p_e_sym[5]['wetting']
+# }
+
+pc_e_sym = {
+    1: p_e_sym[1]['nonwetting'] - p_e_sym[1]['wetting'],
+    2: p_e_sym[2]['nonwetting'] - p_e_sym[2]['wetting'],
+    # 3: -1*p_e_sym[3]['wetting'],
+    # 4: -1*p_e_sym[4]['wetting'],
+    # 5: -1*p_e_sym[5]['wetting']
+}
+
+
+# #### Manufacture source expressions with sympy
+# ###############################################################################
+# ## subdomain1
+# x, y = sym.symbols('x[0], x[1]') # needed by UFL
+# t = sym.symbols('t', positive=True)
+# #f = -sym.diff(u, x, 2) - sym.diff(u, y, 2)  # -Laplace(u)
+# #f = sym.simplify(f)                         # simplify f
+# p1_w = 1 - (1+t**2)*(1 + x**2 + (y-0.5)**2)
+# p1_nw = t*(1-(y-0.5) - x**2)**2 - sym.sqrt(2+t**2)*(1-(y-0.5))
+#
+# #dtS1_w = sym.diff(S_pc_rel_sym[1](p1_nw - p1_w), t, 1)
+# #dtS1_nw = -sym.diff(S_pc_rel_sym[1](p1_nw - p1_w), t, 1)
+# dtS1_w = porosity[1]*sym.diff(sym.Piecewise((S_pc_rel[1](p1_nw - p1_w), (p1_nw - p1_w) > 0), (1, True) ), t, 1)
+# dtS1_nw = -porosity[1]*sym.diff(sym.Piecewise((S_pc_rel[1](p1_nw - p1_w), (p1_nw - p1_w) > 0), (1, True) ), t, 1)
+# print("dtS1_w = ", dtS1_w, "\n")
+# print("dtS1_nw = ", dtS1_nw, "\n")
+#
+# #dxdxflux1_w = -sym.diff(relative_permeability[1]['wetting'](S_pc_rel_sym[1](p1_nw - p1_w))*sym.diff(p1_w, x, 1), x, 1)
+# #dydyflux1_w = -sym.diff(relative_permeability[1]['wetting'](S_pc_rel_sym[1](p1_nw - p1_w))*sym.diff(p1_w, y, 1), y, 1)
+# dxdxflux1_w = -1/viscosity[1]['wetting']*sym.diff(relative_permeability[1]['wetting'](sym.Piecewise((S_pc_rel[1](p1_nw - p1_w), (p1_nw - p1_w) > 0), (1, True) ))*sym.diff(p1_w, x, 1), x, 1)
+# dydyflux1_w = -1/viscosity[1]['wetting']*sym.diff(relative_permeability[1]['wetting'](sym.Piecewise((S_pc_rel[1](p1_nw - p1_w), (p1_nw - p1_w) > 0), (1, True) ))*sym.diff(p1_w, y, 1), y, 1)
+#
+# rhs1_w = dtS1_w + dxdxflux1_w + dydyflux1_w
+# rhs1_w = sym.printing.ccode(rhs1_w)
+# print("rhs_w = ", rhs1_w, "\n")
+# #rhs_w = sym.expand(rhs_w)
+# #print("rhs_w", rhs_w, "\n")
+# #rhs_w = sym.collect(rhs_w, x)
+# #print("rhs_w", rhs_w, "\n")
+#
+# #dxdxflux1_nw = -sym.diff(relative_permeability[1]['nonwetting'](S_pc_rel_sym[1](p1_nw - p1_w))*sym.diff(p1_nw, x, 1), x, 1)
+# #dydyflux1_nw = -sym.diff(relative_permeability[1]['nonwetting'](S_pc_rel_sym[1](p1_nw - p1_w))*sym.diff(p1_nw, y, 1), y, 1)
+# dxdxflux1_nw = -1/viscosity[1]['nonwetting']*sym.diff(relative_permeability[1]['nonwetting'](1-sym.Piecewise((S_pc_rel[1](p1_nw - p1_w), (p1_nw - p1_w) > 0), (1, True) ))*sym.diff(p1_nw, x, 1), x, 1)
+# dydyflux1_nw = -1/viscosity[1]['nonwetting']*sym.diff(relative_permeability[1]['nonwetting'](1-sym.Piecewise((S_pc_rel[1](p1_nw - p1_w), (p1_nw - p1_w) > 0), (1, True) ))*sym.diff(p1_nw, y, 1), y, 1)
+#
+# rhs1_nw = dtS1_nw + dxdxflux1_nw + dydyflux1_nw
+# rhs1_nw = sym.printing.ccode(rhs1_nw)
+# print("rhs_nw = ", rhs1_nw, "\n")
+#
+# ## subdomain2
+# p2_w = 1 - (1+t**2)*(1 + x**2)
+# p2_nw = t*(1- x**2)**2 - sym.sqrt(2+t**2)*(1-(y-0.5))
+#
+# #dtS2_w = sym.diff(S_pc_rel_sym[2](p2_nw - p2_w), t, 1)
