Implement Morris surface tension model

examples/fluid/calibration_water_drop_shape.jl

@@ -0,0 +1,248 @@
+# In this example we try to approach the static shape of a water droplet on a horizontal plane.
+# The shape of a static droplet can be calculated from the Young-Laplace equation.
+using TrixiParticles
+using OrdinaryDiffEq
+
+# ==========================================================================================
+# ==== Resolution
+fluid_particle_spacing = 0.0001
+
+boundary_layers = 4
+spacing_ratio = 1
+
+# ==========================================================================================
+# ==== Experiment Setup
+gravity = 9.81
+tspan = (0.0, 5.0)
+
+# Boundary geometry and initial fluid particle positions
+initial_fluid_size = (0.0, 0.0)
+tank_size = (0.05, 0.01)
+
+fluid_density = 1000.0
+sound_speed = 50
+
+state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
+                                   exponent=1)
+
+tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,
+                       n_layers=boundary_layers, spacing_ratio=spacing_ratio,
+                       faces=(true, true, true, false),
+                       acceleration=(0.0, -gravity), state_equation=state_equation)
+
+sphere_radius = 0.0015
+
+sphere1_center = (0.5, sphere_radius)
+sphere1 = SphereShape(fluid_particle_spacing, sphere_radius, sphere1_center,
+                      fluid_density, sphere_type=VoxelSphere())
+
+# ==========================================================================================
+# ==== Fluid
+# fluid_smoothing_length = 1.2 * fluid_particle_spacing
+# fluid_smoothing_kernel = SchoenbergCubicSplineKernel{2}()
+
+fluid_smoothing_length = 2.5 * fluid_particle_spacing
+fluid_smoothing_kernel = WendlandC2Kernel{2}()
+
+fluid_smoothing_length_2 = 2.5 * fluid_particle_spacing
+fluid_smoothing_kernel_2 = WendlandC2Kernel{2}()
+
+fluid_density_calculator = ContinuityDensity()
+
+# water at 20C
+#nu=0.00089
+nu = 0.00089
+# too much 0.00089 -> 0.00045 -> 0.0002 -> 0.0001
+# no impact 0.00005
+
+# the following term only holds for 2d sims
+# alpha = 8 * nu / (fluid_smoothing_length * sound_speed)
+viscosity = ViscosityAdami(nu=nu)
+# density_diffusion = DensityDiffusionAntuono(sphere2, delta=0.1)
+
+# with increased smoothing length surface_tension is too small
+sphere_surface_tension = WeaklyCompressibleSPHSystem(sphere1, fluid_density_calculator,
+                                                     state_equation, fluid_smoothing_kernel,
+                                                     fluid_smoothing_length,
+                                                     viscosity=viscosity,
+                                                     acceleration=(0.0, -gravity),
+                                                     surface_tension=SurfaceTensionAkinci(surface_tension_coefficient=0.008),
+                                                     correction=AkinciFreeSurfaceCorrection(fluid_density))
+
+# 0.001
+# 0.2 > 90
+# 0.15 > 90
+# 0.125 > 90
+# 0.11 60-90
+# 0.1 ~ 45
+
+# 0.0008
+# 0.11 ~45-50
+# 0.1125 ~90
+# 0.115 ~90
+# 0.12 > 90
+
+# 0.0005
+#0.11 ~100
+
+# 0.00025
+# 0.115 ~60 grad aber zu tief in der mitte 0.006
+# 0.115 and increase nu to 0.0002
+
+# 0.0001 deg 90 (x-axis: 2mm, y-axis: 1.8mm)
+# start x-axis ~1.7mm y-axis: 2.2mm ~110deg distance to boundary to large at adhesion coefficient 1.0
+# increase adhesion coefficient to 1.1 x-axis ~1.8mm y-axis: 2mm ~100deg distance still to high
+# increase adhesion coefficient to 1.2 x-axis ~2.8 y-axis: 1.7 distance still to high
+# decrease adhesion coefficient to 1.15 x-axis 2.4mm y-axis: 1.9 distance high
+# decrease adhesion coefficient to 1.125 x-axis 2-2.2mm y-axis: 1.9 distance high ~90+
