diff --git a/TP-TP-2-patch-test-case/TP-TP-2-patch-alterantive.py b/TP-TP-2-patch-test-case/TP-TP-2-patch-alterantive.py
new file mode 100755
index 0000000000000000000000000000000000000000..5d31e8ecca0577d27e53b3878107f1fabeaf2ce9
--- /dev/null
+++ b/TP-TP-2-patch-test-case/TP-TP-2-patch-alterantive.py
@@ -0,0 +1,486 @@
+#!/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]]
+}
+# 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
+ }
+
+
+solver_tol = 1E-6
+
+############ GRID #######################ü
+mesh_resolution = 30
+timestep_size = 0.001
+number_of_timesteps = 1500
+# 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
+
+Lw = 100/timestep_size
+Lnw=Lw
+
+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}
+}
+
+l_param_w = 25
+l_param_nw = 25
+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)
+
+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 = {
+ 1: p_e_sym[1]['nonwetting'] - p_e_sym[1]['wetting'],
+ 2: p_e_sym[2]['nonwetting'] - p_e_sym[2]['wetting'],
+}
+
+
+# pc_e_sym = {
+# 1: -1*p_e_sym[1]['wetting'],
+# 2: -1*p_e_sym[2]['wetting'],
+# }
+
+# 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 = solver_tol, debug = True)
+simulation.set_parameters(output_dir = "./output/alternative_example/",#
+ 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()
diff --git a/TP-TP-2-patch-test-case/TP-TP-2-patch-test.py b/TP-TP-2-patch-test-case/TP-TP-2-patch-test.py
new file mode 100755
index 0000000000000000000000000000000000000000..239aab1a3dde746075be41fa8c60602339385dac
--- /dev/null
+++ b/TP-TP-2-patch-test-case/TP-TP-2-patch-test.py
@@ -0,0 +1,445 @@
+#!/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 = 40
+timestep_size = 0.0001
+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 = 11
+starttime = 0
+
+Lw = 0.25*1/timestep_size
+Lnw=Lw
+
+l_param_w = 40
+l_param_nw = l_param_w
+
+include_gravity = True
+
+output_string = "./output/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)]
+# 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)
+
+p_e_sym = {
+ 1: {'wetting': (-3 - (1+t*t)*(1 + x*x + y*y)), #*(sym.sin((1+y)/2*sym.pi)*sym.sin((1+x)/2*sym.pi))**2,
+ 'nonwetting': (-1 -t*(1-y - x**2)**2)}, #*(sym.sin((1+y)/2*sym.pi)*sym.sin((1+x)/2*sym.pi))**2},
+ 2: {'wetting': (-3.0 - (1.0 + t*t)*(1.0 + x*x)), #*(sym.sin((1+y)/2*sym.pi)*sym.sin((1+x)/2*sym.pi))**2,
+ 'nonwetting': (-1 -t*(1- x**2)**2 - sym.sqrt(2+t**2)*(1+y)*y**2)}, #*(sym.sin((1+y)/2*sym.pi)*sym.sin((1+x)/2*sym.pi))**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
+}
+
+
+# initialise LDD simulation class
+simulation = ldd.LDDsimulation(tol = 1E-14, LDDsolver_tol = solver_tol, debug = True)
+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()