+# #dtS2_nw = -sym.diff(S_pc_rel_sym[2](p2_nw - p2_w), t, 1)
+# dtS2_w = porosity[2]*sym.diff(sym.Piecewise((sym.Piecewise((S_pc_rel[2](p2_nw - p2_w), (p2_nw - p2_w) > 0), (1, True) ), (p2_nw - p2_w) > 0), (1, True) ), t, 1)
+# dtS2_nw = -porosity[2]*sym.diff(sym.Piecewise((S_pc_rel[2](p2_nw - p2_w), (p2_nw - p2_w) > 0), (1, True) ), t, 1)
+# print("dtS2_w = ", dtS2_w, "\n")
+# print("dtS2_nw = ", dtS2_nw, "\n")
+#
+# #dxdxflux2_w = -sym.diff(relative_permeability[2]['wetting'](S_pc_rel_sym[2](p2_nw - p2_w))*sym.diff(p2_w, x, 1), x, 1)
+# #dydyflux2_w = -sym.diff(relative_permeability[2]['wetting'](S_pc_rel_sym[2](p2_nw - p2_w))*sym.diff(p2_w, y, 1), y, 1)
+# dxdxflux2_w = -1/viscosity[2]['wetting']*sym.diff(relative_permeability[2]['wetting'](sym.Piecewise((S_pc_rel[2](p2_nw - p2_w), (p2_nw - p2_w) > 0), (1, True) ))*sym.diff(p2_w, x, 1), x, 1)
+# dydyflux2_w = -1/viscosity[2]['wetting']*sym.diff(relative_permeability[2]['wetting'](sym.Piecewise((S_pc_rel[2](p2_nw - p2_w), (p2_nw - p2_w) > 0), (1, True) ))*sym.diff(p2_w, y, 1), y, 1)
+#
+# rhs2_w = dtS2_w + dxdxflux2_w + dydyflux2_w
+# rhs2_w = sym.printing.ccode(rhs2_w)
+# print("rhs2_w = ", rhs2_w, "\n")
+# #rhs_w = sym.expand(rhs_w)
+# #print("rhs_w", rhs_w, "\n")
+# #rhs_w = sym.collect(rhs_w, x)
+# #print("rhs_w", rhs_w, "\n")
+#
+# #dxdxflux2_nw = -sym.diff(relative_permeability[2]['nonwetting'](S_pc_rel_sym[2](p2_nw - p2_w))*sym.diff(p2_nw, x, 1), x, 1)
+# #dydyflux2_nw = -sym.diff(relative_permeability[2]['nonwetting'](S_pc_rel_sym[2](p2_nw - p2_w))*sym.diff(p2_nw, y, 1), y, 1)
+# dxdxflux2_nw = -1/viscosity[2]['nonwetting']*sym.diff(relative_permeability[2]['nonwetting'](1-sym.Piecewise((S_pc_rel[2](p2_nw - p2_w), (p2_nw - p2_w) > 0), (1, True) ))*sym.diff(p2_nw, x, 1), x, 1)
+# dydyflux2_nw = -1/viscosity[2]['nonwetting']*sym.diff(relative_permeability[2]['nonwetting'](1-sym.Piecewise((S_pc_rel[2](p2_nw - p2_w), (p2_nw - p2_w) > 0), (1, True) ))*sym.diff(p2_nw, y, 1), y, 1)
+#
+# rhs2_nw = dtS2_nw + dxdxflux2_nw + dydyflux2_nw
+# rhs2_nw = sym.printing.ccode(rhs2_nw)
+# print("rhs2_nw = ", rhs2_nw, "\n")
+#
+#
+# ###############################################################################
+#
+# source_expression = {
+#     1: {'wetting': rhs1_w,
+#         'nonwetting': rhs1_nw},
+#     2: {'wetting': rhs2_w,
+#         'nonwetting': rhs2_nw}
+# }
+#
+# p1_w_00 = p1_w.subs(t, 0)
+# p1_nw_00 = p1_nw.subs(t, 0)
+# p2_w_00 = p2_w.subs(t, 0)
+# p2_nw_00 = p2_nw.subs(t, 0)
+# # p1_w_00 = sym.printing.ccode(p1_w_00)
+#
+# initial_condition = {
+#     1: {'wetting': sym.printing.ccode(p1_w_00),
+#         'nonwetting': sym.printing.ccode(p1_nw_00)},#
+#     2: {'wetting': sym.printing.ccode(p2_w_00),
+#         'nonwetting': sym.printing.ccode(p2_nw_00)}
+# }
+#
+# exact_solution = {
+#     1: {'wetting': sym.printing.ccode(p1_w),
+#         'nonwetting': sym.printing.ccode(p1_nw)},#
+#     2: {'wetting': sym.printing.ccode(p2_w),
+#         'nonwetting': sym.printing.ccode(p2_nw)}
+# }
+#
+# # similary 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.