+# decrease surface tension coefficient from 0.115 to 0.11 x-axis: 2.6mm y-axis: 1.7 -> invalid too much adhesion
+# increase surface tension coefficient from 0.11 to 0.1125 x-axis: 2-2.2mm y-axis: 1.9
+# increase surface tension coefficient from 0.1125 to 0.12 x-axis: 1.9-2mm y-axis: 2.0
+# increase viscosity from 0.0001 to 0.0002 x-axis: 1.8mm y-axis:2.2
+# increase adhesion coefficient from 1.125 to 1.15 x-axis: 1.8mm y-axis:2.1
+# increase adhesion coefficient from 1.15 to 1.2 x-axis: 1.8mm y-axis:2.1
+# increase adhesion coefficient from 1.2 to 1.25 x-axis: 2.6mm y-axis:1.7
+# increase viscosity from 0.0002 to 0.0003 x-axis: 2.1mm y-axis:1.9 <=== (adh=1.25, surft=0.12, nu=0.0003)
+# increase adhesion coefficient from 1.25 to 1.275 x-axis: 2.6mm y-axis:1.7
+# decrease adhesion coefficient from 1.275 to 1.26 x-axis: 2.2mm y-axis:1.8
+# decrease adhesion coefficient from 1.26 to 1.255 x-axis: 1.8-2.6mm y-axis:1.8
+# increase viscosity from 0.0003 to 0.00031 x-axis: 2.2-2.8mm y-axis:1.7
+# increase viscosity from 0.00031 to 0.00032 x-axis: 2.2-2.8mm y-axis:1.7
+
+# 0.0001 deg 75 (x-axis: 2.2mm, y-axis: 1.6mm)
+#(adh=1.25, surft=0.12, nu=0.0003)
+# decrease surft=0.11 x-axis: 2.4-2.6mm y-axis:1.6
+# decrease adh=1.2 x-axis: 2.2-2.4mm y-axis:1.8
+# increase viscosity = 0.00035 x-axis: 2.4-3.2mm y-axis:1.6
+# increase viscosity = 0.0004 x-axis: 2.6-2.8mm y-axis:1.4
+# increase viscosity = 0.00045 x-axis: 2.4-3.2mm y-axis:1.4
+# decrease adh=1.15 x-axis: 1.8mm y-axis:2.1
+# increase adh=1.175 x-axis: 1.8mm y-axis:2.2
+# increase adh=1.19 x-axis: 1.8mm y-axis:2.1
+#(adh=1.25, surft=0.12, nu=0.0003)
+# increase adh=1.3 and  x-axis: 2.4mm y-axis:1.8
+# decrease adh=1.275 and  x-axis: 2.4mm y-axis:1.8
+# decrease adh=1.26 and  x-axis: 2.4mm y-axis:1.8
+# decrease adh=1.255 and  x-axis: 2.4mm y-axis:1.8
+# decrease smoothing length from 4 -> 3.75 x-axis: 2mm y-axis:1.9
+
+# 60deg x=2.4, y=1.3
+# adh = 1.15, surft=0.115, nu=0.0003, h=3.75, x=1.7-1.8, y=2.2
+# adh = 1.2, surft=0.115, nu=0.0003, h=3.75, x=2-2.2, y=1.8
+# adh = 1.25, surft=0.115, nu=0.0003, h=3.75, x=2.2, y=1.8
+# adh = 1.3, surft=0.115, nu=0.0003, h=3.75, x=2.6, y=1.6
+# adh = 1.3, surft=0.12, nu=0.0003, h=3.75, x=2.2-2.6, y=1.6
+# adh = 1.3, surft=0.125, nu=0.0003, h=3.75, x=2, y=1.8
+# adh = 1.35, surft=0.125, nu=0.0003, h=3.75, x=2.5, y=1.6
+# adh = 1.35, surft=0.13, nu=0.0003, h=3.75, x=2-2.2, y=1.8
+# adh = 1.4, surft=0.13, nu=0.0003, h=3.75, x=2-2.2, y=1.8
+# adh = 1.45, surft=0.13, nu=0.0003, h=3.75, x=2.6, y=1.6
+# not possible with smoothing_length -> lower to 3 and switch from C4 to C2
+# adh = 1.45, surft=0.13, nu=0.0003, h=3.0, x=1.4, y=2.6
+# adh = 1.45, surft=0.12, nu=0.0003, h=3.0, x=1.4, y=2.4
+# adh = 1.5, surft=0.12, nu=0.0003, h=3.0, x=1.6, y=2.6
+# adh = 1.5, surft=0.11, nu=0.0003, h=3.0, x=1.4, y=2.4
+# adh = 1.5, surft=0.10, nu=0.0003, h=3.0, x=1.4, y=2.4
+# adh = 1.5, surft=0.05, nu=0.0003, h=3.0, x=3.5, y=2.0
+# adh = 1.4, surft=0.075, nu=0.0003, h=3.0, x=3, y=2.0
+# adh = 1.4, surft=0.075, nu=0.0003, h=2.0, x=3, y=2.0 crap
+# adh = 1.4, surft=0.05, nu=0.0003, h=2.0, x=3, y=2.0 crap
+# adh = 1.4, surft=0.01, nu=0.0003, h=2.0, x=3, y=2.0 crap
+# adh = 1.4, surft=0.01, nu=0.0001, h=2.0, x=3, y=2.0 crap
+# adh = 1.4, surft=0.001, nu=0.00089, h=2.0, x=3, y=2.0 crap
+# adh = 1.2, surft=0.002, nu=0.00089, h=2.0, x=3, y=2.0 crap
+# adh = 1.2, surft=0.002, nu=0.00089, h=2.5, x=2.5, y=1.4 close but too much adh
+# adh = 0.6, surft=0.002, nu=0.00089, h=2.5, x=2.7, y=1.3 close but too much adh
+# adh = 0.3, surft=0.002, nu=0.00089, h=2.5, x=2.8, y=1.2
+# adh = 0.15, surft=0.002, nu=0.00089, h=2.5, x=2.8, y=1.2
+# adh = 0.075, surft=0.002, nu=0.00089, h=2.5, x=3.4, y=1.2
+# adh = 0.075, surft=0.004, nu=0.00089, h=2.5, x=2.8, y=1.4
+# adh = 0.075, surft=0.01, nu=0.00089, h=2.5, x=2.2, y=1.7
+# adh = 0.075, surft=0.007, nu=0.00089, h=2.5, x=2.6, y=1.4 <- okish but form needs more adh
+# adh = 0.075, surft=0.008, nu=0.00089, h=2.5, x=2.6, y=1.4
+# adh = 0.1, surft=0.01, nu=0.00089, h=2.5, x=2.4, y=1.6 <- okish but form needs more adh
+# adh = 0.15, surft=0.01, nu=0.00089, wall_nu=0.00089, h=2.5, x=2.4, y=1.8
+# adh = 0.15, surft=0.01, nu=0.00089, h=2.5, wall_nu=0.5*0.00089, x=2.2, y=1.7 <-lower wall viscosity
+# adh = 0.15, surft=0.01, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.6, y=1.6 <-lower wall viscosity
+# adh = 0.15, surft=0.015, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.2, y=1.8
+# adh = 0.2, surft=0.015, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.2, y=1.8
+# adh = 0.2, surft=0.0125, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.6, y=1.7
+# adh = 0.25, surft=0.014, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.4, y=1.6
+# adh = 0.25, surft=0.014, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.8, y=1.5 diverged for up to 3s
+# adh = 0.25, surft=0.015, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.5, y=1.6 diverged at 2.722 <----
+# adh = 0.25, surft=0.016, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.5, y=1.6
+# adh = 0.1, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=20*0.00089, x=2.2, y=1.7 <- 75d
+# adh = 0.15, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=20*0.00089, x=2.2, y=1.7
+# adh = 0.15, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.3, y=1.7
+# adh = 0.2, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.6, y=1.6
+# adh = 0.15, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=0.00089, x=2.6, y=1.5
+# adh = 0.15, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=5 * 0.00089, x=2.4, y=1.6
+# adh = 0.13, surft=0.011,  nu=0.00089, h=2.5, wall_nu=5*0.00089, x=2.6, y=1.5
+# adh = 0.11, surft=0.011,  nu=0.00089, h=2.5, wall_nu=5*0.00089, x=2.4, y=1.6
+
+# 75deg (x-axis: 2.2mm, y-axis: 1.6mm)
+# adh = 0.20, surft=0.015, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.5, y=1.6
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=20*0.00089, x=2, y=1.7 <- 90d
+# adh = 0.1, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=20*0.00089, x=2.2, y=1.7
+# adh = 0.1, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=7*0.00089, x=2.2, y=1.6
+# adh = 0.08, surft=0.011,  nu=0.00089, h=2.5, wall_nu=7*0.00089, x=2.3, y=1.6
+# adh = 0.07, surft=0.011,  nu=0.00089, h=2.5, wall_nu=7*0.00089, x=2.2, y=1.6
+# adh = 0.07, surft=0.011,  nu=0.00089, h=2.5, wall_nu=5*0.00089, x=2.3, y=1.6
+# adh = 0.05, surft=0.008,  nu=0.00089, h=2.5, wall_nu=2*0.00089, x=2.2, y=1.6
+
+# 90deg (x-axis: 2mm, y-axis: 1.8mm)
+# adh = 0.15, surft=0.015, nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.5, y=1.6
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=0.25*0.00089, x=2.2, y=1.7 after capallariy test
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=0.4*0.00089, x=2.2, y=1.6
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=0.00089, x=2.2, y=1.6
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=2*0.00089, x=2.2, y=1.7
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=4*0.00089, x=2.2, y=1.7
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.2, y=1.7
+# adh = 0.05, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=20*0.00089, x=2, y=1.7 <- does not work in capallariy test
+# adh = 0.04, surft=0.0125,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.1, y=1.7 <- further testing showed surface tension is about 10-20% too high
+# adh = 0.04, surft=0.011,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.2, y=1.7
+# adh = 0.03, surft=0.011,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.2, y=1.7 **** <- seems to be still too high
+# adh = 0.02, surft=0.011,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.2, y=1.8