+# dirichletBC = {
+# #subdomain index: {outer boudary part index: {phase: expression}}
+#     1: { 0: {'wetting': sym.printing.ccode(p1_w),
+#              'nonwetting': sym.printing.ccode(p1_nw)}},
+#     2: { 0: {'wetting': sym.printing.ccode(p2_w),
+#              'nonwetting': sym.printing.ccode(p2_nw)}}
+# }
+
+# turn above symbolic code into exact solution for dolphin and
+# construct the rhs that matches the above exact solution.
+dtS = dict()
+div_flux = dict()
+source_expression = dict()
+exact_solution = dict()
+initial_condition = dict()
+for subdomain, isR in isRichards.items():
+    dtS.update({subdomain: dict()})
+    div_flux.update({subdomain: dict()})
+    source_expression.update({subdomain: dict()})
+    exact_solution.update({subdomain: dict()})
+    initial_condition.update({subdomain: dict()})
+    if isR:
+        subdomain_has_phases = ["wetting"]
+    else:
+        subdomain_has_phases = ["wetting", "nonwetting"]
+
+    # conditional for S_pc_prime
+    pc = pc_e_sym[subdomain]
+    dtpc = sym.diff(pc, t, 1)
+    dxpc = sym.diff(pc, x, 1)
+    dypc = sym.diff(pc, y, 1)
+    S = sym.Piecewise((S_pc_sym[subdomain](pc), pc > 0), (1, True))
+    dS = sym.Piecewise((S_pc_sym_prime[subdomain](pc), pc > 0), (0, True))
+    for phase in subdomain_has_phases:
+        # Turn above symbolic expression for exact solution into c code
+        exact_solution[subdomain].update(
+            {phase: sym.printing.ccode(p_e_sym[subdomain][phase])}
+            )
+        # save the c code for initial conditions
+        initial_condition[subdomain].update(
+            {phase: sym.printing.ccode(p_e_sym[subdomain][phase].subs(t, 0))}
+            )
+        if phase == "nonwetting":
+            dtS[subdomain].update(
+                {phase: -porosity[subdomain]*dS*dtpc}
+                )
+        else:
+            dtS[subdomain].update(
+                {phase: porosity[subdomain]*dS*dtpc}
+                )
+        pa = p_e_sym[subdomain][phase]
+        dxpa = sym.diff(pa, x, 1)
+        dxdxpa = sym.diff(pa, x, 2)
+        dypa = sym.diff(pa, y, 1)
+        dydypa = sym.diff(pa, y, 2)
+        mu = viscosity[subdomain][phase]
+        ka = relative_permeability[subdomain][phase]
+        dka = ka_prime[subdomain][phase]
+        rho = densities[subdomain][phase]
+        g = gravity_acceleration
+
+        if phase == "nonwetting":
+            # x part of div(flux) for nonwetting
+            dxdxflux = -1/mu*dka(1-S)*dS*dxpc*dxpa + 1/mu*dxdxpa*ka(1-S)
+            # y part of div(flux) for nonwetting
+            dydyflux = -1/mu*dka(1-S)*dS*dypc*(dypa - rho*g) \
+                + 1/mu*dydypa*ka(1-S)
+        else:
+            # x part of div(flux) for wetting
+            dxdxflux = 1/mu*dka(S)*dS*dxpc*dxpa + 1/mu*dxdxpa*ka(S)
+            # y part of div(flux) for wetting
+            dydyflux = 1/mu*dka(S)*dS*dypc*(dypa - rho*g) + 1/mu*dydypa*ka(S)
+        div_flux[subdomain].update({phase: dxdxflux + dydyflux})
+        contructed_rhs = dtS[subdomain][phase] - div_flux[subdomain][phase]
+        source_expression[subdomain].update(
+            {phase: sym.printing.ccode(contructed_rhs)}
+            )
+        # print(f"source_expression[{subdomain}][{phase}] =", source_expression[subdomain][phase])
+
+# 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 = 1E-6, debug = False)
+simulation.set_parameters(output_dir = "./output/",#
+    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=True,
+    write2file = write_to_file,#
+    )
+
+simulation.initialise()
+# simulation.write_exact_solution_to_xdmf()
+simulation.run()