+# adh = 0.02, surft=0.011,  nu=0.00089, h=2.5, wall_nu=15*0.00089, x=2.2, y=1.7
+# adh = 0.01, surft=0.011,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.2, y=1.7
+# adh = 0.01, surft=0.008,  nu=0.00089, h=2.5, wall_nu=5*0.00089, x=2.2, y=1.6
+# adh = 0.01, surft=0.008,  nu=0.00089, h=2.5, wall_nu=10*0.00089, x=2.2, y=1.6
+
+# sphere = WeaklyCompressibleSPHSystem(sphere2, fluid_density_calculator,
+#                                      state_equation, fluid_smoothing_kernel_2,
+#                                      fluid_smoothing_length_2, viscosity=viscosity,
+#                                      density_diffusion=density_diffusion,
+#                                      acceleration=(0.0, -gravity))
+
+# ==========================================================================================
+# ==== Boundary
+boundary_density_calculator = AdamiPressureExtrapolation()
+boundary_model = BoundaryModelDummyParticles(tank.boundary.density, tank.boundary.mass,
+                                             state_equation=state_equation,
+                                             boundary_density_calculator,
+                                             fluid_smoothing_kernel_2,
+                                             fluid_smoothing_length_2,
+                                             viscosity=ViscosityAdami(nu=10 * nu))
+
+# adhesion_coefficient = 1.0 and surface_tension_coefficient=0.01 for perfect wetting
+# adhesion_coefficient = 0.001 and surface_tension_coefficient=2.0 for no wetting
+boundary_system = BoundarySPHSystem(tank.boundary, boundary_model,
+                                    adhesion_coefficient=0.01)
+
+# ==========================================================================================
+# ==== Simulation
+semi = Semidiscretization(boundary_system, sphere_surface_tension)
+ode = semidiscretize(semi, tspan)
+
+info_callback = InfoCallback(interval=50)
+saving_callback = SolutionSavingCallback(dt=0.01, output_directory="out", prefix="",
+                                         write_meta_data=true)
+
+callbacks = CallbackSet(info_callback, saving_callback)
+
+# Use a Runge-Kutta method with automatic (error based) time step size control.
+sol = solve(ode, RDPK3SpFSAL35(),
+            abstol=1e-5, # Default abstol is 1e-6
+            reltol=1e-4, # Default reltol is 1e-3
+            dt=1e-4, maxiters=1E12,
+            save_everystep=false, callback=callbacks);

examples/fluid/dam_break_2_phase_2d.jl

@@ -0,0 +1,104 @@
+# 2D dam break simulation with an oil film on top
+
+using TrixiParticles
+using OrdinaryDiffEq
+
+# Size parameters
+H = 0.6
+W = 2 * H
+
+gravity = 9.81
+tspan = (0.0, 2.0)
+
+# Resolution
+fluid_particle_spacing = H / 60
+
+# Numerical settings
+smoothing_length = 3.5 * fluid_particle_spacing
+sound_speed = 100
+
+# physical values
+nu_water = 8.9E-7
+nu_air = 1.544E-5
+nu_ratio = nu_water / nu_air
+
+nu_sim_air = 0.02 * smoothing_length * sound_speed
+nu_sim_water = nu_ratio * nu_sim_air
+
+air_viscosity = ViscosityMorris(nu=nu_sim_air)
+water_viscosity = ViscosityMorris(nu=nu_sim_water)
+
+trixi_include(@__MODULE__, joinpath(examples_dir(), "fluid", "dam_break_2d.jl"),
+              sol=nothing, fluid_particle_spacing=fluid_particle_spacing,
+              viscosity=water_viscosity, smoothing_length=smoothing_length,
+              gravity=gravity, tspan=tspan,
+              density_diffusion=nothing,
+              sound_speed=sound_speed,
+              surface_tension=SurfaceTensionMorris(surface_tension_coefficient=0.08),
+              cfl=0.8,
+              tank_size=(floor(5.366 * H / fluid_particle_spacing) * fluid_particle_spacing,
+                         2.6 * H))
+
+# TODO: broken (fixed?)
+# trixi_include(@__MODULE__, joinpath(examples_dir(), "fluid", "dam_break_2d.jl"),
+# sol=nothing, fluid_particle_spacing=fluid_particle_spacing,
+# viscosity=ViscosityMorris(nu=nu), smoothing_length=smoothing_length,
+# gravity=gravity, tspan=tspan,
+# density_diffusion=DensityDiffusionMolteniColagrossi(delta=0.1),
+# sound_speed=sound_speed,
+# surface_tension=SurfaceTensionAkinci(surface_tension_coefficient=0.02),
+# correction=AkinciFreeSurfaceCorrection(1000.0))
+
+# ==========================================================================================
+# ==== Setup oil layer
+
+oil_size = (tank_size[1], 1.5 * H)
+oil_size2 = (tank_size[1] - W, H)
+oil_density = 1.0
+
+oil = RectangularShape(fluid_particle_spacing,
+                       round.(Int, oil_size ./ fluid_particle_spacing),
+                       zeros(length(oil_size)), density=oil_density)
+
+oil2 = RectangularShape(fluid_particle_spacing,
+                        round.(Int, oil_size2 ./ fluid_particle_spacing),
+                        (W, 0.0), density=oil_density)
+
+# move on top of the water
+for i in axes(oil.coordinates, 2)
+    oil.coordinates[:, i] .+= [0.0, H]
+end
+
+oil = union(oil, oil2)
+
+oil_system = WeaklyCompressibleSPHSystem(oil, fluid_density_calculator,
+                                         StateEquationCole(; sound_speed,
+                                                           reference_density=oil_density,
+                                                           exponent=1,
+                                                           clip_negative_pressure=false),
+                                         smoothing_kernel,
+                                         smoothing_length, viscosity=air_viscosity,
+                                         #density_diffusion=density_diffusion,
+                                         acceleration=(0.0, -gravity))
+
+# oil_system = WeaklyCompressibleSPHSystem(oil, fluid_density_calculator,
+#                                          StateEquationIdealGas(; gas_constant=287.0, temperature=293.0, gamma=1.4),
+#                                          smoothing_kernel,
+#                                          smoothing_length, viscosity=air_viscosity,
+#                                          density_diffusion=density_diffusion,
+#                                          acceleration=(0.0, -gravity))
+
+# ==========================================================================================
+# ==== Simulation
+semi = Semidiscretization(fluid_system, oil_system, boundary_system,
+                          neighborhood_search=GridNeighborhoodSearch{2}(update_strategy=nothing))
+ode = semidiscretize(semi, tspan, data_type=nothing)
+
+# sol = solve(ode, CarpenterKennedy2N54(williamson_condition=false),
+#             dt=1.0, # This is overwritten by the stepsize callback
+#             save_everystep=false, callback=callbacks);
+sol = solve(ode, RDPK3SpFSAL35(),
+            abstol=1e-5, # Default abstol is 1e-6 (may need to be tuned to prevent boundary penetration)
+            reltol=1e-4, # Default reltol is 1e-3 (may need to be tuned to prevent boundary penetration)
+            dtmax=1e-2, # Limit stepsize to prevent crashing
+            save_everystep=false, callback=callbacks);

examples/fluid/dam_break_2d.jl

@@ -35,7 +35,8 @@ tank_size = (floor(5.366 * H / boundary_particle_spacing) * boundary_particle_sp
 fluid_density = 1000.0
 sound_speed = 20 * sqrt(gravity * H)
 state_equation = StateEquationCole(; sound_speed, reference_density=fluid_density,
-                                   exponent=1, clip_negative_pressure=false)
+                                   exponent=1, clip_negative_pressure=false,
+                                   background_pressure=0.0)
 
 tank = RectangularTank(fluid_particle_spacing, initial_fluid_size, tank_size, fluid_density,
                        n_layers=boundary_layers, spacing_ratio=spacing_ratio,
@@ -59,9 +60,11 @@ density_diffusion = DensityDiffusionAntuono(tank.fluid, delta=0.1)
 fluid_system = WeaklyCompressibleSPHSystem(tank.fluid, fluid_density_calculator,
                                            state_equation, smoothing_kernel,
                                            smoothing_length, viscosity=viscosity,
+                                           reference_particle_spacing=fluid_particle_spacing,
                                            density_diffusion=density_diffusion,
                                            acceleration=(0.0, -gravity), correction=nothing,
-                                           surface_tension=nothing)
+                                           surface_tension=nothing,
+                                           reference_particle_spacing=fluid_particle_spacing)
 
 # ==========================================================================================
 # ==== Boundary